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The Project Physics Course 



Models of the Atom 

The Project Physics Course 



5 Models of the Atom 

A Component of the 
Project Physics Course 

Published by 


New York, Toronto 

This publication is one of the many 
instructional materials developed for the 
Project Physics Course. These materials 
include Texts, Handbooks, Teacher Resource 
Books, Readers, Programmed Instruction 
Booklets, Film Loops, Transparencies, 16mm 
films and laboratory equipment. Development 
of the course has profited from the help of 
many colleagues listed in the text units. 

Directors of Harvard Project Physics 

Gerald Holton, Department of Physics, 

Harvard University 
F. James Rutherford, Capuchino High School, 

San Bruno, California, and Harvard University 
Fletcher G. Watson, Harvard Graduate School 

of Education 

Copyright © 1 971 , Project Physics 

All Rights Reserved 

SBN 03-084562-9 

1234 039 98765432 

Project Physics is a registered trademark 

Picture Credits 

Cover drawing; "Relativity," 1953, lithograph by 

M. C. Eschar. Courtesy of the Museum of Modern Art, 

New York City. 

2 4 

5 I 

3 ' 

Picture Credits for frontispiece. 

(1) Photograph by Glen J. Pearcy. 

(2) Jeune fille au corsage rouge lisant by Jean 
Baptiste Camille Corot. Painting. Collection 
Bijhrle, Zurich. 

(3) Harvard Project Physics staff photo. 

(4) Femme lisant by Georges Seurat. Conte crayon 
drawing. Collection C. F. Stoop, London. 

(5) Portrait of Pierre Reverdy by Pablo Picasso. 
Etching. Museum of Modern Art, N.Y.C. 

(6) Lecture au lit by Paul Klee. Drawing. Paul Klee 
Foundation, Museum of Fine Arts, Berne. 

Sources and Acknowledgments 
Project Physics Reader 5 

1. Failure and Success — The Structure of l^olecules 
from The Search, pages 91-96, and 99-104, by 
Charles Percy Snow, reprinted with the permission 
of Charles Scribner's Sons. Copyright 1934 by 
Charles Scribner's Sons; renewal copyright© 

2. The Clock's Paradox in Relativity, pages 976 and 
977, Nature, published by Macmillan (Journals) 
Ltd. Reprinted with permission. 

3. The Island of Research (map) by Ernest Harburg, 
American Scientist, Volume 54, No. 4, 1966. 
Reproduced with permission. 

4. Ideas and Theories from The Story of Quantum 
Mechanics, pages 173-183, by Victor Guillemin, 
copyright © 1968 by Victor Guillemin. Reprinted 
with permission of Charles Scribner's Sons. 

5. Einstein from Quest, pages 254-262 and 285-294, 
by Leopold Infeld, copyright 1941 by Leopold 
Infeld, published by Doubleday & Company, Inc. 
Reprinted with permission of Russell & 
Volkening, Inc. 

6. Mr. Tompkins and Simultaneity from Mr. Tompkins 
In Paperback, pages 19-24, by George Gamow, 
copyright © 1965 by Cambridge University Press. 
Reprinted with permission. 

7. Mathematics: Accurate Language, Shorthand 
Machine and Brilliant Chancellor Relativity: A/eiv 
Science and New Philosophy from Physics for 
the Inquiring Mind: The Methods, Nature and 
Philosophy of Physical Science, pages 468-500, 
by Eric M. Rogers, copyright © 1960 by Princeton 
University Press. Reprinted with permission. 

8. Parable of the Surveyors by Edwin F. Taylor and 
John Archibald Wheeler, from Spacetime Physics, 
copyright © 1966 by W. H. Freeman and 
Company. Reprinted with permission. 

9. Outside and Inside the Elevator from The Evolu- 
tion of Physics: The Growth of Ideas from Early 
Concepts to Relativity and Quanto, pages 214- 
222, by Albert Einstein and Leopold Infeld, 
published by Simon and Schuster, copyright© 
1961 by Estate of Albert Einstein. Reprinted 
with permission. 

10. Einstein and Some Civilized Discontents by Martin 
Klein from Physics Today. 18, No. 1, 38-44 
(January 1965). Reprinted with permission. 

1 1 . The Teacher and the Bohr Theory of the Atom 
from The Search, pages 10-12, by Charles Percy 
Snow, reprinted with the permission of Charles 

Scribner's Sons. Copyright 1934 by Charles 
Scribner's Sons; renewal copyright © 1962. 

12. The New Landscape of Science from The Strange 
Story of the Quantum, pages 1 74-1 99, by Banesh 
Hoffmann, copyright © 1959 by Banesh Hoffmann. 
Published by Dover Publications, Inc. Reprinted 
with permission. 

13. The Evolution of the Physicist's Picture of Nature 
by Paul A. M. Dirac from Scientific American, 
May 1963. Copyright © 1963 by Scientific 
American, Inc. Reprinted with permission. All 
rights reserved. Available separately at 200 each 
as Offprint No. 292 from W. H. Freeman and 
Company, Inc., 660 Market Street, San Francisco, 
California 94104. 

14. Dirac and Born from Quest, pages 202-212, by 
Leopold Infeld, copyright 1941 by Leopold Infeld, 
published by Doubleday & Company, Inc. 
Reprinted with permission of Russell & 
Volkening, Inc. 

15. / Am This Whole World: Erwin Schrodinger 
from A Comprehensible World, pages 100-109, 
by Jeremy Bernstein, copyright © 1965 by Jeremy 
Bernstein. Published by Random House, Inc. 
Reprinted with permission. This article originally 
appeared in The New Yorker. 

16. The Fundamental Idea of Wave Mechanics by 
Erwin Schrodinger from Nobel Prize Lectures in 
Physics, 1922-1941, Elsevier Publishing Com- 
pany, Amsterdam, 1965. Reprinted with 

17. The Sentinel from Expedition To Earth by Arthur 
C. Clarke, copyright 1953 by Arthur C. Clarke. 
Published by Ballantine Books, Inc. Reprinted by 
permission of the author and his agents: Scott 
Meredith Literary Agency, Inc., New York, and 
David Higham Associates, Ltd., London. 

18. The Sea-Captain's Box from Science: Sense and 
Nonsense by John L. Synge, copyright 1951. 
Reprinted with permission of W. W. Norton & 
Company, Inc., and Jonathan Cape Ltd. 

19. Space Travel: Problems of Physics and Engineer- 
ing by the Staff of Harvard Project Physics. 

20. Looking for a New Law from The Character of 
Physical Law, pages 156-173, by Richard P. 
Feynman, copyright © 1965 by Richard P. 
Feynman. Published by the British Broadcasting 
Corporation and The M.I.T. Press. Reprinted 
with permission. 

21. A Portfolio of Computer-made Drawings courtesy 
of California Computer Products. Inc., Lloyd 
Sumner, and Darel Esbach, Jr. 


This Is not a physics textbook. Rather, It Is o physics 
reader, a collection of some of the best articles and 
book passages on physics. A few are on historic events 
In science, others contain some particularly memorable 
description of what physicists do; still others deal with 
philosophy of science, or with the impact of scientific 
thought on the imagination of the artist. 

There are old and new classics, and also some little- 
known publications; many have been suggested for in- 
clusion because some teacher or physicist remembered 
an article with particular fondness. The majority of 
articles Is not drawn from scientific papers of historic 
importance themselves, because material from many of 
these Is readily available, either as quotations in the 
Project Physics text or In special collections. 

This collection is meant for your browsing. If you follow 
your own reading interests, chances are good that you 
will find here many pages that convey the joy these 
authors have in their work and the excitement of their 
Ideas. If you want to follow up on Interesting excerpts, 
the source list at the end of the reader will guide you 
for further reading. 

Reader 5 
Table of Contents 

1 Failure and Success 1 

Charles Percy Snow 

2 The Clock Paradox in Relativity 10 

C. G. Darwin 

3 The Island of Research 12 

Ernest Harburg 

4 Ideas and Theories 13 

V. Guillemin 

5 Einstein 25 

Leopold Infeld 

6 Mr. Tompkins and Simultaneity 43 

George Gamow 

7 Mathematics and Relativity 49 

Eric M. Rogers 

8 Parable of the Surveyors 83 

Edwin F. Taylor and John Archibald Wheeler 

9 Outside and Inside the Elevator 89 

Albert Einstein and Leopold Infeld 

1 Einstein and some Civilized Discontents 99 

Martin Klein 

1 1 The Teacher and the Bohr Theory of the Atom 1 05 

Charles Percy Snow 

12 The New Landscape of Science 109 

Banesh Hoffmann 

1 3 The Evolution of the Physicist's Picture of Nature 1 31 

Paul A. M. Dirac 


14 Dirac and Born 141 

Leopold Infeld 

15 I am this Whole World: Erwin Schrodinger 151 

Jeremy Bernstein 

16 The Fundamental Idea of Wave Mechanics 161 

Erwin Schrodinger 

17 The Sentinel 173 

Arthur C. Clarke 

18 The Sea-Captain's Box 183 

John L. Synge 

1 9 Space Travel: Problems of Physics and Engineering 1 97 

Harvard Project Physics Staff 

20 Looking for a New Law 221 

Richard P. Feynman 

21 A Portfolio of Computer-made Drawings 239 


This author describes fhe frustrations and joy that 
can accompany a scientific discovery. The book is 
based on Snow's early experiences as a physical 

Failure and Success 

Charles Percy Snow 

An excerpt from his novel The Search. 
published in 1934 and 1958. 

Almost as soon as I took up the problem again, it struck 
me in a new light. All my other attempts have been absurd, 
I thought: if I turn them down and make another guess, 
then what? The guess didn't seem probable; but none of 
the others was any good at all. According to my guess, the 
structure was very different from anything one would have 
imagined ; but that must be true, since the obvious structure 
didn't fit any of my facts. Soon I was designing structures 
with little knobs of plasticine for atoms and steel wires to 
hold them together; I made up the old ones, for comparison's 
sake, and then I built my new one, which looked very odd, 
very different from any structure I had ever seen. Yet I was 
excited — "I think it works," I said, "I think it works." 

For I had brought back to mind some calculations of the 
scattering curves, assuming various models. None of the 
values had been anything like the truth. I saw at once that 
the new structure ought to give something much nearer. 
Hurriedly I calculated : it was a long and tiresome and com- 
plicated piece of arithmetic, but I rushed through it, making 
mistakes through impatience and having to go over it 
again. I was startled when I got the answer: the new model 
did not give perfect agreement, but it was far closer than 
any of the others. So far as I remember, the real value at 
one point was 1.32, my previous three models gave i.i, 
1.65 and 1.7, and the new one just under 1.4. 'I'm on 
it, at last,' I thought. 'It's a long shot, but I'm on it at 

For a fortnight I sifted all the evidence from the experi- 
ments since I first attacked the problem. There were a great 
many tables of figures, and a pile of X-ray photographs 
(for in my new instrument in Cambridge I was using a 
photographic detector); and I had been through most of 
them so often that I knew them almost by heart. But I went 
through them again, more carefully than ever, trying to 
interpret them in the Hght of the new structure. 'If it's 
right,' I was thinking, 'then these figures ought to run 
up to a maximum and then run down quickly.' And they 
did, though the maximum was less sharp than it should 
have been. And so on through experiments which repre- 
sented the work of over a year; they all fitted the structure, 
with an allowance for a value a shade too big here, a trifle 
too small there. There were obviously approximations to 
make, I should have to modify the structure a little, but 
that it was on the right lines I was certain. I walked to my 
rooms to lunch one morning, overflowing with pleasure; 
I wanted to tell someone the news; I waved violently to a 
man whom I scarcely knew, riding by on a bicycle: I 
thought of sending a wire to Audrey, but decided to go and 
see her on the following day instead: King's Parade seemed 
a particularly admirable street, and young men shouting 
across it were all admirable young men. I had a quick 
lunch; I wanted to bask in satisfaction, but instead I 
hurried back to the laboratory so that I could have it all 
finished with no loose ends left, and then rest for a while. 
I was feeling the after-taste of effort. 

There were four photographs left to inspect. They had 
been taken earlier in the week and I had looked over them 
once. Now they had to be definitely measured and entered, 
and the work was complete. I ran over the first, it was ever)'- 
thing I expected. The structure was fitting even better than 
in the early experiments. And the second : I lit a cigarette. 
Then the third : I gazed over the black dots. All was well — 
and then, with a thud of the heart that shook me, I saw behind 
each distinct black dot another fainter speck. The bottom 
had fallen out of everything : I was wrong, utterly wrong. 
I hunted round for another explanation: the film might be 
a false one, it might be a fluke experiment; but the look 
of it mocked me: far from being false, it was the only experi- 
ment where I had arrived at precisely the right conditions. 

Failure and Success 

Could it be explained any other way? I stared down at the 
figures, the sheets of results which I had forced into my 
scheme. My cheeks flushing dry, I tried to work this new 
photograph into my idea. An improbable assumption, 
another improbable assumption, a possibility of experi- 
mental error — I went on, fantastically, any sort of criticism 
forgotten. Still it would not fit. I was wrong, irrevocably 
wrong. I should have to begin again. 

Then I began to think: If I had not taken this photo- 
graph, what would have happened? Very easily I might 
not have taken it. I should have been satisfied with my 
idea: everyone else would have been. The evidence is over- 
whelming, except for this. I should have pulled off a big 
thing. I should be made. Sooner or later, of course, someone 
would do this experiment, and I should be shown to be 
wrong : but it would be a long time ahead, and mine would 
have been an honourable sort of mistake. On my evidence 
I should have been right. That is the way everyone would 
have looked at it. 

I suppose, for a moment, I wanted to destroy the photo- 
graph. It was all beyond my conscious mind. And I was 
swung back, also beyond my conscious mind, by all the 
forms of — shall I call it "conscience" — and perhaps more 
than that, by the desire which had thrown me into the 
search. For I had to get to what I myself thought was the 
truth. Honour, comfort and ambition were bound to move 
me, but I think my own desire went deepest. Without any 
posturing to myself, without any sort of conscious thought, 
I laughed at the temptation to destroy the photograph. 
Rather shakily I laughed. And I wrote in my note-book: 

Mar. 30 .• Photograph 3 alone has secondary dots, concentric with 
major dots. This removes all possibility of the hypothesis of structure 
B. The interpretation from Mar. 4—30 must accordingly be dis- 

From that day I understood, as I never had before, the 
frauds that creep into science every now and then. Some- 
times they must be quite unconscious: the not-seeing of 
facts because they are inconvenient, the delusions of one's 
own senses. As though in my case I had not seen, because 
my unconscious self chose not to see, the secondary ring of 

dots. Sometimes, more rarely, the fraud must be nearer 
to consciousness; that is, the fraud must be reaUsed, even 
though the man cannot control it. That was the point of 
my temptation. It could only be committed by a man in 
whom the scientific passion was weaker for the time than 
the ordinary desires for place or money. Sometimes it would 
be done, impulsively, by men in whom no faith was strong; 
and they could forget it cheerfully themselves and go on 
to do good and honest work. Sometimes it would be done 
by a man who reproached himself all his life. I think I 
could pick out most kinds of fraud from among the mis- 
takes I have seen; after that afternoon I could not help 
being tolerant towards them. 

For myself, there was nothing left to do but start again. 
I looked over the entry in my note-book; the ink was still 
shining, and yet it seemed to have stood, final, leaving me 
no hope, for a long time. Because I had nothing better to 
do, I made a list of the structures I had invented and, in 
the end, discarded. There were four of them now. Slowly, I 
devised another, I felt sterile. I distrusted it; and when I 
tried to test it, to think out its properties, I had to force 
my mind to work. I sat until six o'clock, working profitlessly ; 
and when I walked out, and all through the night, the 
question was gnawing at me: 'What is this structure? 
Shall I ever get it? Where am I going wrong?' 

I had never had two sleepless nights together before that 
week. Fulfilment deferred had hit me; I had to keep from 
reproaching myself that I had already wasted months over 
this problem, and now, just as I could consohdate my work, 
I was on the way to wasting another year. I went to bed 
late and heard the Cambridge clocks, one after another, 
chime out the small hours; I would have ideas with the 
uneasy clarity of night, switch on my light, scribble in my 
note-book, look at my watch, and try to sleep again; I 
would rest a little and wake up with a start, hoping that it 
was morning, to find that I had slept for twenty minutes: 
until I lay awake in a grey dawn, with all my doubts pressing 
in on me as I tried with tired eyes to look into the future. 
'What is the structure? What line must I take?' And 
then, as an under-theme, 'Am I going to fail at my first 
big job? Am I always going to be a competent worker 
doing little problems?' And another, 'I shah be twenty- 

Failure and Success 

six in the winter: I ought to be estabUshed. But shall I be 
getting anywhere?' My ideas, that seemed hopeful when 
I got out of bed to write them, were ridiculous when I 
saw them in this cold light. 

This went on for three nights, until my work in the day- 
time was only a pretence. Then there came a lull, when I 
forgot my worry for a night and slept until mid-day. But, 
though I woke refreshed, the questions began to whirl 
round again in my mind. For days it went on, and I could 
find no way out. I walked twenty miles one day, along the 
muddy fen-roads between the town and Ely, in order to 
clear my head ; but it only made me very tired, and I drank 
myself to sleep. Another night I went to a play, but I was 
listening not to the actors' words, but to others that formed 
themselves inside me and were giving me no rest. 


I started. My thoughts had stopped going back upon 
themselves. A5 I had been watching Audrey's eyes, an idea 
had flashed through the mist, quite unreasonably, illogically. 
It had no bearing at all on any of the hopeless attempts I had 
been making; I had explored every way, I thought, but 
this was new; and, too agitated to say even to myself that I 
beheved it, I took out some paper and tried to work it out. 
Audrey was staring with intent eyes. I could not get very far. 
I wanted my results and tables. But everything I could put 
down rang true. 

"An idea's just come to me," I explained, pretending to 
be calm. "I don't think there's anything in it. But there 
might be a little. But anyway I ought to try it out. And I 
haven't my books. Do you mind if we go back pretty soon? " 
I fancy I was getting up from the table, for Audrey smiled. 

"I'm glad you had some excuse for not listening," she 

She drove back very fast, not speaking. I made my 
plans for the work. It couldn't take less than a week, I 
thought. I sat hunched up, telhng myself that it might all 
be wrong again; but the structure was taking shape, and a 
part of me was beginning to laugh at my caution. Once I 
turned and saw Audrey's profile against the fields; but after 
a moment I was back in the idea. 

When I got out at the Cavendish gateway, she stayed in 
the car. "You'd better be alone," she said. 

"And you?" 

"I'll sit in Green Street." She stayed there regularly on 
her week-end visits. 

I hesitated. "It's " 

She smiled. "I'll expect you to-night. About ten 
o'clock," she said. 

I saw very little of Audrey that week-end. When I went 
to her, my mind was active, my body tired, and despite 
myself it was more comfort than love I asked of her. I re- 
member her smihng, a little wryly, and saying: "When this 
is over, we'll go away. Right away." I buried my head 
against her knees, and she stroked my hair. When she left me 
on the Monday morning, we clung to each other for a long 

For three weeks I was thrusting the idea into the mass of 
facts. I could do nothing but calculate, read up new facts, 
satisfy myself that I had made no mistakes in measuring up 
the plates: I developed an uncontrollable trick of not being 
sure whether I had made a particular measurement cor- 
rectly: repeating it: and then, after a day, the uncertainty 
returned, and to ease my mind I had to repeat it once 
more. I could scarcely read a newspaper or write a letter. 
Whatever I was doing, I was not at rest unless it was taking 
me towards the problem ; and even then it was an unsettled 
rest, like lying in a fever half-way to sleep. 

And yet, for all the obsessions, I was gradually being 
taken over by a calm which was new to me. I was beginning 
to feel an exultation, but it was peaceful, as different from 
wild triumph as it was from the ache in my throbbing 
nerves. For I was beginning to feel in my heart that I was 
near the truth. Beyond surmise, beyond doubt, I felt that 
I was nearly right; even as I lay awake in the dawn, or 
worked irritably with flushed cheeks, I was approaching 
a serenity which made the discomforts as trivial as those of 
someone else's body. 

It was after Easter now and Cambridge was almost 
empty. I was glad; I felt free as I walked the deserted 
streets. One night, when I left the laboratory, after an 

Failure and Success 

evening when the new facts were falUng into Hne and 
making the structure seem more than ever true, it was good 
to pass under the Cavendish ! Good to be in the midst of the 
great days of science! Good to be adding to the record 
of those great days ! And good to walk down King's Parade 
and see the Chapel standing against a dark sky without 
any stars! 

The mingling of strain and certainty, of personal worry 
and deeper peace, was something I had never known before. 
Even at the time, 1 knew I was living in a strange happiness. 
Or, rather, I knew that when it was over I should covet its 

And so for weeks I was alone in the laboratory, taking 
photographs, gazing under the red lamp at films which still 
dripped water, carrying them into the Hght and studying 
them until I knew every grey speck on them, from the 
points which were testing my structures down to flaws and 
scratches on the surface. Then, when my eyes tired, I put 
down my lens and turned to the sheets of figures that 
contained the results, the details of the structure and the 
predictions I was able to make. Often I would say — if this 
structure is right, then this crystal here will have its oxygen 
atom 1.2 a.u. from the nearest carbon; and the crystal will 
break along this axis, and not along that; and it will be 
harder than the last crystal I measured, but not so hard as the 
one before, and so on. For days my predictions were not 
only vaguely right, but right as closely as I could measure. 

I still possess those lists of figures, and I have stopped 
writing to look over them again. It is ten years and more 
since I first saw them and yet as I read : 

Predicted Observed 

1-435 1-44 

2.603 2.603 

and so on for long columns, I am warmed with something 
of that first glow. 

At last it was almost finished. I had done everything I 
could; and to make an end of it I thought out one prediction 
whose answer was irrefutable. There was one more substance 
in the organic group which I could not get in England, 
which had only been made in Munich; if my general 

structure was right, the atoms in its lattice could only have 
one pattern. For any other structure the pattern would be 
utterly different. An X-ray photograph of the crystal 
would give me all I wanted in a single day. 

It was tantaUsing, not having the stuff to hand, I could 
write and get some from Munich, but it would take a week, 
and a week was very long. Yet there seemed nothing else 
to do. I was beginning to write in my clumsy scientist's 
German — and then I remembered Liithy, who had returned 
to Germany a year ago. 

I cabled to him, asking if he would get a crystal and 
photograph it on his instrument. It would only take him a 
morning at the most, I thought, and we had become friendly 
enough for me to make the demand on him. Later in the 
afternoon I had his answer: "I have obtained crystal will 
telegraph result to-morrow honoured to assist. Liithy." I 
smiled at the "honoured to assist", which he could not 
possibly have left out, and sent off another cable: "Predict 
symmetry and distances. ..." 

Then I had twenty-four hours of waiting. Moved by some 
instinct to touch wood, I wanted to retract the last cable as 
soon as I had sent it. If — if I were wrong, no one else need 
know. But it had gone. And, nervous as I was, in a way I 
knew that I was right. Yet I slept very litde that night; I 
could mock, with all the detached part of myself, at the 
tricks my body was playing, but it went on playing them. 
I had to leave my breakfast, and drank cup after cup of tea, 
and kept throwing away cigarettes I had just lighted. I 
watched myself do these things, but I could not stop them, 
in just the same way as one can watch one's own body being 

The afternoon passed, and no telegram came. I persuaded 
myself there was scarcely time. I went out for an hour, in 
order to find it at my rooms when I returned. I went through 
all the andcs and devices of waidng. I grew empty with 
anxiety as the evening drew on. I sat trying to read; the 
room was growing dark, but I did not wish to switch on the 
light, for fear of bringing home the passage of the hoiirs. 

At last the bell rang below. I met my landlady on the 
stairs, bringing in the telegram. I do not know whether she 
noticed that my hands were shaking as I opened it. It said: 
"Felicitations on completely accurate prediction which am 

Failure and Success 

proud to confirm apologise for delay due to instrumental 
adjustments. Luthy." I was numbed for a moment; I could 
only see Liithy bowing politely to the postal clerk as he sent 
off the telegram. I laughed, and I remember it had a queer 

Then I was carried beyond pleasure. I have tried to show 
something of the high moments that science gave to me; the 
night my father talked about the stars, Luard's lesson, Austin's 
opening lecfure, the end of my first research. But this was 
different from any of them, different altogether, different 
in kind. It was further from myself My own triumph and 
delight and success were there, but they seemed insignificant 
beside this tranquil ecstasy. It was as though I had looked 
for a truth outside myself, and finding it had become for a 
moment part of the truth I sought; as though all the world, 
the atoms and the stars, were wonderfully clear and close to 
me, and I to them, so that we were part of a lucidity more 
tremendous than any mystery. 

I had never known that such a moment could exist. Some 
of its quality, perhaps, I had captured in the delight which 
came when I brought joy to Audrey, being myself content; 
or in the times among friends, when for some rare moment, 
maybe twice in my life, I had lost myself in a common 
purpose; but these moments had, as it were, the tone of the 
experience without the experience itself. 

Since then I have never quite regained it. But one effect 
will stay with me as long as I live ; once, when I was young, 
I used to sneer at the mystics who have described the experi- 
ence of being at one with God and part of the unity of things. 
After that afternoon, I did not want to laugh again; for 
though I should have interpreted the experience differently, 
I thought I knew what they meant. 

One of the most intriguing results of relativity theory, 
explained in a few paragraphs using only elementary 

The Clock Paradox in Relativity 

C. G. Darwin 

An article in the scientific journal, Nature, 1957. 

The Clock Paradox in Relativity 

In the course of reasoning on this subject with 
some of my more recalcitrant friends, I have come 
across a numerical example which I think makes the 
matter easier to follow than would any mathe- 
matical formulae, and perhaps this might interest 
some readers of Nature. 

There is no doubt whatever that the accepted 
theory of relativity is a complete and self-con- 
sistent theory (at any rate up to a range of 
knowledge far beyond the present matter), and 
it quite definitely implies that a space-traveller 
will return from his journey younger than his 
stay-at-home twin brother. We all of us have an 
instinctive resistance against this idea, but it has 
got to be accepted as an essential part of the 
theory. If Prof. H. Dingle should be correct in 
his disagreement, it would destroy the whole of 
relativity theory as it stands at present. 

Some have found a further difficulty in under- 
standing the matter. When two bodies are moving 
away from each other, each sees the occurrences 
on the other slowed down according to the 
Doppler effect, and relativity requires that they 
should both appear to be slowed down to exactly 
the same degree. Thus if there are clock-dials on 
each body visible from the other, both will appear 
to be losing time at the same rate. Conversely, 
the clocks will appear to be gaining equally as 
they approach one another again. At first sight 
this might seem to suggest that there is an exact 
symmetry between the two bodies, so that the 
clock of neither ought in the end to record a time 
behind that of the other. The present example 
will show how this argument fails. 

In order to see how a time-difference will arise, 
it suffices to take the case of special relativity 
without complications from gravitation. Two 
space-ships, S,, and S,, are floating together in 
free space. By firing a rocket S, goes off to a 

distant star, and on arrival there he fires a 
stronger rocket so as to reverse his motion, and 
finally by means of a third rocket he checks his 
speed so as to come to rest alongside Sg, who has 
stayed quietly at home all the time. Then they 
compare their experiences. The reunion of the 
two ships is an essential of the proceedings, be- 
cause it is only through it that the well-known 
difficulties about time-in-other-places are avoided. 

The work is to be so arranged that it can be 
done by ordinary ships' navigators, and does not 
require the presence in the crews of anyone 
cognizant of the mysteries of time-in-other-places. 
To achieve this, I suppose that the two ships are 
equipped with identical caesium clocks, which are 
geared so as to strike the hours. On the stroke of 
every hour each ship sends out a flash of light. 
These flashes are seen by the other ship and 
counted, and they are logged against the hour 
strokes of its own clock. Finally the two logs 
will be compared. 

In the first place it must be noted that Sj's 
clock may behave irregularly during the short 
times of his three accelerations. This trouble can 
be avoided by instructing him to switch the clock 
off before firing his rockets, and only to start it 
again when he has got up to a uniform speed, 
which he can recognize from the fact that he 
will no longer be pressed against one wall of his 
ship. The total of his time will be affected by 
this error, but it will be to the same extent 
whether he is going to the Andromeda Nebula, 
or merely to Mars. Since the time that is the 
subject under dispute is proportional to the total 
time of his absence, this direct effect of accelera- 
tion can be disregarded. 

I choose as the velocity of S,'s travel v = ic, 
because in this special case there are no tiresome 
irrationalities to consider. I take the star to be 4 
light-years away from S,,. The journey there and 
back will therefore take 10 years according to S„. 


The Clock Paradox in Relativity 

Immediately after the start each will observe the 
other's flashes slowed down by the Doppler effect. 
The formula for this in relativity theory is 
y/(c + v)/(c — v) , which in the present case gives 
exactly 3. That is to say, each navigator will 
log the other's flashes at a rate of one every three 
hours of his own clock's time. Conversely, when 
they are nearing one another again, each will log 
the other's flashes at a rate of three an hour. 

So far everything is perfectly s3Tnmetrical 
between the ships, but the question arises, for 
each ship respectively, how soon the slow flashes 
will change over into fast ones. First take the 
case of Sj. During his outward journey he will 
get slow flashes, but when he reverses direction at 
the star, they will suddenly change to fast ones. 
Whatever his clock shows at this time it is 
certainly just half what it will show when he gets 
home. Thus for half the journey he will get flashes 
at the rate of i per hour, and for the other half at a 
rate of 3 per hour. The average for the whole jour- 
ney will thus be at a rate i(i + 3) — l per hour. 

During this time S^ will have sent out 10 years' 
worth of flashes, and so in the end S^'s clock will 
record 4 X 10 = 6 years, which, of course, he can 
verify directly from his detailed log. 

Sg's log will be quite different. He will start 
with slow flashes and end with fast ones, but the 
changeover is determined by S/s reversal, which 
is occurring 4 light years away from him. Con- 
sequently, he will get slow flashes for 5 -|- 4 = 9 

years, and therefore fast flashes for only 1 year. The 
total number he will count isiX9 + 3Xl = 6 
years' worth. His nine years of slow flashes and 
one of fast are in marked contrast with S^'s 
experience of three years of each. Thus when 
the navigators compare their logs together they 
will be completely different, but both will agree 
that Sg's clock went for ten years and S^'s for 
only six. 

It may be that Sq will suggest that for some 
reason S-^'s clock was going slow during the 
motion, but S-^ will point out that there was no 
sign of anything wrong with it, and that anyhow 
his heart-beat and other bodily functions matched 
the rate of his clock and he may even direct atten- 
tion to the fact that his forehead is perceptibly 
less wrinkled than that of his twin brother. In 
fact — as the relativist knows — he is now actually 
four years younger than his brother. 

In giving this example, I have assumed S^ at 
rest for the sake of simplicity, but it is not hard 
to verify that the two logs will be exactly the 
same if a uniform motion of any amount is super- 
posed on the system. However, to show this would 
go beyond the scope of this communication. 

C. G. Darwin 

Newnham Grange, 


Sept. 30. 


One rule: Do not block the path of inquiry. 

3 The Island of Research 

Ernest Harburg 


Discussion of ways in which fields and quanta are related to 
one another in cases ranging from electrostatics to gravitation. 

4 Ideas and Theories 

V. Guillemin 

A chapter from his textbook, The Story of Quantum Mechanics, 1 968. 


The size of particles compares to that of atoms as atoms 
compare to the scale of things in the world of familiar objects; 
both involve roughly a hundred-thousand-fold ratio in magni- 
tude. A tiny grain of sand, perhaps a thousandth (10~^) centi- 
meter across, behaves in every way like an object of the large- 
scale world. But a downward plunge to a hundred millionth 
(10~*) centimeter leads to a realm in which everything existing 
in space and happening in time is a manifestation of changing 
patterns of matter waves. Things arrange themselves in sequences 
of discrete configurations, changes occur in abrupt quantum 
jumps and the pertinent laws of motion determine only the 
probabilities of events, not the individual events themselves. 
These profound changes in behavior are due primarily to differ- 
ences in the relative size of objects and their de Broglie waves. 
Large objects are enormous compared to their associated waves; 
atoms and their waves are similar in size. 

In the second downward plunge of minuteness, from a scale of 
10 ~^ to one of 10 ~^^ centimeter, a contrast of this sort does not 
exist. Here the matter waves are again comparable in size to the tiny 
regions in which particle events occur. Their radically new char- 
acteristics must be laid to other causes, in part to the change of 
scale itself. By quantum-mechanical principles the wave packets 


associated with events restricted to tiny regions of space must be 
constituted of very short matter waves; and because of the de 
Broghe relation between wavelength and momentum, this implies 
large values of velocity and energy and brief interaction times. 
Therefore, particle phenomena are necessarily rapid and violent, 
so violent that mass and energy interchange freely, and matter 
loses the stability it displays under less drastic conditions. 

Atoms are a "half-way stopover" between the things of every- 
day experience and the weird realm of particles. They could still 
be treated to some extent in terms of familiar concepts. Thus, the 
Bohr atom model is frankly a mechanism operating in a familiar, 
albeit altered, manner. Particles are, however, conceptually more 
remote from atoms than are atoms from sticks and stones. 

It is hardly surprising that attempts to extend the methods of 
quantum mechanics, so sucessful in dealing with atomic phe- 
nomena, to the realm of particles have met with difficulties. To 
make progress, it has been necessary to devise different methods 
of attack for various kinds of problems, for the properties of par- 
ticles, for their groupings, their interactions, and so forth. There 
exists, however, one generally recognized theoretical method of 
dealing with particle phenomena, the quantum field theory, which 
is adequate, in principle, to cope with all aspects of particle 
physics. As the name implies, it is concerned with the relations of 
quanta and fields. 

Electric and magnetic fields have already been discussed briefly 
as regions in which charges experience electric and magnetic 
forces. To physicists in the mid-nineteenth century, fields had a 
more tangible meaning. They were assumed to be conditions of 
strain in an ether, a tenuous elastic "jelly" filling all space. Where 
there is a field, the ether jelly is under a strain of tension or com- 
pression, different from its normal relaxed state; and these strains 
were thought to produce the forces acting upon electric charges. 
There was also the luminiferous ether, possibly different from the 
electric and magnetic ethers which, when set into oscillation at 
one point, could transmit the oscillatory strains as a light wave. 

Maxwell began the development of his monumental synthesis 
of electromagnetism and optics (page 48) by constmcting an 
elaborate model of a mechanical ether, presumably capable of 
transmitting the various field effects. But after having built the 


Ideas and Theories 

electromagnetic theory of light, in which light appears as a com- 
bination of oscillating electric and magnetic fields propagated 
together through space, he saw that his mathematical equations 
contained everything of importance. In the publication of his 
studies On a Dynamical Theory of the Electro-magnetic Field 
( 1864 ) , he presented only the mathematical theory with no men- 
tion of the ether model. Although he had thus made the ether 
unnecessary, neither he nor his contemporaries thought of casting 
it aside. Even up to the beginning of the twentieth century almost 
all physicists continued to believe in the reality of the ether or at 
least in the need of retaining it as an intuitive conception. But 
in 1905, in his famous publication on the theory of relativity, 
Einstein showed that the idea of an entity filling all space and 
acting as a stationary reference, relative to which all motions 
could be described in an absolute manner, is untenable, that only 
the relative motions of objects have meaning. After the ether had 
thus been abolished, the fields remained, like the grin of the van- 
ished Cheshire cat. 

Yet fields, in particular the traveling electromagnetic fields of 
the light waves, still retained a measure of reality. These carried 
energy and momentum and could cause electric charges to oscil- 
late. Again, it was Einstein who robbed them of these trappings 
of reality when, by postulating the photons, he relegated the light 
waves to a mere ghostly existence as nothing more than mathe- 
matical abstractions determining the gross average propagation of 
flocks of photons. 

Quantum field theory has wrought a curious revival in the 
status of fields. Although they are still largely mathematical con- 
ceptions, they have acquired strong overtones of reality. In fact, 
this theory asserts that fields alone are real, that they are the sub- 
stance of the universe, and that particles are merely the momen- 
tary manifestations of interacting fields. 

The way in which particles are derived from fields is analogous 
to the construction of atoms out of patterns of matter waves in 
Schrodinger's original conception of wave mechanics. Here the 
properties of atoms, and their interactions with each other and 
with photons, are described in terms of the configurations and 
changes of these wave patterns. Similarly, the solution of the 
quantum field equations leads to quantized energy values which 


manifest themselves with all the properties of particles. The activi- 
ties of the fields seem particlelike because fields interact very 
abruptly and in very minute regions of space. Nevertheless, even 
avowed quantum field theorists are not above talking about 
"particles" as if there really were such things, a practice which 
will be adopted in continuing this discussion. 

The ambitious program of explaining all properties of particles 
and all of their interactions in terms of fields has actually been 
successful only for three of them: the photons, electrons and posi- 
trons. This limited quantum field theory has the special name of 
quantum electrodynamics. It results from a union of classical 
electrodynamics and quantum theory, modified to be compatible 
with the principles of relativity. The three particles with which 
it deals are well suited to theoretical treatment because they are 
stable, their properties are well understood arid they interact 
through the familiar electromagnetic force. 

Quantum electrodynamics was developed around 1930, largely 
through the work of Paul Dirac. It yielded two important results: 
it showed that the electron has an alter ego, the positron, and it 
gave the electron its spin, a property which previously had to be 
added arbitrarily. When it was applied to the old problem of the 
fine structure of the hydrogen spectrum (the small differences 
between the observed wavelengths and those given by the Bohr 
theory), it produced improved values in good agreement with 
existing measures. However, in 1947 two experimenters, Willis 
Lamb and Robert Retherford, made highly precise measurements 
of the small differences in energy levels, using instead of photons 
the quanta of radio waves, which are more delicate probes of 
far lower energy. Their results, which showed distinct discrep- 
ancies from Dirac's theory, stimulated renewed theoretical efforts. 
Three men, Sin-Itiro Tomonaga of Tokyo University, Richard 
Feynman of the University of California and Julian Schwinger of 
Harvard, working independently, produced an improved theory 
which at long last gave precise agreement with experiment. For 
this work the three shared the 1965 Nobel Prize in physics. 

The study of particles by the methods of quantum field theory 
was begun at a time when only a few were known. Since the field 
associated with a particle represents all of its properties, there 
had to be a distinct kind of field for each kind of particle; and as 


Ideas and Theories 

their number increased, so did the number of different fields, a 
complication which pleased no one. Actually, little further progress 
was made in the two decades following the success of quantum 
electrodynamics. Attempts to deal with the strongly interact- 
ing particles, the mesons and baryons, were frustrated by seem- 
ingly insurmountable mathematical difficulties. Still, the idea of 
developing a basic and comprehensive theory of particles con- 
tinued to have strong appeal. In the mid-1960's the introduction 
of powerful new mathematical techniques has yielded results 
which indicate that this may yet be accomplished. 


The interaction of the electromagnetic fields, whose energy 
is carried by photons, and the electron fields, which manifest 
themselves as electrons, is already familiar in the production of 
photons by the activity of atomic electrons. It is, however, not 
apparent how photons, which travel through space with the 
highest possible velocity, might be involved in static electric fields 
such as those which hold electrons close to the atomic nucleus. 
Here a new concept is needed, that of virtual photons. Their 
existence is due in a remarkable, yet logical manner to the Heisen- 
berg uncertainty principle. One form of this principle (page 99) 
asserts that the uncertainty A£ in the energy possessed by a sys- 
tem and the uncertainty Af in the time at which it has this energy 
are related by the formula: 

AE X A^ ^ /i/27r 

Because of the relativistic correspondence between energy and 
mass, this relation applies as well to the uncertainty Am in mass, 
which is AE/c^. Applied to an electron, this means physically that 
its mass does not maintain one precise value; rather, it fluctuates, 
the magnitude of the fluctuations being in inverse proportion to 
the time interval during which they persist. Electrons effect their 
mass or equivalent energy fluctuations by emitting photons, but 
these exist only on the sufferance of the uncertainty principle. 
When their time M is up, they must vanish. They cannot leave the 
electron permanently, carrying off energy, nor can they deliver 


energy to any detection device, including the human eye. It is 
impossible for them to be seen or detected; therefore they are 
called virtual, not real. Yet there is a warrant for their existence; 
theories in which they are postulated yield results in agreement 
with experimental observation. In the language of quantum field 
theory the interaction of electron and photon fields brings about 
a condition in which by permission of the uncertainty principle 
virtual photons are continually created and destroyed. 

Virtual photons of greater energy exist for shorter times and 
travel shorter distances away from the electron before they are 
annihilated; those of lesser energy reach out farther. In fact, they 
travel a distance equal to the length of their associated waves 
( radio waves, light waves and others ) , which may vary over the 
whole range of values from zero to infinity. This swarm of virtual 
photons darting outward from the central electron in all directions 
constitutes the electric field surrounding the electron. Calculations 
based on this concept show that the field is strongest close by and 
drops off in inverse proportion to the square of the distance, in 
agreement with Coulomb's law of electric force ( page 27 ) . Virtual 
photons are the quanta of all electrostatic fields. For large charged 
objects they are so numerous that they produce a sensibly smooth 
and continuous effect, identical with the classical field. 

Two electrically charged objects exchange virtual photons. This 
produes an exchange force between them, a result which follows 
directly from the principles of quantum electrodynamics, but 
which has unfortunately no analogy in classical physics and can- 
not be visualized in terms of familiar experience. The theory shows 
that the force between charges of like sign is one of repulsion, 
that for opposite signs it is an attraction, again in agreement with 

There are, however, further complications. The virtual photons, 
produced by the electron, interact with the electron field to pro- 
duce additional virtual electrons, which in turn yield virtual 
photons, and so on. Thus the theory, starting with one electron, 
ends up with an infinite number of them. Fortunately, the magni- 
tudes of the successive steps in this sequence drop off rapidly so 
that after much effort the results of all this complex activity could 
be calculated very precisely. 

This production of secondary virtual electrons manifests itself 
in the hydrogen atom as a slight alteration of energy levels. It was 


Ideas and Theories 

this eflFect which Tomonaga, Feynman and Schwinger succeeded 
in determining correctly. 

For situations in which sufficient energy is made available, one 
of the virtual photons surrounding an electron may be "promoted" 
to a real one. This explains real photon emission when atoms 
release energy by making transitions to lower energy states. 

This discussion implies that electrostatic fields are created by 
the activity of virtual photons. The point of view of field theory is 
rather the other way about, the photons being thought of merely 
as the way in which electric fields interact with electron fields. 
It is quite in order, however, to use either concept, depending on 
which is more appropriate to the problem at hand. 


A FEW years after it had been found that atomic nuclei are 
built of protons and neutrons, Hideki Yukawa, working toward his 
Ph.D. at Osaka University, undertook a theoretical study of the 
force which binds nucleons together. The successful description 
of the electromagnetic force in terms of virtual photons suggested 
to him that the strong nuclear force might be accounted for in a 
similar manner. 

It was known that this force does not decrease gradually toward 
zero with increasing distance; rather, its range ends abruptly at 
about 10 ~^^ centimeter. Yukawa concluded that the virtual parti- 
cles associated with the strong-force field should be all of one 
mass. Assuming that they dart out at velocities close to that of 
light, he could estimate that they exist for about 10 ~^^ second; 
and from this value of Af he calculated that their mass Am, as 
given by the uncrtainty principle, is somewhat greater than two 
hundred electron masses. Since particles having a mass interme- 
diate between the electron and proton were unheard of at the 
time this prediction was made, it was received with considerable 

The way in which Yukawa's prediction was verified has already 
been discussed (page 144). The pions discovered in cosmic-ray 
studies are the real particles, not the predicted virtual ones. As is 
true of photons, virtual pions may be promoted to the real state 
if sufficient energy is provided. In this manner pions are produced 


in considerable numbers in the violent collisions of protons or 

Further studies of the strong-force field have shown that its 
quanta include not only the three kinds of pions, but the other 
mesons, the kaons and eta particles, as well. Just as electrons are 
centers surrounded by virtual photons, so protons and neutrons, 
and all the other baryons, are to be pictured as centers of darting 
virtual mesons. A proton is constantly fluctuating between being 
just a proton and being a proton plus a neutral pion or a neutron 
plus a positive pion. Similarly, a neutron may be just a neutron 
or a neutron plus a neutral pion or a proton plus a negative pion. 
These fluctuations may be indicated thus: 

p+ < — > p+ -{- TT^ n < — ^ n -\- TT^ 
p+ < — > n + 7r+ n < — > p+ + 7r~ 

The double-headed arrows imply that the interactions proceed in 
both directions. Similarly, an antineutron may be at times a nega- 
tive antiproton plus a positive pion. 

The neutron, in fact, must be in the form of a proton plus a 
negative pion a good part of the time, for it acts as if it were a 
tiny magnet. Since magnetic effects are produced only by moving 
electric charge, the neutron cannot be devoid of charge; rather, 
it must have equal amounts of both kinds spinning together about 
a common axis. The idea that both the proton and the neutron 
consist part of the time of central particles surrounded by charged 
pions is supported by experimental measurements of their mag- 
netic effects, which are due mainly to the whirling pions. In the 
protons, where this whirling charge is positive, the magnet and 
the mechanical spin point in the same direction; in the neutron 
with its negative pions the two are opposed. 

Direct evidence for the complex structure of protons and neu- 
trons has been obtained through bombardment experiments with 
high-energy electrons (page 135). The proton experiments are 
carried out by bombarding ordinary hydrogen while the observa- 
tions on neutrons are made with heavy hydrogen, whose atoms 
have nuclei which are proton-neutron pairs (since free neutrons 
in quantity are not available). From observations on the scatter- 
ing of the bombarding electrons, it is possible to determine the 
distribution of electric charge within the bombarded particles. It 


Ideas and Theories 

is found that the pions have a range of about 10"^^ centimeter, in 
agreement with Yukawa's theory. This theory gives only the range 
of the strong force and yields no information about its strength 
or details of its nature. 

Attempts have been made to formulate a theory of the weak- 
force field, involving yet another kind of unknown virtual particle. 
All attempts to track down this W-particle experimentally have 
been unsuccessful. Finally, the gravitational field is thought to be 
mediated by virtual gravitons which, like photons, must be mass- 
less since the gravitational field, like the electrical field, has a 
long range. There is at present no expectation of observing real 
gravitons, for their creation in sensible amounts would require the 
violent agitation of huge masses. The particles related to the four 
kinds of fields are the only ones not constrained by number con- 
servation laws; all four may be created and destroyed freely in 
any numbers. 

Force fields consisting of darting virtual photons and mesons 
are again a radically new conception regarding the nature of mat- 
ter. Material particles do not simply exist statically; they are 
centers of intense activity, of continual creation and annihilation. 
Every atom is a seat of such activity. In the nucleus there is a 
constant interplay of mesons, and the space around it is filled with 
swarms of virtual photons darting between the nucleus and the 


Quantum field theory is, from one viewpoint, an attack on 
a problem of ancient origin, the problem of action at a distance. 
The natural philosophers of Aristotle's Lyceum may well have 
been puzzled to observe that a piece of rubbed amber exerts an 
attraction on bits of straw over a short intervening space, a phe- 
nomenon which eighteenth-century physicists would ascribe to 
the electric field in the vicinity of the charge on the amber. But 
they were no doubt more concerned with the analogous but more 
conspicuous observation of the downward pull experienced by all 
objects on the surface of the earth. Classical physics attributed this 
pull to the gravitational field which surrounds all pieces of matter 


but is of appreciable magnitude only near very large pieces such 
as the earth. To say that a stone held in the hand is pulled down- 
ward because it is in the earth's gravitational field, however, is 
merely puting a name to ignorance. It does not detract a whit from 
the mystery that the stone "feels" a pull with no visible or tangible 
agent acting upon it. 

Isaac Newton, who formulated the law of action of the gravita- 
tional force, was well aware of this mystery. In one of his letters 
to the classical scholar and divine Richard Bentley he expressed 
himself thus: 

. . . that one body may act upon another at a distance through 
a vacuum without the mediation of anything else, by and through 
which their action and force may be conveyed from one to 
another, is to me so great an absurdity that, I believe, no man who 
has in philosophic matters a competent faculty of thinking could 
ever fall into it. 

Newton saw clearly that his universal law of gravitation is a 
description not an explanation. The German philosopher and 
mathematician Baron Gottfried von Leibnitz ( 1646-1716), among 
others of Newton's contemporaries, criticized his work on this 
account, holding that his famous formula for the gravitational 
force (F = Gm^mjir) is merely a rule of computation not worthy 
of being called a law of nature. It was compared adversely with 
existing "laws," with Aristotle's animistic explanation of the stone's 
fall as due to its "desire" to return to its "natural place" on the 
ground, and with Descartes's conception of the planets caught up 
in huge ether whirlpools carrying them on their orbits around 
the sun. 

This unjust valuation of his work was repudiated in many of 
Newton's writings, as in the following passage from his Optics: 

To tell us that every species of thing is endow'd with an occult 
specific quality, by which it acts and produces manifest effects, 
is to tell us nothing. But to derive two or three general principles 
of motion from phenomena, and afterwards to tell us how the 
properties and actions of all corporeal things follow from these 
principles would be a very great step in philosophy, though the 
causes of those principles were not yet discovered. 


Ideas and Theories 

Concerning his law of gravitation, which he discussed in the 
Principia, Newton made his position clear: 

I have not yet been able to discover the cause of these proper- 
ties of gravity from phenomena, and I frame no hypotheses. . . . 
It is enough that gravity does really exist and acts according to 
the laws I have explained, and that it abundantly serves to 
account for all the motions of celestial bodies. 

This quotation shows how thoroughly Newton espoused the exper- 
imental philosophy. He clearly expected that, if ever the "cause" 
of gravity is found, it will be deduced "from phenomena," that is, 
from experimental observations, and that in the meantime it is 
advisable to "frame no hypotheses." 

The conception of fields of force as streams of virtual particles 
supplies the means "through which their action and force may be 
conveyed," which Newton so urgently demanded. It mitigates 
the problem of action at a distance, for with virtual photons 
producing the electric field, what happens to the electron happens 
at the electron. 

Here is a lesson about the need for caution as to what "makes 
sense" in science. Nothing would seem more sensible than the 
observation that a stone tossed into the air falls back to earth; it 
would be surprising if the stone failed to do so. Yet upon closer 
study this simple event is seen to involve the metaphysical diffi- 
culties of action at a distance, difficulties which achieve a measure 
of intuitive resolution only in terms of the strange conception of 
virtual gravitons. This may serve as a warning that what passes 
for an understanding of simple things may well be no more than 
a tacit consensus to stop asking questions. 


A noted Polish theoretical physicist and co-worker of 
Albert Einstein takes us into the study of the great 
twentieth -century physicist. 


Leopold Infeld 

Excerpts from his book. Quest, The Evolution of a Scientist, 
published in 1941. 


CAME TO PRINCETON on a Saturday, lived through a dead 
Sunday and entered the office of Fine Hall on Monday, to make 
my first acquaintances. I asked the secretary when I could see 
Einstein. She telephoned him, and the answer was: 


"Professor Einstein wants to see you right away." 

I knocked at the door of 209 and heard a loud ''''herein^' When 
I opened the door I saw a hand stretched out energetically. It 
was Einstein, looking older than when I had met him in Berlin, 
older than the elapsed sixteen years should have made him. His 
long hair was gray, his face tired and yellow, but he had the 
same radiant deep eyes. He wore the brown leather jacket in 
which he has appeared in so many pictures. (Someone had given 
it to him to wear when sailing, and he had liked it so well that 
he dressed in it every day.) His shirt was without a collar, his 
brown trousers creased, and he wore shoes without socks. I 
expected a brief private conversation, questions about my cross- 
ing, Europe, Born, etc. Nothing of the kind: 

"Do you speak German?" 

"Yes," I answered. 

"Perhaps I can tell you on what I am working." 

Quietly he took a piece of chalk, went to the blackboard and 
started to deliver a perfect lecture. The calmness with which 
Einstein spoke was striking. There was nothing of the restless- 
ness of a scientist who, explaining the problems with which he 
has lived for years, assumes that they are equally familiar to the 
listener and proceeds quickly with his exposition. Before going 
into details Einstein sketched the philosophical background 
for the problems on which he was working. Walking slowly and 
with dignity around the room, going to the blackboard from 
time to time to write down mathematical equations, keeping a 
dead pipe in his mouth, he formed his sentences perfectly. 
Everything that he said could have been printed as he said it and 
every sentence would make perfect sense. The exposition was 
simple, profound and clear. 

I listened carefully and understood everything. The ideas be- 
hind Einstein's papers are aways so straightforward and funda- 
mental that I believe I shall be able to express some of them in 
simple language. 

There are two fundamental concepts in the development of 
physics: field and matter. The old physics which developed 
from Galileo and Newton, up to the middle of the nineteenth 



century, is a physics of matter. The old mechanical point of 
view is based upon the belief that we can explain all phenomena 
in nature by assuming particles and simple forces acting among 
them. In mechanics, while investigating the motion of the plan- 
ets around the sun, we have the most triumphant model of the 
old view. Sun and planets are treated as particles, with the forces 
among them depending only upon their relative distances. The 
forces decrease if the distances increase. This is a typical model 
which the mechanist would like to apply, with some unessential 
changes, to the description of all physical phenomena. 

A container with gas is, for the physicist, a conglomeration of 
small particles in haphazard motion. Here— from the planetary 
system to a gas— we pass in one great step from "macrophysics" 
to "microphysics," from phenomena accessible to our immediate 
observation to phenomena described by pictures of particles 
with masses so small that they lie beyond any possibility of di- 
rect measurement. It is our "spiritual" picture of gas, to which 
there is no immediate access for our senses, a microphysical pic- 
ture which we are forced to form in order to understand ex- 

Again this picture is of a mechanical nature. The forces among 
the particles of a gas depend only upon distances. In the motions 
of the stars, planets, gas particles, the human mind of the nine- 
teenth century saw the manifestation of the same mechanical 
view. It understood the world of sensual impressions by forming 
pictures of particles and assuming simple forces acting among 
them. The philosophy of nature from the beginning of physics 
to the nineteenth century is based upon the belief that to under- 
stand phenomena means to use in their explanation the concepts 
of particles and forces which depend only upon distances. 

To understand means always to reduce the complicated to the 
simple and familiar. For the physicists of the nineteenth century, 
to explain meant to form a mechanical picture from which the 
phenomena could be deduced. The physicists of the past century 
believed that it is possible to form a mechanical picture of the 
universe, that the whole universe is in this sense a great and com- 
pHcated mechanical system. 


Through slow, painful struggle and progress the mechanical 
view broke down. It became apparent that the simple concepts 
of particles and forces are not sufficient to explain all phenomena 
of nature. As so often happens in physics, in the time of need 
and doubt, a great new idea was born: that of the field. The old 
theory states: particles and the forces between them are the 
basic concepts. The new theory states: changes in space, spread- 
ing in time through all of space, are the basic concepts of our 
descriptions. These basic changes characterize the field. 

Electrical phenomena were the birthplace of the field concept. 
The very words used in talking about radio w2Lves—se?jt, spread, 
received— imply changes in space and therefore field. Not par- 
ticles in certain points of space, but the whole continuous space 
forms the scenery of events which change with time. 

The transition from particle physics to field physics is un- 
doubtedly one of the greatest, and, as Einstein believes, the 
greatest step accomplished in the history of human thought. 
Great courage and imagination were needed to shift the respon- 
sibility for physical phenomena from particles into the previ- 
ously empty space and to formulate mathematical equations 
describing the changes in space and time. This great change in 
the history of physics proved extremely fruitful in the theory of 
electricity and magnetism. In fact this change is mostly respon- 
sible for the great technical development in modern times. 

We now know for sure that the old mechanical concepts are 
insufficient for the description of physical phenomena. But are 
the field concepts sufficient? Perhaps there is a still more primi- 
tive question: I see an object; how can I understand its exis- 
tence.^ From the point of view of a mechanical theory the 
answer would be obvious: the object consists of small particles 
held together by forces. But we can look upon an object as upon 
a portion of space where the field is very intense or, as we say, 
where the energy is especially dense. The mechanist says: here 
is the object localized at this point of space. The field physicist 
says: field is everywhere, but it diminishes outside this portion 
so rapidly that my senses are aware of it only in this particular 
portion of space. 



Basically, three views are possible: 

1. The mechanistic: to reduce everything to particles and 
forces acting among them, depending only on distances. 

2. The field view: to reduce everything to field concepts con- 
cerning continuous changes in time and space. 

3. The dualistic view: to assume the existence of both matter 
and field. 

For the present these three cases exhaust the possibiUties of a 
philosophical approach to basic physical problems. The past 
generation believed in the first possibility. None of the present 
generation of physicists believes in it any more. Nearly all physi- 
cists accept, for the present, the third view, assuming the ex- 
istence of both matter and field. 

But the feeling of beauty and simplicity is essential to all 
scientific creation and forms the vista of future theories; where 
does the development of science lead? Is not the mixture of 
field and matter something temporary, accepted only out of 
necessity because we have not yet succeeded in forming a con- 
sistent picture based on the field concepts alone? Is it possible to 
form a pure field theory and to create what appears as matter 
out of the field? 

These are the basic problems, and Einstein is and always has 
been interested in basic problems. He said to me once: 

"I am really more of a philosopher than a physicist." 

There is nothing strange in this remark. Every physicist is a 
philosopher as well, although it is possible to be a good ex- 
perimentalist and a bad philosopher. But if one takes physics 
seriously, one can hardly avoid coming in contact with the fun- 
damental philosophic questions. 

General relativity theory (so called in contrast to special 
relativity theory, developed earlier by Einstein) attacks the 
problem of gravitation for the first time since Newton. New- 
ton's theory of gravitation fits the old mechanical view perfectly. 
We could say more. It was the success of Newton's theory that 
caused the mechanical view to spread over all of physics. But 
with the triumphs of the field theory of physics a new task ap- 
peared: to fit the gravitational problem into the new field frame. 


This is the work which was done by Einstein. Formulating the 
equations for the gravitational field, he did for gravitational 
theory what Faraday and Maxwell did for the theory of elec- 
tricity. This is of course only one aspect of the theory of rela- 
tivity and perhaps not the most important one, but it is a part of 
the principal problems on which Einstein has worked for the 
last few years and on which he is still working. 

Einstein finished his introductory remarks and told me why he 
did not like the way the problem of a unitary field theory had 
been attacked by Born and me. Then he told me of his unsuc- 
cessful attempts to understand matter as a concentration of the 
field, then about his theory of "bridges" and the difficulties 
which he and his collaborator had encountered while developing 
that theory during a whole year of tedious work. 

At this moment a knock at the door interrupted our conver- 
sation. A very small, thin man of about sixty entered, smiling and 
gesticulating, apologizing vividly with his hands, undecided in 
what language to speak. It was Levi-Civita, the famous Italian 
mathematician, at that time a professor in Rome and invited to 
Princeton for half a year. This small, frail man had refused some 
years before to swear the fascist oath designed for university 
professors in Italy. 

Einstein had known Levi-Civita for a long time. But the form 
in which he greeted his old friend for the first time in Princeton 
was very similar to the way he had greeted me. By gestures 
rather than words Levi-Civita indicated that he did not want 
to disturb us, showing with both his hands at the door that he 
could go away. To emphasize the idea he bent his small body in 
this direction. 

It was my turn to protest: 

"I can easily go away and come some other time." 

Then Einstein protested: 

*'No. We can all talk together. I shall repeat briefly what I 
said to Infeld just now. We did not go very far. And then we 
can discuss the later part." 

We all agreed readily, and Einstein began to repeat his intro- 
ductory remarks more briefly. This time "English" was chosen 



as the language of our conversation. Since I had heard the first 
part before, I did not need to be very attentive and could enjoy 
the show. I could not help laughing. Einstein's English was very 
simple, containing about three hundred words pronounced in a 
peculiar way. He had picked it up without having learned the 
language formally. But every word was understandable because 
of his quietness, slow tempo and the distinct, attractive sound 
of his voice. Levi-Civita's English was much worse, and the 
sense of his words melted in the Italian pronunciation and vivid 
gestures. Understanding was possible between us only because 
mathematicians hardly need words to understand each other. 
They have their symbols and a few technical terms which are 
recomizable even when deformed. 

I watched the calm, impressive Einstein and the small, thin, 
broadly gesticulating Levi-Civita as they pointed out formulae 
on the blackboard and talked in a language which they thought 
to be English. The picture they made, and the sight of Einstein 
pulling up his baggy trousers every few seconds, was a scene, 
impressive and at the same time comic, which I shall never for- 
get. I tried to restrain myself from laughing by saying to myself: 

"Here you are talking and discussing physics with the most 
famous scientist in the world and you want to laugh because he 
does not wear suspenders!" The persuasion worked and I man- 
aged to control myself just as Einstein began to talk about his 
latest, still unpublished paper concerning the work done during 
the preceding year with his assistant Rosen. 

It was on the problem of gravitational waves. Again I believe 
that, in spite of the highly technical, mathematical character of 
this work, it is possible to explain the basic ideas in simple words. 

The existence of electromagnetic waves, for example, light 
waves, X rays or wireless waves, can be explained by one theory 
embracing all these and many other phenomena: by Maxwell's 
equations governing the electromagnetic field. The prediction 
that electromagnetic waves Tnust exist was prior to Hertz's ex- 
periment showing that the waves do exist. 

General relativity is a field theory and, roughly speaking, it 
does for the problem of gravitation what Maxwell's theory did 


for the problem of electromagnetic phenomena. It is therefore 
apparent that the existence of gravitational waves can be de- 
duced from general relativity just as the existence of electro- 
magnetic waves can be deduced from Maxwell's theory. Every 
physicist who has ever studied the theory of relativity is con- 
vinced on this point. In their motion the stars send out gravi- 
tational waves, spreading in time through space, just as oscillat- 
ing electrons send out electromagnetic waves. It is a common 
feature of all field theories that the influence of one object on 
another, of one electron or star on another electron or star, 
spreads through space with a great but finite velocity in the form 
of waves. A superficial mathematical investigation of the struc- 
ture of gravitational equations showed the existence of gravita- 
tional waves, and it was always believed that a more thorough 
examination could only confirm this result, giving some finer 
features of the gravitational waves. No one cared about a deeper 
investigation of this subject because in nature gravitational 
waves, or gravitational radiation, seem to play a very small role. 
It is different in Maxwell's theory, where the electromagnetic 
radiation is essential to the description of natural phenomena. 

So everyone believed in gravitational waves. In the previous 
two years Einstein had begun to doubt their existence. If we in- 
vestigate the problem superficially, they seem to exist. But Ein- 
stein claimed that a deeper analysis flatly contradicts the pre- 
vious statement. This result, if true, would be of a fundamental 
nature. It would reveal something which would astound every 
physicist: that field theory and the existence of waves are not 
as closely connected as previously thought. It would show us 
once more that the first intuition may be wrong, that deeper 
mathematical analysis may give us new and unexpected results 
quite diff^erent from those foreseen when only scratching the 
surface of gravitational equations. 

I was very much interested in this result, though somewhat 
skeptical. During my scientific career I had learned that you may 
admire someone and regard him as the greatest scientist in the 
world but you must trust your own brain still more. Scientific 
creation would become sterile if results were authoritatively or 



dogmatically accepted. Everyone has his own intuition. Every- 
one has his fairly rigidly determined level of achievement and 
is capable only of small up-and-down oscillations around it. 
To know this level, to know one's place in the scientific world, 
is essential. It is good to be master in the restricted world of your 
own possibilities and to outgrow the habit of accepting results 
before they have been thoroughly tested by your mind. 

Both Levi-Civita and I were impressed by the conclusion re- 
garding the nonexistence of gravitational waves, although there 
was no time to develop the technical methods which led to this 
conclusion. Levi-Civita indicated that he had a luncheon ap- 
pointment by gestures so vivid that they made me feel hungry. 
Einstein asked me to accompany him home, where he would give 
me the manuscript of his paper. On the way we talked physics. 
This overdose of science began to weary me and I had difficulty 
in following him. Einstein talked on a subject to which we re- 
turned in our conversations many times later. He explained why 
he did not find the modern quantum mechanics aesthetically 
satisfactory and why he believed in its provisional character 
which would be changed fundamentally by future development. 

He took me to his study with its great window overlooking 
the bright autumn colors of his lovely garden, and his first and 
only remark which did not concern physics was: 

"There is a beautiful view from this window." 

Excited and happy, I went home with the manuscript of Ein- 
stein's paper. I felt the anticipation of intense emotions which 
always accompany scientific work: the sleepless nights in which 
imagination is most vivid and the controlling criticism weakest, 
the ecstasy of seeing the light, the despair when a long and 
tedious road leads nowhere; the attractive mixture of happiness 
and unhappiness. All this was before me, raised to the highest 
level because I was working in the best place in the world. 



.HE PROGRESS OF MY WORK with Einstcin brought an in- 
creasing intimacy between us. More and more often we talked of 
social problems, politics, human relations, science, philosophy, 
life and death, fame and happiness and, above all, about the 
future of science and its ultimate aims. Slowly I came to know 
Einstein better and better. I could foresee his reactions; I under- 
stood his attitude which, although strange and unusual, was 
always fully consistent with the essential features of his per- 

Seldom has anyone met as many people in his life as Einstein 
has. Kings and presidents have entertained him; everyone is 
eager to meet him and to secure his friendship. It is compara- 
tively easy to meet Einstein but difficult to know him. His mail 
brings him letters from all over the world which he tries to an- 
swer as long as there is any sense in answering. But through all 
the stream of events, the impact of people and social life forced 
upon him, Einstein remains lonely, loving solitude, isolation and 
conditions which secure undisturbed work. 

A few years ago, in London, Einstein made a speech in Albert 
Hall on behalf of the refugee scientists, the first of whom had 
begun to pour out from Germany all over the world. Einstein 
said then that there are many positions, besides those in universi- 
ties, which would be suitable for scientists. As an example he 
mentioned a lighthouse keeper. This would be comparatively 
easy work which would allow one to contemplate and to do 
scientific research. His remark seemed funny to every scientist. 
But it is quite understandable from Einstein's point of view. One 
of the consequences of loneliness is to judge everything by one's 
own standards, to be unable to change one's co-ordinate sys- 



tern by putting oneself into someone else's being. I always 
noticed this difficulty in Einstein's reactions. For him loneliness, 
life in a lighthouse, would be most stimulating, would free him 
from so many of the duties which he hates. In fact it would be 
for him the ideal life. But nearly every scientist thinks just the 
opposite. It was the curse of my life that for a long time I was 
not in a scientific atmosphere, that I had no one with whom to 
talk physics. It is commonly known that stimulating environ- 
ment strongly influences the scientist, that he may do good 
work in a scientific atmosphere and that he may become sterile, 
his ideas dry up and all his research activity die if his environ- 
ment is scientifically dead. I knew that put back in a gymnasium, 
in a provincial Polish town, I should not publish anything, and 
the same would have happened to many another scientist better 
than I. But genius is an exception. Einstein could work any- 
where, and it is difficult to convince him that he is an exception. 

He regards himself as extremely lucky in life because he never 
had to fight for his daily bread. He enjoyed the years spent in 
the patent office in Switzerland. He found the atmosphere more 
friendly, more human, less marred by intrigue than at the uni- 
versities, and he had plenty of time for scientific work. 

In connection with the refugee problem he told me that he 
would not have minded working with his hands for his daily 
bread, doing something useful like making shoes and treating 
physics only as a hobby; that this might be more attractive than 
earning money from physics by teaching at the university. 
Again something deeper is hidden behind this attitude. It is the 
"religious" feeling, bound up with scientific work, recalling that 
of the early Christian ascetics. Physics is great and important. 
It is not quite right to earn money by physics. Better to do 
something different for a living, such as tending a lighthouse or 
making shoes, and keep physics aloof and clean. Naive as it may 
seem, this attitude is consistent with Einstein's character. 

I learned much from Einstein in the realm of physics. But 
what I value most is what I was taught by my contact with him 
in the human rather than the scientific domain. Einstein is the 
kindest, most understanding and helpful man in the world. But 


again this somewhat commonplace statement must not be taken 

The feeling of pity is one of the sources of human kindness. 
Pity for the fate of our fellow men, for the misery around us, 
for the suffering of human beings, stirs our emotions by the 
resonance of sympathy. Our own attachments to life and people, 
the ties which bind us to the outside world, awaken our emo- 
tional response to the struggle and suffering outside ourselves. 
But there is also another entirely different source of human 
kindness. It is the detached feeling of duty based on aloof, clear 
reasoning. Good, clear thinking leads to kindness and loyalty 
because this is what makes life simpler, fuller, richer, diminishes 
friction and unhappiness in our environment and therefore also 
in our lives. A sound social attitude, helpfulness, friendliness, 
kindness, may come from both these different sources; to express 
it anatomically, from heart and brain. As the years passed I 
learned to value more and more the second kind of decency that 
arises from clear thinking. Too often I have seen how emotions 
unsupported by clear thought are useless if not destructive. 

Here again, as I see it, Einstein represents a limiting case. I had 
never encountered so much kindness that was so completely 
detached. Though only scientific ideas and physics really matter 
to Einstein, he has never refused to help when he felt that his 
help was needed and could be effective. He wrote thousands of 
letters of recommendation, gave advice to hundreds. For hours 
he talked with a crank because the family had written that 
Einstein was the only one who could cure him. Einstein is kind, 
smiling, understanding, talkative with people whom he meets, 
waiting patiently for the moment when he will be left alone to 
return to his work. 

Einstein wrote about himself: 

My passionate interest in social justice and social responsibility 
has always stood in curious contrast to a marked lack of desire for 
direct association with men and women. I am a horse for single 
harness, not cut out for tandem or teamwork. I have never belonged 
wholeheartedly to country or state, to my circle of friends or even 
to my own family. These ties have always been accompanied by a 



vague aloofness, and the wish to withdraw into myself increases 
with the years. 

Such isolation is sometimes bitter, but I do not regret being cut 
off from the understanding and sympathy of other men. I lose 
something by it, to be sure, but I am compensated for it in being 
rendered independent of the customs, opinions and prejudices of 
others and am not tempted to rest my peace of mind upon such 
shifting foundations. 


For scarcely anyone is fame so undesired and meaningless as 
for Einstein. It is not that he has learned the bitter taste of fame, 
as frequently happens, after having desired it. Einstein told me 
that in his youth he had always wished to be isolated from the 
struggle of life. He was certainly the last man to have sought 
fame. But fame came to him, perhaps the greatest a scientist has 
ever known. I often wondered why it came to Einstein. His ideas 
have not influenced our practical life. No electric light, no tele- 
phone, no wireless is connected with his name. Perhaps the only 
important technical discovery which takes its origin in Ein- 
stein's theoretical work is that of the photoelectric cell. But 
Einstein is certainly not famous because of this discovery. It is 
his work on relativity theory which has made his name known 
to all the civilized world. Does the reason lie in the great influ- 
ence of Einstein's theory upon philosophical thought? This 
again cannot be the whole explanation. The latest developments 
in quantum mechanics, its connection with determinism and in- 
determinism, influenced philosophical thought fully as much. 
But the names of Bohr and Heisenberg have not the glory that 
is Einstein's. The reasons for the great fame which diffused 
deeply among the masses of people, most of them removed from 
creative scientific work, incapable of estimating his work, must 
be manifold and, I believe, sociological in character. The ex- 
planation was suggested to me by discussions with one of my 
friends in England. 

It was in 19 19 that Einstein's fame began. At this time his great 
achievement, the structure of the special and general relativity 
theories, was essentially finished. As a matter of fact it had been 
completed five years before. One of the consequences of the 


general relativity theory may be described as follows: if we 
photograph a fragment of the heavens during a solar eclipse 
and the same fragment in normal conditions, we obtain slightly 
different pictures. The gravitational field of the sun slightly dis- 
turbs and deforms the path of light, therefore the photographic 
picture of a fragment of the heavens will vary somewhat during 
the solar eclipse from that under normal conditions. Not only 
qualitatively but quantitatively the theory of relativity predicted 
the difference in these two pictures. English scientific expedi- 
tions sent in 19 19 to different parts of the world, to Africa 
and South America, confirmed this prediction made by Einstein. 

Thus began Einstein's great fame. Unlike that of film stars, 
politicians and boxers, the fame persists. There are no signs of 
its diminishing; there is no hope of relief for Einstein. The fact 
that the theory predicted an event which is as far from our 
everyday life as the stars to which it refers, an event which 
follows from a theory through a long chain of abstract argu- 
ments, seems hardly sufficient to raise the enthusiasm of the 
masses. But it did. And the reason must be looked for in the 
postwar psychology. 

It was just after the end of the war. People were weary of 
hatred, of killing and international intrigues. The trenches, 
bombs and murder had left a bitter taste. Books about war did 
not sell. Everyone looked for a new era of peace and wanted to 
forget the war. Here was something which captured the imagi- 
nation: human eyes looking from an earth covered with graves 
and blood to the heavens covered with stars. Abstract thought 
carrying the human mind far away from the sad and disappoint- 
ing reality. The mystery of the sun's eclipse and the penetrating 
power of the human mind. Romantic scenery, a strange glimpse 
of the eclipsed sun, an imaginary picture of bending light rays, 
all removed from the oppressive reality of life. One further 
reason, perhaps even more important: a new event was pre- 
dicted by a German scientist Einstein and confirmed by English 
astronomers. Scientists belonging to two warring nations had 
collaborated again! It seemed the beginning of a new era. 

It is difficult to resist fame and not to be influenced by it. But 



fame has had no effect on Einstein. And again the reason lies in 
his internal isolation, in his aloofness. Fame bothers him when 
and as long as it impinges on his life, but he ceases to be con- 
scious of it the moment he is left alone. Einstein is unaware of 
his fame and forgets it when he is allowed to forget it. 

Even in Princeton everyone looks with hungry, astonished 
eyes at Einstein. During our walks we avoided the more 
crowded streets to walk through fields and along forgotten by- 
ways. Once a car stopped us and a middle-aged woman got out 
with a camera and said, blushing and excited: 

"Professor Einstein, will you allow me to take a picture of 

"Yes, sure." 

He stood quiet for a second, then continued his argument. 
The scene did not exist for him, and I am sure after a few min- 
utes he forgot that it had ever happened. 

Once we went to a movie in Princeton to see the Life of ^mile 
'Lola. After we had bought our tickets we went to a crowded 
waiting room and found that we should have to wait fifteen 
minutes longer. Einstein suggested that we go for a walk. When 
we went out I said to the doorman: 

"We shall return in a few minutes." 

But Einstein became seriously concerned and added in all 

"We haven't our tickets any more. Will you recognize us?" 

The doorman thought we were joking and said, laughing: 

"Yes, Professor Einstein, I will." 

Einstein is, if he is allowed to be, completely unaware of his 
fame, and he furnishes a unique example of a character un- 
touched by the impact of the greatest fame and publicity. But 
there are moments when the aggressiveness of the outside world 
disturbs his peace. He once told me: 

"I envy the simplest working man. He has his privacy." 

Another time he remarked': 

"I appear to myself as a swindler because of the great pub- 
licity about me without any real reason." 


Einstein understands everyone beautifully when logic and 
thinking are needed. It is much less easy, however, where emo- 
tions are concerned; it is difficult for him to imagine motives 
and emotions other than those which are a part of his life. Once 
he told me: 

"I speak to everyone in the same way, whether he is the 
garbage man or the president of the university." 

I remarked that this is difficult for other people. That, for 
example, when they meet him they feel shy and embarrassed, 
that it takes time for this feeling to disappear and that it was so 
in my case. He said: 

"I cannot understand this. Why should anyone be shy with 

If my explanation concerning the beginning of Einstein's fame 
is correct, then there still remains another question to be an- 
swered: why does this fame cling so persistently to Einstein in a 
changing world which scorns today its idols of yesterday? I do 
not think the answer is difficult. 

Everything that Einstein did, everything for which he stood, 
was always consistent with the primary picture of him in the 
minds of the people. His voice was always raised in defense of 
the suppressed; his signature always appeared in defense of lib- 
eral causes. He was like a saint with two halos around his head. 
One was formed of ideas of justice and progress, the other of 
abstract ideas about physical theories which, the more abstruse 
they were, the more impressive they seemed to the ordinary 
man. His name became a symbol of progress, humanity and 
creative thought, hated and despised by those who spread hate 
and who attack the ideas for which Einstein's name stands. 

From the same source, from the desire to defend the op- 
pressed, arose his interest in the Jewish problem. Einstein himself 
was not reared in the Jewish tradition. It is again his detached 
attitude of sympathy, the rational idea that help must be given 
where help is needed, that brought him near to the Jewish 
problem. Jews have made splendid use of Einstein's gentle atti- 
tude. He once said: 



"I am something of a Jewish saint. When I die the Jews will 
take my bones to a banquet and collect money." 

In spite of Einstein's detachment I had often the impression 
that the Jewish problem is nearer his heart than any other social 
problem. The reason may be that I met him just at the time 
when the Jewish tragedy was greatest and perhaps, also, because 
he believes that there he can be most helpful. 

Einstein also fully realized the importance of the war in Spain 
and foresaw that on its outcome not only Spain's fate but the 
future of the world depended. I remember the gleam that came 
into his eyes when I told him that the afternoon papers carried 
news of a Loyalist victory. 

"That sounds like an angel's song," he said with an excite- 
ment which I had hardly ever noticed before. But two minutes 
later we were writing down formulae and the external world 
had again ceased to exist. 

It took me a long time to realize that in his aloofness and isola- 
tion lie the simple keys leading to an understanding of many 
of his actions. I am quite sure that the day Einstein received the 
Nobel prize he was not in the slightest degree excited and that 
if he did not sleep well that it was because of a problem which 
was bothering him and not because of the scientific distinction. 
His Nobel prize medal, together with many others, is laid aside 
among papers, honorary degrees and diplomas in the room 
where his secretary works, and I am sure that Einstein has no 
clear idea of what the medal looks like. 

Einstein tries consciously to keep liis aloofness intact by 
small idiosyncrasies which may seem strange but which increase 
his freedom and further loosen his ties with the external world. 
He never reads articles about himself. He said that this helps 
him to be free. Once I tried to break his habit. In a French 
newspaper there was an article about Einstein which was repro- 
duced in many European papers, even in Poland and Lithuania. 
I have never seen an article which was further from the truth 
than this one. For example, the author said that Einstein wears 
glasses, lives in Princeton in one room on the fifth floor, comes to 
the institute at 7 a.m., always wears black, keeps many of his 


technical discoveries secret, etc. The article could be character- 
ized as the peak of stupidity if stupidity could be said to have a 
peak. Fine Hall rejoiced in the article and hung it up as a curi- 
osity on the bulletin board at the entrance. I thought it so funny 
that I read it to Einstein, who at my request listened carefully 
but was little interested and refused to be amused. I could see 
from his expression that he failed to understand why I found it 
so funny. 

One of my colleagues in Princeton asked me: 
"If Einstein dislikes his fame and would like to increase his 
privacy, why does he not do what ordinary people do? Why 
does he wear long hair, a funny leather jacket, no socks, no 
suspenders, no collars, no ties?" 

The answer is simple and can easily be deduced from his 
aloofness and desire to loosen his ties with the outside world. 
The idea is to restrict his needs and, by this restriction, increase 
his freedom. We are slaves of millions of things, and our slavery 
progresses steadily. For a week I tried an electric razor— and one 
more slavery entered my life. I dreaded spending the summer 
where there was no electric current. We are slaves of bathrooms, 
Frigidaires, cars, radios and millions of other things. Einstein 
tried to reduce them to the absolute minimum. Long hair mini- 
mizes the need for the barber. Socks can be done without. One 
leather jacket solves the coat problem for many years. Suspend- 
ers are superfluous, as are nightshirts and pajamas. It is a mini- 
mum problem which Einstein has solved, and shoes, trousers, 
shirt, jacket, are the very necessary things; it would be difficult 
to reduce them further. 

I like to imagine Einstein's behavior in an unusual situation. 
For example: Princeton is bombed from the air; explosives fall 
over the city, people flee to shelter, panic spreads over the town 
and everyone loses his head, increasing the chaos and fear by his 
behavior. If this situation should find Einstein walking through 
the street, he would be the only man to remain as quiet as before. 
He would think out what to do in this situation; he would do it 
without accelerating the normal speed of his motions and he 
would still keep in mind the problem on which he was thinking. 
There is no fear of death in Einstein. He said to me once: 
"Life is an exciting show. I enjoy it. It is wonderful. But if I 
knew that I should have to die in three hours it would impress 
me very little. I should think how best to use the last three hours, 
then quietly order my papers and lie peacefully down." 


Mr, Tompkins takes a holiday trip in a physically 
possible science-fiction land. In solving a murder 
case there he learns the meaning of the concept of 
simultaneity in the theory of relativity. 

Mr. Tompkins and Simultaneity 

George Gamow 

Excerpt from his book, Mr. Tompkins In Paperback, published in 1965 

Mr Tompkins was very amused about his adventures in the 
relativistic city, but was sorry that the professor had not been with 
him to give any explanation of the strange things he had observed: 
the mystery of how the railway brakeman had been able to pre- 
vent the passengers from getting old worried him especially. 
Many a night he went to bed with the hope that he would see this 
interesting city again, but the dreams were rare and mostly un- 
pleasant; last time it was the manager of the bank who was firing 
him for the uncertainty he introduced into the bank accounts . . . 
so now he decided that he had better take a holiday, and go for a 
week somewhere to the sea. Thus he found himself sitting in a 
compartment of a train and watching through the window the 
grey roofs of the city suburb gradually giving place to the green 
meadows of the countryside. He picked up a newspaper and tried 
to interest himself in the Vietnam conflict. But it all seemed to be 
so dull, and the railway carriage rocked him pleasantly .... 

When he lowered the paper and looked out of the window 
again the landscape had changed considerably. The telegraph 
poles were so close to each other that they looked like a hedge, 
and the trees had extremely narrow crowns and were like Italian 
cypresses. Opposite to him sat his old friend the professor, look- 
ing through the window with great interest. He had probably 
got in while Mr Tompkins was busy with his newspaper. 

' We are in the land of relativity,' said Mr Tompkins, ' aren't we.'* ' 

* Oh ! ' exclaimed the professor, ' you know so much already ! 
Where did you learn it from.''' 

' I have already been here once, but did not have the pleasure of 
your company then.' 


' So you are probably going to be my guide this time,' the old 
man said. 

'I should say not,' retorted Mr Tompkins. 'I saw a lot of 
unusual things, but the local people to whom I spoke could not 
understand what my trouble was at all.' 

' Naturally enough,' said the professor. ' They are bom in this 
world and consider all the phenomena happening around them as 
self-evident. But I imagine they would be quite surprised if they 
happened to get into the world in which you used to live. It would 
look so remarkable to them.' 

'May I ask you a question.''' said Mr Tompkins. 'Last time 
I was here, I met a brakeman from the railway who insisted that 
owing to the fact that the train stops and starts again the passengers 
grow old less quickly than the people in the city. Is this magic, or 
is it also consistent with modern science?' 

'There is never any excuse for putting forward magic as an 
explanation,' said the professor. 'This follows directly from the 
laws of physics. It was shown by Einstein, on the basis of his 
analysis of new (or should I say as-old-as-the-world but newly 
discovered) notions of space and time, that all physical processes 
slow down when the system in which they are taking place is 
changing its velocity. In our world the effects are almost un- 
observably small, but here, owing to the small velocity of light, 
they are usually very obvious. If, for example, you tried to boil 
an egg here, and instead of letting the saucepan stand quietly on 
the stove moved it to and fro, constantly changing its velocity, it 
would take you not five but perhaps six minutes to boil it properly. 
Also in the human body all processes slow down, if the person is 
sitting (for example) in a rocking chair or in a train which changes 
its speed ; we live more slowly under such conditions. As, how- 
ever, all processes slow down to the same extent, physicists prefer 
to say that in a non-uniformly moving system time flows more slowly.^ 

'But do scientists actually observe such phenomena in our 
world at home.''* 


Mr Tompkins and Simultaneity 

'They do, but it requires considerable skill. It is technically 
very difficult to get the necessary accelerations, but the conditions 
existing in a non-uniformly moving system are analogous, or 
should I say identical, to the result of the action of a very large 
force of gravity. You may have noticed that when you are in an 
elevator which is rapidly accelerated upwards it seems to you that 
you have grown heavier; on the contrary, if the elevator starts 
downward (you realize it best when the rope breaks) you feel as 
though you were losing weight. The explanation is that the gravi- 
tational field created by acceleration is added to or subtracted 
from the gravity of the earth. Well, the potential of gravity on the 
sun is much larger than on the surface of the earth and all processes 
there should be therefore slightly slowed down. Astronomers do 
observe this.' 

'But they cannot go to the sun to observe it.-^' 

* They do not need to go there. They observe the light coming 
to us from the sun. This light is emitted by the vibration of dif- 
ferent atoms in the solar atmosphere. If all processes go slower 
there, the speed of atomic vibrations also decreases, and by com- 
paring the light emitted by solar and terrestrial sources one can 
see the difference. Do you know, by the way' — the professor 
interrupted himself — ' what the name of this little station is that 
we are now passing.'' ' 

The train was rolling along the platform of a little countryside 
station which was quite empty except for the station master and a 
young porter sitting on a luggage trolley and reading a news- 
paper. Suddenly the station master threw his hands into the air 
and fell down on his face. Mr Tompkins did not hear the sound of 
shooting, which was probably lost in the noise of the train, but the 
pool of blood forming round the body of the station master left 
no doubt. The professor immediately pulled the emergency cord 
and the train stopped with a jerk. When they got out of the 
carriage the young porter was running towards the body, and a 
country policeman was approaching. 


* shot through the heart,' said the policeman after inspecting the 
body, and, putting a heavy hand on the porter's shoulder, he went 
on: 'I am arresting you for the murder of the station master.' 

'I didn't kill him,' exclaimed the unfortunate porter. 'I was 
reading a newspaper when I heard the shot. These gentlemen from 
the train have probably seen all and can testify that I am innocent.' 

' Yes,' said Mr Tompkins, ' I saw with my own eyes that this 
man was reading his paper when the station master was shot. 
I can swear it on the Bible.' 

* But you were in the moving train,' said the policeman, taking 
an authoritative tone, 'and what you saw is therefore no evidence 
at all. As seen from the platform the man could have been shoot- 
ing at the very same moment. Don't you know that simultaneous- 
ness depends on the system from which you observe it.'* Come 
along quietly,' he said, turning to the porter. 

' Excuse me, constable,' interrupted the professor, ' but you are 
absolutely wrong, and I do not think that at headquarters they will 
like your ignorance. It is true, of course, that the notion of simul- 
taneousness is highly relative in your country. It is also true that 
two events in different places could be simultaneous or not, 
depending on the motion of the observer. But, even in your 
country, no observer could see the consequence before the cause. 
You have never received a telegram before it was sent, have you.-* 
or got drunk before opening the bottle.-^ As I understand you, you 
suppose that owing to the motion of the train the shooting would 
have been seen by us much later than its effect and, as we got out 
of the train immediately we saw the station master fall, we still had 
not seen the shooting itself. I know that in the police force you 
are taught to believe only what is written in your instructions, but 
look into them and probably you will find something about it.' 

The professor's tone made quite an impression on the police- 
man and, pulling out his pocket book of instructions, he started to 
read it slowly through. Soon a smile of embarrassment spread out 
across his big, red face. 


Mr Tompkins and Simultaneity 

'Here it is,' said he, 'section 37, subsection 12, paragraph e: 
"As a perfect alibi should be recognized any authoritative proof, 
from any moving system whatsoever, that at the moment of the 
crime or within a time interval ± cd (c being natural speed limit 
and ^the distance from the place of the crime) the suspect was seen 
in another place.'" 

* You are free, my good man,' he said to the porter, and then, 
turning to the professor: 'Thank you very much. Sir, for saving 
me from trouble with headquarters, I am new to the force and 
not yet accustomed to all these rules. But I must report the 
murder anyway,' and he went to the telephone box. A minute 
later he was shouting across the platform. 'All is in order now ! 
They caught the real murderer when he was running away from 
the station. Thank you once more ! ' 

'I may be very stupid,' said Mr Tompkins, when the train 
started again, 'but what is all this business about simultaneous- 
ness.'^ Has it really no meaning in this country .''' 

'It has,' was the answer, 'but only to a certain extent; other- 
wise I should not have been able to help the porter at all. You see, 
the existence of a natural speed limit for the motion of any body or 
the propagation of any signal, makes simultaneousness in our 
ordinary sense of the word lose its meaning. You probably will 
see it more easily this way. Suppose you have a friend living in a 
far-away town, with whom you correspond by letter, mail train 
being the fastest means of communication. Suppose now that 
something happens to you on Sunday and you learn that the same 
thing is going to happen to your friend. It is clear that you cannot 
let him know about it before Wednesday. On the other hand, if 
he knew in advance about the thing that was going to happen to 
you, the last date to let you know about it would have been the 
previous Thursday. Thus for six days, from Thursday to next 
Wednesday, your friend was not able either to influence your fate 
on Sunday or to learn about it. From the point of view of causality 
he was, so to speak, excommunicated from you for six days.' 


'what about a telegram?' suggested Mr Tompkins. 

* Well, I accepted that the velocity of the mail train was the 
maximum possible velocity, which is about correct in this 
country. At home the velocity of light is the maximum velocity 
and you cannot send a signal faster than by radio.' 

'But still,' said Mr Tompkins, 'even if the velocity of the mail 
train could not be surpassed, what has it to do with simultaneous- 
ness? My friend and myself would still have our Sunday dinners 
simultaneously, wouldn't we?' 

'No, that statement would not have any sense then; one ob- 
server would agree to it, but there would be others, making their 
observations from different trains, who would insist that you eat 
your Sunday dinner at the same time as your friend has his Friday 
breakfast or Tuesday lunch. But in no way could anybody 
observe you and your friend simultaneously having meals more 
than three days apart.' 

'But how can all this happen?' exclaimed Mr Tompkins un- 

'In a very simple way, as you might have noticed from my 
lectures. The upper limit of velocity must remain the same as 
observed from different moving systems. If we accept this we 
should conclude that . . . .' 

But their conversation was interrupted by the train arriving at 
the station at which Mr Tompkins had to get out. 


Rogers, a noted physics teacher, introduces the 
fundamental concepts of the theory of relativity 
and illustrates the relation of mathematics to 

Mathematics and Relativity 

Eric M. Rogers 

Chapter from his textbook. Physics for the Inquiring Mind, 1960. 

Mathematics as Language 

The scientist, collecting information, formulating 
schemes, building knowledge, needs to express him- 
self in clear language; but ordinary languages are 
much more vague and imreliable than most people 
think. "I love vegetables" is so vague that it is almost 
a disgrace to a civilized language — a few savage 
cries could make as full a statement. "A thermometer 
told me the temperatme of the bath water." Ther- 
mometers don't "tell." All you do is try to decide on 
its reading by staring at it — and you are almost cer- 
tainly a little wrong. A thermometer does not show 
the temperature of the water; it shows its own tem- 
perature. Some of these quarrels relate to the physics 
of the matter, but they are certainly not helped by 
the wording. We can make our statements safer by 
being more careful; but our science still emerges 
with wording that needs a series of explanatory 
footnotes. In contrast, the language of mathematics 
says what it means with amazing brevity and hon- 
esty. When we write 2x^ — 3x -f 1 = we make a 
very definite, though very dull, statement about x. 
One advantage of using mathematics in science is 
that we can make it write what we want to say with 
accuracy, avoiding vagueness and unwanted extra 
meanings. The remark "Au/A* = 32" makes a clear 
statement without dragging in a long, wordy de- 
scription of acceleration, y = 16t^ tells us how a 
rock falls without adding any comments on mass or 

Mathematics is of great use as a shorthand, both in 
stating relationships and in carrying out complicated 

arguments, as when we amalgamate several relation- 
ships. We can say, for uniformly accelerated motion, 
"the distance travelled is the sum of the product of 
initial velocity and time, and half the product of the 
acceleration and the square of the time," but it is 
shorter to say, "s = Oo* + ^ at^" If we tried to oper- 
ate with wordy statements instead of algebra, we 
should still be able to start viath two accelerated- 
motion relations and extract a third one, as when 
we obtained v^ = v^' -\- 2as in Chapter 1, Appendix 
A; but, without the compact shorthand of algebra, 
it would be a brain-twister argument. Going still 
further, into discussions where we use the razor- 
sharp algebra called calculus, arguing in words 
would be impossibly complex and cumbersome. In 
such cases mathematics is like a sausage-machine that 
operates with the rules of logical argument instead 
of wheels and pistons. It takes in the scientific in- 
formation we provide — facts and relationships from 
experiment, and schemes from our minds, dreamed 
up as guesses to be tried — and rehashes them into 
new form. Like the real sausage-machine, it does not 
always deliver to the new sausage all the material 
fed in; but it never delivers anything that was not 
supplied to it originally. It cannot manufacture 
science of the real world from its own machinations. 

Mathematics: the Good Servant 

Yet in addition to routine services mathematics 
can indeed perform marvels for science. As a lesser 
marvel, it can present the new sausage in a form 
that suggests further uses. For example, suppose 


you had discovered that falling bodies have a con- 
stant acceleration of 32 ft/sec/sec, and that any 
downward motion they are given to start vvath is just 
added to the motion gained by acceleration. Then 
the mathematical machine could take your experi- 
mental discovery and measurement of "g" and pre- 
dict the relationship 5 = v^^t -f %(32)f^ Now suppose 
you had never thought of including upward-thrown 
things in your study, had never seen a ball rise and 
fall in a parabola. The mathematical machine, not 
having been warned of any such restriction, would 
calmly offer its prediction as if unrestricted. Thus 
you might try putting in an upward start, giving Uq 
a negative value in the formula. At once the formula 
tells a different-looking story. In that case, it says, 

^-' ^^^ ^ ^3^^ 





Fic. 31-1. 

the stone would fly up slower and slower, reach a 
highest point, and then fall faster and faster. This 
is not a rash guess on the algebra's part. It is an 
unemotional routine statement. The algebra-ma- 
chine's defense would be, "You never told me v^ 
had to be downward. I do not know whether the 
new prediction is right. All I can say is that IF an 
upward throw follows the rules I was told to use for 
downward throws, THEN an upward throum hall 
will rise, stop, fall." It is we who make the rash guess 
that the basic rules may be general. It is we who 
welcome the machine's new hint; but we then go out 
and try it.' To take another example from projectile 
mathematics, the following problem, which you met 
earlier, has two answers. 

Problem : 

"A stone is thrown upward, 

with initial speed 64 ft/sec, 

at a bird in a tree. How 

long after its start will the 

stone hit the bird, which is . ■ 

48 feet above the thrower?" // t v 


1 second or 3 seconds. 




Fic. 31-2. 

This shows algebra as a very honest, if rather dumb, 
servant. There are two answers and there should 
be, for the problem as presented to the machine. 
The stone may hit the bird as it goes up ( 1 sec from 
start), or as it falls down again (after 3 sees). 
The machine, if blamed for the second answer 
would complain, "But you never told me the stone 
had to hit the bird, still less that it must hit it on the 
way up. I only calculated when the stone would be 
48 feet above the thrower. There are two such 
times." Looking back, we see we neither wrote any- 
thing in the mathematics to express contact between 
stone and bird nor said which way the stone was to 
be moving. It is our fault for giving incomplete in- 
structions, and it is to the credit of the machine that 
it politely tells us all the answers which are possible 
within those instructions. 

If the answer to some algebra problem on farm- 
ing emerges as 3 cows or 2% cows, we rightly reject 
the second answer, but we blame ourselves for not 
telling the mathematical machine an important fact 
about cows. In physics problems where several 
answers emerge we are usually unwise to throw 
some of them away. They may all be quite true; or, 
if some are very queer, accepting them provisionally 
may lead to new knowledge. If you look back at the 
projectile problem, No. 7 in Chapter 1, Appendix B, 
you may now see what its second answer meant. 
Here is one like it: 


A man throws a stone down 
a well which is 96 feet deep. 
It starts with downward 
velocity 16 ft/sec. When 
will it reach the bottom? 

Fig. 31-4. 

' Tliis is a simple example, chosen to use physics you are 
familiar with — unfortunately so simple that you know the 
answer before you let the machine suggest it. There are many 
cases where the machine can produce suggestions that are 
quite unexpected and do indeed send us rushing to experi- 
ment. E.g., mathematical treatment of the wave theory of 
light suggested that when light casts a sharp shadow of a 
disc there will be a tiny bright spot of light in the middle 
of the shadow on a wall: "There is a hole in every coin." 

Point spune 


Fic. 31-3. 



Mathematics and Relativity 

Assign suitable + and — signs to the data, substi- 
tute them in a suitable relation for free fall, and 
solve the equation. You will obtain two answers: 
One a sensible time with -|- sign (the "right" an- 
swer), the other a negative time. Is the negative 
answer necessarily meaningless and silly? A time 
such as "—3 seconds" simply means, "3 seconds be- 
fore the clock was started." The algebra-machine is 
not told that the stone loas flung down by the man. 
It is only told that when the clock started at zero 
the stone was moving DOWN with speed 16 ft/sec, 
and thereafter fell freely. For all the algebra knows, 
the stone may have just skimmed through the man's 
hand at time zero. It may have been started much 
earlier by an assistant at the bottom of the well who 
hurled it upward fast enough to have just the right 
velocity at time zero. So, while our story runs, 
"George, standing at the top of the well, hurled the 
stone down . . . ," an answer — 3 seconds suggests an 
alternative story: "Alfred, at the bottom of the well, 
hurled the stone up with great speed. The stone rose 
up through the well and into the air above, with 
diminishing speed, reached a highest point, fell with 
increasing speed, moving down past George 3 sec- 
onds after Alfred threw it. George missed it (at 
f =r 0), so it passed him at 16 ft/sec and fell on 
down the well again." According to the algebra, the 
stone will reach the bottom of the well one second 
after it leaves George, and it might have started 
from the bottom 3 seconds before it passes George. 
Return to Problem 7 of Chapter 1, Appendix B 
and try to interpret its two answers. 


Problem 7: 

A man standing on the top 
of a tower throws a stone 
up into the air with initial 
velocity 32 feet/sec up- 
ward. The man's hand is 
48 feet above the ground. 

How long wall the stone //)////////7///////7'A 
take to reach the ground? 

In these problems mathematics shows itself to 
be the completely honest servant — rather like the 
honest boy in one of G. K. Chesterton's "Father 
Brown" stories. (There, a slow-witted village lad 
delivered a telegram to a miser. The miser meant to 
tip the boy with the smallest English coin, a bright 
bronze farthing (%(*), but gave him a golden pound 
($3) by mistake. What was the boy to do when he 
discovered the obvious mistake? Keep the pound, 
trading on the mistake dishonestly? Or bring it back 
with unctuous virtue and embarrass the miser into 

saying "Keep it, my boy"? He did neither. He simply 
brought the exact change, 19 shillings and 11% 
pence. The miser was delighted, saying, "At last I 
have found an honest man"; and he bequeathed to 
the boy all the gold he possessed. The boy, in 
wooden-headed honesty, interpreted the miser's will 
literally, even to the extent of taking gold fillings 
from his teeth.) 

Mathematics: the Clever Servant 

As a greater marvel, mathematics can present the 
new sausage in a form that suggests entirely new 
viewpoints. With vision of genius the scientist may 
see, in something new, a faint resemblance to some- 
thing seen before — enough to suggest the next step 
in imaginative thinking and trial. If we tried to do 
without mathematics we should lose more than a 
clear language, a shorthand script for argument and 
a powerful tool for reshaping information. We 
should also lose an aid to scientific vision on a higher 

With mathematics, we can codify present science 
so clearly that it is easier to discover the essential 
simplicity many of us seek in science. That is no 
crude simplicity such as finding all planetary orbits 
circles, but a sophisticated simplicity to be read 
only in the language of mathematics itself. For ex- 
ample, imagine we make a hump in a taut rope by 
slapping it (Fig. 31-6). Using Newton's Law II, we 

Newton IT „ .^ Itc nsio 

X. (jeometry 

Fig. 31-6. Wave Travels Along a Rope 

can codify the behavior of the hump in compact 
mathematical form. There emerges, quite uninvited, 
the clear mathematical trademark of wave motion.* 
The mathematical form predicts that the hump will 
travel along as a wave, and tells us how to compute 
the wave's speed from the tension and mass of the 

• The wave-equation reduces to the essential form: 
V^V = (1/c') d'V/dt- 
For an\j wave of constant pattern that travels with speed c. 
( If you are familiar with calculus, ask a physicist to show 
you this remarkable piece of general mathematical physics. ) 
This equation connects a spreading-in-space with a rate-of- 
change in time. V'V would be zero for an inverse-square 
field at rest in space: but here it has a value that looks like 
some acceleration. In the electromagnetic case, we may trace 
the dy Idf back to an accelerating electron emitting the 


rope. Another example: A century ago, Maxwell re- 
duced the experimental laws of electromagnetism to 
especially simple forms by boiling them down math- 
ematically. He removed the details of shape and 
size of apparatus, etc., much as we remove the shape 
and size of the sample when we calculate the den- 
sity of a metal from some weighing and measuring. 
Having thus removed the "boundary conditions," he 
had electrical laws that are common to all apparatus 
and all circumstances, just as density is common to 
all samples of the same metal. His rules were boiled 
down by the calculus-process of differentiation to a 
final form called differential equations. You can in- 
spect their form without understanding their termi- 
nology. Suppose that at time t there are fields due to 
electric charges and magnets, whether moving or 
not; an electric field of strength E, a vector with 
components £,, E^, E^, and a magnetic field H with 
components H^, H^, H,. Then, in open space (air 
or vacuum ), the experimental laws known a century 
ago reduce to the relations shown in Fig. 31-7. 




di^ dz- 

dl' dz' = ' 

dH^ dUy dH, 
dx (iy iz 




dzj " " dt 



dx J " dt 



dEA dH,, 
dxf J' " dt 


Fic. 31-7. Maxwell's Equations (incomplete) 
The constant Ka relates to magnetic fields. It appears 
in the expression for the force exerted by a magnetic 

field on an electric current. ( See the discussion in 

this chapter and in Ch. 37. ) There is a corresponding 

electric constant, Kb, which appears in 

Coulomb's Law (See Ch. 33). 

Look at IV and compare it with HI. The equations 
of IV look incomplete, spoiling the general sym- 
metry.' Maxwell saw the defect and filled it by in- 
venting an extra electric current, a spookv one in 
space, quite unthought-of till then, but later ob- 

' In completing IV, you will need to insert a constant K% 
corresponding to Kh in III. The minus sign is obviously un- 
necessary in the present form of IV, and when IV is com- 
pleted it will spoil the symmetry somewhat; but the experi- 
mental facts produce it, conservation of energy requires it, 
and without it there would be no radio waves. 

served experimentally. How would i/cm change IV 
to match III if told that part of the algebra had been 
left out because it was then unknown? Try this. 

The addition was neither a lucky guess nor a 
mysterious inspiration. To Maxwell, fuUy aware of 
the state of developing knowledge, it seemed covtv- 
pulsory, a necessary extension of symmetry — that is 
the difference between the scientific advance of the 
disciphned, educated expert and the free invention 
of the enthusiastic amateur. 

Having made his addition, fantastic at the time, 
Maxwell could pour the whole bunch of equations 
into the mathematical sausage-machine and grind 
out a surprising equation which had a familiar look, 
the same trademark of wave-motion that appears for 
a hump on a rope. That new equation suggested 
strongly that changes of electric and magnetic fields 
would travel out as waves with speed v = l/y/K^K^. 
Here Ke is a constant involved in the magnetic 
effects of moving charges; and Ke is the correspond- 
ing electrostatic constant inserted by Maxwell in his 
improvement.' (Ke is involved in the inverse-square 
force between electric charges. ) 

An informal fanciful derivation is sketched near 
the end of Chapter 37. 

To Maxwell's delight and the wonder of his con- 
temporaries, the calculated v agreed with the speed 
of light, which was already known to consist of 
waves of some sort. This suggested that light might 
be one form of Maxwell's predicted electromagnetic 

It was many years before Maxwell's prediction 
was verified directly by generating electromagnetic 
waves with electric ciurents. As a brilliant intuitive 
guess, a piece of synthetic theory, Ma.xwell's work 
was one of the great developments of physics — its 
progeny, new guesses along equally fearless lines, 
are making the physics of todav. 

One of the great contributions of mathematics to 
physics is Relativity, which is both mathematics and 
physics: you need good knowledge of both mathe- 
matics and physics to understand it. We shall give 
an account of Einstein's "Special Relativity" and 
then return to comments on mathematics as a 

' In this course we use diflFerent s\Tnbols. See Ch. 33 
and Ch. 37. We write the force between electric charges 
F = B (Qi Q,)ld'. Comparison with Maxwell's form shows 
our B is the same as 1/Ke. Again, we write the force between 
two short pieces of current-carrying wire, due to magnetic- 
field effects, F = B' (C, L,) (C, U)/d; and our B' is the 
same as Kb. Then Maxwell's prediction, u = 1 /\'K» Ka , be- 
comes, in our tenninoli)g\-. t> = 1/\(1/B) (B') = \'B/B' . 
So, if you measure B and B' you can predict the speed of 
electromagnetic waves. The arithmetic is easy. Try it and com- 
pare the result with the measured speed of light, 3.0 X 10* 
meters/sec. (B = 9.00 X 10* and B' = 10 -^ in our units). 


Mathematics and Relativity 


The theory of Relativity, which has modified our 
mechanics and clarified scientific thinking, arose 
from a simple question: "How fast are we moving 
through space?" Attempts to answer that by experi- 
ment led to a conflict that forced scientists to think 
out their system of knowledge afresh. Out of that 
reappraisal came Relativity, a brilliant apphcation 
of mathematics and philosophy to our treatment of 
space, time, and motion. Since Relativity is a piece 
of mathematics, popular accounts that try to explain 
it without mathematics are almost certain to fail. 
To understand Relativity you should either follow 
its algebra through in standard texts, or, as here, 
examine the origins and final results, taking the 
mathematical machine-work on trust. 

What can we find out about space? Where is its 
fixed framework and how fast are we moving 
through it? Nowadays we find the Copernican view 
comfortable, and picture the spinning Earth moving 
around the Sun with an orbital speed of about 70,000 
miles/hour. The whole Solar system is moving to- 
wards the constellation Hercules at some 100,000 
miles/hour, while our whole galaxy. . . . 

We must be careering along a huge epicycloid 
through space without knowing it. Without know- 
ing it, because, as GaUleo pointed out, the mechan- 
ics of motion — projectiles, collisions, . . . , etc. — is 
the same in a steadily moving laboratory as in a 
stationary one.* Galileo quoted thought-e.xperiments 
of men walking across the cabin of a sailing ship 
or dropping stones from the top of its mast. We il- 
lustrated this "Galilean relativity" in Chapter 2 by 
thought-experiments in moving trains. Suppose one 
train is passing another at constant velocity without 
bumps, and in a fog that conceals the countryside. 
Can the passengers really say which is moving? Can 
mechanical experiments in either train tell them? 
They can onlv observe their relative motion. In fact, 
we developed the rules of vectors and laws of mo- 
tion in earthly labs that are moving; yet those state- 
ments show no effect of that motion. 

We give the name inertial frame to anv frame of 
reference or laboratory in which Newton's Laws 

* Though the Earth's velocity changes around its orbit, we 
think of it as steady enough during any short experiment. In 
fact, the steadiness is perfect, because any changes in the 
Earth's velocity exactly compensate the effect of the Sun's 
gravitation field that "causes" those changes. We see no 
effect on the Earth as a whole, at its center; but we do see 
differential effects on outlying parts — solar tides. The Earth's 
rotation does produce effects that can be seen and measured 
— Foucault's pendulum changes its line of swing, g shows 
differences between equator and poles, &c. — but we can 
make allowances for these where they matter. 

seem to describe nature truly: objects left alone 
without force pursue straight lines with constant 
speed, or stay at rest; forces produce proportional 
accelerations. We find that any frame moving at 
constant velocity relative to an inertial frame is also 
an inertial frame — Newton's Laws hold there too. 
In all the following discussion that concerns GaU- 
lean relativity and Einstein's special Relativity, we 
assume that every taborutory we discuss is an in- 
ertial frame — as a laboratory at rest on Earth is, to 
a close approximation.* In our later discussion of 
General Relativity, we consider other laboratory 
frames, such as those which accelerate. 

We are not supplied by nature with an obvious 
inertial frame. The spinning Earth is not a perfect 
inertial frame (because its spin imposes central ac- 
celerations), but if we could ever find one perfect 
one then our relativity view of nature assures us we 
could find any number of other inertial frames. 
Every frame moving with constant velocity relative 
to our first inertial frame proves to be an equally 
good inertial frame — Newton's laws of motion, 
which apply by definition in the original frame, 
apply in all the others. When we do experiments on 
force and motion and find that Newton's Laws seem 
to hold, we are, from the point of view of Relativity, 
simply showing that our earthly lab does provide 
a practically perfect inertial frame. Any experiments 
that demonstrate the Earth's rotation could be taken 
instead as showing the imperfection of our choice of 
frame. However, by saying "the Earth is rotating" 
and blaming that, we are able to imagine a perfect 
frame, in which Newton's Laws would hold exactly. 

We incorporate Galilean Relativity in our formu- 
las. When we write, s ^= vj -j- ^iat^ for a rocket ac- 
celerating horizontally, we are saying, "Start the 
rocket with v^, and its efi^ect will persist as a plain 
addition, vj, to the distance travelled." 

Fic. 31-8. 

This can be reworded: "An experimenter e, starts a 
rocket from rest and observes the motion: s = Hat^. 
Then another experimenter, e', running away with 
speed Va will measure distances-travelled given by 
s' = vj + l^t-. He will include Vgt due to his own 

We are saying that the effects of steady motion 


and accelerated motion do not disturb each other; 
they just add. 

e and C" have the following statements for the 
distance the rocket travels in time t. 


5 = ^^t- 


Ns' r= Cot + 'Aat^ 

Both statements say that the rocket travels with 
constant acceleration.' 

Both statements say the rocket is at distance zero 
( the origin ) att = 0. 

The first statement says e sees the rocket start 
from rest. When the the clock starts at t = the 
rocket has no velocity relative to him. At that instant, 
the rocket is moving with his motion, if any — so he 
sees it at rest — and he releases it to accelerate. 

The difference between the two statements says 
the relative velocity between e and e' is t>o. There 
is no information about absolute motion, e may be 
at rest, in which case e' is running backward with 
speed Vo- Or e' may be at rest, and e running for- 
ward Vo ( releasing the rocket as he runs, at t = ) . 
Or both e and e' may be carried along in a moving 
train with terrific speed V, still with e moving ahead 
vidth speed t>o relative to e'. In every case, v^ is the 
relative velocity between the observers; and nothing 
in the analysis of their measurements can tell us 
(or them) who is "really" moving. 



rjTj — 

'III/ 1 / II ////// n// / //////////////J y /////// 

Fic. 31-9. 

" The first statement is simpler because it belongs to the 
observer who releases the rocket from rest relative to him, at 
the instant the clock starts, t = 0. 








t ' 

X * 



Adding n^^ only shifts the graph of s vs. t. It does 
not affect estimates of acceleration, force, etc. Then, 
to tfie question, "How fast are we moving through 
space?" simple mechanics replies, "No experiments 
with weights, springs, forces, . . . , can reveal our ve- 
locity. Accelerations could make themselves known, 
but uniform velocity would be unfelt." We could 
only measure our relative velocity — relative to some 
other object or material framework. 

Each crnv 
iflifS otfvfr 
OTIV is 
tMvma ymt 
at speed V 


Fic. 3i-Il. 

Observers in two laboratories, one moving with constant 

velocity d relative to the other, will find the 

same mechanical laws. 

Yet we are still talking as if there is an absolute 
motion, past absolute landmarks in space, however 
hard to find. Before exploring that hope into greater 
disappointments, v/e shall codify rules of relative 
motion in simple algebraic form. 


Mathematics and Relativity 

Galilean Transformation for Coordinates 

We can put the comparison between two such 
observers in a simple, general way. Suppose an ob- 
server e records an event in his laboratory. Another 


Fig. 31-12a 

Observer ready to observe an event at time t and 

place I, y, z. 

observer, e', flies through the laboratory with con- 
stant velocity and records the same event as he goes. 
As sensible scientists, e and e' manufactiu-e identical 
clocks and meter-sticks to measure with. Each car- 
ries a set of x-y-z-axes with him. For convenience, 
they start their clocks (t = and f -— 0) at the 
instant they are together. At that instant their co- 
ordinate origins and axes coincide. Suppose £ re- 
cords the event as happening at time t and place 
{x, y, z) referred to his axes-at-rest-with-him.' The 
same event is recorded by observer e' using his in- 
struments as occurring at f and (x', y', z') referred 
to the axes-he-carries-with-him. How will the bA'o 
records compare? Common sense tells us that time 


Fig. 31-12b. 

Another observer, moving at constant velocity o 

relative to the first, also makes observations. 

is the same for both, so f' = f. Suppose the relative 
velocity between the two observers is t; meters/sec 

'For example: he files a bullet along OX from the origin 
at t = with speed 1000 m./sec. Then the event of the 
bullet reaching a target 3 meters away might be recorded 
as X = 3 meters, y = 0,z = 0,t := 0.003 sec. 

along OX. Measurements of y and z are the same 
for both: y' z= y and z' = z. But since e' and his 
coordinate framework travel ahead of £ by vt meters 
in t seconds, all his x'-measurements will be vt 
shorter. So every x' must = x — vt. Therefore: 

x' z= X — vt v' = y z' ^ z f = t 


Fig. 31-12C. 

For measurements along direction of relative motion v, 

the second observer measures x'; the first measures x. 

Then it seems obvious that x' ■=. x — vt. 

These relations, which connect the records made by 
£' and £, are called the Galilean Transformation. 

The reverse transformation, connecting the rec- 
ords of £ and £', is: 

x = x' -{-vt y = y' z = z' t = t' 

These two transformations treat the two observers 
impartially, merely indicating their relative velocity, 
-f- u for e' — £ and — o for £ — e'. They contain our 
common-sense knowledge of space and time^ written 
in algebra. 

Velocity of Moving Object 

If £ sees an object moving forward along the x 
direction, he measures its velocity, u, by Ax/ At. 
Then e' sees that object moving with velocitv u' 
given by his Ax'/Af. Simple algebra, using the 
Gahlean Transformation, shows that t/ = u — v. 
(To obtain this relation for motion with constant 
velocity, just divide x' = x — vthyt.) For example: 
suppose £ stands beside a railroad and sees an 
express train moving with u = 70 miles/hour. An- 
other observer, e', rides a freight train moving 30 
miles/hour in the same direction. Then e' sees the 
express moving with 

«' =r u — o = 70 — 30 = 40 miles/hour. 

(If e' is mo'.ong the opposite way, as in a head-on 
collision, t; = —30 miles/hour, and e' sees the ex- 
press approaching with speed 

W = 70 — (-50) = 100 miles/hour.) 


Ax wv tunc M 

Fic. 31-13. 

Each experimenter calculates the velocity of a moving 

object from his observations of time taken and 

distance travelled. 

Fig. 31-14. 

Stationary experimenter £ observes the velocities shown 

and calculates the relative velocity that moving 

experimenter e' should observe. 

This is the "common sense" way of adding and 
subtracting velocities. It seems necessarily true, and 
we have taken it for granted in earlier chapters. Yet 
we shall find we must modify it for very high speeds. 

PAbsolute Motion? 

If we discover our laboratory is in a moving train, 
we can add the train's velocity and refer our experi- 
ments to the solid ground. Finding the Earth mov- 
ing, we can shift our "fixed" axes of space to the Sun, 
then to a star, then to the center of gravity of all 
the stars. If these changes do not affect our knowl- 
edge of mechanics, do they really matter? Is it 
honest to worry about finding an absolutely fixed 
framework? Curiosity makes us reply, "Yes. If we 
are moving through space it would be interesting 
to know how fast." Though mechanical experiments 
cannot tell us, could we not find out by electrical 
experiments? Electromagnetism is summed up in 
Maxwell's equations, for a stationary observer. Ask 

what a moving observer should find, by changing 
X to x', etc., with the Galilean Transformation: then 
Maxwell's equations take on a different, more com- 
plicated, form. An experimenter who trusted that 
transformation could decide which is really moving, 
himself or his apparatus: absolute motion would be 
revealed by the changed form of electrical laws. 
An easy way to look for such changes would 
be to use the travelling electric and magnetic fields 
of light waves — the electromagnetic waves pre- 
dicted by Maxwell's equations. We might find our 
velocity through space by timing flashes of light. 
Seventy-five years ago such experiments were being 
tried. When the experiments yielded an unexpected 
result — failure to show any effect of motion — there 
were many attempts to produce an explanation. 
Fitzgerald in England suggested that whenever any 
piece of matter is set in motion through space it 
must contract, along the direction of motion, bv a 
fraction that depended only on its speed. With the 
fraction properly chosen, the contraction of the 
apparatus used for timing light signals would pre- 
vent their reveaUng motion through space. This 
strange contraction, which would make even meas- 
uring rods such as meter-sticks shrink hke everv- 
thing else when in motion, was too surprising to be 
welcome; and it came with no suggestion of mecha- 
nism to produce it. Then the Dutch physicist Lo- 
rentz (also Larmor in England) worked out a suc- 
cessful electrical "explanation." 

The Lorentz Transfomuttion 

Lorentz had been constructing an electrical theory 
of matter, with atoms containing small electric 
charges that could move and emit hght waves. The 
experimental discover)' of electron streams, soon 
after, had supported his speculations; so it was 
natural for Lorentz to try to explain the unex- 
pected result with his electrical theory. He found 
that if Maxwell's equations are not to be changed in 
form by the motion of electrons and atoms of mov- 
ing apparatus, then lengths along the motion must 
shrink, in changing from x to x', by the modifying 






He showed thai this shrinkage (the same as Fitz- 
gerald's) of the apparatus would just conceal any 
motion through absolute space and thus explain the 
experimental result. But he also gave a reason for 
the change: he showed how electrical forces — in the 


Mathematics and Relativity 

new form he took for Maxwell's equations — would 
compel the shrinkage to take place. 

It was uncomfortable to have to picture matter in 
motion as invisibly shrunk — invisibly, because we 
should shrink too — but that was no worse than the 
previous discomfort that physicists with a sense of 
mathematical form got from the uncouth effect of 
the Galilean Transformation on Maxwell's equa- 
tions. Lorentz's modifying factor has to be apphed 
to f as well as i', and a strange extra term must be 
added to f. And then Maxwell's equations maintain 
their same simple symmetrical form for all observ- 
ers moving with any constant velocity. You will see 
this "Lorentz Transformation'' put to use in Rela- 
tivity; but first see how the great experiments were 
made with light signals. 

Measuring Our Speed through "Space"? 

A century ago, it was clear that light consists of 
waves, which travel with very high speed through 
glass, water, air, even "empty space" between the 
stars and us. Scientists imagined space filled with 
"ether"' to carry light waves, much as air carries 
sound waves. Nowadays we think of light (and all 
other radio waves ) as a travelling pattern of electric 
and magnetic fields and we need no "ether"; but be- 
fore we reached that simple view a tremendous 
contradiction was discovered. 

Experiments with light to find how fast we are 
moving through the "ether" gave a surprising result: 
"no comment." These attempts contrast with suc- 
cessful measurements with sound waves and air. 

Sound travels as a wave in air. A trumpet-toot is 
handed on by air molecules at a definite speed 
through the air, the same speed whether the trum- 
pet is moving or not. But a moving observer finds 
his motion added to the motion of sound waves. 
When he is running towards the trumpet, the toot 
passes by him faster. He can find how fast he is 
moving through air by timing sound signals passing 


n n/iTj Dii iinnniuiuinnii iin'iujinn 

Fic. 31-15. 

Experimenter running towards source of sound finds the 

speed of sound 1120 ft/sec, in excess of normal 

by his own speed. 

^ This ether or a?ther was named after the universal sub- 
stance that Greek philosophers had pictured filling all space 
beyond the atmosphere. 

A moving observer wiU notice another effect if 
he is out to one side, listening with a direction- 
finder. He will meet the sound slanting from a new 


Fic. 31-16. 

Observer running across the line-of-travel of sound 

notices a change of apparent direction of source. 

direction if he nms. Again he can estimate his run- 
ning speed if he knows the speed of sound. 

In either case, his measurements would tell him 
his speed relative to the air. A steady wind blowing 
would produce the same effects and save him the 
trouble of running. Similar experiments with light 
should reveal our speed relative to the "ether," 
which is our only remaining symbol of absolute 
space. Such experiments were tried, with far- 
reaching results. 

Aberration of Starlight 

Soon after Newton's death, the astronomer Brad- 
ley discovered a tiny yearly to-and-fro motion of 
all stars that is clearly due to the Earth's motion 
around its orbit. Think of starhght as rain shower- 
ing down (at great speed) from a star overhead. If 
you stand in vertical rain holding an umbrella up- 
right, the rain will hit the umbrella top at right 
angles. Drops falling through a central gash will hit 
your head. Now run quite fast. To you the rain will 
seem slanting. To catch it squarely you must tilt 
the umbrella at the angle shown by the vectors in the 
sketch. Then drops falling through the gash will still 
hit your head. If you run around in a circular orbit, 
or to-and-fro along a line, you must wag the um- 
brella this way and that to fit your motion. This is 
what Bradley found when observing stars precisely 
with a telescope." Stars near the ecliptic seemed to 
slide to-and-fro, their directions swine;ing through 
a small angle. Stars up near the pole of the ecliptic 

' This aberration is quite distinct from parallax, the ap- 
parent motion of near stars against the background of re- 
moter stars. Aberration makes a star seem to move in the 
same kind of pattern, but it applies to all stars; and it is 
dozens of times bigger than the parallax of even the nearest 
stars. (, a star's aberration, which gins with thr F.artli's 
velocity, is three months out of phase with its parallax. ) 


f?> "fvS. <S- 

^ 1' 




Vofocify ijf mnncr 


I ' ,;• 

move in small circles in the course of a year. The 
telescope following the star is like the tilting um- 
brella. In six months, the Earth's velocity around 
the Sun changes from one direction to the reverse, 
so the telescope tilt must be reversed in that time. 
From the tiny measured change in 6 months, Brad- 
ley estimated the speed of light. It agreed with the 
only other estimate then available — based on the 
varying delays of seeing eclipses of Jupiter's moons, 
at varying distances across the Earth's orbit.' 

Man standi ittd 
' / in ww\i ' / / 

^A Velac^j oj 



Fic. 31-17. "Aberration" of R.'MN 

of 1VI 

Fig. 31-18. "Aberration" of Rain Falling in Wind 
If you stand still but a steady wind carries the air 
past you, you should still tilt the umbrella. 

To catch rain drops fair and square, you must 
tilt your umbrella if you are running or if there 
is a steady wind, but not if you are running and 
there is also a wind carrying the air and raindrops 
along with you — if you just stand in a shower inside 
a closed railroad coach speeding along, you do not 
tilt the umbrella. Therefore, Bradley's successful 
measurement of aberration showed that as the Earth 
runs around its orbit it is moving through the "ether" 
in changing directions, moving through space if you 
like, nearly 20 miles/sec. 

An overall motion of the solar system towards 
some group of stars would remain concealed, since 
that would give a permanent slant to star directions, 

' It was another century before terrestrial experiments 


(~ 1600): Galileo recorded an attempt with experimenters 
signalling by lantern flashes between two moun- 
tain tops. E' sent a flash to ej who immediately 
returned a flash to Ei. At first ej was clumsy and 
they obtained a medium speed for light. As they 
improved with practice, the estimated speed 
grew greater and greater, towards "infinity" — 
light travels too fast to clock by hand. 

(~ 1700): Newton knew only Roemer's estimate from Jupi- 
ter's moons. 
(1849): Fizeau succeeded, by using a distant mirror to 
return the light and a spinning toothed wheel as 
a chopper to make the flashes and catch them 
one tooth later on their return. His result con- 
firmed the astronomical estimate. His and all 
later terrestrial methods use some form of 
chopper — as in some methods for the speeds of 
bullets, and electrons. 
Tlie result: speed of light is 300,000,000 meters/sec or 

186,000 miles /sec. 


Mathematics and Relativity 

"Partuia" of STARLIGHT comt in vertiialli) 
I I I I I > 


: I 
' [ ' 1 

mmviq aSma orfir around sun 


Fig. 31-19. Aberration of Starlight 

whereas Bradley measured changes of slant from 
one season to anotlier. 

The Michehon-Morley Experiment 

Then, seventy-five years ago, new experiments 
were devised to look for our absolute motion in 
space. One of the most famous and decisive was 
devised and carried out by A. A. Michelson and 
E. W. Morley in Cleveland; this was one of the first 
great scientific achievements in modem physics in 
the New World. In their experiment, two flashes of 
light travelling in different directions were made to 
pace each other. There was no longer a moving 
observer and fixed source, as with Bradley and a 
star. Both source and observer were carried in a 
laboratory, but the experimenters looked for motion 
of the intervening ether that carried the light waves. 


HaCf- iUvemC mUrvr 







Fig. 31-20. The Michelson-Mobley Experiment 

A semi-transparent mirror split the light into two 
beams, one travelling, say, North-South and the 
other East-West. The two beams were returned 
along their paths by mirrors and rejoined to form an 
interference pattern. The slightest change in trip- 
time for one beam compared with the other would 
shift the pattern. Now suppose at some season the 
whole apparatus is moving upward in space: an 
outside observer would see the light beams tilted up 
or down by the "ether-wind" the same tilt for both 
routes. At another season, suppose the whole Earth 
is moving due North horizontally in space, then the 
N-S light beam would take longer for its round trip 
than the E-W one. You wiU find the experiments 
described in standard texts, with the algebra to 
show that if the whole laboratory is sweeping 
through the ether, light must take longer on the 
trip along the stream and back than on the trip 
across and back. 

You can see that this is so in the following ex- 
ample. Instead of light, consider a bird flying across 
a cage and back, when the cage is moving relative 


Fig. 31-21. Giant Birdcage fn Wind 

•1- • o * ' 



: •^i^ > 


■ 'tl ■ 

-^ — 


• • • • 

Fig. 31-22. 
Bird flies either against the wind and back; 
or across the wind and back across the wind. 

to the air. Either (a) drag the cage steadily along 
through still air, or ( b ) keep the cage still and have 
an equal wind blow through it the opposite way. 
We shall give the wind version; but you can re-tell 
the story for a moving cage, with the same results. 
Suppose the bird has air-speed 5 ft/sec, the cage is 
40 ft square, and the wind blows through at 3 ft/sec. 
To fly across-stream from side to side and back takes 




^ Jjt/i« 

^ .y 


r^ I^ vWINO 



' 1 - 

iji /sec 








y^sir- — ■ — 



„^ — , _^ _—- 

Fig. 31-23. 

Cage moving 3 ft/sec through still air has same eflFect 

on bird's flight as wind blowing 3 ft/sec through 

stationary cage. 

the bird 10 sec + 10 sec,* or 20 sec for the round 
trip. To fly from end to end, upstream and back, takes 

40 ft 40 ft 

(5 — 3)ft/sec (5 + 3)ft/sec 

or 20 sec + 5 sec, a much longer time.'" Put a bird 
in a cage like this and compare his round-trip times 
E-W and N-S, and you will be able to tell how fast 
the cage is moving through the air; or use twin birds 
and compare their returns. Twist the cage to dif- 
ferent orientations, and returns of the tvdns will 
tell you which way the cage is travelling through 
air and how fast. A similar experiment with sound 
waves in an open laboratory moving through air 
would tell us the laboratory's velocity. Let a trum- 
peter stand in one corner and give a toot. The ar- 
rivals of returning echoes will reveal general motion, 
of lab or wind. (Of course, if the moving laboratory 

" This requires some geometrical thinking. The bird must 
fly a 50-ft hypotenuse to cross the 40-ft cage while the 
wind carries him 30 ft downstream. The simple answer 
8 -|- 8 sec, which is incorrect, is even shorter. 

(rt) ; : 

Bird jUes ^ ^ i ' ■' ; WIND 

5-jr/5fc — < ■_ -' -^ 

m cur -^T = ■f^\it:t?*3)Jr,sa'^ j f^ yc 

< ^oji > 

(b) Birds ixfw of - — -j. 

(n>5i■fC^o(\^ . ' \ WIND 
x/ I \Jy <^ 

/ • - ^ _ _ N . 

/Air rnoti(fy\ rcdnUi 
r<(atiMC to caac ^ 

Fic. 31-24. Details of Flights 
Bird flies 5 ft/sec. Steady wind 3 ft/sec. 

>o If you are still not convinced and feel sure the trips up 
and downstream should average out, try a thought-expieri- 
ment with the wind blowing faster, say 6 ft /sec. Then the 
bird could never make the trip upstream — that time would 
be infinite! 


: ■ I I ■ . 


Mathematics and Relativity 

is closed and carries its air with it, the echoes will 
show no motion.) 

The corresponding test with light-signals is diflB- 
cult, but the interference pattern affords a very deli- 
cate test of trip-timing. When it was tried by Michel- 
son and Morley, and repeated by Miller, it gave a 
surprising answer: no motion through the "ether." 
It was repeated in different orientations, at different 
seasons: always the same answer, no motion. If 
you are a good scientist you will at once ask, "How 
big were the enor-boxes? How sensitive was the 
experiment?" The answer: "It would have shown 
reliably Vi of the Earth's orbital speed around the 
Sun, and in later'' work, Vw. Yet aberration shows 
us moving through the "ether" with ^%o of that 
speed. Still more experiments added their testimony, 
some optical, some electrical. Again and again, the 
same "null result." Here then was a confusing con- 

formation, electrical experiments would show rela- 
tive velocity (as they do), but would never reveal 
uniform absolute motion. But then the Lorentz 
Transformation made mechanics suffer; it twisted 
F= Ma and 5 r= Uo + ^^t^ into unfamiliar forms 
that contradicted Galileo's common-sense relativity 
and Newton's simple law of motion. 

Some modifications of the Michelson-Morley ex- 
periment rule out the Fitzgerald contraction as a 
sufiBcient "explanation." For example, Kennedy and 
Thomdike repeated it with unequal lengths for the 
two perpendicular trips. Their null result requires 
the Lorentz change of time-scale as well as the 
shrinkage of length. 

Pour these pieces of information into a good logic 
machine. The machine puts out a clear, strong con- 
clusion: "Inconsistent." Here is a very disturbing 
result. Before studying Einstein's solution of the 
problem it posed, consider a useful fable. 

OF Starlight 

Light from star 
to telescope 

showed change 

of tilt in 6 


THROUGH "ether" 



Light signals compared for 
perpendicular round trips: 
pattern showed no change 
when apparatus was rotated 
or as seasons changed. 


THROUGH "ether"; OT 




Growing electrical theory added confusion, be- 
cause Maxwell's equations seemed to refer to 
currents and fields in an absolute, fixed, space 
(= ether). Unlike Newton's Laws of Motion, they 
are changed by the Galilean Transforma'tion to a 
different form in a moving laboratory. However, 
the modified transformation devised by Lorentz 
kept the form of Maxwell's equations the same for 
moving observers. This seemed to fit the facts — in 
"magnets and coils experiments" (Experiment C in 
Ch. 41 ), we get the same effects whether the magnet 
moves or the coil does. With the Lorentz Trans- 
it The latest test (Townes, 1958) made by timing micro- 
waves in a resonant cavity, gave a null result when it would 
have shown a velocity as small as 1/1000 of the Earth's 
orbital speed. 

A Fable 

[This is an annoying, untrue, fable to warn you 
of the diflBculty of accepting Relativity. Counting 
items is an absolute process that no change of view- 
point can alter, so this fable is very distressing to 
good mathematical physicists with a strong sense of 
nature — take it with a grain of tranquilizer. You will 
find, however, that what it alleges so impossibly for 
adding up balls does occur in relativistic adding of 

I ask you to watch a magic trick. I take a black 
cloth bag and convince you it is empty. I then put 
into it 2 white balls. You count them as they go 
in — one, two — and then two more — three, four. 
Now I take out 5 white balls, and the bag is empty. 

ifaHs in 


Fic. 31-26. 

Pour this record into the logic machine and it will 
say, "Inconsistent." What is your solution here? 
First, "It's an illusion." It is not. You are allowed 
to repeat the game yourself. (Miller repeated the 
Michelson-Morley experiment with great precision. ) 
Next, "Let me re-examine the bag for concealed 
pockets." There are none. Now let us re-state the 
record. The bag is simple, the balls are solid, the 



M-U-hcCion- f\AorQu - MURt experimnits 
(and extpnuans iry KermecCj and 
othcn); ludC muCc 

Adcrmhcm of itariighc 

The mnfianicaC [atvs of Gaii(eo and Nev>'Uni 

ikchTmaqncnc LUvs -^ 


I Tfxt rwmiaL "conunon icnse" neks of 

I antkmetu m\d actmvtni cipp(i), indudatw tfv 

I Gaidean TnxnsfnnuUu/n for moticn : 



INSTRUCTIONS Vi . . /-. , . ^^ 



Fig. 31-25 

X'X-vt w'=i< :''Z. 



tally is true: 2 + 2 go in and 5 come out. What 
can you say now? If you cannot refute tried and 
true observations, you must either give up science — 
and go crazy — or attack the rules of logic, includ- 
ing the basic rules of arithmetic. Short of neurotic 
lunacy, you would have to say, "In some cases, 
2 + 2 do not make 4." Rather than take neurotic 
refuge in a catch-phrase such as "It all adds up to 
anything," you might set yourself to cataloguing 
events in which 2 and 2 make 4 — e.g. adding beans 
on a table, coins in a purse; and cataloguing events 
for which 2 + 2 make something else.^- 

^^ There are cases where 2 -f 2 do not make 4. Vectors 
2 -f 2 may make anything between and 4. Two quarts of 
alcohol -1- two quarts of water mix to make less than 4 quarts. 
In the circuit sketched, all the resistors, R, are identical but 
the heating effects do not add up. Two currents each deliv- 
ering 2 joules/sec add to one delivering 8 joules/sec. 




Fic. 31-27. 

In studying Nature, scientists have been seeking and 
selecting quantities that do add simply, such as masses of 
Lquids rather than volumes, copper-plating by currents rather 
than heating. The essence of the "exceptions" is that they are 
cases where the items to be added interact; they do not just 
act independently so that their efiFects can be superposed. 

In this fable, you have three explanations to 
choose from: 

(a) "It is witchcraft." That way madness lies. 

(b) "There is a special invisible mechanism": 
hardly any better — it turns science into a horde 
of demons. 

(c) "The rules of arithmetic must be modified." 

However unpleasant (c) looks, you had better 
try it — desperate measures for desperate cases. 
Think carefully what you would do, in this plight. 

You are not faced with that arithmetical paradox 
in real life, but now turn again to motion through 
space. Ruling out mistaken experimenting, there 
were similar choices: blame witchcraft, invent spe- 
cial mechanisms, or modify the physical rules of 
motion. At first, scientists invented mechanisms, 
such as electrons that squash into ellipsoids when 
moving, but even these led to more troubles. 
Poincar^ and others prepared to change the rules for 
measuring time and space. Then Einstein made 
two brilliant suggestions: an honest viewpoint, and 
a single hypothesis, in his Theory of Relativity. 

The Relativitv viewpoint is this: scientific think- 
ing should be built of things that can be observed 
in real experiments; details and pictures that cannot 
be observed must not be treated as real, questions 
about such details are not only unanswerable, they 
are improper and unscientific. On this view, fixed 
space (and the "ether" thought to fill it) must be 


Mathematics and Relativity 

thrown out of our scientific thinking if we become 
convinced that all experiments to detect it or to 
measure motion through it are doomed to failure. 
This viewpoint merely says, "let's be realistic," on 
a ruthless scale. 

All attempts like the Michelson-Morley-Miller ex- 
periment failed to show any change of light's speed. 
Aberration measurements did not show light moving 
with a new speed, but only gave a new direction to 
its apparent velocity. So, the Relativity hypothesis 
is this: The measured speed of light (electromag- 
netic waves) will be the same, whatever the motion 
of observer or source. This is quite contrary to com- 

mon sense; we should expect to meet light faster or 
slower by running against it or with it. Yet this is 
a clear apphcation of the reahstic viewpoint to the 
experimental fact that all experiments with light 
fail to show the observer's motion or the motion of 
any "ether wind." Pour this hypothesis into the 
logic machine that previously answered, "Incon- 
sistent"; but remove the built-in "geometry rules" of 
space-&-time and motion, with their Galilean Trans- 
formation. Ask instead for the (simplest) new rules 
that uHll make a consistent scheme. However, since 
Newtonian mechanics has stood the test of time, 
in moving ships and trains, in the Solar System, etc.. 



EXCLPr -^ I 


OaLiUan T-mnsprmAtu'n praciicAUij 
comet at [<nv speeds (v « c) 

Ai-ernuum oj starQ^(ic 

MxcktCsorv, Mor{e\j, MiCCer experiments: 
nulL nsuLc 




i r - r r ? 

I vehcitij Of scuru cr ctseryer. | 

f (Ai\d tfiercJvTC \\<.v>.McU''s i:iiuAtunis sfu.'uiJ | 

^ t(\kc tfu same form jcr a{[ c(<SCr\Tr} 1 | 



Wfiat tra»t5/i»7nflti'(7n {jsiheynes 
of qevmctru and. mirtiipnj wxii 
fit insteaji cf 



x' = x - Vt 





ihx. o*dij naicna$[e scfione 

cf anfmetru and ttwtwn- 
that will fit 15 



V ' - X - vt 


^ - y 


(coding tc prcdictuni that, whe>\ an ohen-e>- is 
movMiJ at conihuif vcL'Uty rcLuive to tfa oMjaratus, 
hii meanovmcnti cf 


win difftr noticea^kf, at hi^fi syccds. from tfwsc of 
an observer mcvin^ »vi£h tiie apparatus. 

Fig. 31-28. 

t = 

1 = 2 




the new rules must reduce to the Galilean Trans- 
formation at low speeds.'^ The logic machine re- 
plies: "There is only one reasonable scheme: the 
transformation suggested by Lorentz and adopted 
by Einstein." 

Instead of the Galilean Transformation 

x' = X — vt y' = y z' = z t' 





= z t' 

t — xv/c^ 

and these turn into the reverse transformation, with 
relative velocity t; changing to — v 

X = 

x'-f uf 

y = y' 

f -f x'f/'c= 

VI — «Vc' - ^ - Vl-uVc' 

where c is the speed of light in vacuum. That speed 
is involved essentially in the new rules of measure- 
ment, because the new transformation was chosen 
to make all attempts to measure that speed yield 
the same answer. And the symmetrical form shows 
that absolute motion is never revealed by experi- 
ment. We can measure relative motion of one ex- 
perimenter past another, but we can never say which 
is really moving. 

Of course the new transformation accounts for 
the Michelson-Morley-Miller null result — it was 
chosen to do so. It also accounts for aberration, pre- 
dicting the same aberration whether the star moves 
or we do. But it modifies Newtonian mechanics. In 
other words, we have a choice of troubles: the old 
transformation upsets the form of electromagnetic 
laws; the new transformation upsets the form of 
mechanical laws. Over the full range of experiment, 
high speeds as well as low, the old electromagnetic 
laws seem to remain good simple descriptions of na- 
ture; but the mechanical laws do fail, in their classi- 
cal form, at high speeds. So we choose the new 
transformation, and let it modify mechanical laws, 
and are glad to find that the modified laws describe 
nature excellently when mechanical experiments 
are made with improved accuracy. 

The new transformation looks unpleasant" be- 
cause it is more complicated, and its implications are 
less pleasant. To maintain his Galilean relativity, 
Newton could assume that length, mass, and time are 
independent of the observer and of each other. He 

coidd assert that mechanical experiments will fail 
to reveal uniform motion through "space."" When 
Einstein extended the assertion of failure to experi- 
ments with hght, he found it necessary to have 
measurements of length and time, and therefore 
mass, different for observers with different motions. 
We shall not show the steps of the logic machine 
grinding out the transformation and its implications, 
but you may trust them as routine algebra.^' We 
shall follow custom and call it the Lorentz Trans- 

Implications of the Lorentz Transformation 

Take the new modified geometry that will fit the 
experimental information, and argue from it how 
measurements by different observers will compare. 




© 6 

^ V 

RElATfVE to £' 

V ^ 

R£IAT/V£ tt> S 



Fig. 31-29 

One experimenter is moving with constant velocity relative 

to the other. They arrange to use standard 

measuring instruments of identical construction. 

RetTirn to our two observers e and e', who operate 
with identical meter sticks, clocks, and standard 
kilograms, e' and his coordinate framework are mov- 
ing with speed t; relative to e; and e is moving 
backward with speed v relative to e'. The trans- 

" This is an application of Bolir's great "Principle of Cor- 
respondence": in any extreme case where the new require- 
ment is trivial — here, at low speeds — the new theory must 
leduce to the old. 

•This transformation may seem more reasonable if vou 
see that it represents a rotation of axes in space-&-time. For 
that, see later in this chapter, page 495. 

'* When an experiment leads us to beheve Newton's 
Laws I and II are valid, it is really just telhng us that we are 
lucky enough to be in a laboratory that is (practically) an 
inertial frame. If we had always experimented in a tossing 
ship, we should not have formulated those simple laws. 

" For details, see standard texts. There is a simple version 
in Relativity . . . A Popular Exposition by A. Einstein (pub- 
lished by Methuen, London, 15th edn., 1955). 


Mathematics and Relativity 

formations e — » e' and e' -^ e are completely sym- 
metrical, and show only the relative velocity t; — the 
same in both cases — with no indication of absolute 
motion, no hint as to which is "really moving." 

The results of arguing from the transformation 
differ strangely from earlier common sense, but only 
at exceedingly high speeds. An observer flying past 
a laboratory in a plane, or rocket, would apply 
Galilean Transformations safely. He would agree 
to the ordinary rules of vectors and motion, the 
Newtonian laws of mechanics. 
The speed of light, c, is huge: 

c r= 300,000,000 meters/sec = 186,000 miles/sec 
^ a billion ft/sec =« 700 million miles/hour 
« 1 ft/nonasecond, in the latest terminology. 

For relative motion with any ordinary velocity v, 
the fraction v/c is tiny, uVc' still smaller. The factor 
Vl — V'/c'^ is 1 for all practical purposes, and 
the time-lag xv/c'^ is negligible — so we have the 
Galilean Transformation. 

Now suppose e' moves at tremendous speed rela- 
tive to e. Each in his own local lab will observe the 
same mechanical laws; and any beam of light pass- 
ing through both labs will show the same speed, 
universal c, to each observer. But at speeds like 
20,000 miles/sec, 40,000, 60,000 and up towards the 
speed of light, experimenter e would see surprising 
things as e' and his lab whizz past, e would say, 
"The silly fellow e' is using inaccurate apparatus. 

/ / 

B. Hj) '^•" 

"fixed. Lalacfmur^" 

Fic. 31-30. 

Each experimenter finds, by using his own standard 

instruments, that the other experimenter is using 

incorrect instruments: a shrunken meter stick, a 

clock that runs too slowly, and a standard, mass 

that is too big. 

His meter stick is shrunken — less than my true 
meter. His clock is running slow — taking more than 
one of my true seconds for each tick." Meanwhile 
e' finds nothing wrong in his own laboratory, but 
sees £ and his lab moving away backwards, and 
says, "The silly fellow e ... his meter stick is 
shrunken . . . clock running slow." 

Suppose e measures and checks the apparatus 
used by e' just as they are passing, e finds the meter 
stick that e' holds as standard shrunk to \/l — v^/c^ 
meter, e finds the standard clock that e' holds to tick 
seconds is ticking longer periods, of l/\/l — v^/c^ 
second. And e finds the 1 kg standard mass that t 
holds is greater, 1/Vl — v^c^ kg. These are 
changes that a "stationary" observer sees in a mov- 
ing laboratory; but, equally, a moving observer 
watching a "stationary" laboratory sees the same 
peculiarities: the stationary meter stick shorter, 
clock running slower, and masses increased. The 
Lorentz Transformations e' — e and £ — e' are sym- 
metrical. If ^ and £ compare notes they will quarrel 
hopelessly, since each imputes the same errors to 
the other! Along the direction of relative motion, 
each sees all the other's apparatus shrunk, even 
electrons. Each sees all the other's clocks running 
slowly, even the vibrations of atoms. (Across the 
motion, in y- and z-directions, £ and £' agree.) In 
this symmetrical "relativity" we see the same thing 
in the other fellow's laboratory, whether he is mov- 
ing or we are. Only the relative motion between us 
and apparatus matters— we are left without any hint 
of being able to distinguish absolute motion through 

The shrinkage-factor and the slowing-factor are 
the same, l/\/l — v-/c^. This factor is practically 
1 for all ordinary values of v, the relative speed 
between the two observers. Then the transformation 
reduces to Gahlean form where geometry follows our 
old "common sense." Watch a supersonic 'plane fly- 
ing away from you 1800 miles/hour ( = ii mile/sec). 
For that speed, the factor is 

V / /2 mile/ sec V 

~ \ 186,000 miles/sec/ 

or 1.000000000004 

The plane's length would seem shrunk, and its clock 
ticking slower, by less than half a billionth of 1%. 
At 7,000,000 miles/hour (nearly 1/100 of c) the 
factor rises to 1.00005. At 70,000,000 miles/hour it 
is 1.005, making a 1-^% change in length. 

Until this century, scientists never experimented 
with speeds approaching the speed of light — except 
for light itself, where the difference is paramount. 


IS sjieciC oj [iMfiC, I SO, 000 miles/sec 

d) Lngtd of 

moviM mtttr- ititk. . 

Hi titimaui 

6ij sttttumary 


iii) Un^td cf 
ai tititnattcC 
bit mownj 
therver ' 

(ui) Turu between 
ticks cf tnevui^ 
standard duck, 
as eitinuUed ^ 

(iv) Mass if 
irufvinj Standard 
titimatid Sy 

Sjieti cj twnna stitk 



Sptei (f observer 

Spied of mpvma dock. 

Speed of iwvuij morSS 

Fic. 31-31. 
Chances of Measurement I*bedicted by Relativity 

Nowadays we have protons hurled out from small 
cyclotrons at 2/10 of c, making the factor 1.02; 
electrons hitting an X-ray target at 6/10 of c, making 
the factor 1.2; beta-rays flung from radioactive 
atoms with 98/100 of c, making the factor 5; and 
billion-volt electrons from giant accelerators, with 
.99999988 c, factor 2000. 

Among cosmic rays we find some very energetic 
particles, mu-mesons; some with energy about 1000 
million electron • volts moving with 199/200 of the 
speed of light. For them 
1/Vl - «Vc' = l/Vl - 199V200' = 1/ \/^= 10. 

Now these mesons are known to be unstable, with 
lifetime about 2 X 10"" sec (2 microseconds). Yet 
they are manufactured by collisions high up in the 
atmosphere and take about 20 X 10* seconds on the 
trip down to us. It seemed puzzling that they could 
last so long and reach us. Relativity removes the 
puzzle: we are looking at the flying meson's internal 
life-time-clock. To us that is slowed by a factor of 
10. So the flying meson's lifetime should seem to us 
20 X 10' seconds. Or, from the meson's own point 
of view, its lifetime is a normal 2 microseconds, but 
the thickness of our atmosphere, which rushes past 
it, is foreshortened to 1/10 of our estimate — so it 
can make the shrunk trip in its short lifetime. 

Measuring Rods and Clocks 

We used to think of a measuring rod such as a 
meter stick as an unchanging standard, that could 
be moved about to step ofiE lengths, or pointed in 
different directions, without any change of length. 
True, this was an idealized meter stick that would 
not warp with moisture or expand with some tem- 
peratiure change, but we felt no less confident of its 
properties. Its length was invariant. So was the time 
between the ticks of a good clock. ( If we distrusted 
pendulum-regulated clocks, we could look forward 
to completely constant atomic clocks.) Now, Rela- 
tivity warns us that measuring rods are not com- 
pletely rigid with invariant length. The whole idea 
of a rigid body — a harmless and useful idealization 
to 19th-century physicists — now seems misleading. 
And so does the idea of an absolutely constant 
stream of time flowing independently of space. In- 
stead, our measurements are affected by our motion, 
and only the speed of light, c, is invariant. A broader 
view treats c as merely a constant scale-factor for 
our choice of units in a compound space-&-time, 
which different observers sHce differently. 

Changes of Mass 

If length- and time-measurements change, mass 
must change too. We shall now find out how mass 
must change, when a moving observer estimates it, 
by following a thought-experiment along lines sug- 
gested by "Tolman. We shall assume that the con- 
servation of momentum holds true in any (inertial 
frame) laboratory whatever its speed relative to the 
observer — we must cling to some of our working 
rules or we shall land in a confusion of unnecessary 

Consider e and e' in their labs, moving with rela- 
tive velocity v in the x-direction. Suppose they make 
two platinum blocks, each a standard kilogram, 
that they know are identical — they can count the 


Mathematics and Relativity 

atoms if necessary. Each places a 1-kg block at rest 
in his lab on a frictionless table. Just as they are 
passing each other e and e' stretch a long light 
.«:piral spring between their blocks, along the {/-direc- 
tion. They let the spring tug for a short while and 
then remove it, leaving each block with some y- 
momentum. Then each experimenter measures the 
y-velocity of his block and calculates its momentum. 



fUiatwi vtCocuu i' - i = V *■ 

/ ^ ^ -i— "^ mcmxentcvnj 



Fig. 31-32. Two Observers Measuring Masses 

( A thought-experiment to find how mass depends on speed of 

object relative to observer.) £ says: 1 have 1 kg, 

moving across my lab with velocity 3 meters/sec. 

I know e' has 1 kg, and I see that he records its 

velocity as 3 meters/sec; but I know his clock is 

ticking slowly, so that the velocity of his lump 

is less than 3 meters/sec. Therefore his lump 

has mass more than 1 kg. 

They compare notes: each records 3 meters/sec for 
his block in his own framework. They conclude: 
equal and opposite velocities; equal and opposite 
momenta. They are pleased to adopt Newton's Law 
III as a workable rule. Then e, watching e' at work, 
sees that e' uses a clock that runs slowly (but they 
agree on normal meter sticks in the {/-directions). So 
e sees that when e' said he measured 3 meters travel 
in 1 sec, it was "really" 3 meters in more-than-l- 
second as e would measure it by his clock. There- 
fore E computes that velocity as smaller than 3 
meters/ second by \/l — v^/c^. Still believing in 
Newton III and momentum-conservation, e con- 
cludes that, since his own block acquired momentum 
1 kg • 3 meters ^sec, the other, which he calculates 
is moving slower must have greater mass'' — in- 

creased by the factor l/\/i~— t^/P. While that 
block is drifting across the table after the spring's 
tug, e also sees it whizzing along in the x-direction, 
table and all, with great speed v. Its owner, e', at 
rest with the table, calls his block 1 kg. But e, who 
sees it whizzing past, estimates its mass as greater, by 

This result applies to all moving masses: mass, a? 
we commonly know it, has different values for 
different observers. Post an observer on a moving 
body and he will find a standard value, the "rest- 
mass," identical for every electron, the same for 
every proton, standard for every pint of water, etc. 
But an observer moving past the body, or seeing 
it move past him, will find it has greater mass 

m = — -■ Again, the factor l/\/l — « Vc* 

Vl - «Vc' 
makes practically no difference at ordinary speeds. 
However, in a cyclotron, accelerated ions increase 
their mass significantly. They take too long on their 
wider trips, and arrive late unless special measures 
are taken. Electrons from billion-volt accelerators 
are so massive that they practically masquerade as 

For example, an electron from a 2-million-volt 
gun emerges with speed about 294,000,000 meters/ 
sec or 0.98 c. The factor l/\/l — (.98c)Vc' is 
l/Vl — (98/100)2 ^ l/\/47T00 = 5. To a sta- 
tionary observer the electron has 5 times its rest- 
mass." (Another way of putting this is: that elec- 
tron's kinetic energy is 2 million electron • volts; the 
energy associated with an electron's rest-mass is 
half a million ev, and therefore this electron has 
K.E. that has mass 4 rest-masses; and that with the 
original rest-mass makes 5 rest-masses.) 

This dependence on speed has been tested by 
deflecting very fast electrons ( beta-rays ) with elec- 
tric and magnetic fields, and the results agree ex- 
cellently with the prediction. Another test: in a 
cloud chamber a very fast electron hitting a sta- 
tionary electron ("at rest" in some atom of the wet 
air) does not make the expected 90° fork. In the 
photograph of Fig. 31-34c, the measured angles 

18 Suppose e and e' are passing each other with relative 
velocity 112,000 miles/sec. Then e sees the clock used by 
e' running slow, ticking once e\'ery 1.2 seconds. So he knows 
the block belonging to c' has velocity 3 meters/ 1.2 sees or 
2.5 meters/sec. His ovrt\ block has momentum 1 kg • 3 m./ 
sec. To preserve momentum conservation, he must say tliat 
the other block has momentum 1.2 kg. • 2.5 m./scc. So he 
estimates the mass of the other block as 1.2 kg, a 20S increase. 

• To the moving electron, or to a neighbor flying along 
beside it, its mass is the normal rest-mass; and it is the ex- 
perimenter rushing towards it who has 5 times his normal 
rest-mass and is squashed to '5 his normal thickness. 


MASS of 
movun^ jiarticU 
S\j itatunumj 



A man's walk to a rocket's 



Tfvqratik fcCcw Cs tde etuii jpart c>f tfte ^rajok aSove maanijitd ),ooo,ooo tunts in (uniwntaC scde. 



CAR $0 



INCREASE 00000000002*^ 

INCREASE 000000000 







" VELOCITY O.lmlU/sec 

36O miUs/funur 

: O.z mi 
; 720 

0.060001 c 



Fic. 31-33. Chances of Mass of Objects Moving Relative to Observeb 

(The graphs of Fig. 31-31 cover the whole range of speeds from zero to the speed of light; and they may give a 

mistaken impression of noticeable increase of mass at ordinary speeds. This graph is a copy of the 

mass-graph there, with comments.) 

agree well with those predicted by Relativity for a 
moving mass 12.7m hitting a stationary mass m, in 
an elastic collision. The tracks are curved be- 
cause there was a strong magnetic field perpen- 
dicular to the picture. Measurements of the curva- 
tures give the momentum of each electron after 
collision, and the momentum of the bombarding 

electron before collision. Measurements of the 
angles shown in the sketch confirm the proportions 
of these momenta. If non-relativistic mechanics 
[K.E. = Hmt;', etc.] is used to calculate the masses, 
assuming an elastic collision, the projectile's mass 
appears to be about four times the target particle's 
mass. Yet the tracks look like those of an electron- 


Mathematics and Relativity 

Fig. 31-34. Relattvistic Mass in Elastic Collisions 


(a) NucUi I A' ' 


(6) Etectnms 

s(w e 



Jh^tr e 

(a) Collision of alpha-particle with stationary atom. Even 
with its high energy, an alpha-particle from a radioactive 
atom has a speed that is less than 0.1 c, so its mass is not 
noticeably increased. It makes the expected 90° fork when 
it hits a stationary particle (He) of its own mass. With a 
hydrogen atom as target, it shows its greater mass. 

(b) When a slow electron hits a stationary one, the fork 
shows the expected 90°. When a fast electron hits a sta- 
tionary one, the angles show that the fast one has much 
greater mass. 


(c) cbud cfuunftr ffwtB 

(c) Cloud-chamber photograph of very fast electron collid- 
ing with a stationary one. ( Photograph by H. R. Crane, 
University of Michigan.) 

(d) MeasMrments 

\ (3) 


(d) Measurements of original photograph, (c), gave the 
following radii: (1) 0.15 ± 0.01 meter, (2) 0.105 m., 
(3) 0.050 m. Magnetic field strength was 1,425,000 (in 
our units for H in F = lO'''{Qv)(H), discussed in Ch. 37.) 

electron collision; and we do not expect 4m and m 
classically for two electrons. So we try assuming 
relativistic mechanics [K.E. = (m — m„)c^, 

MOMENTUM = mv, with m = mj/Vl — v-/c'] . 

Then we find a consistent story: from the magnetic 
field and our measurements of curvature we find: 

BEFORE collision: 

projectile had mass 12.7rMo, speed 0.9969 c; 
Since the track is short and only slightly curved, its 
radius cannot be measured very precisely; so the 
projectile's momentum, and thence mass, is uncer- 
tain within about 6%. We should say 
mass = 12.7 mo ±: 6% or mass r= 12.7 mg ± 0.8 m„ 


projectile had mass 8.9 m„, speed 0.9936 c; 

target particle had mass 4.3 m„, speed 0.9728 c, 
where tMo is the standard rest-mass of an electron 
and c is the speed of light. Before collision the total 
mass was 13.7 mo (including the target); after col- 
lision it was 13.2 mo. Mass is conserved in this col- 
lision — within the 6% experimental uncertainty — 
and so is energy, now measured by mc^. 

A Meaning for Mass Change 

There is an easy physical interpretation of the 
change of mass: the extra mass is the mass of the 
body's kinetic energy. Try some algebra, using the 
binomial theorem to express the V as a series, 
for fairly low speeds: 

VI — uVc' 



powers of — I 
at low speeds J 

^m^l'+'V + ^g^^-- 
_which are very small 

= mo -|- V2"*ot'Vc^ + negligible terms at low 
I / 2 speeds 

= REST-MASS + K.E./C* '^ 


Maximum Speed: c 

As a body's speed grows nearer to the speed of 
light, it becomes increasingly harder to accelerate — 
the mass sweeps up towards infinite mass at the 


speed of light. Experimenters using "linear accel- 
erators" ( which drive electrons straight ahead ) find 
that at high energies their victims approach the 
speed of light but never exceed it. The electrons gain 
more energy at each successive push ( and therefore 
more mass ) but hardly move any faster ( and there- 
fore the accelerating "pushers" can be spaced evenly 
along the stream — a welcome simplification in de- 

Mass growing towards infinity at the speed of 
light means imaccelerability growing to infinity. Our 
efForts at making an object move faster seem to nm 
along the level of constant mass, till it reaches very 
high speeds; then they climb a steeper and steeper 
mountain towards an insurmountable wall at the 
speed of light itself. No wonder Relativity predicts 
that no piece of matter can move faster than light; 
since in attempting to accelerate it to that speed we 
should encounter more and more mass and thereby 
obtain less and less response to our accelerating 

Adding Velocities, Relativistically 

Faster than light? Surely that is possible: moimt 
a gun on a rocket that travels with speed %c and 
have the gun fire a bullet forward with muzzle 
velocity ^. The bullet's speed should he '¥ic -\- %c 
or IViC. No: that is a Galilean addition of velocities. 
We must find the relativistic rule. 



— »• u 

---*■ u' 

_ u- V 


-/ i 

Fic. 31-35a. Observers Measure a Velocity 

Two experimenters observe the same moving object. How do 

their estimates of its velocity compare? The Lorcntz 

transformation leads to the relation shown, between u 

as measured by e and u' as measured by e'. 

Suppose £ sees an object moving in his laboratory 
with velocity u, along the x-direction. What speed 
will e' measure for the object? As measured by e, 
u = ^x/At. As measured by e', u' = Ax'/Af and 
simple algebra leads from the Lorentz Transforma- 
tion to 



instead of the Galilean u' = {u — v] 
verse relation rvms: 

(ti' + c) 

And the in- 


The factor in [ ] is practically 1 for all ordinary 
speeds, and then the relations reduce to Galilean 
form. Try that on a bullet fired by an ordinary rifle 
inside an ordinary express train, ef, riding in the 
train, sees the rifle fire the bullet with speed u'. 
e, sitting at the side of the track, sees the bullet 
move with speed u. He sees the train passing him 
with speed v. Then u = (u' + f)/[l]- The Galilean 
version fits closely: 






y^ f ifti ^wi&f ipeed u = u'-h V 

Fig. 31-35b. Adding VELOcrriES at Ordinary Speeds 

Two experimenters observe the same bullet, shot from a 

gun in a moving train. With such speeds, the Lorentz 

transformation leads to the simple Galilean relations : 

u' = u — V and u = u' -}- t;. 

Now return to the gim on a Vtc rocket firing a %c 
bullet forward, e' rides on the rocket and sees the 
bullet emerge with u' = Vk;. e on the ground sees e' 
and his rocket moving with speed ^4c; and e learns 
from e' how' fast the gun fired the bullet. Then, using 
the relativity-formula above, e predicts the bullet- 
speed that he will observe, thus: 

Fig. 31-36. Adding Velocities at Very High Speeds 

1 '/z spnc[<^(i:j(\t 


(a) Experimenter e on ground observes a rocket moving at ^'\C. 

Experimenter e' riding on the rocket fires a bullet at Vi c 

relative to the rocket. What will be the speed of the 

bullet, as measured by £ on the ground? 


Mathematics and Relativity 

u' -\-v 




1 + u'v/c^ I + Vic- y^c/c" 1 + 

(y4)c 10 „ . , 

— — — = —c, sbll just less than c. 

( 78; il 









Have another try at defeating the limit of velocity, 
c. Run two rockets head on at each other, with 
speeds %c and Vtc. e on the ground sees e' riding on 

? , 

( b ) Experimenter e on ground sees two rockets approaching 

each other, one with speed %c, the other with speed %c. 

What speed of approach will experimenter e' riding on 

the first rocket see? 

one rocket with velocity t; = %c and the other rocket 
travelling with u =: — Vtc; and he thinks they must 
be approaching each other with relative velocity 
l%c. e', riding on the first rocket, sees the second 
rocket moving with predicted speed 



-\y^c 10 

1 + % 


Their rate of approach is less than c. Whate"er we 
do, we cannot make a material object move faster 
than Hght — as seen by any observer. 

Speed of Light 

Finally, as a check on our velocity-addition for- 
mula, make sure it does yield the same speed of light 
for observers with different speeds. Take a flash of 
light travelling with speed u =: c, sls observed by e. 
Observer e' is travelling with speed t; relative to e, 
in the same direction, e' observes the flash moving 
with speed 

,_ u — v c — v c(l — v/c) 

1 — tit;/c^ " 1 — cv/c^ ~ (1 — v/c) ~ 
Every observer measures the same speed c for light. 


Fig. 31-37. 

Two experimenters measure the speed of the same sample of 

light. Experimenter e sees that e' is running with 

velocity v in the direction the hght is travelling. 

( No wonder, since the Lorentz Transformation was 
chosen to produce this. ) This certainly accounts for 
the Michelson-Morley-Miller null results. 


We rebuild the Newtonian view of energy to fit 
Relativity as follows. Define momentum as mv, 
where m is the observed mass of the body in motion: 
m = mo/Vl — v^/cr^. Define force, F, as A(inv)/At. 
Define change from potential energy to K.E. as 
WORK, F • As. Combine these to calculate the K.E. 
of a mass m moving with speed v. We shall give the 
result, omitting the calculus derivation. 

_ "'o fp^t of Lorentz"! 

yyi v^/c^ [TransformationJ 

F = 


A(K.E.) = F- AS 

= F -v At 
K.E. = if i; = 

[Newton Law III 
Relativity form J 

of K.E. J 



K.E. = mc^ — mgC" 

We assign the body a permanent store of "rest- 
energy" moC'— locked up in its atomic force-fields, 
perhaps. We add that to the K.E.; then the total en- 
ergy, E, of the body is m^c^ ^ (mc^ — m^c^ ) = mc^. 
Therefore total E = mc^. This applies whatever its 
speed — but remember that m itself changes with 
speed. At low speeds, mc' reduces" to 

(rest-energy m„c^) + (K.E. ^unv^). 

For a short, direct derivation oi E = mc^, see the 
note at the bottom of the next page. 

This view that energy and mass go together ac- 
cording to £ = mc^ has been given many successful 
tests in nuclear physics. Again and again we find 
some mass of material particles disappears in a 

^' See the discussion above, with the binomial theorem. 


nuclear break-up; but then we find a release of 
energy — radiation in some cases, K.E. of flying frag- 
ments in others — and that energy carries the miss- 
ing mass. 

The expression for mass, m = rrjo/Vl — v^/c' 
follows from the Lorentz Transformation and con- 
servation of momentum. So E = mc^ follows from 
Newton's Laws II and III combined with the Lo- 
rentz Transformation. 

Then if an observer assigns to a moving body a 
mass m, momentum mv, and total energy mc^ he 
finds that, in any closed system, mass is conserved, 
momentum is conserved (as a vector sum), and 
energy is conserved. In all this he must use the 
observed mass m, which is m^/y/l — v^/c'^ for any 
body moving with speed v relative to him. Then 
he is doubling up his claim of conservation because, 
if the sum of all the masses (mj + m, -f- . . .), is 
constant, the total energy (m^c^ + m^c^ + . . .) 
must also be constant. If energy is conserved, mass 
must also be conserved. One rule will cover both. 
That is why some scientists say rather carelessly, 
"mass and energy are the same, but for a factor c*." 
In fact, since c^ is universally constant, there is little 
harm in saying that mass and energy are the same 
thing, though commonly measured in different units. 
But there is also little harm if you prefer to think 
of them still with quite different flavors as physical 
concepts. And a very important distinction remains 
between matter and radiation (and other forms of 
energy). Matter comes in particles, whose total 
number remains constant if we count the produc- 
tion or destruction of a [particle -|- anti-particle] 

pair as no change. Radiation comes in photons; and 
the total number of photons does change when one 
is emitted or absorbed by matter. 


Finally, Einstein treated momentum as a vector 
with three components in space-&-time, and kinetic J 
energy with them as a fourth, time-Uke, component i 
of a "supervector." Thus, conservation rules for 
mass, momentum, and energy can be rolled into 
one great formula in relativistic mechanics. The 
Lorentz Transformation gives this formula the same 
form with respect to any (steadily moving) set of 
axes whatever their velocity. We say such a formula 
or relation is "covariant." We put great store by 
covariance: covariant laws have the most general 
form possible and we feel they are the most perfect 
mathematical statement of natural laws. "We lose 
a frame of reference, but we gain a universally vaUd 
symbolic form."" 

"A Wrong Question" 

The physical laws of mechanics and electromag- 
netism are covariant: they give no hope of telling 
how fast we move through absolute space. This 
brings us back to Einstein's basic principle of being 
realistic. Where the answer is "impossible," the 
question is a foolish one. We are unscientific to 
imply there is an absolute space, as we do when 
we ask "How fast . . . through space?" We are 
begging the question, inside our own question, bv 
mentioning space. We are asking a wrong question, 

" Frederic Keffer. 

NOTE: Derivation of E = mc' 

This short derivation, due to Einstein, uses the experi- 
mental knowledge that when radiation with energy E joulej 
is absorbed by matter, it delivers momentum E/c kg-m./sf.c. 
(Experiment shows that pressube of radiation on an absorb- 
ing wall is ENEBCY-PER-UNiT-voLUME of radiation-beam. 
Suppose a beam of area A falls on an absorbing surface 
head-on. In time At, a length of beam c • M arrives. Then 
MOMENTUM delivered in At 



= (enercy/A ' c ' At) ' A' At 
= energy/ c 

This also follows from Maxwell's equations). 

JE fE t£ , -jE 


We take two views of the same thought-experiment: 

(A) Place a block of matter at rest on a frictionless table. 
Give it some energy £ by firing two chunks of radiation at 
it, /i£ from due East, )iE from due West. The block absorbs 
the radiation and gains energy E; but its net gain of mo- 
mentum is zero: it stays at rest. (B) Now let a running 
observer watch the same event. He runs with speed t> due 
North; but according to Relativity he can equally well think 
he is at rest and see the table, etc. moving towards him 
with speed v due South. Then he sees the block moNing 
South with momentum .\/i;. He sees the two chunks of radia- 
tion moving towards the block, each with speed c but in 
directions, slanted southward with slope v/c. ( This is like 
the aberration of starlight. ) In his view, each chunk 
has momentum (Vi£/c) with a southward component 
(V^£/c) (u/c). Thinking himself at rest, he sees total south- 
ward momentum Mv + 2( V&£/c) (c/c). After the block 
has absorbed the radiation, he still sees it moving South 
with the same speed v — since in version (A) we saw that 
the block gained no net momentum. However, the block 
may gain some mass, say m. Find out how big m is by 
trusting conservation of momentum: 

Mv + 2(^E/c)(«/c) = (M -t- m)v 
m = E/c' or £ = mc", 

where m is the mass gained when energy £ is gained. 


Mathematics and Relativity 

like the lawyer who says, "Answer me 'yes' or 'no.' 
Have you stopped beating your wife?" The answer 
to that is, "A reasonable man does not answer un- 
reasonable questions." And Einstein might suggest 
that a reasonable scientist does not ask unreasonable 


The observers e and e' do not merely see each 
other's clocks running slowly; worse still, clocks at 
different distances seem to disagree. Suppose each 
observer posts a series of clocks along the x-direc- 
tion in his laboratory and sets them all going to- 
gether. And when e and e' pass each other at the 
origin, they set their central clocks in agreement. 
Then each will blame the other, saying: "His clocks 
are not even synchronized. He has set his distant 
clocks wrong by liis own central clock — the greater 
the distance, the worse his mistake. The farther I 
look down his corridor, along the direction he is 
moving, the more he has set his clocks there back — 
they read early, behind my proper time. And look- 
ing back along his corridor, opposite to the direc- 
tion of his motion, I see his clocks set more and 
more forward, to read later than my correct time." 
(That judgment, which each makes of the other's 
clocks, is not the result of forgetting the time-delay 
of seeing a clock that is far away. Each observer 
allows for such delays — or reads one of his own 
clocks that is close beside the other's — and then 
finds the disagreement. This disagreement about 
setting of remote clocks belongs with the view that 
each observer takes of clock rates. Each claims that 
all the other's clocks are running too slowly; so they 
should not be surprised to find that their central 
clocks, originally synchronized at the origin, dis- 
agree after a while. Each says: "His central clock, 
that was opposite me, has moved ahead and was 
running too slowlv all the while; so no wonder its 
hands have not moved around as fast as my clock." ) 

e observes his own row of clocks ticking simul- 
taneously all in agreement. But e' does not find those 
ticks simultaneous. Events that are simultaneous 
for e are not simultaneous for e'. This is a serious 
change from our common-sense view of universal 
time; but it is a part of the Lorentz Transformation. 
In fact, the question of simultaneity played an es- 
sential role in the development of relativitv bv 
Poincare and Einstein. Arguing with thought-ex- 
periments that keep "c" constant, you can show this 
change is necessary. The following example il- 
lustrates this. 

Suppose E and e' have their laboratories in two 
transparent railroad coaches on parallel tracks, one 
moving with speed v relative to the other. Just 


I © © ©'0 




'set 'set "set 'sec 

back" comaiij" aheud" AHEAD' 



£ with fwi 

Fic. 31-38. "Si.MULTANEous" Clock Settings 
Each experimenter sets his own clocks all in agreement 
( allowing carefully for the time taken by any light 
signals he uses in looking at them). Each experimenter 
finds that the other man's clocks disagree among 
themselves, progressively with distance. ( That is, after 
he has allowed carefully for the time taken by the light 
signals he uses in checking the other man's clocks 
against his own. ) The sketch shows a series of clocks 
all fixed in the framework belonging to £. As adjusted 
and observed by e, they all agree: they are synchro- 
nized. As investigated by e' those clocks disagree with 
each other. The lower sketch shows what e' finds by 
comparing those clocks simultaneously ( as he, e', 
thinks) with his own clock. The two sketches of clocks 
disagree because each experimenter thinks he com- 
pares them all simultaneously but disagrees with the 
other man's idea of simultaneity. 

as the coaches are passing, e and e' lean out of their 
center windows and shake hands. They happen to 
be electrically charged, + and — , so there is a flash 
of light as they touch. Now consider the light from 
this flash. Some of it travels in each coach starting 
from the mid-point where the experimenter is 
standing, e finds it reaches the front and hind ends 
of his coach simultaneously. And e' finds it reaches 
the ends of his coach simultaneously. Each con- 
siders he is in a stationary coach with light travelling 
out from the center with constant speed c. But e 
can also observe the light flash reaching the ends 
of the other coach that carries z'. He observes the 
events that e' observes; but he certainly does not 
find them simultaneous, as e' claims. Bv the time 


the flash has travelled a half-length of the e' coach, 
that coach has moved forward past e. As e sees it, 
the light travels farther to reach the front end of 
that moving coach, and less to the hind end. So e 
sees the flash hit the hind end first, v^'hile e' claims 
the hits are simultaneous." (Reciprocally, e' sees 
the light reach the ends of the coach carrying e at 
different instants, while e claims they are simul- 
taneous.) You will meet no such confusion in ordi- 
nary life, because such disagreements over priority 
arise only when the events are very close in time, 
or very far apart in distance. Where events P and 
Q are closer in time than the travel-time for light 
between them, observers with different motions may 
take different views: one may find P and Q simul- 
taneous, while another finds P occurs before Q, and 

/ / 


- > 

^ — 

bird's eye view 




^lasW itarts 
a,i £ and I'tnttt 

O o Ci 


t ices jtoik fuc kU 

coach s endi sinudcaneowi^ 

Cseti^k fuchii 
codcfi s eruU sunuUaneoM^ 

€ ienfioifi fiit iotfi endi of fm couch sunuimneomUj, 

iut tht endi of t' ccach ac different timei. {Sunlkr^ for £') 

Fic. 31-39. Thought-Experiment 

To show that events that are simultaneous for one observer 

are not simultaneous for an observer moving with a 

different velocity. 

still another finds P later than Q. To maintain Ein- 
stein's Relativity, we must regard time as interlocked 
with space in a compound space-time, whose slicing 
into separate time and space depends somewhat on 
the observer's motion. If we accept this compound 
space-time system, we must modify our philosophy 
of cause and effect. 

Cause and Effect 

Earher science was much concerned with cau- 
sality. Greeks looked for "first causes"; later scien- 
tists looked for immediate causes — "the heating 
caused the rock to melt"; "the pressure caused the 
hquid to flow"; "the alpha-particle caused the ions 
to be formed." It is diflBcult to define cause and 
effect. "P causes Q": what does that mean? The 
best we can say is that cause is something that pre- 
cedes the effect so consistently that we tliink there is 
a connection between them. 

Even in common cases ( like stress and strain or 
P.O. and current), we prefer to say P and Q go to- 
gether: we still look for relationships to codify our 
knowledge, but we treat P and Q as cousins rather 
than as parent and child. 

And now Relativity tells us that some events can 
show a different order in time for different observ- 
ers — and all observers are equally "right." The 
sketches of Fig. 31-40(e), below, show how various 
observers at an event P, here-new, must classify 
some other events (e.g., Q, ) as in the absolute 
future; some other events ( e.g., Q, ) in the absolute 
past; and some events (e.g., Q., ) in the absolute 
elsewhere (as Eddington named it) where observ- 
ers with different motions at P may disagree over 
the order of events P and Q. 

^' Note that the disagreement over simultaneity is not due 
to forgetting the time taken by light signals to bring the in- 
formation to either observer. We treat the problem as if each 
observer had a whole gang of perfectly trained clockwatchers 
ranged along his coach to make observations without signal 
delays and then report at leisure. The observers compare 
notes (e.g. by radio). Then each has an obvious explanation 
of the other man's claim that he saw the light flash reach the 
ends of his own coach simultaneously: "Why, the silly fellow 
has set his clocks askew. He has a clock at each end of his 
coach, and when the light flash hit those end clocks they 
both showed the same instant of time — I saw that, too. But 
he is wrong in saying his end clocks are set in agreement: 
I can see that he has set his front-end clock back by my 
standard, and his hind-end clock ahead. / can see that 
the flash had to travel farther to reach his front end. And 
my clocks tell me it arrived there later, as I know it should. 
But since his clock is mis-set, early by mine, the late- 
ness of arrival did not show on it. Those mistakes of his 
in setting his clocks just cover up the difference of transit- 
time for what I can see are different travel-distances to the 
ends of his coach." As in all such relativistic compariscns, 
each observer blames the other for making exactly the same 
kind of mistake. 


Mathematics and Relativity 






R c, 









i ^ 

£ 14 mcvin^ tvitfi 1 /. \', ni 
vcfcci'ty V ) mcasmrX 


^/ S and C' at t -c 


2 o'dcck iitic 

[ : / / o'docii. Rnc 

tr =0 
t = <7 

TWO OBSERVERS m oving very fast relative to each other RECORD EVENTS P si d 

t at z c'dcck £' i-tf 2 c'cfcck 




z o'ciodc fwif fin- 

i and £ 

; / 






€ and 


t 'C 

t' and t 
SiUnc for 
£ aiui e' 


iMice X, \ 


«, r 



Fig. 31-40. Charts of Space (One Dimension) and Time 

*- t = 

(These fanciful sketches are highly restricted: all the events 
shown occur in one straight Une, in a one-dimensional space, 
along an i-axis. 

In the Lorentz picture, a very high relative velocity between 
e and e' is assumed. The distortion of the i' and f system in 
the Lorentz picture shows the view taken by £. Of course, e' 
himself would take an "undistorted" view of his own system, 
but e' would find the x and ( system "distorted." 

It is not possible to show the essential symmetry here; so the 
Lorentz picture should only be taken as a suggestion: taken 
literally, it would be misleading. ) 

(a) An event that occurs on the straight hne (x-axis) is 
shown by a point on this chart. Distance along shows where 
the event occurs on the line. Distance up shows when it 
occurs. Event P precedes event Q in time. It may be sensi- 
ble to say that P causes Q, for some types of event. 

(b) A moving experimenter carries his origin for distance 
with him. On the Galilean system he uses the same time- 
scale as a stationary experimenter. 

(c) With a Galilean transformation between two experi- 
menters, the lines for each hour by the clock are the same 
for both observers, and parallel to the axis, ( = 0. 

(d) The Lorentz transformation between two experimenters 
tilts one coordinate system of space-&-time relative to the 
other (through a negligible angle, except when speed of 
e' relative to £ approaches c). 

Then an event Q that follows event P in time for one ex- 
perimenter may precede P for another — but only if the 
events are so far apart that a light signal from one event 
could not travel to the place of the otfier event and reach 
it before the other event occurred there. 


z stcs .. 


FUTURE for t 

NOW ^ t 

PASTi/or C 

^^ ^-- --^Jaoo,ooo\ 

^ ^ MOW J" 

Fig. 31-40(e) [after Eddington] 

Observer £ is at the origin; and so is e' who is moving fast 
along X-axis relative to g. The line seen-now has equation 
X = -ct, and marks all events that e (or z') sees at this 
instant now. e, knowing the value of c, allows for travel- 
time and marks his axis of events that happen now along the 
X-axis. However, e' will make a different allowance from the 
same seen-now line and will mark a tilted "now" line as his 
x'-axis. The hnes continuing seen-now in the forward direc- 
tion of time mark the maximum tilt that e' could have for 
his NOW line — because e' can never have relative velocity 
greater than c; so his x'-axis can never tilt as much as those 

> NOW (tT t 


"hght-lines" which have slope c. Rotate the picture around 
tlie t-axis and the light-lines make a double cone. Suppose 
an event P occurs at the origin, here-now, and another 
event at Q. If Q is within the upper light-cone ( Qi ) , it is 
definitely in the future of P for all observers. Similarly, all 
events in the lower Ught-cone (Qj) are in the absolute past, 
earlier than P for all observers. But Q» in the space between 
the cones may be in the future for e and yet be in the past 
for an observer e' whose x'-axis tilts above it. So we label 
that intermediate region absolute elsewhere. If Q falls 
tht.°, neither P rwr Q cari cause the other — they simply 
occur at different places. 

So now we must be more careful. We may keep 
cause and eflFect in simple cases such as apples and 
stomach-ache, or alpha-particles and ions; but we 
must be wary with events so close in time, for their 
distance apart, that they fall in each other's abso- 

In atomic physics you will meet other doubts con- 
cerning cause and efiFect. Radioactive changes ap- 
pear to be a matter of pure chance — the future life- 
time of an individual atom being unpredictable. In 
the final chapter you will see that nature enforces 
partial unpredictability on all our knowledge, hedg- 
ing individual atomic events with some unavoidable 
uncertainty, making it unwise to insist on exact 
"eflFects" from exact "causes." 

The Lorentz Transformation as a Rotation 

The sketches of Fig. 31-40 suggest we can throw light 
on the Lorentz transformation if we look at the effect 
of a simple rotation of the axes of a common x-, t/-graph. 
Try the algebra, and find the "transformation" connect- 
ing the old coordinates of a point, x, y, with the new 
coordinates, x', tf of the same point, thus: 







Refer a point in a plane to i-(/-axes. Then rotate the 
axes through an angle A (around the z-axis). The point, 
remaining at its old position in space, has coordinates 
x', y' referred to the new axes. Use the svmbol s for the 
slope of the new x-axis, so that s is tan A. Then, as the 
diagram shows 

x' = (x + b) cos A = (x + y tan A) cos A 
= (i + sy)/sec A = {x + sy)/\/(l -h tan^A) 
.-. X' = (x-|-«/)/V(l + s') 
Similarly, y' = (y — sx)/\/{l + s-) 

This transformation for a simple rotation of axes shows 
a square root playing much the same role as in the 
Lorentz transformation. In fact we obtain the Lorentz 
form if we replace i/ by a time coordinate, thus: instead 
of y, use t multiplied by constant c and by « the square 
root of (—1). And instead of slope s use i{v'c). Then, 
with y — ict and y' = ict' and 5 = iv/c, the simple 
rotation-transformation is the Lorentz transformation. 
Try that. That shows how the Lorentz transformation 
can be regarded as a sUcing of space-&-time with a dif- 
ferent slant for different observers. 

The Invariant "IntervaT' between Two Events 

We can define the "interval" R between two events 
(i,, tj) and (Xj, t.^) by the Pythagorean form 

R» = (x, -X.J' + (icf, - ict^)' 

Then we can also write the expression that gives R', the 
"interval" for another observer who records the same 
two events at (x/, t/) and (x^', t^) on his coordinates. 
If we then use the Lorentz transformabon to express R' 
in terms of the first observer's coordinates, we find that 
R' is the same as R. The Lorentz transformation keeps 
that "interval" invariant. That states the Relativitv' as- 
sumption — measured c is always the same — in a dif- 
ferent way. 

John A. Wheeler suggests a fable to illustrate the role 
of c. Suppose the inhabitants of an island do their 
surveying with rectangular coordinates, but measure 
North-South distances in miles and East-West ones in 
feet. Then a sudden, permanent shift of magnetic North 
through an angle A makes them turn their system of 
axes to the new direction. Thev again measure in miles 
along the new N'-S' direction, and in feet E'-W. They 
trv to compute the distance R between two points by 
Pythagoras: R' = (Ax)» -I- (Ai/)'; and they find that 
R takes a different value with the new coordinates. 


Mathematics and Relativity 

Then they find that they obtain the same value for R 
(and a useful one) with both sets of coordinates if they 
define R by: R^ = (Ax)^ + (5280ai/)2. 

Their "mysterious essential factor," 5280, corresponds 
to c in the relativistic "interval" in the paragraph above. 
Moral: c is not so much a mysterious limiting velocity 
as a unit-changing factor, which suggests that time and 
space are not utterly different: they form one con- 
tinuum, with both of them measurable in meters. 

Is There a Framework of Fixed Space? 

Thus we have devised, in special Relativity, a new 
geometry and physics of space-&-time with our 
clocks and measuring scales (basic instruments of 
physics), conspiring, by their changes when we 
change observers, to present us wath a universally 
constant velocity of light, to limit all moving matter 
to lesser speeds, to reveal physical laws in the same 
form for all observers moving with constant veloc- 
ities; and thus to conceal from us forever any abso- 
lute motion through a fixed framework of space; in 
fact, to render meaningless the question whether 
such a framework exists. 


Mathematical Form and Beauty 

As a language, algebra may be very truthful or 
accurate, and even fruitful, but is it not doomed to 
remain dull, uninteresting prose and never rise to 
poetry? Most mathematicians will deny that doubt 
and claim there is a great beauty in mathematics. 
One can learn to enjoy its form and elegance as 
much as those of poetry. As an example, watch a 
pair of simultaneous equations being polished up 
into elegance. Start with 

2x -f 3t/ = 9 
4x — 2y =: 10. 

Then with some juggling we can get rid of y and 
find X == 3; and then «/ ^ 1. But these are lopsided, 
individual equations. Let us make them more gen- 
eral, replacing the coeflBcients, 2, 3, 9, etc., by letters 
a, b, c, etc., thus 

ax -|- bt/ = c 

After heavier juggling we find x 

dx -f- ey = f 
;c — fb 

ae — db 

more juggling is needed to find y. These solutions 
enable us to solve the earlier equations and others 
like them by substituting the number coefficients 
for a, b, c, etc. But unless we had many equations 
to solve that would hardly pay; and we seem no 
nearer to poetry. But now let us be more system- 
atic. We are dealing with x and y as much the 
same things; so we might emphasize the similarity 

by calling them x, and Xj. To match that change, 
we use aj, a^, ao instead of a, b, c and vvaite: 
a^Xj + ^2*2 = ^0- But then we have the second 
equation's coeflBcients. We might call them a/, etc., 
but even so the two equations do not look quite 
symmetrical. To be fairer still, we call the first lot 
a/ etc. and the second lot a/' etc. Then: 

^1 ^1 ~r ^2 ^2 ^ ^0 

and a/'Xi -f a/'x^ = a." 
These look neat, but is their neatness much use? 

Solve for x. We obtain x^ = 

an a, — a^ 


is a gain: we need not solve for Xj or y. Symmetry 
will show us the answer straight away. Note that x^ 
and Xj (the old x and y) and their coeflBcients are 
only distinguished by the subscripts i and j- If we 
interchange the subscripts ^ and ^ throughout, we 
get the same equations again, and therefore we must 
have the same solutions. We make that interchange 

in the solution above and x^ = 


^i ^- 

—. Now we have the an- 

swer for Xj (the old y), free of charge. The economy 
of working may seem small; but think of the in- 
creased complexity if we had, say, five unknowns 
and five simultaneous equations. With this sym- 
metrical system of writing, we just solve for one 
unknown, and then write down the other four solu- 
tions by symmetry. Here is form playing a part 
that is useful for economy and pleasant in appear- 
ance to the mathematical eye. More than that, the 
new form of equations and answers is general and 
universal — in a sense this is a case of covariance. 
This is the kind of symmetrical form that appealed 
to Maxwell and Einstein. 

This is only a little way towards finding poetry 
in the language of mathematics — about i>.s far as 
well-metered verse. The next stage would be to use 
symmetrical methods rather than symmetrical forms, 
e.g. "determinants." As the professional mathema- 
tician develops the careful arguments which back 
up his methods, he builds a structure of logic and 
form which to his eye is as beautiful as the finest 

Geometry and Science: Truth and General Relatimty 

Thus, mathematics goes far beyond working 
arithmetic and sausage-grinding algebra. It even 
abandons pert definitions and some of the restric- 
tions of logic, to encourage full flowering of its 
growth; but yet its whole scheme is based on its 
own starting points; the views its founders take of 


numbers, points, parallel lines, vectors, .... Pure 
mathematics is an ivory-tower science. The results, 
being derived by good logic, are automatically true 
to the original assumptions and definitions. Whether 
the real world fits the assumptions seems at first a 
matter for experiment. We certainly must not trust 
the assumptions just because they seem reasonable 
and obvious. However, they may be more like defini- 
tions of procedure, in which case mathematics, still 
true to those definitions, might interpret any world 
in terms of them. 

We used to think that when the mathematician 
had developed his world of space and numbers, we 
then had to do experiments to find out whether the 
real world agrees with him. For example, EucHd 
made assumptions regarding points and lines, etc. 
and proved, or argued out, a consistent geometry. 
On the face of it, by rough comparison with real 
circles and triangles drawn on paper or surveyed 
on land, the results of his system seemed true to 
nature. But, one felt, more and more precise experi- 
ments were needed to test whether Euclid had 
chosen the right assumptions to imitate nature 
exactly; whether, for example, the three angles of a 
triangle do make just 180 degrees. ^° Relativity- 
mechanics and astronomical thinking about the uni- 
verse have raised serious questions about the most 
fitting choice of geometry. Mathematicians have 
long known that Euclid's version is only one of 
several devisable geometries which agree on a 
small scale but diflFer radically on a large scale in 
their physical and philosophical nature. 

Special Relativity deals with cases where an ob- 
server is moving with constant velocity relative to 
apparatus or to another observer. Einstein then de- 
veloped General Relativity to deal with measure- 
ment in systems that are accelerating. 

What is General Relativity, and how does it affect 
our views of physics — and of geometry? 

*o It probably seems obvious to you that they do. This may 
be because you have swallowed Euclid's proof whole — au- 
thoritarian deduction. Or you may have assured yourself in- 
ductively by making a paper triangle, tearing off the corners 
and assembling them. Suppose, however, we lived on a huge 
globe, without knowing it. Small triangles, confined to the 
schoolroom would have a 180° sum. But a huge triangle 
would have a bigger sum. For example, one with a 90° apex 
at the N-pole would have right angles at its base on the 

Fic. 31-41. 
( a ) Tearing a paper triangle. ( b ) Triangle on a sphere. 

Einstein's Principle of Equivalence 

Einstein was led to General Relativity by a single 
question: "Could an observer in a falling elevator 
or accelerating train really know he is accelerating?" 
Of course he would notice strange forces ( as in the 
case of truck-and-track experiments to test F = Ma 
in an accelerating railroad coach.' There strange 
forces act on the truck and make F = Ma untrue). 
But could he decide by experiment between ac- 
celeration of his frame of reference and a new gravi- 
tational field? (If a carpenter builds a correctly 
tilted laboratory in the accelerating coach, the ob- 
server will again find F =. Ma holds, but he will 
find "g" different.)" Therefore, Einstein assumed 
that no local experiments — mechanical, electrical or 
optical — could decide: no experiments could tell an 
observer whether the forces he finds are due to his 
acceleration or to a local "gravitational" field. Then, 
Einstein said, the laws of physics must take the same 
essential form for ALL observers, even those who 
are accelerating. In other words, Einstein required 
all the laws of physics to be covariant for all trans- 
formations from one frame of reference (or labora- 
tory) to another. That is the essential basis of Gen- 
eral Relativity: all physical laws to keep the same 

It was obvious long ago that for mechanical be- 
havior a gravitational field and an accelerating 
frame of reference are equivalent. Einstein's great 
contribution was his assumption that they are com- 
pletely equivalent, that even in optical and electrical 
experiments a gravitational field would have the 
same effect as an accelerated frame of reference. 
"This assertion supplied the long-sought-for link 
between gravitation and the rest of physics. . . ."-' 

Accelerating Local Observer ^ "Gravitational Field" 

The Principle of Equivalence influences our view 
of matter motion and geometry in several ways: 

(1) Local Physics for Accelerated Observers. If 
the Principle of Equivalence is true, all the strange 
effects observed in an accelerating laboratory can 
be ascribed to an extra force-field. If the labora- 
tory's acceleration is a meters 'sec', we may treat the 
laboratory as at rest instead if we give every mass 
m kg an extra force —ma newtons, presumably due 
to a force-field of strength — a newtons kg. Then, 
with this field included, the ordinary rules of 
mechanics should apply — or rather the Lorentz 
modification of Newtonian mechanics and Euclid- 
ean geometry, just as in Special Relativity. 

• See Chapter 7, Problems 30 and 31. 

" Sir Edmund Whittaker, in From Euclid to Eddington 
(Cambridge University Press, 1949): now in Dover paper- 
back edition. 


Mathematics and Relativity 


(i) Experimenters in a railroad coach that is ac- 
celerating — or in a rocket that is being driven 
by its fuel — will find Newton's Laws of motion 
applying at low speeds, provided they add to 
all visible forces on each mass m the extra 
(backward) force, —ma, due to the equivalent 
force-field.^^ Objects moving through the labo- 
ratory at very high speeds would seem to have 
increased mass, etc., just as we always expect 
from Special Relativity. 

(ii) An experimenter weighing himself on a spring 
scale in an elevator moving with downward 
acceleration a would obtain the scale reading 
that he would expect in a gravitational field 
of strength (g — a). (See Ch. 7, Problem 10.) 

(iii) In a freely falling box the force exerted by the 
equivalent force-field on a mass m would be 
mg upward. Since this would exactly balance 
the weight of the body, mg downward, every- 
thing would appear to be weightless. The 
same applies to experiments inside a rocket 
when its fuel has stopped driving it, or to ex- 
periments on any satellite pursuing an orbit 
around the Earth: the pull of the Earth's con- 
trolling gravity is not felt, because the whole 
laboratory is accelerating too. 

(iv) In a rotating laboratory, adding an outward 
force-field of strength v-/R would reduce the 
local mechanical behavior to that of a sta- 
tionary lab. 

( 2 ) Interpreting Gravity. All ( real ) gravitational 
fields can be reinterpreted as local modifications of 
space-&-time by changing to appropriate accelerat- 
ing axes so that the field disappears. This change 
gives us no help in mechanical calculations, but it 
leads to a new meaning for gravity, to be discussed 
in the next section. 

(3) "Removing Gravity." If a gravitational field 
is reallv equivalent to an accelerating frame, we can 
remove it by giving our laboratory an appropriate 
acceleration. Common gravity, the pull of the Earth, 
pulls vertically down. It is equivalent to an accelera- 

-' Over 200 years ago, the French philosopher and mathe- 
matician d'Alembert stated a general principle for solving 
problems that involve accelerated motion: add to all the 
known forces acting on an accelerating mass m an extra 
force —ma; then treat m as in equilibrium. By adding such 
"d'Alembert forces" to all the bodies of a complex system 
of masses in motion we can convert the dynamical problem 
of predicting forces or motion into a statical problem of 
forces in equilibrium. This is now common practice among 
professional physicists, but it is an artificial, sophisticated 
notion that is apt to Ije misleading; so we avoid it in ele- 
mentary teaching. It is the basis of the "engineer's headache- 
cure" mentioned in Opinion III of centrifugal force, in 
Chapter 21. 

tion of our frame, g vertically up. If we then let 
our lab fall through our frame of reference with 
acceleration g vertically down, we observe no effects 
of gravity. Our lab has two accelerations, the "real" 
one of falling and the opposite one that replaces the 
gravitational field. The two just cancel and we have 
the equivalent of a stationary lab in zero gravita- 
tional field. That just means, "let the lab fall freely, 
and gravity is not felt in it." We do that physically 
when we travel in a space ship, or in a freely-falling 
elevator. Our accelerating framework removes all 
sign of the gravitational field of Earth or Sun-' on 
a small local scale. Then we can leave a body to 
move with no forces and watch its path. We call its 
path in space-&-time a straight line and we expect 
to find simple mechanical laws obeyed. We have an 
inertial frame in our locality. 

( 4 ) Artificial Gravity. Conversely, by imposing a 
large real acceleration we can manufacture a strong 
force-field. If we trust the Principle of Equivalence 
we expect this force-field to treat matter in the same 
way as a very strong gravitational field. On this 
view, centrifuging increases available "g" many 

(5) Myth-and-Symbol Experiment. To an ob- 
server with acceleration a every mass m" seems to 
suffer an opposite force of size m°a, in addition to 
the pushes and pulls exerted on it by known agents. 
In a gravitational field of strength g every mass m^ 
is pulled with a force m^g. Here, we are using m' 
for inertial mass, the m in F = ma, and m^ for gravi- 
tational mass, the m in F = GMm/d^. The Principle 
of Equivalence says that gravitational field of 
strength g can be replaced in effect by an opposite 
acceleration g of the observer. 

.". m^g must be the same as m^g .'. m^ ^ m" 

The Principle of Equivalence requires gravitational 
mass and inertial mass to be the same; and the 
Myth-and-Symbol Experiment long ago told us that 
they are. As you will see in the discussion that fol- 
lows, Einstein, in his development of General Rela- 
tivity, gave a deeper meaning for this equality of 
the two kinds of mass. 

General Relativity and Geometry 

Over small regions of space-&-time, the Earth's 
gravity is practically uniform — and so is any other 

" That is why the Sun's gravitational pull produces "no 
noticeable field" as we move with the Earth around its yearly 
orbit. (That phrase in the table of field values on p. 116 was 
a quibble! ) Only if inertial mass and gravitational mass 
failed to keep exactly the same proportion for different sub- 
stances would any noticeable effect occur. Minute differences 
of such a kind are being looked for — if any are discovered, 
they will have a profound effect on our theory. 


gravitational field. So we can "remove" gravity for 
local experiments by having our lab accelerate 
freely; and it will behave like an inertial frame with 
no gravitational field: an object left alone will stay 
at rest or move in a straight line; and with forces ap- 
plied we shall find F ^=. ma. However, on a grander 
scale, say all around the Earth or the Sun, we should 
have to use many different accelerations for our 
local labs to remove gravity. In fitting a straight 
line defined in one lab by Newton's Law I to its 
continuation in a neighboring lab, also accelerating 
freely, we should find we have to 'Tjend" our straight 
line to make it fit. The demands of bending would 
get worse as we proceeded from lab to lab around 
the gravitating mass. How can we explain that? 
Instead of saying "we have found there is gravity 
here after all" we might say "Euclidean geometry 
does not quite fit the real world near the massive 
Earth or Sun." The second choice is taken in develop- 
ing General Relativity. As in devising Special Rela- 
tivity, Einstein looked for the simplest geometry to 
fit the new assumption that the laws of physics 
should always take the same form. He arrived at a 
General-Relativity geometry in which gravity disap- 
pears as a strange force reaching out from matter; 
instead, it appears as a distortion of space-&-time 
around matter. 

"From time immemorial the physicist and the 
pure mathematician had worked on a certain agree- 
ment as to the shares which they were respectively 
to take in the study of nature. The mathematician 
was to come first and analyse the properties of space 
and time building up the primary sciences of geome- 
try and kinematics (pure motion); then, when the 
stage had thus been prepared, the physicist was to 
come along with the dramatis personae — material 
bodies, magnets, electric charges, light and so forth — 
and the play was to begin. But in Einstein's revolu- 
tionary conception the characters created the stage 
as they walked about on it: geometry was no longer 
antecedent to physics but indissolubly fused with 
it into a single discipline. The properties of space 
in General Relativity depend on the material bodies 
and the energy that are present. . . ."^* 

Is this new geometry right and the old wrong? 
Let us return to our view of mathematics as the 
obedient servant. Could we not use any system of 
geometry to carry out our description of the physical 
world, stretching the world picture to fit the geome- 
try, so to speak? Then our search would not be to 
find the right geometry but to choose the simplest 
or most convenient one which would describe the 

" Sir Edmund Whittaker, From Euclid to Eddington, 
op.cit., p. 117. 

world with least stretching." If we do, we must 
realize that we choose our geometry but we have 
our universe; and if we ruthlessly make one fit the 
other by pushing and pulling and distorting, then 
we must take the consequences. 

For example, if all the objects in our world con- 
sisted of some pieces of the elastic skin of an orange, 
the easiest geometrical model to fit them on would 
be a ball. But if we were brought up with an un- 
dying belief in plane geometry, we could press the 
peel down on a flat table and glue it to the surface, 
making it stretch where necessary to accommodate 
to the table. We might find the cells of the peel 
larger near the outer edge of our flattened piece, 
but we should announce that as a law of nature. We 
might find strange forces trying to make the middle 
of the patch bulge away from the table — again, a 
"law of nature." If we sought to simplify our view of 
nature, the peel's behavior would tempt us to use a 
spherical surface instead of a flat one, as our model 
of "surface-space." All this sounds fanciful, and it 
is; but just such a discussion on a three- or four- 
dimensional basis, instead of a two-dimensional one, 
has been used in General Relativity. The strange 
force of gravity may be a necessary result of trying 
to interpret nature with an unsuitable geometry — 
the system Euclid developed so beautifully. If we 
choose a different geometry, in which matter dis- 
torts the measurement system around it, then gravi- 
tation changes from a surprising set of forces to a 
mere matter of geometry. A cannon ball need no 
longer be regarded as being dragged by gravity in 
what the old geometry would call a "curve" in space. 
Instead, we may think of it as sailing serenely along 
what the new geometry considers a straight line in 
its space-&-time, as distorted by the neighboring 

This would merely be a change of view (and 
as scientists we should hardly bother much about 
it), unless it could open our eyes to new knowledge 
or improve our comprehension of old knowledge. 
It can. On such a new geometrical view, the 
"curved" paths of freely moving bodies are inlaid 
in the new geometry of space-&-time and all pro- 
jectiles, big and small, with given speed must follow 
the same path. Notice how the surprise of the 
Myth-and-Symbol fact disappears. The long-stand- 
ing mystery of gravitational mass being equal to 
inertial mass is solved. Obviously a great property 
of nature, this equality was neglected for centuries 
until Einstein claimed it as a pattern property im- 
posed on space-&-time by matter. 

-' You can have your coffee served on any tray, but on 
some trays it wobbles less. 


Mathematics and Relativity 

Even a light ray must follow a curve, just as much 
as a bullet moving at light speed. Near the Earth 
that curve would be imperceptible, but starlight 
streaming past the Sun should be deflected by an 
angle of about 0.0005 degrees, just measurable by 
modern instruments. Photographs taken during total 
eclipses show that stars very near the edge of the 
Sun seem shifted by about 0.0006°. On the tradi- 
tional ("classical") view, the Sun has a gravitational 
field that appears to modify the straight-line law for 
light rays of the Euclidean geometrical scheme. On 
the General Relativity view, we replace the Sun's 
gravitational field by a crumpling of the local ge- 
ometry from simple Euclidean form into a version 
where light seems to us to travel slower. Thus the 
light beam is curved slightly around as it passes the 
Sun — the reverse of the bending of light by hot air 
over a road, when it makes a mirage. 

Finding this view of gravitation both simple and 
fruitful — when boiled down to simplest mathemati- 
cal form — we would like to adopt it. In any ordinary 
laboratory experiments we find Euclid's geometry 
gives simple, accurate descriptions. But in astro- 
nomical cases with large gravitational fields we 
must either use a new geometry ( in which the mesh 
of "straight lines" in space-&-time seems to us 
slightly crumpled) or else we must make some com- 
plicating changes in the laws of physics. As in 
Special Relativity, the modern fashion is to make 
the change in geometry. This enables us to polish 
up the laws of physics into simple forms which hold 
universally; and sometimes in doing that we can see 
the possibility of new knowledge. 

In specifying gravitation on the new geometrical 
view, Einstein found that his simplest, most plausi- 
ble form of law led to slightly different predictions 
from those produced by Newton's inverse-square 
law of gravitation. He did not "prove Newton's Law 
wrong" but offered a refining modification — though 
this involved a radical change in viewpoint. We 
must not think of either law as right because it is 
suggested by a great man or because it is enshrined 
in beautiful mathematics. We are offered it as a 
brilliant guess from a great mind unduly sensitive 
to the overtones of evidence from the real universe. 
We take it as a promising guess, even a likely one, 
but we then test it ruthlessly. The changes, from 
Newton's predictions to Einstein's, though funda- 
mental in nature, are usually too small in effect to 
make any difference in laboratory experiments or 
even in most astronomical measurements. But there 
should be a noticeable effect in the rapid motion 
of the planet Mercury around its orbit. Newton pre- 


^J OCiij' "^ ""If 

Fic. 31-42. Motion of Planet Mebcury 

dieted a simple ellipse, with other planets producing 
perturbations which could be calculated and ob- 
served. General Relativity theory predicts an extra 
motion, a very slow slewing around of the long axis 
of the ellipse by 0.00119 degree per century. When 
Einstein predicted it, this tiny motion was already 
known, discovered long before by Leverrier. The 
measured value, 0.001 17 "/century was waiting to 
test the theory. 

Accepting this view of gravity, astronomers can 
speculate on the geometry of all space and ask 
whether the universe is infinite or bounded by its 
own geometric curvature ( as a sphere is ) . We may 
yet be able to make some test of this question. 

There are still difficulties and doubts about Gen- 
eral Relativity. Even as we use it confidently to deal 
with Mercury's motion, or the light from a massive 
star, we may have to anchor our calculations to some 
frame of reference, perhaps the remotest regions of 
space far from gravitating matter, or perhaps the 
center of gravity of our universe. So space as we 
treat it, m^y have some kind of absolute milestones. 
This doubt, this threat to a powerful theory, does 
not irritate the wise scientist: he keeps it in mind 
with hopes of an interesting future for his thoughts. 

New Mathematics for Nuclear Physics 

In atomic and nuclear physics, mathematics now 
takes a strong hand. Instead of sketching a model 
with sharp bullet-like electrons whirling round an 
equally sharp nucleus, we express our knowledge 
of atoms in mathematical forms for which no picture 
can be drawn. These forms use unorthodox rules of 
algebra, dreamed up for the purpose; and some 
show the usual mathematical trademark of waves. 
Yet, although they remain mathematical forms, they 
yield fruitful predictions, ranging from the strength 
of metal wires and chemical energies to the behavior 
of radioactive nuclei. 

We now see mathematics, pure thought and argu- 
ment, again offering to present physics in clearer 
forms which help our thinking; but now far from a 
servant, it is rather a Lord Chancellor standing be- 
hind the throne of ruling Science to advise on law. 
Or, we might describe mathematics as a master 
architect designing the building in which science 
can grow to its best. 


Invarrance Is central to the theory of relativity as to 
all modern physics. The story told here Introduces 
many of the Important fundamental concepts of rela- 
tivity theory. 

8 Parable of the Surveyors 

Edwin F. Taylor and John Archibald Wheeler 

Excerpt from their book, Spacetime Physics, W. H. Freeman and Company 
Copyright ©1966. 

Once upon a time there was a Daytime surveyor who measured off the king's 
lands. He took his directions of north and east from a magnetic compass 
needle. Eastward directions from the center of the town square he measured in 
meters (x in meters). Northward directions were sacred and were measured in 
a different unit, in miles {y in miles). His records were complete and accurate 
and were often consulted by the Daytimers. 

Nighttimers used the services of another surveyor. His north and east 
directions were based on the North Star. He too measured distances eastward 
from the center of the town square in meters {x' in meters) and sacred distances 
north in miles {y' in miles). His records were complete and accurate. Every 
corner of a plot appeared in his book with its two coordinates, x' and y' . 

One fall a student of surveying turned up with novel openmindedness. 
Contrary to all previous tradition he attended both of the rival schools 
operated by the two leaders of surveying. At the day school he learned from 
one expert his method of recording the location of the gates of the town and 
the corners of plots of land. At night school he learned the other method. As 
the days and nights passed the student puzzled more and more in an attempt 
to find some harmonious relationship between the rival ways of recording 
location. He carefully compared the records of the two surveyors on the loca- 
tions of the town gates relative to the center of the town square: 

Daytime surveyor uses 
magnetic north 

Nighttime surveyor 
uses North Star north 

Table 1. Two different sets of records for the same points. 


Daytime surveyor's axes oriented 

to magnetic north 

(x in meters; y in miles) 

Nighttime surveyor's axes 
oriented to the North Star 
(x' in meters , >' in miles) 

Town square 
Gate A 
Gate B 
Other gales 







In defiance of tradition, the student took the daring and heretical step to 
convert northward measurements, previously expressed always in miles, into 
meters by multiplication with a constant conversion factor, k. He then dis- 
covered that the quantity [{xj^f + (^>'a)^]"^ based on Daytime measurements 
of the position of gate A had exactly the same numerical value as the quantity 




Fig. 1 . The town and its gates, showing coordi- 
nate axes used by two different surveyors. 

[(x^y -f {kyAyy^ computed from the readings of the Nighttime surveyor for 
gate A. He tried the same comparison on the readings computed from the re- 
corded positions of gate B, and found agreement here too. The student's 
excitement grew as he checked his scheme of comparison for all the other 
town gates and found everywhere agreement. He decided to give his dis- 
covery a name. He called the quantity 

Discovery: invariance (1) [(x)- -\- (kyyy^ 

of distance 

the distance of the point (x, y) from the center of town. He said that he had 

discovered the principle of the invariance of distance; that one gets exactly the 
same distances from the Daytime coordinates as from the Nighttime coordi- 
nates, despite the fact that the two sets of surveyors' numbers are quite 

This story illustrates the naive state of physics before the discovery of 
special relativity by Einstein of Bern, Lorentz of Leiden, and Poincare of 
Paris. How naive? 

1. Surveyors in this mythical kingdom measured northward distances in a 
sacred unit, the mile, different from the unit used in measuring eastward 
distances. Similarly, people studying physics measured time in a sacred 
unit, the second, different from the unit used in measuring space. No one 
thought of using the same unit for both, or of what one could learn by 
squaring and combining space and time coordinates when both were 
measured in meters. The conversion factor between seconds and meters, 
namely the speed of light, c = 2.997925 X 10* meters per second, was 
regarded as a sacred number. It was not recognized as a mere conversion 
factor like the factor of conversion between miles and meters— a factor 
that arose out of historical accidents alone, with no deeper physical 

2. In the parable the northbound coordinates, y and >•', as recorded by the 
two surveyors did not differ very much because the two directions of 
north were separated only by the small angle of 10 degrees. At first our 
mythical student thought the small differences between y and y' were due 
to surveying error alone. Analogously, people have thought of the time 
between the explosion of two firecrackers as the same, by whomever 
observed. Only in 1905 did we learn that the time difference between 
the second event and the first, or "reference event," really has dif- 


Parable of the Surveyors 

ferent values, / and /', for observers in different states of motion. 
Think of one observer standing quietly in the laboratory. The other 
observer zooms by in a high-speed rocket. The rocket comes in through 
the front entry, goes down the middle of the long corridor and out the 
back door. The first firecracker goes off in the corridor ("reference 
event") then the other ("event A"). Both observers agree that the 
reference event establishes the zero of time and the origin for distance 
measurements. The second explosion occurs, for example, 5 seconds 
later than the first, as measured by laboratory clocks, and 12 meters 
further down the corridor. Then its time coordinate is /a = 5 seconds 
and its position coordinate is X\ = 12 meters. Other explosions and 
events also take place down the length of the corridor. The readings of 
the two observers can be arranged as in Table 2. 

One observer uses 
laboratory frame 

Another observer uses 
rocket frame 

Table 2. Space and time coordinates of the same events as seen by two 
observers in relative motion. For simplicity the y and z co- 
ordinates are zero, and the rocket is moving in the x direction. 

Coordinates as measured by observer who is 


(x in meters: t in seconds) 

moving by in rocket 
{x' in meters; t' in seconds) 

Reference event 

Event A 



Event B 



Other events 

X \ 


The mythical student's discovery of the concept of distance is matched by 
the Einstein-Poincare discovery in 1905 of the idea of interval. The in- 
terval as calculated from the one observer's measurements 


interval = [(c/a)' - (xa)'] 


agrees with the interval as calculated from the other observer's measure- 


interval = [{cuj - {x^'n''' 

even though the separate coordinates employed in the two calculations 
do not agree. The two observers will find different space and time coordi- 
nates for events A, B, C, . . . relative to the same reference event, but 
when they calculate the Einstein intervals between these events, their 
results will agree. The invariance of the interval— its independence from 
the choice of the reference frame— forces one to recognize that time can- 
not be separated from space. Space and time are part of the single 
entity, spacetime. The geometry of spacetime is truly four-dimensional. 
In one way of speaking, the "direction of the time axis" depends upon the 
state of motion of the observer, just as the directions of the y axes 
employed by the surveyors depend upon their different standards of 

Discovery: invariance. 
of interval 


The rest of this chapter is an elaboration of the analogy between surveying 
in space and relating events to one another in spacetime. Table 3 is a preview 
of this elaboration. To recognize the unity of space and time one follows the 
procedure that makes a landscape take on meaning— he looks at it from several 
angles. This is the reason for comparing space and time coordinates of an 
event in two different reference frames in relative motion. 

Table 3. Preview: Elaboration of the parable of the surveyors. 

Parable of the surveyors : 
geometry of space 

Analogy to physics : 
geometry of spacetime 

The task of the surveyor is to locate the posi- 
tion of a point (gate A) using one of two co- 
ordinate systems that are rotated relative to 
one another. 

The task of the physicist is to locate the posi- 
tion and time of an event (firecracker explo- 
sion A) using one of two reference frames 
which are in motion relative to one another. 

The two coordinate systems: oriented to 
magnetic north and to North-Star north. 

The two reference frames: the laboratory 
frame and the rocket frame. 

For convenience all surveyors agree to make 
position measurements with respect to a 
common origin (the center of the town 

For convenience all physicists agree to make 
position and time measurements with re- 
spect to a common reference event (explo- 
sion of the reference firecracker). 

The analysis of the surveyors' results is sim- 
plified if X and v coordinates of a point are 
both measured in the same units, in meters. 

The separate coordinates x\ and y^ of gate 
A do not have the same values respectively in 
two coordinate systems that are rotated 
relative to one another. 

The analysis of the physicists' results is sim- 
plified if the X and t coordinates of an event 
are both measured in the same units, in 

The separate coordinates x?, and fx of event 
A do not have the same values respectively in 
two reference frames that are in uniform 
motion relative to one another. 

Invariance of distance. The distance (.va^ + 
>'A^)"' between gate A and the town square 
has the same value when calculated using 
measurements made with respect to either of 
two rotated coordinate systems (x\ and va 
both measured in meters). 

Invariance of the interval. The interval (t\^ — 
ata^)"^ between event A and the reference 
event has the same value when calculated 
using measurements made with respect to 
either of two reference frames in relative 
motion (x\ and t\ both measured in meters). 

Euclidean transformation. Using Euclidean 
geometry, the surveyor can solve the follow- 
ing problem: Given the Nighttime coordin- 
ates xa.' and y\' of gate A and the relative 
inclination of respective coordinate axes, 
find the Daytime coordinates .va and va of 
the same gate. 

Lorentz transformation. Using Lorentz 
geometry, the physicist can solve the follow- 
ing problem: Given the rocket coordinates 
.ya' and t\' of event A and the relative 
velocity between rocket and laboratory 
frames, find the laboratory coordinates xk 
and /a of the same event. 

Measure time in meters 

The parable of the surveyors cautions us to use the same unit to measure 
both distance and time. So use meters for both. Time can be measured in 
meters. When a mirror is mounted at each end of a stick one-half meter 
long, a flash of light may be bounced back and forth between these two mir- 


Parable of the Surveyors 

rors. Such a device is a clock. This clock may be said to "tick" each time the 
light flash arrives back at the first mirror. Between ticks the light flash has 
traveled a round-trip distance of 1 meter. Therefore the unit of time between 
ticks of this clock is called 1 meter of light-travel time or more simply / meter 
of time. (Show that 1 second is approximately equal to 3 X ICH meters of 
light-travel time.) 

One purpose of the physicist is to sort out simple relations between events. 
To do this here he might as well choose a particular reference frame with 
respect to which the laws of physics have a simple form. Now, the force of Simplify: Pick freely 
gravity acts on everything near the earth. Its presence complicates the laws of falling laboratory 
motion as we know them from common experience. In order to eliminate this 
and other complications, we will, in the next section, focus attention on a 
freely falling reference frame near the earth. In this reference frame no gravi- 
tational forces will be felt. Such a gravitation-free reference frame will be 
called an inertial reference frame. Special relativity deals with the classical 
laws of physics expressed with respect to an inertial reference frame. 

The principles of special relativity are remarkably simple. They are very 
much simpler than the axioms of Euclid or the principles of operating an auto- 
mobile. Yet both Euclid and the automobile have been mastered— perhaps 
with insufficient surprise— by generations of ordinary people. Some of the 
best minds of the twentieth century struggled with the concepts of relativity, 
not because nature is obscure, but simply because man finds it difficult to out- 
grow established ways of looking at nature. For us the battle has already been 
won. The concepts of relativity can now be expressed simply enough to make 
it easy to think correctly— thus "making the bad difficult and the good easy."t 
The problem of understanding relativity is no longer one of learning but one of 
intuition— a. practiced way of seeing. With this way of seeing, a remarkable 
number of otherwise incomprehensible experimental results are seen to be 
perfectly natural, t 

tEinstein, in a similar connection, in a letter to the architect Le Corbusier. 

tFor a comprehensive set of references to introductory literature concerning the special theory 
of relativity, together with several reprints of articles, see Special Relativity Theory, Selected 
Reprints, published for the American Association of Physics Teachers by the American Insti- 
tute of Physics, 335 East 45th Street, New York 17, New York, 1963. 


The father of the general theory of relativity and his 
associate illustrate one of the central ideas of the 
theory through the commonplace experience of riding 
in an elevator. (Note: The initials C. S. mean 
"coordinate system" in this selection.) 

3 Outside and Inside the Elevator 

Albert Einstein and Leopold Infeld 

Excerpt from their book. The Evolution of Physics. 1938 and 1961. 

The law of inertia marks the first great advance in 
physics; in fact, its real beginning. It was gained by the 
contemplation of an idealized experiment, a body mov- 
ing forever with no friction nor any other external 
forces acting. From this example and later from many 
others, we recognized the importance of the idealized 
experiment created by thought. Here again, idealized 
experiments will be discussed. Although these may 
sound very fantastic they will, nevertheless, help us to 
understand as much about relativity as is possible by 
our simple methods. 

We had previously the idealized experiments with a 
uniformly moving room. Here, for a change, we shall 
have a falling elevator. 

Imagine a great elevator at the top of a skyscraper 
much higher than any real one. Suddenly the cable 
supporting the elevator breaks, and the elevator falls 
freely toward the ground. Observers in the elevator 
are performing experiments during the fall. In describ- 
ing them, we need not bother about air resistance or 
friction, for we may disregard their existence under 
our idealized conditions. One of the observers takes a 
handkerchief and a watch from his pocket and drops 
them. What happens to these two bodies? For the out- 


side observer, who is looking through the window of 
the elevator, both handkerchief and watch fall toward 
the ground in exactly the same way, with the same 
acceleration. We remember that the acceleration of a 
falling body is quite independent of its mass and that 
it was this fact which revealed the equaUty of gravita- 
tional and inertial mass (p. 37). We also remember that 
the equality of the two masses, gravitational and in- 
ertial, was quite accidental from the point of view 
of classical mechanics and played no role in its struc- 
ture. Here, however, this equality reflected in the equal 
acceleration of all falling bodies is essential and forms 
the basis of our whole argument. 

Let us return to our falling handkerchief and watch; 
for the outside observer they are both falling with the 
same acceleration. But so is the elevator, with its walls, 
ceiling, and floor. Therefore: the distance between the 
two bodies and the floor will not change. For the in- 
side observer the two bodies remain exactly where 
they were when he let them go. The inside observer 
may ignore the gravitational field, since its source lies 
outside his CS. He finds that no forces inside the ele- 
vator act upon the two bodies, and so they are at 
rest, just as if they were in an inertial CS. Strange 
things happen in the elevator! If the observer pushes 
a body in any direction, up or down for instance, it 
always moves uniformly so long as it does not collide 
with the ceiling or the floor of the elevator. Briefly 
speaking, the laws of classical mechanics are valid for 
the observer inside the elevator. All bodies behave in 
the way expected by the law of inertia. Our new CS 
rigidly connected with the freely falling elevator dif- 
fers from the inertial CS in only one respect. In an 


Outside and Inside the Elevator 

inerrial CS, a moving body on which no forces are 
acting will move uniformly forever. The inertial CS as 
represented in classical physics is neither hmited in 
space nor time. The case of the observer in our elevator 
is, however, different. The inertial character of his CS 
is limited in space and time. Sooner or later the uni- 
formly moving body will collide with the wall of the 
elevator, destroying the uniform motion. Sooner or 
later the whole elevator will collide with the earth 
destroying the observers and their experiments. The 
CS is only a "pocket edition" of a real inertial CS. 

This local character of the CS is quite essential. If 
our imaginary elevator were to reach from the North 
Pole to the Equator, with the handkerchief placed over 
the North Pole and the watch over the Equator, then, 
for the outside observer, the two bodies would not 
have the same acceleration; they would not be at rest 
relative to each other. Our whole argument would 
fail! The dimensions of the elevator must be limited 
so that the equality of acceleration of all bodies rela- 
tive to the outside observer may be assumed. 

With this restriction, the CS takes on an inertial 
character for the inside observer. We can at least indi- 
cate a CS in which all the physical laws are valid, even 
though it is limited in time and space. If we imagine 
another CS, another elevator moving uniformly, rela- 
tive to the one falling freely, then both these CS will 
be locally inertial. All laws are exactly the same in both. 
The transition from one to the other is given by the 
Lorentz transformation. 

Let us see in what way both the observers, outside 
and inside, describe what takes place in the elevator. 

The outside observer notices the motion of the ele- 


vator and of all bodies in the elevator, and finds them 
in agreement with Newton's gravitational law. For 
him, the motion is not uniform, but accelerated, be- 
cause of the action of the gravitational field of the 

However, a generation of physicists bom and 
brought up in the elevator would reason quite differ- 
ently. They would believe themselves in possession of 
an inertial system and would refer all laws of nature to 
their elevator, stating with justification that the laws 
take on a specially simple form in their CS. It would 
be natural for them to assume their elevator at rest and 
their CS the inertial one. 

It is impossible to settle the differences between the 
outside and the inside observers. Each of them could 
claim the right to refer all events to his CS. Both de- 
scriptions of events could be made equally consistent. 

We see from this example that a consistent descrip- 
tion of physical phenomena in two different CS is pos- 
sible, even if they are not moving uniformly, relative 
to each other. But for such a description we must take 
into account gravitation, building so to speak, the 
"bridge" which effects a transition from one CS to the 
other. The gravitational field exists for the outside ob- 
server; it does not for the inside observer. Accelerated 
motion of the elevator in the gravitational field exists 
for the outside observer, rest and absence of the gravi- 
tational field for the inside observer. But the "bridge," 
the gravitational field, making the description in both 
CS possible, rests on one very important pillar: the 
equivalence of gravitational and inertial mass. Without 
this clew, unnoticed in classical mechanics, our present 
argument would fail completely. 


Outside and Inside the Elevator 

Now for a somewhat different idealized experiment. 
There is, let us assume, an inertial CS, in which the 
law of inertia is valid. We have already described what 
happens in an elevator resting in such an inertial CS. 
But we now change our picture. Someone outside has 
fastened a rope to the elevator and is pulling, with a 
constant force, in the direction indicated in our draw- 
ing. It is immaterial how this is done. Since the laws of 
mechanics are valid in this CS, the whole elevator 
moves with a constant acceleration in the direction of 
the motion. Again we shall listen to the explanation of 

phenomena going on in the elevator and given by both 
the outside and inside observers. 

The outside observer: My CS is an inertial one. The 
elevator moves with constant acceleration, because a 
constant force is acting. The observers inside are in 
absolute motion, for them the laws of mechanics are 
invalid. They do not find that bodies, on which no 
forces are acting, are at rest. If a body is left free, it 
soon collides with the floor of the elevator, since the 
floor moves upward toward the body. This happens 


exactly in the same way for a watch and for a handker- 
chief. It seems very strange to me that the observer 
inside the elevator must always be on the "floor" be- 
cause as soon as he jumps, the floor will reach him 

The inside observer: I do not see any reason for be- 
lieving that my elevator is in absolute motion. I agree 
that my CS, rigidly connected with my elevator, is not 
really inertial, but I do not believe that it has anything 
to do with absolute motion. My watch, my handker- 
chief, and all bodies arc falling because the whole ele- 
vator is in a gravitational field. I notice exactly the 
same kinds of motion as the man on the earth. He 
explains them very simply by the action of a gravita- 
tional field. The same holds good for me. 

These two descriptions, one by the outside, the other 
by the inside, observer, are quite consistent, and there is 
no possibility of deciding which of them is right. We 
may assume either one of them for the description of 
phenomena in the elevator: either nonuniform mo- 
tion and absence of a gravitational field with the out- 
side observer, or rest and the presence of a gravitational 
field with the inside observer. 

The outside observer may assume that the elevator 
is in "absolute" nonuniform motion. But a motion 
which is wiped out by the assumption of an acting 
gravitational field cannot be regarded as absolute mo- 

There is, possibly, a way out of the ambiguity of two 
such different descriptions, and a decision in favor of 
one against the other could perhaps be made. Imagine 
that a light ray enters the elevator horizontally through 
a side window and reaches the opposite wall after a 


Outside and Inside the Elevator 

very short time. Again let us see how the path of the 
light would be predicted by the two observers. 

The outside observer, believing in accelerated mo- 
tion of the elevator, would argue: The light ray enters 
the window and moves horizontally, along a straight 
line and with a constant velocity, toward the opposite 
wall. But the elevator moves upward and during the 
time in which the light travels toward the wall, the 
elevator changes its position. Therefore, the ray will 
meet a point not exactly opposite its point of entrance, 
but a little below. The difference will be very slight, 
but it exists nevertheless, and the light ray travels, rela- 
tive to the elevator, not along a straight, but along a 


slightly curved Hne. The difference is due to the dis- 
tance covered by the elevator during the time the ray 
is crossing the interior. 

The inside observer, who believes in the gravitational 
field acting on all objects in his elevator, would say: 
there is no accelerated motion of the elevator, but only 
the action of the gravitational field. A beam of light is 
weightless and, therefore, will not be affected by the 
gravitational field. If sent in a horizontal direction, it 
will meet the wall at a point exactly opposite to that at 
which it entered. 


It seems from this discussion that there is a possibility 
of deciding between these two opposite points of view 
as the phenomenon would be different for the two ob- 
servers. If there is nothing illogical in either of the 
explanations just quoted, then our whole previous ar- 
gument is destroyed, and we cannot describe all phe- 
nomena in two consistent ways, with and without a 
gravitational field. 

But there is, fortunately, a grave fault in the reason- 
ing of the inside observer, which saves our previous 
conclusion. He said: "A beam of light is weightless 
and, therefore, it will not be affected by the gravita- 
tional field." This cannot be right! A beam of light 
carries energy and energy has mass. But every inertial 
mass is attracted by the gravitational field as inertial 
and gravitational masses are equivalent. A beam of light 
will bend in a gravitational field exactly as a body 
would if thrown horizontally with a velocity equal to 
that of light. If the inside observer had reasoned cor- 
rectly and had taken into account the bending of light 
rays in a gravitational field, then his results would have 
been exactly the same as those of an outside observer. 

The gravitational field of the earth is, of course, too 
weak for the bending of light rays in it to be proved 
directly, by experiment. But the famous experiments 
performed during the solar eclipses show, conclu- 
sively though indirectly, the influence of a gravitational 
field on the path of a light ray. 

It follows from these examples that there is a well- 
founded hope of formulating a relativistic physics. But 
for this we must first tackle the problem of gravitation. 

We saw from the example of the elevator the con- 
sistency of the two descriptions. Nonuniform motion 


Outside and Inside the Elevator 

may, or may not, be assumed. We can eliminate "abso- 
lute" motion from our examples by a gravitational field. 
But then there is nothing absolute in the nonuniform 
motion. The gravitational field is able to wipe it out 

The ghosts of absolute motion and inertial CS can 
be expelled from physics and a new relativistic physics 
built. Our idealized experiments show how the prob- 
lem of the general relativity theory is closely con- 
nected with that of gravitation and why the equiv- 
alence of gravitational and inertial mass is so essential 
for this connection. It is clear that the solution of the 
gravitational problem in the general theory of rela- 
tivity must differ from the Newtonian one. The laws 
of gravitation must, just as all laws of nature, be formu- 
lated for all possible CS, whereas the laws of classical 
mechanics, as formulated by Newton, are valid only 
in inertial CS. 


What lessons can be learned from the life and 
philosophy of a "high-school drop-out" named 
Albert Einstein? Martin Klein, a physicist and 
historian of science, discusses the possibility of 

inadequacies in our present education Dollcies. 

10 Einstein and some Civilized Discontents 

Martin Klein 

Article from the journal, Physics Today, January 1965. 

The French novelist Stendhal began his most 
brilliant novel with this sentence: "On May 15, 
1796, General Bonaparte made his entrance into 
Milan at the head of that youthful army which 
had just crossed the bridge of Lodi, and taught 
the world that after so many centuries Caesar 
and Alexander had a successor." In its military 
context, the quotation is irrelevant here, but it 
can be paraphrased a bit: almost exactly a cen- 
tury later Milan saw the arrival of another young 
foreigner who would soon teach the world that 
after so many centuries Galileo and Newton had 
a successor. It would, however, have taken super- 
human insight to recognize the future intellectual 
conqueror in the boy of fifteen who had just 
crossed the Alps from Munich. For this boy, 
Albert Einstein, whose name was to become a 
symbol for profound scientific insight, had left 
Munich as what we would now call a high-school 

He had been a slow child; he learned to speak 
at a much later age than the average, and he 
had shown no special ability in elementary school 
—except perhaps a talent for day-dreaming. The 
education offered at his secondary school in Mu- 
nich, one of the highly praised classical gymnasia, 
did not appeal to him. The rigid, mechanical 
methods of the school appealed to him even less. 
He had already begun to develop his own intel- 
lectual pursuits, but the stimulus for them had 
not come from school. The mystery hidden in 
the compass given to him when he was five, the 
clarity and beauty of Euclidean geometry, discov- 
ered by devouring an old geometry text at the 
age of twelve— it was these things that set him on 
his own road of independent study and thought. 
The drill at school merely served to keep him 
from his own interests. ^Vhen his father, a small 
and unsuccessful manufacturer, moved his busi- 
ness and his family from Munich to Milan, Albert 
Einstein was left behind to finish his schooling 
and acquire the diploma he would need to insure 

his future. After some months, however, Einstein 
was fed up with school, and resolved to leave. 
His leaving was assisted by the way in which his 
teachers reacted to his attitude toward school. 
"You will never amount to anything, Einstein," 
one of them said, and another actually suggested 
that Einstein leave school because his very pres- 
ence in the classroom destroyed the respect of the 
students. This suggestion was gratefully accepted 
by Einstein, since it fit so well with his own 
decisions, and he set off to join his family in 
Milan. The next months were spent gloriously 
loafing, and hiking around northern Italy, enjoy- 
ing the many contrasts with his homeland. With 
no diploma, and no prospects, he seemed a very 
model dropout. 

It is sobering to think that no teacher had 
sensed his potentialities. Perhaps it suggests why 
I have chosen this subject in talking to this gath- 
ering of physics teachers seriously devoted to 
improving education in physics, and devoted in 
particular to a program aimed at the gifted 
student of our science— at his early detection and 
proper treatment. For what I really want to do 
is to highlight some aspects of Einstein's career 
and thought that stand in sharp contrast to a 
number of our accepted ideas on education and 
on the scientific career. The first matter we must 
reckon with is Einstein's own education and the 
way it affected him; but let me carry the story 
a little further before raising some questions. 

Einstein had dropped out of school, but he had 
not lost his love for science. Since his family's 
resources, or lack of them, would make it neces- 
sary for him to become self-supporting, he decided 
to go on with his scientific studies in an official 
way. He, therefore, presented himself for admis- 
sion at the renowned Swiss Federal Institute of 
Technology in Zurich. Since he had no high- 
school diploma he was given an entrance exam- 
ination—and he failed. He had to attend a Swiss 
high school for a year in order to make up his 


deficiencies in almost everything except mathe- 
matics and physics, the subjects of his own private 
study. And then, when he was finally admitted 
to the Polytechnic Institute, did he settle down 
and assume what we would consider to be his 
rightful place at the head of the class? Not at all. 
Despite the fact that the courses were now almost 
all in mathematics and physics, Einstein cut most 
o£ the lectures. He did enjoy working in the lab- 
oratory, but he spent most of his time in his 
room studying the original works of the masters 
of nineteenth-century physics, and pondering 
what they set forth. 

The lectures on advanced mathematics did not 
hold him, because in those days he saw no need 
or use for higher mathematics as a tool for grasp- 
ing the structure of nature. Besides, mathematics 
appeared to be split into so many branches, each 
of which could absorb all one's time and energy, 
that he feared he could never have the insight 
to decide on one of them, the fundamental one. 
He would then be in the position of Buridan's 
ass, who died of hunger because he could not 
decide which bundle of hay he should eat. 

Physics presented no such problems to Einstein, 
even then. As he wrote many years later: "True 
enough, physics was also divided into separate 
fields, each of which could devour a short working 
life without having satisfied the hunger for deeper 
knowledge. . . . But in physics I soon learned to 
scent out the paths that led to the depths, and 
to disregard everything else, all the many things 
that clutter up the mind, and divert it from the 
essential. The hitch in this was, of course, the 
fact that one had to cram all this stuff into one's 
mind for the examination, whether one liked it 
or not." 

That was indeed the rub. Einstein had recon- 
ciled himself to being only an average scholar 
at the Polytechnic. He knew that he did not have 
and could not, or perhaps would not, acquire 
the traits of the outstanding student: the easy 
facility in comprehension, the willingness to con- 
centrate one's energies on all the required sub- 
jects, and the orderliness to take good notes and 
work them over properly. Fortunately, however, 
the Swiss system required only two examinations. 
Even more fortunately Einstein had a close friend, 
Marcel Grossmann, who possessed just the qual- 
ities that Einstein lacked, and who generously 
shared his excellent systematic notes with his non- 
conforming comrade. So Einstein was able to 
follow his own line of study, and still succeed in 
the exams by doing some appropriate cramming 
from Grossmann's notes. This success left more 

photo by lotte Jacob! 

than a bad taste in his mouth. As he put it, 
"It had such a deterring effect upon me that, 
after I had passed the final examination, I found 
the consideration of any scientific problems dis- 
tasteful to me for an entire year." .\nd he went 
on to say, "It is little short of a miracle that 
modern methods of instruction have not already 
completely strangled the holy curiosity of inquiry, 
because what this delicate little plant needs most, 
apart from initial stimulation, is freedom; with- 
out that it is surely destroyed ... I believe that 
one could even deprive a healthy beast of prey 
of its voraciousness, if one could force it with a 
whip to eat continuously whether it were hungry 
or not. . . ." 

This is strong language. Should we take it 
personally? Could it be meant for us, for the 
teachers responsible for an educational system of 
achievement tests, preliminary college boards, col- 
lege boards, national scholarships, grade point 
averages, graduate record exams, PhD qualifying 


Einstein and some Civilized Discontents 

exams— a system that starts earlier and earlier 
and ends later and later in our students' careers? 
Could this system be dulling the appetites of our 
young intellectual tigers? Is it possible that our 
students need more time to day-dream rather than 
more hours in the school day? That the relentless 
pressure of our educational system makes every- 
thing only a step toward something else and 
nothing an end in itself and an object of pleasure 
and contemplation? 

For almost two years after his graduation from 
the Polytechnic in 1900 Einstein seemed to be 
headed for no more success than his earlier history 
as a dropout might have suggested. He applied 
for an assistantship, but it went to someone else. 
During this period he managed to subsist on the 
odd jobs of the learned world: he substituted for 
a Swiss high-school teacher who was doing his 
two months of military service, he helped the 
professor of astronomy with some calculations, he 
tutored at a boys' school. Finally, in the spring 
of 1902, Einstein's good friend Marcel Grossmann, 
"the irreproachable student", came to his rescue. 
Grossmann's father recommended Einstein to the 
director of the Swiss Patent Office at Berne, and 
after a searching examination he was appointed 
to a position as patent examiner. He held this 
position for over seven years and often referred 
to it in later years as "a kind of salvation". It 
freed him from financial worries; he found the 
work rather interesting; and sometimes it served 
as a stimulus to his scientific imagination. And 
besides, it occupied only eight hours of the day, 
so that there was plenty of time left free for 
pondering the riddles of the universe. 

In his spare time during those seven years at 
Berne, the young patent examiner wrought a 
series of scientific miracles: no weaker word is 
adequate. He did nothing less than to lay out 
the main lines along which twentieth-century 
theoretical physics has developed. A very brief 
list will have to suffice. He began by working 
out the subject of statistical mechanics quite inde- 
pendently and without knowing of the work of 
J. VVillard Gibbs. He also took this subject seri- 
ously in a way that neither Gibbs nor Boltzmann 
had ever done, since he used it to give the theo- 
retical basis for a final proof of the atomic 
nature of matter. His reflections on the problems 
of the Maxwell-Lorentz electrodynamics led him 
to create the special theory of relativity. Before 
he left Berne he had formulated the principle 
of equivalence and was struggling with the prob- 

lems of gravitation which he later solved with 
the general theory of relativity. And, as if these 
were not enough, Einstein introduced another 
new idea into physics, one that even he described 
as "very revolutionary", the idea that light con- 
sists of particles of energy. Following a line of 
reasoning related to but quite distinct from 
Planck's, Einstein not only introduced the light 
quantum hypothesis, but proceeded almost at 
once to explore its implications for phenomena 
as diverse as photochemistry and the temperature 
dependence of the specific heat of solids. 

What is more, Einstein did all this completely 
on his own, with no academic connections what- 
soever, and with essentially no contact with the 
elders of his profession. Years later he remarked 
to Leopold Infeld that until he was almost thirty 
he had never seen a real theoretical physicist. To 
which, of course, we should add the phrase (as 
Infeld almost did aloud, and as Einstein would 
never have done) , "except in the mirror!" 

I suppose that some of us might be tempted 
to wonder what Einstein might have done during 
those seven years, if he had been able to work 
"under really favorable conditions", full time, at 
a major university, instead of being restricted to 
spare-tirrie activity while earning his living as a 
minor civil servant. We should resist the tempta- 
tion: our speculations would be not only fruitless, 
but completely unfounded. For not only did 
Einstein not regret his lack of an academic post 
in these years, he actually considered it a real 
advantage. "For an academic career puts a young 
man into a kind of embarrassing position," he 
wrote shortly before his death, "by requiring him 
to produce scientific publications in impressive 
quantity— a seduction into superficiality which 
only strong characters are able to withstand. Most 
practical occupations, however, are of such a 
nature that a man of normal ability is able to 
accomplish what is expected of him. His day-to- 
day existence does not depend on any special 
illuminations. If he has deeper scientific interests 
he may plunge into his favorite problems in ad- 
dition to doing his required work. He need not 
be oppressed by the fear that his efforts may lead 
to no results. I owed it to Marcel Grossmann 
that I was in such a fortunate position." 

These were no casual remarks: forty years 
earlier Einstein had told Max Born not to worry 
about placing a gifted student in an academic 
position. Let him be a cobbler or a locksmith; 
if he really has a love for science in his blood 
and if he's really worth anything, he will make 
his own way. (Of course, Einstein then gave what 


help he could in placing the young man.) Einstein 
was even a little reluctant about accepting a re- 
search professorship at Berlin, partly because 
Prussian rigidity and academic bourgeois life 
were not to his Bohemian taste. But he was also 
reluctant because he kncAv very well that such 
a research professor was expected to be a sort of 
prize hen, and he did not want to guarantee 
that he would lay any more golden eggs. 

It will not have escaped your notice that 
Einstein's views on research and the nature of a 
scientific career differ sharply from those which 
are standard in the scientific community. No 
doubt some of this difference in attitude reflects 
only Einstein's uniquely solitary nature. It is hard 
to imagine anyone else seriously suggesting as he 
did, that a position as lighthouse keeper might 
be suitable for a scientist. Most scientists feel the 
need to test their ideas on their peers, and often 
to form these ideas in the give and take of dis- 
cussions, as among their most urgent needs. One 
may still question the necessity of as many meet- 
ings as we find announced in Physics Today, and 
one may question even more insistently the ne- 
cessity of reporting on each and publishing its 
proceedings as if it were the first Solvay Congress 

More serious is the attitude that every young 
man of scientific ability can claim the right to 
a position as prize hen. "Doing research" has 
become the hallowed activity in the academic 
world, and, as Jacques Barzun has put it, "To 
suggest that practice, or teaching, or reflection 
might be preferred is blasphemy." I do not need 
to re-emphasize Einstein's remark on the publish- 
or-perish policy that corrupts one aspect of aca- 
demic life. I would, however, like to remark 

parenthetically that I am always astonished when 
college administrators and department heads 
claim that it is terribly difficult, virtually impos- 
sible, to judge the quality of a man's teaching, 
but never doubt their ability to evaluate the 
results of his research. This is astonishing because 
any honest undergraduate can give a rather canny 
and usually accurate appraisal of the teaching he 
is subjected to, but judging the quality of a sci- 
entific paper generally increases in difficulty with 
the originality of the work reported. Einstein's 
hypothesis of light quanta, for example, was con- 
sidered as wildly off the mark, as at best a par- 
donable excess in an otherwise sound thinker, 
even by Planck a decade after it was introduced. 
The way in which physics is taught is deeply 
influenced by our views of how and why physics 
is done. Einstein, who was skeptical about the 
professionalization of research, was unswerving in 
his pursuit of fundamental understanding; he 
was a natural philosopher in the fullest sense of 
that old term, and he had no great respect for 
those who treated science as a game to be played 
for one's personal satisfaction, or those who 
solved problems to demonstrate and maintain 
their intellectual virtuosity. If physics is viewed 
in Einstein's way, it follows that it should be 
taught as a drama of ideas and not as a battery 
of techniques. It follows too that there should 
be an emphasis on the evolution of ideas, on 
the history of our attempts to understand the 
physical world, so that our students acquire some 
perspective and realize that, in Einstein's words, 
"the present position of science can have no last- 
ing significance." Do we keep this liberal view 
of our science, or is it lost in what we call nec- 
essary preparation for graduate work and research? 

One last theme that cannot be ignored when 
we speak of Einstein is that of the scientist as 
citizen. Einstein's active and courageous role in 
public affairs is widely known, and it absorbed a 
substantial fraction of his efforts for forty years. 
He stepped onto the public stage early and in 
characteristic style. In October 1914, two months 
after the outbreak of the First World War, a 
document was issued in Berlin bearing the gran- 
diose title. Manifesto to the Civilized World; it 
carried the signatures of almost a himdred of 
Germany's most prominent scientists, artists, men 
of letters, clergymen, etc. This manifesto pro- 
claimed its signers' fidl support of Germany's war 
effort, denoimced the opponents of the fatherland, 
and defiantly asserted that German militarism and 
German culture formed an inseparable unity. 


Einstein and some Civilized Discontents 

Not all German intellectuals approved this chau- 
vinistic document, but among the very few who 
were willing to sign a sharply worded answer, 
calling for an end to war and an international 
organization, was Albert Einstein. The highly un- 
popular stand that he took in 1914 expressed a 
deeply felt conviction, one on which he acted 
throughout his life, regardless of the consequences 
to himself. During the succeeding decades Einstein 
devoted a great deal of his energy to the causes 
in which he believed, lending his name to many 
organizations which he felt could further these 
causes. Contrary to the view held in some circles, 
however, Einstein carefully considered each signa- 
ture that he inscribed on a petition, each political 
use that he made of the name that had become 
renowned for scientific reasons, and often refused 
his support to organizations that attempted to 
solicit it. 

His public statements became even more fre- 
quent and more outspoken in the years after the 
Second World War, as he put all his weight 
behind the effort to achieve a world government 
and to abolish war once and for all. Einstein 
was among those who have been trying to impress 
upon the world the very real likelihood that an- 
other war would destroy civilization and perhaps 
humanity as well. He was not overly optimistic 

about his efforts, but they had to be made. He 
also felt that he had to speak out, loudly and 
clearly, during the McCarthy era, urging intel- 
lectuals to adopt the method of civil disobedience 
as practiced earlier by Gandhi (and later by 
Martin Luther King) . .As he wrote in an open 
letter, "Every intellectual who is called before one 
of the committees ought to refuse to testify; i.e., 
he must be prepared for jail and economic ruin, 
in short, for the sacrifice of his personal welfare 
in the interest of the cultural welfare of his 
country." If such a program were not adopted 
then, wrote Einstein, "the intellectuals of this 
country deserve nothing better than the slavery 
which is intended for them." 

It is quite evident that Einstein approached 
political and social questions as a man who con- 
sidered himself outside the Establishment. He had 
a very strong sense of responsibility to his con- 
science, but he did not feel obliged to accept 
all the restrictions that society expects of a "re- 
sponsible spokesman". This approach is neither 
possible nor appropriate for today's leading sci- 
entists who are constantly serving as scientific 
statesmen— as advisers to the AEC, or the Depart- 
ment of Defense, or major corporations, or even 
the President. Such men are in no position to 
adopt Einstein's critical stance, even if they 
wanted to. At this time, when science requires 
and receives such large-scale support, it seems that 
we have all given more hostages to fortune than 
we may realize. 

One of Einstein's last public statements was 
made in answer to a request that he comment 
on the situation of scientists in America. He 
wrote: "Instead of trying to analyze the problem 
I should like to express my feeling in a short 
remark. If I were a young man again and had 
to decide how to make a living, I would not 
try to become a scientist or scholar or teacher. 
I would rather choose to be a plumber or a 
peddler, in the hope of finding that modest degree 
of independence still available under present 

We may wonder how literally he meant this 
to be taken, but we cannot help feeling the force 
of the affront to our entire institutionalized life 
of the intellect. 

As we pride ourselves on the success of physics 
and physicists in today's world, let us not forget 
that it was just that success and the way in 
which it was achieved that was repudiated by 
Einstein. And let us not forget to ask why: it 
may tell us something worth knowing about our- 
selves and our society. 



We visit, in this brief passage, on elementary 
science class hearing for the first time about 
the Bohr theory of the atom. 

11 The Teacher and the Bohr Theory of the Atom 

Charles Percy Snow 

An excerpt from his novel The Search, published in 1934 
and 1958. 

Then one day, just before we broke up for Christmas, 
Luard came into the class-room almost brightly. 

"We're not going into the laboratory this morning," 
he said. "I'm going to talk to you, my friends." He used to 
say "my friends " whenever he was lashing us with his tongue, 
but now it sounded half in earnest. "Forget everything 
you know, will you? That is, if you know anything at all." 
He sat on the desk swinging his legs. 

"Now, what do you think all the stuff in the world is 
made of? Every bit of us, you and me, the chairs in this 
room, the air, everything. No one knows? Well, perhaps 
that's not surprising, even for nincompoops like you. Because 
no one did know a year or two ago. But now we're beginning 
to think we do. That's what I want to tell you. You won't 
understand, of course. But it'll amuse me to tell you, and it 
won't hurt you, I suppose — and anyway I'm going to." 

Someone dropped a ruler just then, and afterwards the 
room was very quiet. Luard took no notice and went on: 
"Well, if you took a piece of lead, and halved it, and halved 
the half, and went on like that, where do you think you'd 
come to in the end? Do you think it would be lead for ever? 
Do you think you could go down right to the infinitely small 
and still have tiny pieces of lead? It doesn't matter what you 
think. My friends, you couldn't. If you went on long enough, 
you'd come to an atom of lead, an atom, do you hear, 
an atom, and if you split that up, you wouldn't have lead 
anymore. What do you think you would have? The answer 
to that is one of the oddest things you'll ever hear in your 
life. If you split up an atom of lead, you'd get — pieces of 
positive and negative electricity. Just that. Just positive 
and negative electricity. That's all matter is. That's all you 
are. Just positive and negative electricity — and, of course, 
an immortal soul." At the time I was too busy attending to 
his story to observe anything else ; but in the picture I have 
formed later of Luard, I give him here the twitch of a smile. 
"And whether you started with lead or anything else it 


wouldn't matter. That's all you'd come to in the end. Posi- 
tive and negative electricity. How do things differ then? 
Well, the atoms are all positive and negative electricity and 
they're all made on the same pattern, but they vary among 
themselves, do you see? Every atom has a bit of positive 
electricity in the middle of it — the nucleus, they call it — 
and every atom has bits of negative electricity going round 
the nucleus — like planets round the sun. But the nucleus 
is bigger in some atoms than others, bigger in lead than it is 
in carbon, and there are more bits of negative electricity 
in some atoms than others. It's as though you had different 
solar systems, made from the same sort of materials, some 
with bigger suns than others, some with a lot more planets. 
That's all the difference. That's where a diamond's different 
from a bit of lead. That's at the bottom of the whole of this 
world of ours." He stopped and cleaned his pince-nez, and 
talked as he swung them : 

"There you are, that's the way things are going. Two 
people have found out about the atoms: one's an English- 
man, Rutherford, and the other's a Dane called Bohr. And 
I tell you, my friends, they're great men. Greater even than 
Mr. Miles" — I flushed. I had come top of the form and this 
wa5 his way of congratulating me — "incredible as that may 
seem. Great men, my friends, and perhaps, when you're 
older, by the side of them your painted heroes, your Cassars 
and Napoleons, will seem like cocks crowing on a dung- 

I went home and read everything I could discover about 
atoms. Popular exposition was comparatively slow at that 
time, however, and Rutherford's nucleus, let alone Bohr's 
atom, which could only have been published a few months 
before Luard's lesson, had not yet got into my Encyclopaedia. 
I learned something of electrons and got some idea of size ; I 
was fascinated by the tininess of the electron and the 
immensity of the great stars : I became caught up in light- 
years, made time-tables of a journey to the nearest star (in the 


The Teacher and the Bohr Theory of the Atom 

Encyclopaedia there was an enthralling picture of an express 
train going off into space at the speed of light, taking years 
to get to the stars). Scale began to impress me, the in- 
finitesimal electronic distances and the vastness of Aldebaran 
began to dance round in my head; and the time of an elec- 
tronic journey round the nucleus compared itself with the 
time it takes for light to travel across the Milky Way. Distance 
and time, the infinitely great and the infinitely small, electron 
and star, went reeling round my mind. 

It must have been soon after this that I let myself seep 
in the fantasies that come to many imaginative children 
nowadays. Why should not the electron contain worlds 
smaller than itself, carrying perhaps inconceivably minute 
replicas of ourselves? 'They wouldn't know they're small. 
They wouldn't know of us,' I thought, and felt serious and 
profound. And why should not our world be just a part of 
an electron in some cosmic atom, itself a part of some 
gargantuan world? The speculations gave me a pleasant 
sense of philosophic agoraphobia until I was about sixteen 
and then I had had enough of them. 

Luard, who had set me alight by half an hour's talk, did 
not repeat himself Chemistry lessons relapsed once more into 
exercises meaningless to me, definitions of acids and bases 
which I learned resentfully, and, as we got further up the 
school, descriptions of the properties of gases, which always 
began "colourless, transparent, non-poisonous." Luard, 
who had once burst into enthusiasm, droned out the de- 
finitions or left us to a text-book while he sat by himself 
at the end of the laboratory. Once or twice there would be a 
moment of fire; he told us about phlogiston — "that should 
be a lesson to you, my friends, to remember that you can 
always fall back on tradition if only you're dishonest 
enough " and Faraday — "there never will be a better scientist 
than he was ; and Davy tried to keep him out of the Royal 
Society because he had been a laboratory assistant. Davy 
was the type of all the jumped-up second-raters of all time." 


Educated as we are in classical physics, we may 
be unprepared to comprehend the world of quantum 
mechanics. This book tries to introduce us to this 
new view of the world. 

12 The New Landscape of Science 

Banesh Hoffmann 

Chapter from his book. The Strange Story of the Quantum. 1959. 

Let us now gather the loose threads of our thoughts and see 
what pattern they form when knit together. 

We seem to ghmpse an eerie shadow world lying beneath 
our world of space and time; a weird and cryptic world which 
somehow rules us. Its laws seem mathematically precise, and 
its events appear to unfold with strict causality. 

To pry into the secrets of this world we make experiments. 
But experiments are a clumsy instrument, afflicted with a fatal 
indeterminacy which destroys causality. And because our 
mental images are formed thus clumsily, we may not hope to 
fashion mental pictures in space and rime of what transpires 
within this deeper world. Abstract mathematics alone may try 
to paint its likeness. 

With indeterminacy corrupting experiment and dissolving 
causality, all seems lost. We must wonder how there can be a 
rational science. We must wonder how there can be any- 
thing at all but chaos. But though the detailed workings of the 
indeterminacy lie hidden from us, we find therein an astound- 
ing uniformity. Despite the inescapable indeterminacy of 
experiment, we find a definite, authentic residue of exactitude 
and determinacy. Compared with the detailed determinacy 
claimed by classical science, it is a meager residue indeed. But 
it is precious exactitude none the less, on which to build a 
science of natural law. 

The very nature of the exactitude seems a paradox, for it is 
an exactitude of probabilities; an exactitude, indeed, of wave- 
like, interfering probabilities. But probabilities are potent 


things — if only they are applied to large numbers. Let us see 
what strong reliance may be placed upon them. 

When we toss a coin, the result may not be predicted, for 
it is a matter of chance. Yet it is not entirely undetermined. 
We know it must be one of only two possibilities. And, more 
important even than that, if we toss ten thousand coins we 
know we may safely predict that about half will come down 
heads. Of course we might be wrong once in a very long 
while. Of course we are taking a small risk in making such a 
prediction. But let us face the issue squarely, for we really 
place far more confidence in the certainty of probabilities than 
we sometimes like to admit to ourselves when thinking of them 
abstractly. If someone offered to pay two dollars every time a 
coin turned up heads provided we paid one dollar for every 
tails, would we really hesitate to accept his offer? If we did 
hesitate, it would not be because we mistrusted the probabili- 
ties. On the contrary, it would be because we trusted them so 
well we smelled fraud in an offer too attractive to be honest. 
Roulette casinos rely on probabilities for their gambling prof- 
its, trusting to chance that, in the long run, zero or double 
zero will come up as frequently as any other number and thus 
guarantee them a steady percentage of the total transactions. 
Now and again the luck runs against them and they go broke 
for the evening. But that is because chance is still capricious 
when only a few hundred spins are made. Insurance companies 
also rely on probabilities, but deal with far larger numbers. One 
does not hear of their ever going broke. Tliey make a hand- 
some living out of chance, for when precise probabilities can 
be found, chance, in the long run, becomes practical certainty. 
Even classical science built an elaborate and brilliantly suc- 
cessful theory of gases upon the seeming quicksands of prob- 

In the new world of the atom we find both precise proba- 
bilities and enormous numbers, probabilities that follow exact 
mathematical laws, and vast, incredible numbers compared 
with which the multitude of persons carr}ing insurance is as 
nothing. Scientists have determined the weight of a single 


The New Landscape of Science 

electron. Would a million electrons weigh as much as a feather, 
do you think? A million is not large enough. Nor even a billion. 
Well, surely a million billion then. No. Not even a bilHon 
billion electrons would outweigh the feather. Nor }et a million 
billion billion. Not till we have a billion billion billion can we 
talk of their weight in such everyday terms. Quantum 
mechanics having discovered precise and wonderful laws gov- 
erning the probabilities, it is with numbers such as these that 
science overcomes its handicap of basic indeterminacy. It is 
by this means that science boldly predicts. Tliough now hum- 
bly confessing itself powerless to foretell the exact behavior 
of individual electrons, or photons, or other fundamental 
entities, it yet can tell with enomious confidence how such 
great multitudes of them must behave precisely. 

But for all this mass precision, we are only human if, on 
first hearing of the breakdown of determinacy in fundamental 
science, we look back longingly to the good old classical days, 
when waves were waves and particles particles, when the work- 
ings of nature could be readily visualized, and the future was 
predictable in every individual detail, at least in theory. But 
the good old days were not such happy days as nostalgic, rose- 
tinted retrospect would make them seem. Too many contradic- 
tions flourished unresolved. Too many well-attested facts played 
havoc with their pretensions. Those were but days of scientific 
childhood. There is no going back to them as they were. 

Nor may we stop with the world we -have just described, if 
we are to round out our story faithfully. To stifle nostalgia, we 
pictured a world of causal law lying beneath our world of space 
and rime. While important scientists seem to feel that such 
a world should exist, many others, pointing out that it is not 
demonstrable, regard it therefore as a bit of homely mysticism 
added more for the sake of comfort than of cold logic. 

It is difficult to decide where science ends and mysricism 
begins. As soon as we begin to make even the most elementary 
theories we are open to the charge of indulging in metaphysics. 
Yet theories, however provisional, are the very lifeblood of 
scientific progress. We simply cannot escape metaphysics, 


though we can perhaps overindulge, as well as have too little. 
Nor is it feasible always to distinguish good metaphysics from 
bad, for the "bad" may lead to progress where the "good" 
would tend to stifle it. When Columbus made his historic 
voyage he believed he was on his westward way to Japan. Even 
when he reached land he thought it was part of Asia; nor did 
he live to learn otherwise. Would Columbus have embarked 
upon his hazardous journey had he known what was the true 
westward distance of Japan? Quantum mechanics itself came 
partly from the queer hunches of such men as Maxwell and 
Bohr and de Broglie. In talking of the meaning of quantum 
mechanics, physicists indulge in more or less mysticism accord- 
ing to their individual tastes. Just as different artists instinc- 
tively paint different likenesses of the same model, so do 
scientists allow their different personalities to color their inter- 
pretations of quantum mechanics. Our story would not be 
complete did we not tell of the austere conception of quantum 
mechanics hinted at above, and also in our parable of the coin 
and the principle of perversity, for it is a view held by many 

These physicists are satisfied with the sign-language rules, 
the extraordinary precision of the probabilities, and the strange, 
wavelike laws which they obey. They realize the impossibility 
of following the detailed workings of an indeterminacy through 
which such bountiful precision and law so unaccountably seep. 
They recall such incidents as the vain attempts to build models 
of the ether, and their own former naive beliefs regarding 
momentum and position, now so rudely shattered. And, recall- 
ing them, they are properly cautious. They point to such 
things as the sign-language rules, or the probabilities and the 
exquisite mathematical laws in multidimensional fictional space 
which govern them and which have so eminently proved them- 
selves in the acid test of experiment. And they say that these 
are all we may hope and reasonably expect to know; that 
science, which deals with experiments, should not probe too 
deeply beneath those experiments for such things as cannot be 
demonstrated even in theory. 


The New Landscape of Science 

The great mathematician John von Neumann, who accom- 
plished the Herculean labor of cleaning up the mathematical 
foundations of the quantum theory, has even proved mathe- 
matically that the quantum theory is a complete system in 
itself, needing no secret aid from a deeper, hidden world, and 
offering no evidence whatsoever that such a world exists. Let 
us then be content to accept the world as it presents itself 
to us through our experiments, however strange it may seem. 
This and this alone is the image of the world of science. After 
casrigaring the classical theorists for their unwarranted assump- 
tions, however seemingly innocent, would it not be foolish and 
foolhardy to invent that hidden world of exact causality of 
which we once thought so fondly, a worid which by its very 
nature must lie beyond the reach of our experiments? Or, 
indeed, to invent anything else which cannot be demonstrated, 
such as the detailed occurrences under the Heisenberg micro- 
scope and all other pieces of .comforring imagery wherein we 
picture a wavicle as an old-fashioned particle preliminary to 
proving it not one? 

All that talk of exactitude somehow seeping through the 
indeterminacy was only so much talk. We must cleanse our 
minds of previous pictorial notions and start afresh, taking the 
laws of quantum mechanics themselves as the basis and the 
complete outline of modem physics, the full delineation of 
the quantum worid beyond which there is nothing that may 
properiy belong to physical science. As for the idea of strict 
causality, not only does science, after all these years, suddenly 
find it an unnecessary concept, it even demonstrates that 
according to the quantum theory strict causality is funda- 
mentally and intrinsically undemonstrable. Therefore, strict 
causahty is no longer a legitimate scienrific concept, and must 
be cast out from the official domain of present-day science. As 
Dirac has written, 'The onJy object of theoTeticaJ physics is to 
calculate results that can be compared with experiment, and it 
is quite unnecessary that any satisfying description of the 
whole course of the phenomena should be given." The italics 
here are his. One cannot escape the feeling that it might have 


been more appropriate to italicize the second part of the state- 
ment rather than the first! 

Here, then, is a more restricted pattern which, paradoxically, 
is at once a more cautious and a bolder view of the world of 
quantum physics; cautious in not venturing beyond what is 
well established, and bold in accepting and being well content 
with the result. Because it does not indulge too freely in specu- 
lation it is a proper view of present-day quantum physics, and 
it seems to be the sort of view held by the greatest number. 
Yet, as we said, there are many shades of opinion, and it is 
sometimes difficult to decide what are the precise views of 
particular individuals. 

Some men feel that all this is a transitional stage through 
which science will ultimately pass to better things — and they 
hope soon. Others, accepting it with a certain discomfort, 
have tried to temper its awkwardness by such devices as the 
introduction of new types of logic. Some have suggested that 
the observer creates the result of his observation by the act 
of observation, somewhat as in the parable of the tossed coin. 
Many nonscientists, but few scientists, have seen in the new 
ideas the embodiment of free will in the inanimate world, and 
have rejoiced. Some, more cautious, have seen merely a revived 
possibility of free will in ourselves now that our physical proc- 
esses are freed from the shackles of strict causality. One could 
continue endlessly the list of these speculations, all testifying 
to the devastating potency of Planck's quantum of action h, a 
quantity so incredibly minute as to seem uttedy inconse- 
quential to the uninitiated. 

That some prefer to swallow their quantum mechanics plain 
while others gag unless it be strongly seasoned with imager^' 
and metaphysics is a matter of individual taste behind which 
lie certain fundamental facts which may not be disputed; hard, 
uncompromising, and at present inescapable facts of experi- 
ment and bitter experience, agreed upon by all and directly 
opposed to the classical way of thinking: 

There is simply no satisfactory way at all of picturing the 
fundamental atomic processes of nature in terms of space and 
time and causality. 


The New Landscape of Science 

The result of an experiment on an individual atomic particle 
generally cannot be predicted. Only a list of various possible 
results may be known beforehand. 

Nevertheless, the statistical result of performing the same 
individual experiment over and over again an enormous num- 
ber of times may he predicted with virtual certainty. 

For example, though we can show there is absolutely no con- 
tradiction involved, we cannot visualize how an electron which 
is enough of a wave to pass through two holes in a screen and 
interfere with itself can suddenly become enough of a particle 
to produce a single scintillation. Neither can we predict where 
it will scintillate, though we can say it may do so only in certain 
regions but not in others. Nevertheless when, instead of a 
single electron, we send through a rich and abundant stream we 
can predict with detailed precision the intricate interference 
pattern that will build up, even to the relative brightness of its 

various parts. 

Our inability to predict the individual result, an inability 
which, despite the evidence, the classical view was unable to 
tolerate, is not only a fundamental but actually a plausible 
characteristic of quantum mechanics. So long as quantum 
mechanics is accepted as wholly valid, so long must we accept 
this inability as intrinsically unavoidable. Should a way ever 
be found to overcome this inability, that event would mark the 
end of the reign of quantum mechanics as a fundamental 
pattern of nature. A new, and deeper, theory would have to 
be found to replace it, and quantum mechanics would have to 
be retired, to become a theory emeritus with the revered, if 
faintly irreverent title "classical." 

Now that we are accustomed, a little, to the bizarre new 
ideas we may at last look briefly into the quantum mechanical 
significance of something which at first sight seems trivial and 
inconsequential, namely, that electrons are so similar we can- 
not tell one from another. This is true also of other atomic 
particles, but for simplicity let us talk about electrons, with the 
understanding that the discussion is not thereby confined to 
them alone. 


Imagine, then, an electron on this page and another on the 
opposite page. Take a good look at them. You cannot tell 
them apart. Now blink your eyes and take another look at 
them. They are still there, one on this page and one on that. 
But how do you know they did not change places just at the 
moment your eyes were closed? You think it most unlikely? 
Does it not always rain on just those days when you go out 
and leave the windows open? Does it not always happen that 
your shoelace breaks on just those days when you are in a 
special hurry? Remember these electrons are identical twins 
and apt to be mischievous. Surely you know better than to 
argue that the electron interchange was unlikely. You cer- 
tainly could not prove it one way or another. 

Perhaps you are still unconvinced. Let us put it a little 
differently, then. Suppose the electrons collided and bounced 
off one another. Hien you certainly could not tell which one 
was which after the collision. 

You still think so? You think you could keep your eyes 
glued on them so they could not fool you? But, my dear sir, 
that is classical. That is old-fashioned. We cannot keep a 
continual watch in the quantum world. The best we can do is 
keep up a bombardment of photons. And with each impact 
the electrons jump we know not how. For all we know they 
could be changing places all the time. At the moment of 
impact especially the danger of deception is surely enormous. 
Let us then agree that wc can never be sure of the identity of 
each electron. 

Now suppose we wish to write down quantum equations for 
the two electrons. In the present state of our theories, we are 
obliged to deal with them first as individuals, saying that cer- 
tain mathematical co-ordinates belong to the first and certain 
others to the second. Tliis is dishonest though. It goes beyond 
permissible information, for it allows each electron to preserve 
its identity, whereas electrons should belong to the nameless 
masses. Somehow we must remedy our initial error. Somehow 
we must repress the electrons and remove from them their 
unwarranted individuality. This reduces to a simple question 


The New Landscape of Science 

of mathematical symmetries. We must so remold our equations 
that interchanging the electrons has no physically detectable 
effect on the answers they yield. 

Imposing this nonindividuality is a grave mathematical re- 
striction, strongly influencing the behavior of the electrons. Of 
the possible ways of imposing it, two are specially simple math- 
ematically, and it happens that just these two are physically of 
interest. One of them implies a behavior which is actually 
observed in the case of photons, and a particles, and other 
atomic particles. Tlie other method of imposing nonindi- 
viduality turns out to mean that the particles will shun one 
another; in fact, it gives precisely the mysterious exclusion 
principle of Pauli. 

This is indeed a remarkable result, and an outstanding 
triumph for quantum mechanics. It takes on added significance 
when we learn that all those atomic particles which do not 
obey the Pauli principle are found to behave like the photons 
and a particles. It is about as far as anyone has gone toward 
an understanding of the deeper significance of the exclusion 
principle. Yet it remains a confession of failure, for instead 
of having nonindividuality from the start we begin v^th indi- 
viduality and then deny it. The Pauli principle lies far deeper 
than this. It lies at the very heart of inscrutable Nature, Some- 
day, perhaps, we shall have a more profound theory in which 
the exclusion principle will find its rightful place. Meanwhile 
we must be content with our present veiled insight. 

The mathematical removal of individuality warps our equa- 
tions and causes extraordinary effects which cannot be properly 
explained in pictorial terms. It may be interpreted as bringing 
into being strange forces called exchange forces, but these 
forces, though already appearing in other connections in 
quantum mechanics, have no counterpart at all in classical 

We might have suspected some such forces were involved. 
It would have been incredibly naive to have believed that so 
stringent an ordinance against overcrowding as the exclusion 
principle could be imposed without some measure of force, 
however well disguised. 


Is it so sure that these exchange forces cannot be properly 
explained in pictorial terms? After all, with force is associated 
energy. And with energy is associated frequency according to 
Planck's basic quantum law. With frequency we may asso- 
ciate some sort of oscillation. Perhaps, then, if we think not 
of the exchange forces themselves but of the oscillations asso- 
ciated with them we may be able to picture the mechanism 
through which these forces exist. This is a promising idea. But 
if it is clarity we seek we shall be greatly disappointed in it. 

It is true there is an oscillation involved here, but what 
a fantastic oscillation it is: a rhythmic interchange of the elec- 
trons' identities. The electrons do not physically change places 
by leaping the intervening space. That would be too simple. 
Rather, there is a smooth ebb and flow of individuality between 
them. For example, if we start with electron A here and elec- 
tron B on the opposite page," then later on we would here have 
some such mixture as sixty per cent A and forty per cent B, 
vdth forty per cent A and sixty per cent B over there. Later still 
it would be all B here and all A there, the electrons then 
having definitely exchanged identities. The flow would now 
reverse, and the strange oscillation continue indefinitely. It is 
with such a pulsation of identity that the exchange forces of 
the exclusion principle are associated. There is another type 
of exchange which can affect even a single electron, the elec- 
tron being analogously pictured as oscillating in this curious, 
disembodied way between two different positions. 

Perhaps it is easier to accept such curious pulsations if we 
think of the electrons more as waves than as particles, for then 
we can imagine the electron waves becoming tangled up with 
each other. Mathematically this can be readily perceived, but 
it does not lend itself well to visualization. If we stay with the 
particle aspect of the electrons we find it hard to imagine what 
a 60 per cent-40 per cent mixture of A and B would look like 
if we observed it. We cannot observe it, though. Tlic act of 
observation would so jolt the electrons that we would find 
cither pure A or else pure B, but never a combination, the 
percentages being just probabilities of finding cither one. It 


The New Landscape of Science 

is really our parable of the tossed coin all over again. In mid- 
air the coin fluctuates rhythmically from pure heads to pure 
tails through all intermediate mixtures. When it lands on the 
table, which is to say when we observe it, there is a jolt 
which yields only heads or tails. 

Though we can at least meet objections, exchange remains 
an elusive and difficult concept. It is still a strange and awe-in- 
spiring thought that you and I are thus rhythmically exchang- 
ing particles with one another, and with the earth and the 
beasts of the earth, and the sun and the moon and the stars, 
to the uttermost galaxy. 

A striking instance of the power of exchange is seen in 
chemical valence, for it is essentially by means of these mys- 
terious forces that atoms cling together, their outer electrons 
busily shuttling identity and position back and forth to weave 
a bond that knits the atoms into molecules. 

Such are the fascinating concepts that emerged from the 
quantum mechanical revolution. TTie days of tumult shook 
science to its deepest foundations. They brought a new charter 
to science, and perhaps even cast a new light on the significance 
of the scientific method itself. The physics that survived the 
revolution w^as vastly changed, and strangely so, its whole out- 
look drastically altered. Where once it confidently sought a 
clear-cut mechanical model of nature for all to behold, it now 
contented itself with abstract, esoteric forms which may not 
be clearly focused by the unmathematical eye of the imagina- 
tion. Is it as strongly confident as once it seemed to be in 
younger days, or has internal upheaval undermined its health 
and robbed it of its powers? Has quantum mechanics been an 
advance or a retreat? 

If it has been a retreat in any sense at all, it has been a 
strategic retreat from the suffocating determinism of classical 
physics, which channeled and all but surrounded the advancing 
forces of science. Whether or not science, later in its quest, 
may once more encounter a deep causaUty, the determinism of 
the nineteenth century, for all the great discoveries it sired, was 
rapidly becoming an impediment to progress. When Planck 


first discovered the infinitesimal existence of the quantum, it 
seemed there could be no proper place for it anywhere in the 
whole broad domain of physical science. Yet in a brief quarter 
century, so powerful did it prove, it thrust itself into every 
nook and cranny, its influence growing to such undreamed-of 
proportions that the whole aspect of science was utterly trans- 
formed. With explosive violence it finally thrust through the 
restraining walls of determinism, releasing the pent-up forces 
of scientific progress to pour into the untouched fertile plains 
beyond, there to reap an untold harvest of discovery while still 
retaining the use of those splendid edifices it had created 
within the classical domain. The older theories were made 
more secure than ever, their triumphs unimpaired and their 
failures mitigated, for now their validity was established 
wherever the influence of the quantum might momentarilv be 
neglected. Their failures were no longer disquieting perplexi- 
ties which threatened to undermine the whole structure and 
bring it toppling down. With proper diagnosis the classical 
structures could be saved for special purposes, and their very 
weaknesses turned to good account as strong corroborations of 
the newer ideas; ideas which transcended the old without 
destroying their limited effectiveness. 

True, the newer theory baffled the untutored imagination, 
and was formidably abstract as no physical theory had ever 
been before. But this was a small price to i^ay for its extraor- 
dinary accomplishments. Newton's theory too had once 
seemed almost incredible, as also had that of Maxwell, and 
strange though quantum mechanics might appear, it was 
firmly founded on fundamental experiment. Here at long last 
was a theory which could embrace that primitive, salient fact 
of our material universe, that simple, everyday fact on which 
the Maxwellian theory so spectacularly foundered, the endur- 
ing stability of the different elements and of their physical and 
chemical properties. Nor was the new theory too rigid in this 
regard, but could equally well embrace the fact of radioactive 
transformation. Here at last was a theory which could yield the 
precise details of the enormously intricate data of spectroscopy. 
The photoelectric effect and a host of kindred phenomena suc- 


The New Landscape of Science 

cumbed to the new ideas, as too did the wavehke interference 
effects which formerly seemed to contradict them. With the 
aid of relativity, the spin of the electron was incorporated with 
remarkable felicity and success. Pauli's exclusion principle 
took on a broader significance, and through it the science of 
chemistry acquired a new theoretical basis amounting almost 
to a new science, theoretical chemistry, capable of solving 
problems hitherto beyond the reach of the theorist. The theory 
of metallic magnetism was brilliantly transformed, and stagger- 
ing difficulties in the theory of the flow of electricity through 
metals were removed as if by magic thanks to quantum 
mechanics, and especially to Pauli's exclusion principle. The 
atomic nucleus was to yield up invaluable secrets to the new 
quantum physics, as will be told; secrets which could not be 
revealed at all to the classical theory, since that theory was too 
primitive to comprehend them; secrets so abstruse they may 
not even be uttered except in quantum terms. Our understand- 
ing of the nature of the tremendous forces residing in the 
atomic nucleus, incomplete though it be, would be meager 
indeed wdthout the quantum theory to guide our search and 
encourage our comprehension in these most intriguing and 
mysterious regions of the universe. This is no more than a 
glimpse of the unparalleled achievements of quantum me- 
chanics. The wealth of accomplishment and corroborative 
evidence is simply staggering. 

"Daddy, do scientists really know what they are talking 

To still an inquiring child one is sometimes driven to regret- 
table extremes. Was our affirmative answer honest in this 
particular instance? 

Certainly it was honest enough in its context, immediately 
following the two other questions. But what of this same ques- 
tion now, standing alone? Do scientists really know what 
they are talking about? 

If we allowed the poets and philosophers and priests to 
decide, they would assuredly decide, on lofty grounds, against 
the physicists — quite irrespective of quantum mechanics. But 
on sufficiently lofty grounds the poets, philosophers, and priests 


themselves may scarcely claim they know whereof they talk, 
and in some instances, far from lofty, science has caught both 
them and itself in outright error. 

True, the universe is more than a collection of objective 
experimental data; more than the complexus of theories, 
abstractions, and special assumptions devised to hold the data 
together; more, indeed, than any construct modeled on this 
cold objectivity. For there is a deeper, more subjective world, 
a world of sensation and emotion, of aesthetic, moral, and 
religious values as yet beyond the grasp of objective science. 
And towering majestically over all, inscrutable and inescapable, 
is the awful mystery of Existence itself, to confound the mind 
with an eternal enigma. 

But let us descend from these to more mundane levels, for 
then the quantum physicist may make a truly impressive 
case; a case, moreover, backed by innumerable interlocking 
experiments forming a proof of stupendous cogency. Where 
else could one find a proof so overwhelming? How could one 
doubt the validity of so victorious a system? Men are hanged on 
evidence which, by comparison, must seem small and incon- 
sequential beyond measure. Surely, then, the quantum physi- 
cists know what they are talking about. Surely their present 
theories are proper theories of the workings of the universe. 
Surely physical nature cannot be markedly different from what 
has at last so painfully been revealed. 

And yet, if this is our belief, surely our whole story has been 
told in vain. Here, for instance, is a confident utterance of the 
year 1889: 

"The wave theory of light is from the point of view of human 
beings a certainty," 

It was no irresponsible visionary who made this bold asser- 
tion, no fifth-rate incompetent whose views might be lightly 
laughed away. It was the very man whose classic experiments, 
more than those of any other, established the electrical charac- 
ter of the waves of light; none other than the great Heinrich 
Hertz* himself, whose own seemingly incidental observation 
contained the seed from which there later was to spring the 
revitalized particle theory. 


The New Landscape of Science 

Did not the classical physicists point to overwhelming evi- 
dence in support of their theories, theories which now seem to 
us so incomplete and superficial? Did they not generally believe 
that physics was near its end, its main problems solved and its 
basis fully revealed, with only a little sweeping up and polish- 
ing left to occupy succeeding generations? And did they not 
believe these things even while they were aware of such 
unsolved puzzles as the violet catastrophe, and the photo- 
electric effect, and radioactive disintegration? 

The experimental proofs of science are not ultimate proofs. 
Experiment, that final arbiter of science, has something of the 
aspect of an oracle, its precise factual pronouncements couched 
in muffled language of deceptive import. While to Bohr such 
a thing as the Balmer ladder meant orbits and jumps, to 
Schrodmgcr it meant a smeared-out essence of <A; neither view 
is accepted at this moment. Even the measurement of the 
speed of light in water, that seemingly clear-cut experiment 
specifically conceived to decide between wave and particle, 
yielded a truth whose import was misconstrued. Science 
abounds with similar instances. Each change of theory demon- 
strates anew the uncertain certainty of experiment. One would 
be bold indeed to assert that science at last has reached an 
ultimate theory, that the quantum theory as we know it now 
will sundve with only superficial alteration. It may be so, but 
we are unable to prove it, and certainly precedent would seem 
to be against it. The quantum physicist does not know whether 
he knows what he is talking about. But this at least he does 
know, that his talk, however incorrect it may ultimately prove 
to be, is at present immeasurably superior to that of his 
classical forebears, and better founded in fact than ever before. 
And that is surely something well worth knowing. 

Never had fundamental science seen an era so explosively 
triumphant. With such revolutionary concepts as relativity 
and the quantum theory developing simultaneously, physics 
experienced a turmoil of upheaval and transformation without 
parallel in its history. The majestic motions of the heavens and 
the innermost tremblings of the atoms alike came under the 


searching scrutiny of the new theories. Man's concepts of time 
and space, of matter and radiation, energy, momentum, and 
causahty, even of science and of the universe itself, all were 
transmuted under the electrifying impact of the double revolu- 
tion. Here in our story we have followed the frenzied fortunes 
of the quantum during those fabulous years, from its first 
hesitant conception in the minds of gifted men, through 
precarious early years of infancy, to a temporary lodgment in 
the primitive theory of Bohr, there to prepare for a bewilder- 
ing and spectacular leap into maturity that was to turn the 
orderly landscape of science into a scene of utmost confusion. 
Gradually, from the confusion we saw a new landscape emerge, 
barely recognizable, serene, and immeasurably extended, and 
once more orderly and neat as befits the landscape of science. 

The new ideas, when first they came, were wholly repugnant 
to the older scientists whose minds were firmly set in tradi- 
tional ways. In those days even the flexible minds of the 
younger men found them startling. Yet now the physicists of 
the new generation, like infants incomprehensibly enjoying 
their cod-liver oil, lap up these quantum ideas with hearty 
appetite, untroubled by the misgivings and gnawing doubts 
which so sorely plagued their elders. Thus to the already bur- 
densome list of scientific corroborations and proofs may now 
be added this crowning testimony out of the mouths of babes 
and sucklings. TTie quantum has arrived. The tale is told. Let 
the final curtain fall. 

But ere the curtain falls we of the audience thrust forward, 
not yet satisfied. We are not specialists in atomic physics. We 
are but plain men who daily go about our appointed tasks, and 
of an evening peer hesitantly over the shoulder of the scientific 
theorist to glimpse the enchanted pageant that passes before 
his mind. Is all this business of wavicles and lack of causality 
in space and time something which the theorist can now accept 
with serenit}'? Can wc ourselves ever learn to welcome it with 
any deep feeling of acceptance? When so alien a world has 
been revealed to us we cannot but shrink from its vast unfriend- 
liness. It is a world far removed from our everyday experience. 


The New Landscape of Science 

It offers no simple comfort. It beckons us without warmth. 
We are saddened that science should have taken this curious, 
unhappy turn, ever away from the beliefs we most fondly 
cherish. Surely, we console ourselves, it is but a temporary 
aberration. Surely science will someday find the tenuous road 
back to normalcy, and ordinary men wall once more under- 
stand its message, simple and clear, and untroubled by abstract 

But we must remember that men have always felt thus when 
a bold new idea has arisen, be the idea right or wrong. When 
men first proclaimed the earth was not flat, did they not 
propose a paradox as devilish and devastating as any we have 
met in our tale of the quantum? How utterly fantastic must 
such a belief at first have appeared to most people; this belief 
which is now so readily and blindly accepted by children, 
against the clearest evidence of their immediate senses, that 
they are quick to ridicule the solitary crank who still may claim 
the earth is flat; their only concern, if any, is for the welfare 
of the poor people on the other side of this our round earth 
who, they so vividly reason, are fated to live out their lives 
walking on their heads. Let us pray that political wisdom and 
heaven-sent luck be granted us so that our children's children 
may be able as readily to accept the quantum horrors of today 
and laugh at the fears and misgivings of their benighted 
ancestors, those poor souls who still believed in old-fashioned 
waves and particles, and the necessity for national sovereignty, 
and all the other superstitions of an outworn age. 

It is not on the basis of our routine feelings that we should 
try here to weigh the value and significance of the quantum 
revolution. It is rather on the basis of its innate logic. 

"What!" you will exclaim. "Its innate logic? Surely that is 
the last thing we could grant it. We have to concede its over- 
whelming experimental support. But innate logic, a sort of 
aura to compel our belief, experiment or no experiment? No, 
that is too much. The new ideas are not innately acceptable, 
nor will talking ever make them so. Experiment forced them 
on us, but we cannot feel their inevitability. We accept them 


only laboriously, after much obstinate struggle. We shall never 
see their deeper meaning as in a flash of revelation. Though 
Nature be for them, our whole nature is against them. Innate 
logic? No! Just bitter medicine." 

But there is yet a possibiUty. Perhaps there is after all some 
innate logic in the quantum theory. Perhaps we may yet see 
in it a profoundly simple revelation, by whose light the ideas 
of the older science may appear as laughable as the doctrine 
that the earth is flat. We have but to remind ourselves that our 
ideas of space and time came to us through our ever)'day experi- 
ence and were gradually refined by the careful experiment of 
the scientist. As experiment became more precise, space and 
time began to assume a new aspect. E\en the relatively super- 
ficial experiment of Michelson and Morley, back in 1887, 
ultimately led to the shattering of some of our concepts of 
space and time by the theory of relativity. Nowadays, through 
the deeper techniques of the modern physicist we find that 
space and time as we know them so familiarly, and even space 
and time as relativity knows them, simply do not fit the more 
profound pattern of existence revealed by atomic experiment. 

What, after all, are these mystic entities space and time? 
We tend to take them for granted. We imagine space to be so 
smooth and precise we can define within it such a thing as a 
point — something having no size at all but only a continuing 
location. Now, this is all very well in abstract thought. Indeed, 
it seems almost an unavoidable necessity. Yet if we examine 
it in the light of the quantum discoveries, do we not find the 
beginning of a doubt? For how would we try to fix such a dis- 
embodied location in actual physical space as distinct from 
the purely mental image of space we have within our minds? 
What is the smallest, most delicate instrument we could use 
in order to locate it? Certainly not our finger. That could 
suffice to point out a house, or a pebble, or even, with difficulty, 
a particular grain of sand. But for a point it is far too gross. 

What of the point of a needle, then? Better. But far from 
adequate. Look at the needle point under a microscope and the 


The New Landscape of Science 

reason is clear, for it there appears as a pitted, tortured land- 
scape, shapeless and useless. WTiat then? We must try smaller 
and ever smaller, finer and ever finer indicators. But try as we 
will we cannot continue indefinitely. The ultimate point will 
always elude us. For in the end we shall come to such things as 
individual electrons, or nuclei, or photons, and beyond these, 
in the present state of science, we cannot go. WTiat has 
become, then, of our idea of the location of a point? Has it not 
somehow dissolved away amid the swirling wa\icles? True, we 
have said that we may know the exact position of a wavicle if 
we will sacrifice all knowledge of its motion. Yet even here 
tliere happen to be theoretical reasons connected with Comp- 
ton's experiment which limit the precision with which this 
position may be known. Even supposing the position could be 
known with the utmost exactitude, would wc then have a point 
such as we have in mind? Xo. For a point has a continuing 
location, while our location would be evanescent. Wc would 
still have merely a sort of abstract wavicle rather than an 
abstract point. Whether we think of an electron as a wavicle, 
or whether we think of it as a particle buffeted by the photons 
under a Heisenberg microscope, we find that the physical 
notion of a precise, continuing location escapes us. Though we 
have reached the present theoretical limit of refinement w^e 
have not yet found location. Indeed, we seem to be further 
from it than when we so hopefully started out. Space is not so 
simple a concept as we had naively thought. 

It is much as if we sought to obser\-e a detail in a newspaper 
photograph. We look at the picture more closely but the 
tantalizing detail still escapes us. Annoyed, we bring a magnify- 
ing glass to bear upon it, and lo! our eager optimism is shat- 
tered. We find ourselves far worse off than before. \Miat 
seemed to be an eye has now dissolved away into a meaning- 
less jumble of splotches of black and white. Tlie detail we had 
imagined simply was not there. Yet from a distance the picture 
still looks perfect. 

Perhaps it is the same with space, and with time too. Instinc- 


tively we feel they have infinite detail. But when we bring to 
bear on them our most refined techniques of observation and 
precise measurement we find that the infinite detail we had 
imagined has somehow vanished away. It is not space and time 
that are basic, but the fundamental particles of matter or 
energy themselves. Without these we could not have formed 
even the picture we instinctively have of a smooth, un- 
blemished, faultless, and infinitely detailed space and time. 
These electrons and the other fundamental particles, they do 
not exist in space and time. It is space and time that exist 
because of them. These particles — wavicles, as we must regard 
them if we wish to mix in our inappropriate, anthropomorphic 
fancies of space and time — these fundamental particles precede 
and transcend the concepts of space and time. They are deeper 
and more fundamental, more primitive and primordial. It is 
out of them in the untold aggregate that we build our spatial 
and temporal concepts, much as out of the multitude of seem- 
ingly haphazard dots and splotches of the newspaper photo- 
graph we build in our minds a smooth, unblemished portrait; 
much as from the swift succession of quite motionless pictures 
projected on a motion-picture screen we build in our minds the 
illusion of smooth, continuous motion. 

Perhaps it is this which the quantum theory is striving to 
express. Perhaps it is this which makes it seem so paradoxical. 
If space and time are not the fundamental stuff of the universe 
but merely particular average, statistical effects of crowds of 
more fundamental entities lying deeper down, it is no longer 
strange that these fundamental entities, when imagined as 
existing in space and time, should exhibit such ill-matched 
properties as those of wave and particle. There may, after all, 
be some innate logic in the paradoxes of quantum physics. 

This idea of average effects which do not belong to the 
individual is nothing new to science. Temperature, so real and 
definite that we can read it with a simple thermometer, is 
merely a statistical effect of chaotic molecular motions. Nor 
are we at all troubled that it should be so. TTie air pressure in 
our automobile tires is but the statistical effect of a ceaseless 


The New Landscape of Science 

bombardment by tireless air molecules. A single molecule has 
neither temperature nor pressure in any ordinary sense of those 
terms. Ordinary temperature and pressure are crowd effects. 
When we try to examine them too closely, by observing an 
individual molecule, they simply vanish away. Take the smooth 
flow of water. It too vanishes away when we examine a single 
water molecule. It is no more than a potent myth created out 
of the myriad motions of water molecules in enormous 

So too may it well be with space and time themselves, 
though this is something far more difficult to imagine even 
tentatively. As the individual water molecules lack the every- 
day qualities of temperature, pressure, and fluidity, as single 
letters of the alphabet lack the quality of poetry, so perhaps 
may the fundamental particles of the universe individually lack 
the quality of existing in space and time; the very space and 
time which the particles themselves, in the enormous aggregate, 
falsely present to us as entities so pre-eminently fundamental 
we can hardly conceive of any existence at all without them. 
Sec how it all fits in now. The quantum paradoxes are of our 
own making, for wc have tried to follow the motions of indi- 
vidual particles through space and time, while all along these 
individual particles have no existence in space and time. It is 
space and time that exist through the particles. An individual 
particle is not in two places at once. It is in no place at all. 
Would we feel amazed and upset that a thought could be in 
two places at once? A thought, if wc imagine it as something 
outside our brain, has no quality of location. If we did wish to 
locate it hypothctically, for any particular reason, we would 
expect it to transcend the ordinary limitations of space and 
time. It is only because we have all along regarded matter as 
existing in space and time that wc find it so hard to renounce 
this idea for the individual particles. But once we do renounce 
it the paradoxes vanish away and the message of the quantum 
suddenly becomes clear: space and time are not fundamental. 

Speculation? Certainly. But so is all theorizing. While 
nothing so drastic has yet been really incorporated into the 


mathematical fabric of quantum mechanics, this may well be 
because of the formidable technical and emotional problems 
involved. Meanwhile quantum theorists find themselves more 
and more strongly thrust toward some such speculation. It 
would solve so many problems. But nobody knows how to set 
about giving it proper mathematical expression. If something 
such as this shall prove to be the true nature of space and time, 
then relativity and the quantum theory as they now stand 
would appear to be quite irreconcilable. For relativity, as a field 
theory, must look on space and time as basic entities, while 
the quantum theory, for all its present technical inability to 
emancipate itself from the space-time tyranny, tends ver)' 
strongly against that view. Yet there is a deal of truth in both 
relativity and the present quantum theory, and neither can 
wholly succumb to the other. Where the two theories meet 
there is a vital ferment. A process of cross-fertilizarion is under 
way. Out of it someday will spring a new and far more potent 
theory, bearing hereditary traces of its two illustrious ancestors, 
which uill ultimately fall heir to all their rich possessions and 
spread itself to bring their separate domains under a single 
rule. What will then survive of our present ideas no one can 
say. Already we have seen waves and particles and causality and 
space and time all undermined. Let us hasten to bring the 
curtain down in a rush lest something really serious should 


An account of how physical theory has developed 
m the past and how It might be expected to develop 
in the future. 

13 The Evolution of the Physicist's Picture of Nature 

Paul A. M. Dirac 

Popular article published in 1963. 

In this article I should like to discuss 
the development of general physical 
theory: how it developed in the past 
and how one may expect it to develop in 
the future. One can look on this con- 
tinual development as a process of evo- 
lution, a process that has been going on 
for several centuries. 

The first main step in this process of 
evolution was brought about by Newton. 
Before Newton, people looked on the 
world as being essentially two-dimen- 
sional—the two dimensions in which one 
can walk about— and the up-and-down 
dimension seemed to be something es- 
sentially different. Newton showed how 
one can look on the up-and-down direc- 
tion as being symmetrical with the other 
two directions, by bringing in gravita- 
tional forces and showing how they take 
their place in physical theory. One can 
say that Newton enabled us to pass from 
a picture with two-dimensional sym- 
metry to a picture with three-dimension- 
al symmetry. 

Einstein made another step in the 
same direction, showing how one can 
pass from a picture with three-dimen- 
sional symmetry to a picture with four- 
dimensional symmetry. Einstein brought 
in time and showed how it plays a role 
that is in many ways symmetrical with 
the three space dimensions. However, 
this symmetry is not quite perfect. With 

Einstein's picture one is led to think of 
the world from a four-dimensional point 
of view, but the four dimensions are not 
completely symmetrical. There arc some 
directions in the four-dimensional pic- 
ture that are different from others: di- 
rections that arc called null directions, 
along which a ray of light can move; 
hence the four-dimensional picture is not 
completely symmetrical. Still, there is a 
great deal of symmetry among the four 
dimensions. The only lack of ss mmctry, 
so far as concerns the ccjuations of phys- 
ics, is in the appearance of a minus sign 
in the ecjuations with respect to the time 
dimension as compared with the three 
space dimensions [sec top equation on 
page 8]. 

We have, then, the development from 
the three-dimensional picture of the 
world to the four-dimensional picture. 
The reader will probablv not be happy 
with this situation, because the world 
still appears three-dimensional to his 
consciousness. How can one bring this 
appearance into the four-dimensional 
picture that Einstein re(juires the physi- 
cist to have? 

What appears to our consciousness is 
really a three-dimensional section of the 
four-dimensional picture. We must take 
a three-dimensional section to give us 
what appears to our consciousness at one 
time; at a later time we shall have a 

different three-dimensional section. The 
task of the physicist consists largely of 
relating events in one of sections to 
events in another section referring to a 
later time. Thus the picture with four- 
dimensional symmetry does not give us 
the whole situation. This becomes par- 
ticularly important when one takes into 
account the developments that have 
been brought about by (juantum theory. 
Quantum theory has taught us that we 
have to take the process of observation 
into account, and observations usually 
require us to bring in the three-dimen- 
sional sections of the four-dimensional 
picture of the universe. 

The special theory of relativity, which 
Einstein introduced, re<juires us to put 
all the laws of ph\sics into a form that 
displays four-dimensional svmmetrv. But 
when we use these laws to get results 
about observations, we have to bring in 
something additional to the four-dimen- 
sional symmetry, namely the three-di- 
mensional sections that describe our 
consciousness of the universe at a cer- 
tain time. 

P'^instein made another most important 
'—' contribution to the development of 
our physical picture: he put forward the 
general theory of relativity, which re- 
(|uires us to suppose that the space of 
physics is curved. Before this physicists 


had always worked with a flat space, the 
three-dimensional flat space of Newton 
which was then extended to the four- 
dimensional flat space of special relativ- 
ity. General relativity mads a really im- 
portant contribution to the evolution of 
our physical picture by requiring us to 
go over to cur\'ed space. The general re- 
quirements of this theory mean that aU 
the laws of physics can be formulated in 
curved four-dimensional space, and that 
they show symmetry among the four 
dimensions. But again, when we want to 
bring in observ-ations, as we must if we 
look at things from the point of view of 
quantum theory, we have to refer to a 
section of this foxu-dimensional space. 
With the four-dimensional space curved, 
any section that we make in it also has to 
be curved, because in general we cannot 
give a meaning to a flat section in a 
curved space. This leads us to a picture 
in which we have to take curved three- 
dimensional sections in the curv'ed four- 
dimensional space and discuss obser\a- 
tions in these sections. 

During the past few years people have 
been trying to apply quantum ideas to 

gravitation as well as to the other 
phenomena of physics, and this has led 
to a rather unexpected development, 
namely that when one looks at gravita- 
tional theor>- from the point of view of 
the sections, one finds that there are 
some degrees of freedom that drop out 
of the theory. The gravitational field is 
a tensor field with 10 components. One 
finds that six of the components are ade- 
quate for describing everything of physi- 
cal importance and the other four can be 
dropped out of the equations. One can- 
not, however, pick out the six important 
components from the complete set of 10 
in any way that does not destroy the 
four-dimensional s>'mmetr\-. Thus if one 
insists on preserving four-dimensional 
symmetr>' in the equations, one cannot 
adapt the theory of gravitation to a dis- 
cussion of measurements in the way 
quantum theory requires without being 
forced to a more complicated description 
than is needed by the physical situation. 
This result has led me to doubt how 
fundamental the four-dimensional re- 
quirement in physics is. A few decades 
ago it seemed quite certain that one had 

ISAAC NEWTON (1642-1727t, with his law of i^ravitation, rhanfied the physicist's picture 
of nature from one with two-dimensional symmetry to one with threc^limensional symmetry. 
This drawing of him was made in 1760 by James MacarJcl from a painting by Enoch Seeman. 

to express the whole of physics in four- 
dimensional form. But now it seems that 
four-dimensional sv'mmetr>' is not of such 
overriding importance, since the descrip- 
tion of nature sometimes gets simplified 
when one departs from it. 

Now I should like to proceed to the 
developments that have been brought 
about by quantum theorv-. Quantum 
theory is the discussion of very small 
things, and it has formed the main sub- 
ject of physics fcM- the past 60 years. 
During this period physicists have been 
amassing quite a lot of experimental in- 
formation and developing a theorv- to 
correspond to it, and this combination of 
theorv- and experiment has led to im- 
portant developments in the physicist's 
picture of the world. 

The quantum first made its appear- 
ance when Planck discovered the need 
to suppose that the energv- of electro- 
magnetic wav-es can exist only in mul- 
tiples of a certain unit, depending on the 
frequency of the waves, in order to ex- 
plain the law of black-body radiation. 
Then Einstein discovered the same unit 
of energy occurring in the photoelectric 
effect. In this early work on quantum 
theorv' one simply had to accept the unit 
of energv- without being able to incor- 
porate it into a physical picture. 

rilhe first new picture that appeared 
-*• v»-as Bohr's picture of the atom. It was 
a pictiu-e in which we had electrons mov- 
ing about in certain well-defined orbits 
and occasionally making a jump from 
one orbit to another. We could not pic- 
ture how the jump took place. We just 
had to accept it as a kind of discon- 
tinuit>'. Bohr's pictiire of the atom 
worked only for si>ecial examples, essen- 
tially when there was only one electron 
that was of importance for the problem 
under consideration. Thus the picture 
was an incomplete and primitive one. 

The big advance in the quantum 
theorv- came in 1925, v*ith the discovery- 
of quantiun mechanics. This adv-ance 
was brought about independently by tw-o 
men, Heisenberg first and Schrodinger 
soon afterward, working from different 
points of view. Heisenberg worked keep- 
ing close to the experimental evidence 
about spectra that was being amassed at 
that time, and he found out how the ex- 
perimental information could be fitted 
into a scheme that is now known as 
matrix mechanics. All the experimental 
data of spectroscopy fitted beautifully 
into the scheme of matrix mechanics, and 
this led to quite a different picture of the 
atomic world. Schrodinger worked from 
a more mathematical point of view, trv-- 
ing to find a beautiful theory- for describ- 


"■^B Evotution ii *he t^hysicist's O'cture of Mature 

iw^atamme >^'-"« and was l^i ip ed faqr He 
Hkj^m- 's idsea :)( vaves associated -«itii 
QHticiea. 9e «as ibie 'o ^TTPfvi D» 
Bta^ie-.s .deas md to ^ec x "err beantifni 
eqpatiaB. known js Scinodm^er^ '^ 
eqpitiaair fcr iescnfam^ itoimc TUfy- 
itmaa Sei m J uIi Bgeg ^ot tm eeinaxioii 37 
Bpae tfasMg^ loottny for iome beantifiii 
jl " iiliwi at 3tf Jioi^'s ideas, and 

oar faqr hiimMujg cioae o :fae ^xperimeBtei 
ciereiaciiMeat of the snmect ji the wagr 
Ocuvfiiju^ did. 

r laisbt tefl ym tfae- ;n»r I beaad irxma 
S iiiaudlug er « baar, wfaea he Sot 90* 
tfac idea, for rfaa eiipatiaa. be iameduite- 
Ir awpiied it to the b eth t-vi o i of tbe ote e- 
txoo m tfae hrdiogezi jtom. and tfaea be 
got resoits that Hd 3>t i^ree- "vrth ay 
Tbe aisazreenieat iztse be- 
lt :faat "ime t »^ia lot -<iiowti diat 
the steetron aaa v jxnn- That if x>iiise. 
WBS 3 ^jreat -Ti»i»iM»»iininfqTt ~o ?rfnn >- 
. llnu i i * - and it oanaed "'"Tn -o ibasdOD the 
woefc ffw jonae -nonths. Then ae soticed 
that if be amiiied -bs zbeory .n a. 3Kwe 
H^iniiii III va.T. not -a»im» into 3C- 
LU WM l the rednefnents rKjmred '~iy reia.- 
tnriijr, to t*™ -r mim ipproxcnatica bu 
^woafc was in n^rcement '^nth ifaserva:- 
tion. BEe pnbiiahed bis Stst tiaper "^iA 

OldT' this -r mgn i {H iii »iiirr!itin> r» and in 

that '»B.v Schrodin^prs ' eeinatiaB 
was ^reseated "o "he worid- .afterward. 
at- cwmse. vhen peopie fotrnd yat bow to 
take into acconnt ^orreedy the spni at 
the ^iectxoa. the discrepancy between 
the »« " '>« Jt lyuivLUif 5cfaid«iuB5ier's rei- 
atinstic ^onatiDii md tiie sspecnnents 
was comfieEefT :ieaxed up. 

I^tnr-^ these is 3. nioral to this s tot y , 
im^Kifr that it- is more important to 
lanre^ beauty in lXK-'s ^gnarinns tloua to 
bnre^them St espemnent. If Schxbdinger 
hai. been more :»nfident it 'ais -^uiL. be 
coHld published t lOtne months 
xnd. be »nid bave pobiished 1 
F accnrate carnation. That egnaffrni \s 
kiMW»u as the Kleni-Gordon eana- 
iMtwinrh it '*as reailv iiscovered by 
and "in tact -»as iiscovcred 
faqr ■V - iMli ri OT i betore be ibcovpred bis 
lamiiiiiliiiiilii 'reatmetit jt ^be bydio- 
^BK atoaa. Tt ie ems "hat f Tne is .•uikiiuf 
fniiB the poiA of -/iew trf getting beanty 
in Doe's eeinatioaa. and if me bas reaily 
3. setnxt 'iituini ine is tji 1 nrre jne it 
pao^tess. If There is rvx xjmpiete is;ree- 
imuM. latLvt ii u. the -esuits it me s -^ork 
aaid esperiaaent. me siiouid 10C ailow 
mesetf "-o be "00 iSscoura^Bd. beeanse 
"be iiscrenancy -nav veil be ine to 
moair teatnres 'hat ue not propettv 
taioen nto acconnt md that ^tiJL ^et 
Tiieared ip '^nth tnitfaeT' 'Jcv*ifa HW i M f i t ot 
-he theory. 

aLHERT EIN"-TKIN 1879 iVri.'i(. wiUi Ins -ivrtai Itmowy <rf rrlalirrty. <hancrti the pinM- 
iil- uiriurp irom oiip with tbrr«:>tliBnm>MMKii -vnaartry to uae- with f OT^iiimai niri i >>!■- 
raetrv. Thi^ (itwlufti i|rii ui hiiB and iua wife ood tfactr ifaaghler Margn* was 11 dr ia 1!^Z9. 

Tfiat is bow niwiHi i iw laii hum 
iiscovered- It led to a. drastic 
■Jl tlie physicist'o picttire of tiie <n>oaldr 
periiaps the biggest that bas yet taken 
piace. This change comes iiroin yaz basr- 
.ni5 to ?rve ip the ietermimstic pictme 
•»e bad always taicen for granted. We are 
led to 1 tlieory that iocs not predict -^ith 
certainty ^vhat is mtmr to bappen m the 
rntnre brnt '^zves us Tirfi*i»a*T«— ouiy 
about the profaafaiiity of accnxeaee at 
"azioDS events. Tins jiviun up of deter- 
nnnacy bas been a. very controversial 
ndiieet, and some peopie do xxt like 'it at 
aU. T gj »i *i in ix i.MfHftr" iMvcr lifced it. 

.\ithon^ Snstsm was oae ai the atud 
coHtxAHtocs to tfae deretot— fr^ o^ niiai* - 
tCBB neefanncs. be 3^31 was <dwii»s ratb- 
ee luiMiiTi' to tfae fosm tfaat (inantcan 

i i m ioa evolved inlD '>■"■■»■ bis life 

time '"^ "ha* it stiil retains. 

The bostiiity iome peopie have to the 
jitiiiit up It tlie deterministic pactme 
can be centered on a. mnefa diwuwrd. 
paper by Kirntem. Podoisky and Hoaen 
d^MJirt? '«Tch tfae difScoity me bas in 
foiimm^ a. consistent .u ct me that stiil 
^ires resoits according to tlie roles at 
Ij iiii 1 1 neefaanics. The mies oi qnan- 
are Ttnte de&nrts^ Pes^ie 






^^^^^r^^-<,t\* ' 'll 







■ /•.( '-^ . ,/: 


' ^^^L 1 



: V 


m .^ 





NIEIS BOHR (1885 1962) imroduced the idea that the elertron 
moved about the nucleus in well-defined orbits. Thi» photograph 
was made in 1922, nine years after the publication of his paper. 

MAX PLANCK (1858-1917) introduced the idea that electro- 
magnetir radiation consists of quanta, or particles. This photograph 
was made in 1913, 13 years after his original paper was published. 

know how to calculate results and how to 
compare the results of their calculations 
with experiment. Everyone is agreed on 
the formalism. It works so well that no- 
body can afford to disagree with it. But 
still the picture that we are to set up 
behind this formalism is a subject of 

I should like to suggest that one not 
worry too much about this controversy. I 
feel very strongly that the stage physics 
has reached at the present day is not the 
final stage. It is just one stage in the evo- 
lution of our picture of nature, and we 
should expect this process of evolution 
to continue in the future, as biological 
evolution continues into the future. The 
present stage of physical theory is mere- 
ly a steppingstone toward the better 
stages we shall have in the future. One 
can be quite sure that there will be better 
stages simply because of the difficulties 
that occur in the physics of today. 

T should now like to dwell a bit on 
■*- the difficulties in the physics of the 
present day. The reader who is not an 
expert in the subject might get the idea 
that because of all these difficulties 
physical theory is in pretty poor shape 
and that the quantum theory is not much 
good. I should like to correct this impres- 
sion by saying that quantum theory is an 
extremely good theory. It gives wonder- 
ful agreement with observation over a 
wide range of phenomena. There is no 
dovibt that it is a good theory, and the 
only reason physicists talk so much about 

the difficulties in it is that it is precisely 
the difficulties that are interesting. The 
successes of the theory are all taken for 
granted. One does not get anywhere 
simply by going over the successes again 
and again, whereas by talking over the 
difficulties people can hope to make 
some progress. 

The difficulties in quantum theory are 
of two kinds. I might call them Class One 
difficulties and Class Two difficulties. 
Class One difficulties are the difficulties 
I have already mentioned: How can one 
form a consistent picture behind the 
rules for the present quantum theory? 
These Class One difficulties do not really 
worry the physicist. If the physicist 
knows how to calculate results and com- 
pare them with experiment, he is quite 
happy if the results agree with hi^ ex- 
periments, and that is all he needs. It is 
only the philosopher, wanting to have a 
satisfying description of nature, who is 
bothered by Qass One difficulties. 

There are, in addition to the Class One 
difficulties, the Class Two difficulties, 
which stem from the fact that the present 
laws of quantum theory are not always 
adequate to give any results. If one 
pushes the laws to extreme conditions— 
to phenomena involving very high ener- 
gies or very small distances— one some- 
times gets results that are ambiguous or 
not really sensible at all. Then it is clear 
that one has reached the limits of appli- 
cation of the theory and that some fur- 
ther development is needed. The Class 
Two difficulties are important even for 

the physicist, because they put a limita- 
tion on how far he can use the rules of 
quantum theory to get results compara- 
ble with experiment. 

I should like to say a little more about 
the Class One difficulties. I feel that one 
should not be bothered with them too 
much, because they are difficulties that 
refer to the present stage in the de\'elop- 
ment of our physical picture and are 
almost certain to change with future de- 
velopment. There is one strong reason, I 
think, why one can be quite confident 
that these difficulties will change. There 
are some fundamental constants in na- 
ture: the charge on the electron (desig- 
nated c), Planck's constant divided by 
2ir (designated h) and the velocity of 
light (c). From these fundamental con- 
stants one can construct a number that 
has no dimensions: the number hc/e'. 
That number is found by experiment to 
have the value 137, or something very 
close to 137. Now, there is no known 
reason why it should have this value 
rather than some other number. Various 
people have put forward ideas about it, 
but there is no accepted theory. Still, 
one can be fairly sure that someday 
physicists will solve the problem and 
explain why the number has this value. 
There will be a physics in the future that 
works when hc/e^ has the value 137 
and that will not work when it has any 
other value. 

The physics of the future, of course, 
cannot have the three quantities h, e and 
c all as fundamental quantities. Only two 


The Evolution of the Physicist's Picture of Nature 

of them can be fundamental, and the 
third must be derived from those two. It 
is almost certain that c will be one of the 
two fundamental ones. The velocity of 
light, c, is so important in the four- 
dimensional picture, and it plays such a 
fundamental role in the special theory of 
relativity, correlating our units of space 
and time, that it has to be fundamental. 
Then we are faced with the fact that of 
the two quantities h and e, one will be 
fundamental and one will be derived. If 
h is fundamental, e will have to be ex- 
plained in some way in terms of the 
square root of h, and it seems most un- 
likely that any fundamental theory can 
give e in terms of a square root, since 
square roots do not occur in basic equa- 
tions. It is much more likely that e will 
be the fundamental quantity and that h 
will be explained in terms of e^. Then 
there will be no square root in the basic 
equations. I think one is on safe ground 
if one makes the guess that in the physi- 
cal picture we shall have at some future 
stage e and c will be fundamental quan- 
tities and h will be derived. 

If h is a derived quantity instead of a 
fundamental one, our whole set of ideas 
about uncertainty will be altered: h is 
the fundamental quantity that occurs in 
the Heisenberg uncertainty relation con- 
necting the amount of uncertainty in a 
position and in a momentum. This un- 
certainty relation cannot play a funda- 
mental role in a theory in which h itself 
is not a fundamental quantity. I think 
one can make a safe guess that uncertain- 
ty relations in their present form will not 
survive in the physics of the future. 

Of course there will not be a return to 
the determinism of classical physi- 
cal theory. Evolution does not go back- 
ward. It will have to go forward. There 
will have to be some new development 
that is quite unexpected, that we cannot 
make a guess about, which will take us 
still further from classical ideas but 
which will alter completely the discus- 
sion of uncertainty relations. And when 
this new development occurs, people 
will find it all rather futile to have had so 
much of a discussion on the role of ob- 
servation in the theory, because they will 
have then a much better point of view 
from which to look at things. So I shall 
say that if we can find a way to describe 
the uncertainty relations and the in- 
determinacy of present quantum me- 
chanics that is satisfying to our philo- 
sophical ideas, we can count ourselves 
lucky. But if we cannot find such a way, 
it is nothing to be really disturbed 
about. We simply have to take into ac- 
count that we are at a transitional stage 

and that perhaps it is quite impossible to 
get a satisfactory picture for this stage. 
I have disposed of the Class One dif- 
ficulties by saying that they are really 
not so important, that if one can make 
progress with them one can count one- 
self lucky, and that if one cannot it is 
nothing to be genuinely disturbed about. 
The Class Two difficulties are the really 
serious ones. They arise primarily from 
the fact that when we apply our quan- 
tum theory to fields in the way we have 
to if we are to make it agree with special 
relativity, interpreting it in terms of the 
three-dimensional sections I have men- 
tioned, we have equations that at first 
look all right. But when one tries to solve 
them, one finds that they do not have any 
solutions. At this point we ought to say 
that we do not have a theory. But physi- 
cists are very ingenious about it, and 
they have found a way to make prog- 
ress in spite of this obstacle. They find 
that when they try to solve the equations, 
the trouble is that certain quantities 
that ought to be finite are actually in- 
finite. One gets integrals that diverge 
instead of converging to something defi- 
nite. Physicists have found that there is a 

way to handle these infinities according 
to certain rules, which makes it possible 
to get definite results. This method is 
known as the renormalization method. 

I shall merely explain the idea in words. 
We start out with a theory involving 
equations. In these equations there occur 
certain parameters: the charge of the 
electron, e, the mass of the electron, m, 
and things of a similar nature. One then 
finds that these quantities, which appear 
in the original equations, are not equal 
to the measured values of the charge and 
the mass of the electron. The measured 
values differ from these by certain cor- 
recting terms— Ae, A"* and so on— so 
that the total charge is c -1- Ac and 
the total mass m + Am. These changes 
in charge and mass are brought about 
through the interaction of our elemen- 
tary particle with other things. Then one 
says that e + Ac and m + A"», being 
the observed things, are the important 
things. The original e and m are just 
mathematical parameters; they are un- 
observable and therefore just tools one 
can discard when one has got far enough 
to bring in the things that one can com- 

LOUIS DE BROGLIE (1892- ) put forward the idea that particles are associated with 
waves. This photograph was made in 1929, five years after the appearance of his paper. 


pare with observation. This would be a 
quite correct way to proceed if /\e 
and A"* were small (or even if they 
were not so small but finite) corrections. 
According to the actual theory, however, 
Ac and A"" are infinitely great. In spite 
of that fact one can still use the formal- 
ism and get results in terms of c + A^ 
and m + A"*, which one can interpret 
by saying that the original e and m have 
to be minus infinity of a suitable amount 
to compensate for the A« and A"» that 
are infinitely great. One can use the 
theory to get results that can be com- 
pared with experiment, in particular for 
electrodynamics. The surprising thing is 
that in the case of electrodynamics one 
gets results that are in extremely good 
agreement with experiment. The agree- 
ment applies to many significant fig- 
ures—the kind of accuracy that previ- 
ously one had only in astronomy. It 
is because of this good agreement that 
physicists do attach some value to the 
renormalization theory, in spite of its 
illogical character. 

It seems to be quite impossible to put 
this theory on a mathematically sound 
basis. At one time physical theory was all 
built on mathematics that was inherently 

sound. I do not say that physicists always 
use sound mathematics; they often use 
unsound steps in their calculations. But 
previously when they did so it was 
simply because of, one might say, lazi- 
ness. They wanted to get results as 
quickly as possible without doing un- 
necessary work. It was always possible 
for the pvu-e mathematician to come 
along and make the theory sound by 
bringing in further steps, and perhaps by 
introducing quite a lot of cumbersome 
notation and other things that are desir- 
able from a mathematical point of view 
in order to get everything expressed 
rigorously but do not contribute to the 
physical ideas. The earlier mathematics 
could always be made sound in that way, 
but in the renormalization theory we 
have a theory that has defied all the at- 
tempts of the mathematician to make it 
sound. I am inclined to suspect that the 
renormalization theory is something that 
will not survive in the future, and that 
the remarkable agreement between its 
results and experiment should be looked 
on as a fluke. 

This is perhaps not altogether surpris- 
ing, because there have been similar 
flukes in the past. In fact, Bohr's elec- 

ds" = edt^ - c/x^ - dy^ - dz' 

FOUR-DIMENSIONAL SYMMETRY introduced by the special theory of relativity is not 
quite perfect. This equation is the expression for the invariant distance in four-dimensional 
space-time. The symbol s is the invariant distance; c, the speed of light; t, time; x, y and z, 
the three spatial dimensions. The d's are differentials. The lack of complete symmetry lies 
in the fact that the contribution from the time direction (c'-dt-) does not have the same 
sign as the contributions from the three spatial directions ( — dx-, — dy- and — dz-) . 



SCHRODINGER'S FIRST WAVE EQUATION did not fit experimental results because it 
did not take into account the spin of the electron, which was not known at the time. The 
equation is a generalization of De Broglie's equation for the motion of a free electron. The 
symbol e represents the charge on the electron; i, the square root of minus one; h, Planck's 
constant; r, the distance from the nucleus; ^, Schrodinger's wave function; m, the mass of 
the electron. The symbols resembling sixes turned backward are partial derivatives. 

SCHRODINGER'S SECOND WAVE EQUATION is an approximation to the original 
equation, which does not take into account the refinements that are required by relativity. 

tron-orbit theory was found to give very 
good agreement with observation as long 
as one confined oneself to one-electron 
problems. I think people will now say 
that this agreement was a fluke, because 
the basic ideas of Bohr's orbit theory 
have been superseded by something 
radically different. I believe the suc- 
cesses of the renormalization theory will 
be on the same footing as the successes 
of the Bohr orbit theory applied to one- 
electron problems. 

' I ^ he renormalization theory has re- 
moved some of these Class Two dif- 
ficulties, if one can accept the illogical 
character of discarding infinities, but it 
does not remove all of them. There are 
a good many problems left over concern- 
ing particles other than those that come 
into electrodynamics: the new particles- 
mesons of various kinds and neutrinos. 
There the theory is still in a primitive 
stage. It is fairly certain that there will 
have to be drastic changes in our funda- 
mental ideas before these problems can 
be solved. 

One of the problems is the one I have 
already mentioned about accounting for 
the number 137. Other problems are 
how to introduce the fundamental length 
to physics in some natural way, how to 
explain the ratios of the masses of the 
elementary particles and how to explain 
their other properties. I believe separate 
ideas wiU be needed to solve these dis- 
tinct problems and that they will be 
solved one at a time through successive 
stages in the future evolution of physics. 
At this point I find myself in disagree- 
ment with most physicists. They are in- 
clined to think one master idea will be 
discovered that will solve all these prob- 
lems together. I think it is asking too 
much to hope that anyone will be able to 
solve all these problems together. One 
should separate them one from- another 
as much as possible and try to tackle 
them separately. And I believe the fu- 
ture development of physics will consist 
of solving them one at a time, and that 
after any one of them has been solved 
there will still be a great mystery about 
how to attack further ones. 

I might perhaps discuss some ideas 
I have had about how one can possibly 
attack some of these problems. None of 
these ideas has been worked out very 
far, and I do not have much hope for any 
one of them. But I think they are worth 
mentioning briefly. 

One of these ideas is to introduce 
something corresponding to the luminif- 
erous ether, which was so popular among 
the physicists of the 19th century. I said 
earlier that physics does not evolve back- 


The Evolution of the Physicist's Picture of Nature 

ward. \Mien I talk about reintroducing 
the ether, I do not mean to go back to 
the picture of the ether that one had in 
the 19th century, but I do mean to intro- 
duce a new picture of the ether that will 
conform to our present ideas of quantum 
theory. The objection to the old idea of 
the ether was that if you suppose it to 
be a fluid filling up the whole of space, 
in any place it has a definite velocity, 
which destroys the four-dimensional 
symmetry required by Einstein's special 
principle of relativity. Einstein's special 
relativity killed this idea of the ether. 

But with our present quantum theor\- 
we no longer have to attach a definite 
velocity to any given physical thing, be- 
cause the velocity is subject to uncer- 
tainty relations. The smaller the mass of 
the thing we are interested in, the more 
important are the uncertainty relations. 
Now, the ether will certainly have very 
little mass, so that uncertainty relations 
for it will be extremely important. The 
velocity of the ether at some particular 
place should therefore not be pictiu-ed as 
definite, because it will be subject to un- 
certainty relations and so may be any- 
thing over a wide range of values. In that 
way one can get over the difficulties of 
reconciling the existence of an ether with 
the special theory of relativity. 

There is one important change this 
will make in our picture of a vacuum. We 
would like to think of a vacuum as a 
region in which we have complete sym- 
metry between the four dimensions of 
space-time as required by special relativ- 
ity. If there is an ether subject to uncer- 
tainty relations, it will not be possible to 
have this symmetry accurately. We can 
suppose that the velocity of the ether is 
equally likely to be anything within a 
wide range of values that would give the 
symmetry only approximately. We can- 
not in any precise way proceed to the 
limit of allowing all values for the veloc- 
ity between plus and minus the velocity 
of light, which we would have to do in 
order to make the symmetry accurate. 
Thus the vacuum becomes a state that is 
unattainable. I do not think that this is a 
physical objection to the theory. It would 
mean that the vacuum is a state we can 
approach very closely. There is no limit 
as to how closely we can approach it, 
but we can never attain it. I believe 
that would be quite satisfactory to the 
experimental physicist. It would, how- 
ever, mean a departure from the notion 
of the vacuum that we have in the 
quantum theory, where we start off with 
the vacuum state having exactly the 
symmetry required by special relativity. 

That is one idea for the development 
of physics in the future that would 

ERWIN SCHRODINGEH ( 1887-1961 1 devised his nave equation l.y extending Ue Broglie's 
idea that waves are assoriated with particles to the electrons moving around the nucleus. 
This photograph was made in 1929, four years after he had published his second equation. 

change our picture of the vacuum, but 
change it in a way that is not unaccept- 
able to the experimental physicist. It has 
proved difficult to continue with the 
theory, because one would need to set up 
mathematically the uncertainty relations 
for the ether and so far some satisfactory 
theory along these lines has not been dis- 
covered. If it could be developed satis- 
factorily, it would give rise to a new kind 
of field in physical theory, which might 
help in explaining some of the elemen- 
tary particles. 

A nother possible picture I should like 
-^^ to mention concerns the question of 
why all the electric charges that are ob- 
served in natiure should be multiples of 
one elementary unit, e. Why does one 
not have a continuous distribution of 
charge occurring in nature? The picture 
I propose goes back to the idea of 
Faraday lines of force and involves a 
development of this idea. The Faraday 

lines of force are a way of picturing elec- 
tric fields. If we have an electric field in 
any region of space, then according to 
Faraday we can draw a set of lines that 
have the direction of the electric field. 
The closeness of the lines to one another 
gives a measure of the strength of the 
field— they are close where the field is 
strong and less close where the field is 
weak. The Faraday lines of force give 
us a good picture of the electric field in 
classical theory. 

When we go over to quantum theor>', 
we bring a kind of discreteness into our 
basic picture. We can suppose that the 
continuous distribution of Faraday lines 
of force that we have in the classical pic- 
ture is replaced by just a few discrete 
lines of force with no lines of force be- 
tween them. 

Now, the lines of force in the Faraday 
picture end where there are charges. 
Therefore with these quantized Faraday 
lines of force it would be reasonable to 


suppose the charge associated with each 
line, which has to lie at the end if the 
line of force has an end, is always the 
same ( apart from its sign ) , apd is al- 
ways just ^ the electronic charge, — c or 
+ e. This leads us to a picture of discrete 
Faraday lines of force, each associated 
with a charge, — e or + e. There is a di- 
rection attached to each line, so that the 
ends of a line that has two ends are not 
the same, and there is a charge + e at 
one end and a charge — e at the other. 
We may have lines of force extending to 
infinity, of course, and then there is no 

If we suppose that these discrete 
Faraday lines of force are something 
basic in physics and lie at the bottom of 
our picture of the electromagnetic field, 
we shall have an explanation of why 
charges always occur in multiples of e. 
This happens because if we have any 
particle with some lines of force ending 
on it, the number of these lines must be 
a whole number. In that way we get 
a picture that is qualitatively quite rea- 

We suppose these lines of force can 

move about. Some of them, forming 
closed loops or simply extending from 
minus infinity to infinity, will correspond 
to electromagnetic waves. Others will 
have ends, and the ends of these lines 
will be the charges. We may have a line 
of force sometimes breaking. When that 
happens, we have two ends appearing, 
and there must be charges at the two 
ends. This process— the breaking of a line 
of force— would be the picture for the 
creation of an electron (c ) and a posi- 
tron (e+). It would be quite a reason- 
able picture, and if one could develop it, 
it would provide a theory in which e 
appears as a basic quantity. I have not 
yet found any reasonable system of equa- 
tions of motion for these lines of force, 
and so I just put forward the idea as a 
possible physical picture we might have 
in the future. 

There is one very attractive feature 
in this picture. It will quite alter the 
discussion of renormalization. The re- 
normalization we have in our present 
quantum electrodynamics comes from 
starting off with what people call a bare 
electron— an electron without a charge 

WERNER HEISENBERG (l')Ol- l inlr.><lu..<l matrix m.-.liiiiii.-. wlii.h. lik.- lli^ S, hri.- 
dinger theory, arrountrd (or the motions of the t-Ioctron. This pholopraph »:i'- iiiaili- in \'>'2'i. 

on it. At a certain stage in the theory one 
brings in the charge and puts it on the 
electron, thereby making the electron 
interact with the electromagnetic field. 
This brings a perturbation into the equa- 
tions and causes a change in the mass of 
the electron, the A*", which is to be 
added to the previous mass of the elec- 
tron. The procedure is rather roundabout 
because it starts off with the unphysical 
concept of the bare electron. Probably in 
the improved physical picture we shall 
have in the future the bare electron will 
not exist at all. 

Now, that state of affairs is just what 
we have with the discrete lines of force. 
We can picture the lines of force as 
strings, and then the electron in the pic- 
ture is the end of a string. The string it- 
self is the Coulomb force around the 
electron. A bare electron means an elec- 
tron without the Coulomb force around 
it. That is inconceivable with this pic- 
ture, just as it is inconceivable to think of 
the end of a piece of string without think- 
ing of the string itself. This, I think, is the 
kind of way in which we should try to 
develop our physical picture— to bring in 
ideas that make inconceivable the things 
we do not want to have. Again we have a 
picture that looks reasonable, but I have 
not found the proper equations for de- 
veloping it. 

I might mention a third picture with 
which I have been dealing lately. It 
involves departing from the picture of 
the electron as a point and thinking of 
it as a kind of sphere with a finite size. 
Of course, it is really quite an old idea 
to picture the electron as a sphere, but 
previously one had the difficulty of dis- 
cussing a sphere that is subject to ac- 
celeration and to irregular motion. It 
will get distorted, and how is one to deal 
with the distortions? I propose that one 
should allow the electron to have, in 
general, an arbitrary shape and size. 
There will be some shapes and sizes in 
which it has less energy than in others, 
and it will tend to assume a spherical 
shape with a certain size in which the 
electron has the least energy. 

This picture of the extended electron 
has been stimulated by the discovery of 
the mu meson, or muon, one of the new 
particles of physics. The muon has the 
surprising property of being almost iden- 
tical with the electron except in one 
particular, namely, its mass is some 200 
times greater than the mass of the elec- 
tron. Apart from this disparity in mass 
the muon is remarkably similar to the 
electron, having, to an extremely high 
degree of accuracy, the same spin and 
the same magnetic moment in propor- 
tion to its mass as the electron does. This 


The Evolution of the Physicist's Picture of Nature 

leads to the suggestion that the muon 
should be looked on as an excited elec- 
tron. If the electron is a point, picturing 
how it can be excited becomes quite 
awkward. But if the electron is the most 
stable state for an object of finite size, 
the muon might just be the next most 
stable state in which the object under- 
goes a kind of oscillation. That is an idea 
I have been working on recently. There 
are difficulties in the development of this 
idea, in particular the difficulty of bring- 
ing in the correct spin. 

T have mentioned three possible ways 
-•- in which one might think of develop- 
ing our physical picture. No doubt there 
will be others that other people will 
think of. One hopes that sooner or later 
someone will find an idea that really fits 
and leads to a big development. I am 
rather pessimistic about it and am in- 
clined to think none of them will be good 
enough. The future evolution of basic 
physics— that is to say, a development 
that will really solve one of the funda- 
mental problems, such as bringing in the 
fundamental length or calculating the 
ratio of the masses— may require some 
much more drastic change in our physi- 
cal picture. This would mean that in our 
present attempts to think of a new physi- 
cal picture we are setting our imagina- 
tions to work in terms of inadequate 
physical concepts. If that is really the 
case, how can we hope to make progress 
in the future? 

There is one other line along which 
one can still proceed by theoretical 
means. It seems to be one of the funda- 
mental features of nature that funda- 
mental physical laws are described in 
terms of a mathematical theory of great 
beauty and power, needing quite a high 
standard of mathematics for one to un- 
derstand it. You may wonder: Why is 
nature constructed along these lines? 
One can only answer that our present 
knowledge seems to show that nature is 
so constructed. We simply have to accept 
it. One could perhaps describe the situa- 
tion by saying that God is a mathema- 
tician of a very high order, and He used 
very advanced mathematics in construct- 
ing the universe. Our feeble attempts at 
mathematics enable us to understand a 
bit of the universe, and as we proceed 
to develop higher and higher mathe- 
matics we can hope to understand the 
universe better. 

This view provides us with another 
way in which we can hope to make ad- 
vances in our theories. Just by studying 
mathematics we can hope to make a 
guess at the kind of mathematics that 
will come into the physics of the future. 

LINES OF rOR(;L ill :in eleiiromapnotic held, it thi-y arc assumed to be discrete in the 
<|uantum theory, sUKpe-t why elr.tric charpi'b ul^^ay^ occur in multiples of the charge of the 
electron. In Dirac"> view, uh<-n ;; lino of forie liii; tuo enii^. there is a particle with charge 
— e. perhaps an electron, at one end and a panicle with charpc + e, pcrhap«> a positron, at 
the other end. Wlien a closed line of force i# broken, un electron-positron pair materializes. 

A good many people are working on the 
mathematical basis of quantum theory, 
trying to understand the theory better 
and to make it more powerful and more 
beautiful. If someone can hit on the 
right lines along which to make this de- 
velopment, it may lead to a future ad- 
vance in which people will first discover 
the equations and then, after examining 
them, gradually learn how to apply 
them. To some extent that corresponds 
with the line of development that oc- 
curred with Schrodinger's discovery of 
his wave equation. Schrodinger discov- 
ered the equation simply by looking for 
an equation with mathematical beauty. 
When the equation was first discovered, 
people saw that it fitted in certain ways, 
but the general principles according to 
which one should apply it were worked 
out only some two or three years later. It 
may well be that the next advance in 
physics will come about along these 
lines: people first discovering the equa- 

tions and then needing a few years of 
development in order to find the physical 
ideas behind the equations. My own be- 
lief is that this is a more likely line of 
progress than trying to guess at physical 

Of course, it may be that even this line 
of progress will fail, and then the only 
line left is the experimental one. Experi- 
mental physicists are continuing their 
work quite independently of theory, col- 
lecting a vast storehouse of information. 
Sooner or later there will be a new 
Heisenberg who will be able to pick out 
the important features of this informa- 
tion and see how to use them in a way 
similar to that in which Heisenberg used 
the experimental knowledge of spectra 
to build his matrix mechanics. It is in- 
evitable that physics will develop ulti- 
mately along these lines, but we may 
have to wait quite a long time if people 
do not get bright ideas for developing 
the theoretical side. 


Infeld reminisces what it was like to work at 
Cambridge University in England with two great, 
but very different, theoretical physicists. 

14 Dirac and Born 

Leopold Infeld 

Excerpt from Quest. 

The greatest theoretical physicist in Cambridge was P. A. M. 
Dirac, one of the outstanding scientists of our generation, then a 
young man about thirty. He stiJl occupies the chair of math- 
ematics, the genealogy of which can be traced directly to 

I knew nothing of Dirac, except that he was a great math- 
ematical physicist. His papers, appearing chiefly in the Proceed- 
ings of the Royal Society, were written with wonderful clarity 
and great imagination. His name is usually linked with those of 
Heisenberg and Schroedinger as the creators of quantum me- 
chanics. Dirac's book The Principles of Quantum Mechanics is 
regarded as the bible of modem physics. It is deep, simple, lucid 
and original. It can only be compared in its importance and ma- 
turity to Newton's Principia. Admired by everyone as a genius, 
as a great star in the firmament of English physics, he created 
a legend around him. His thin figure with its long hands, 
walking in heat and cold without overcoat or hat, was a familiar 
one to Cambridge students. His loneliness and shyness were 
famous among physicists. Only a few men could penetrate his 
soHtude. One of the fellows, a well-known physicist, told me: 

"I still find it very diflicult to talk with Dirac. If I need his 
advice I try to formulate my question as briefly as possible. 
He looks for five minutes at the ceiling, five minutes at the win- 
dows, and then says *Yes' or 'No.' And he is always right." 

Once— according to a story which I heard— Dirac was lectur- 
ing in the United States and the chairman called for questions 
after the lecture. One of the audience said: 

"I did not understand this and this in your arguments." 

Dirac sat quietly, as though the man had not spoken. A dis- 
agreeable silence ensued, and the chairman turned to Dirac un- 

"Would you not be kind enough, Professor Dirac, to answer 
this question?" 


To which Dirac replied: "It was not a question; it was a state- 

Another story also refers to his stay in the United States. He 
lived in an apartment with a famous French physicist and they 
invariably talked English to each other. Once the French physi- 
cist, finding it difficult to explain something in English, asked 
Dirac, who is half English and half French: 

"Do you speak French?" 

"Yes. French is my mother's tongue," answered Dirac in an 
unusually long sentence. The French professor burst out: 

"And you say this to me now, having allowed me to speak my 
bad, painful English for weeks! Why did you not tell me this 

"You did not ask me before," was Dirac's answer. 

But a few scientists who knew Dirac better, who managed 
after years of acquaintance to talk to him, were full of praise of 
his gentle attitude toward everyone. They believed that his sol- 
itude was a result of shyness and could be broken in time by 
careful aggressiveness and persistence. 

These idiosyncrasies made it difficult to work with Dirac. The 
result has been that Dirac has not created a school by personal 
contact. He has created a school by his papers, by his book, but 
not by collaboration. He is one of the very few scientists who 
could work even on a lonely island if he had a library and could 
perhaps even do without books and journals. 

When I visited Dirac for the first time I did not know how 
difficult it was to talk to him as I did not then know anyone who 
could have warned me. 

I went along the narrow wooden stairs in St John's College 
and knocked at the door of Dirac's room. He opened it silently 
and with a friendly gesture indicated an armchair. I sat down 
and waited for Dirac to start the conversation. Complete silence. 
I began by warning my host that I spoke very little English. A 
friendly smile but again no answer. I had to go further: 

"I talked with Professor Fowler. He told me that I am sup- 
posed to work with you. He suggested that I work on the in- 
ternal conversion effect of positrons." 


Dirac and Born 

No answer. I waited for some time and tried a direct question: 

"Do you have any objection to my working on this subject?" 


At least I had got a word out of Dirac. 

Then I spoke of the problem, took out my pen in order to 
write a formula. Without saying a word Dirac got up and 
brought paper. But my pen refused to write. Silently Dirac took 
out his pencil and handed it to me. Again I asked him a direct 
question to which I received an answer in five words which 
took me two days to disrest. The conversation was finished. I 
made an attempt to prolong it. 

"Do you mind if I bother you sometimes when I come across 


I left Dirac's room, surprised and depressed. He was not for- 
bidding, and I should have had no disagreeable feeling had I 
known what everyone in Cambridge knew. If he seemed peculiar 
to Englishmen, how much more so he seemed to a Pole who had 
polished his smooth tongue in Lwow cafes! One of Dirac's prin- 
ciples is: 

"One must not start a sentence before one knows how to 
finish it." 

Someone in Cambridge generalized this ironically: 

"One must not start a life before one knows how to finish it." 

It is difficult to make friends in England. The process is slow 
and it takes time for one to graduate from pleasantries about the 
weather to personal themes. But for me it was exactly right. I 
was safe because nobody on the island would suddenly ask me: 
"Have you been married?" No conversation would even ap- 
proach my personal problems. The gossipy atmosphere of 
Lwow's cafes belonged to the past. How we worked for hours, 
analyzing the actions and reactions of others, inventing talks and 
situations, imitating their voices, mocking their weaknesses, lift- 
ing gossip to an art and cultivating it for its own sake! I was glad 
of an end to these pleasures. The only remarks which one is 
Hkely to hear from an EngHshman, on the subject of another's 
personahty, are: 


"He is very nice." 

"He is quite nice." 

Or, in the worst case: 

"I believe that he is all right." 

From these few variations, but much more from the subtle 
way in which they are spoken, one can gain a very fair picture 
after some practice. But the poverty of words kills the conversa- 
tion after two minutes. 

The first month I met scarcely anyone. The problem on which 
I worked required tedious calculations rather than a search for 
new ideas. I had never enjoyed this kind of work, but I deter- 
mined to learn its technique. I worked hard. In the morning I 
went to a small dusty Hbrary in the Cavendish Laboratory. Every 
time I entered this building I became sentimental. If someone had 
asked me, "What is the most important place in the world?" I 
would have answered: "The Cavendish Laboratory." Here Max- 
well and J. J. Thomson worked. From here, in the last years 
under Rutherford's leadership, ideas and experiments emerged 
which changed our picture of the external world. Nearly all the 
great physicists of the world have lectured in this shabby old 
auditorium which is, by the way, the worst I have ever seen. 

I studied hard all day until late at night, interrupted only by a 
movie which took the place of the missing English conversation. 
I knew that I must bring results back to Poland. I knew what 
happened to anyone who returned empty-handed after a year on 
a fellowship. I had heard conversations on the subject and I 
needed only to change the names about to have a complete pic- 

A: I saw Infeld today; he is back already. What did he do in Eng- 

B: We have just searched carefully through the science abstracts. 
He didn't publish anything during the whole year. 

A: What? He couldn't squeeze out even one brief paper in twelve 
months, when he had nothing else to do and had the best help 
in the world? 

B: I'm sure he didn't. He is finished now. I am really very sorry for 
him. Loria ought to have known better than to make a fool of 
himself by recommending Infeld for a Rockefeller fellowship. 


Dirac and Born 

A: We can have fun when Loria comes here. We'll ask him what his 
protege did in England. Loria is very talkative. Let's give him a 
good opportunity. 

B: Yes. It will be quite amusing. What about innocently asking 
Infeld to give a lecture about Cambridge and his work there? It 
will be fun to see him dodging the subject of his own work. 

This is the way academic failure was discussed in Poland. I 
should have little right to object. Bitter competition and lack 
of opportunity create this atmosphere. 

When I came to Cambridge, before the academic year began, 
I learned that Professor Born would lecture there for a year. His 
name, too, is well known to every physicist. He was as famous 
for the distinguished work which he did in theoretical physics 
as for the school which he created. Bom was a professor in Goet- 
tingen, the strongest mathematical center of the world before it 
was destroyed by Hitler. Many mathematicians and physicists 
from all over the world went to Goettingen to do research in the 
place associated with the shining names of Gauss in the past and 
Hilbert in the present. Dirac had had a fellowship in Goettingen 
and Heisenberg obtained his docentship there. Some of the most 
important papers in quantum mechanics were written in collab- 
oration by Bom and Heisenberg. Born was the first to present 
the probability interpretation of quantum mechanics, intro- 
ducing ideas which penetrated deeply into philosophy and are 
linked with the much-discussed problem of determinism and 

I also knew that Bom had recently published an interesting 
note in Nature, conceming the generalization of Maxwell's 
theory of electricity, and had announced a paper, dealing at 
length with this problem which would appear shortly in the 
Proceedings of the Royal Society. 

Being of Jewish blood. Professor Bom had to leave Germany 
and immediately received five offers, from which he chose the 
invitation to Cambridge. For the first term he announced a course 
on the theory on which he was working. 

I attended his lectures. The audience consisted of graduate 
students and fellows from other colleges, chiefly research work- 


ers. Born spoke English with a heavy German accent. He was 
about fifty, with gray hair and a tense, inteUigent face with eyes 
in which the suffering expression was intensified by fatigue. In 
the beginning I did not understand his lectures fully. The whole 
general theory seemed to be sketchy, a program rather than a 
finished piece of work. 

His lectures and papers revealed the difference between the 
German and English style in scientific work, as far as general 
comparisons of this kind make any sense at all. It was in the tra- 
dition of the German school to publish results quickly. Papers 
appeared in German journals six weeks after they were sent to 
the editor. Characteristic of this spirit of competition and prior- 
ity quarrels was a story which Loria told me of a professor of his 
in Germany, a most distinguished man. This professor had at- 
tacked someone's work, and it turned out that he had read the 
paper too quickly; his attack was unjustified, and he simply had 
not taken the trouble to understand what the author said. When 
this was pointed out to him he was genuinely sorry that he had 
published a paper containing a severe and unjust criticism. 
But he consoled himself with the remark: "Better a wrong paper 
than no paper at all." 

The English style of work is quieter and more dignified. No 
one is interested in quick publishing, and it matters much less to 
an Englishman when someone else achieves the same results and 
publishes them a few days earlier. It takes sLx months to print a 
paper in the Proceedings of the Royal Society. Priority quarrels 
and stealing of ideas are practically unknown in England. The 
attitude is: "Better no paper at all than a wrong paper." 

In the beginning, as I have said, I was not greatly impressed 
with Born's results. But later, when he came to the concrete 
problem of generalizing Maxwell's equations, I found the sub- 
ject exciting, closely related to the problems on which I had 
worked before. In general terms the idea was: 

Maxwell's theory is the theory of the electromagnetic field, 
and it forms one of the most important chapters in theoretical 
physics. Its great achievement lies in the introduction of the con- 
cept of the field. It explains a wide region of experimental facts 


Dirac and Born 

but, like every theory, it has its limitations. Maxwell's theory does 
not explain why elementary particles like electrons exist, and it 
does not bind the properties of the field to those of matter. 

After the discovery of elementary particles it was clear that 
Maxwell's theory, hke all our theories, captures only part of the 
truth. And again, as always in physics, attempts were made to 
cover, through modifications and generalizations, a wider range 
of facts. Born succeeded in generalizing Maxwell's equations and 
replacing them by new ones. As their first approximation these 
new equations gave the old laws confirmed by experiments. 
But in addition they gave a new solution representing an elemen- 
tary particle, the electron. Its physical properties were deter- 
mined to some extent by the new laws governing the field. The 
aim of this new theory was to form a bridge between two hith- 
erto isolated and unreconciled concepts: field and matter. Born 
called it the Unitary Field Theory, the name indicating the union 
of these two fundamental concepts. 

After one of his lectures I asked Born whether he would lend 
me a copy of his manuscript. He gave it to me with the assur- 
ance that he would be very happy if I would help him. I wanted 
to understand a point which had not been clear to me during the 
lecture and which seemed to me to be an essential step. Born's 
new theory allowed the construction of an elementary particle, 
the electron, with a finite mass. Here lay the essential difference 
between Born's new and Maxwell's old theories. A whole chain 
of argument led to this theoretical determination of the mass of 
the electron. I suspected that something was wrong in this deri- 
vation. On the evening of the day I received the paper the point 
suddenlv became clear to me. I knew that the mass of the elec- 
tron was wrongly evaluated in Born's paper and I knew how to 
find the right value. My whole argument seemed simple and con- 
vincing to me. I could hardly wait to tell it to Born, sure that he 
would see my point immediately. The next day I went to him 
after his lecture and said: 

"I read your paper; the mass of the electron is wrong." 

Born's face looked even more tense than usual. He said: 

"This is very interesting. Show me why." 


Two of his audience were still present in the lecture room. I 
took a piece of chalk and wrote a relativistic formula for the 
mass density. Born interrupted me angrily: 

"This problem has nothing to do with relativity theory. I 
don't like such a formal approach. I find nothing wrong with 
the way I introduced the mass." Then he turned toward the two 
students who were listening to our stormy discussion. 

"What do you think of my derivation?" 

They nodded their heads in full approval. I put down the 
piece of chalk and did not even try to defend my point. 

Born felt a little uneasy. Leaving the lecture room, he said: 

"I shall think it over." 

I was annoyed at Bom's behavior as well as at my own and 
was, for one afternoon, disgusted with Cambridge. I thought: 
"Here I met two great physicists. One of them does not talk. I 
could as easily read his papers in Poland as here. The other talks, 
but he is rude." I scrutinized my argument carefully but could 
find nothing wrong with it. I made some further progress and 
found that new and interesting consequences could be drawn if 
the "free densities" were introduced relativistically. A different 
interpretation of the unitary theory could be achieved which 
would deepen its physical meaning. 

The next day I went again to Bom's lecture. He stood at the 
door before the lecture room. When I passed him he said to me: 

"I am waiting for you. You were quite right. We will talk it 
over after the lecture. You must not mind my being rude. Every- 
one who has worked with me knows it. I have a resistance 
against accepting something from outside. I get angry and swear 
but always accept after a time if it is right." 

Our collaboration had begun with a quarrel, but a day later 
complete peace and understanding had been restored. I told Bom 
about my new interpretation connecting more closely and 
clearly, through the "free densities," the field and particle as- 
pects. He immediately accepted these ideas with enthusiasm. Our 
collaboration grew closer. We discussed, worked together after 
lectures, in Bom's home or mine. Soon our relationship became 
informal and friendly. 


Dirac and Born 

I ceased to work on my old problem. After three months of 
my stay in Cambridge we published together two notes in 
Nature, and a long paper, in which the foundations of the New 
Unitary Field Theory were laid down more deeply and care- 
fully than before, was ready for publication in the Proceedings 
of the Royal Society. 

For the first time in my hfe I had close contact with a famous, 
distinguished physicist, and I learned much through our relation- 
ship. Born came to my home on his bicycle whenever he wished 
to communicate with me, and I visited him, unannounced, when- 
ever I felt like it. The atmosphere of his home was a combination 
of high intellectual level with heavy Germany pedantry. In the 
hall there was a wooden gadget announcing which of the mem- 
bers of the family were out and which were in. 

I marveled at the way in which he managed his heavy corre- 
spondence, answering letters with incredible dispatch, at the 
same time looking through scientific papers. His tremendous col- 
lection of reprints was well ordered; even the reprints from 
cranks and lunatics were kept, under the heading "Idiots." Born 
functioned like an entire institution, combining vivid imagination 
with splendid organization. He worked quickly and in a restless 
mood. As in the case of nearly all scientists, not only the result 
was important but the fact that he had achieved it. This is human, 
and scientists are human. The only scientist I have ever met for 
whom this personal aspect of work is of no concern at all is 
Einstein. Perhaps to find complete freedom from human 
weakness we must look up to the highest level achieved by the 
human race. There was something childish and attractive in 
Bom's eagerness to go ahead quickly, in his restlessness and his 
moods, which changed suddenly from high enthusiasm to deep 
depression. Sometimes when I would come with a new idea he 
would say rudely, "I think it is rubbish," but he never minded if 
I applied the same phrase to some of his ideas. But the great, the 
celebrated Born was as happy and as pleased as a young student 
at words of praise and encouragement. In his enthusiastic atti- 
tude, in the vividness of his mind, the impulsiveness with which 
he grasped and rejected ideas, lay liis great charm. Near his bed 


he had always a pencil and a piece of paper on which to scribble 
his inspirations, to avoid turning them over and over in his mind 
during sleepless nights. 

Once I asked Born how he came to study theoretical physics. 
I was interested to know at what age the first impulse to choose 
a definite path in life crystalizes. Born told me his story. His 
father was a medical man, a university professor, famous and 
rich. When he died he left his son plenty of money and good 
advice. The money was sufficient, in normal times, to assure his 
son's independence. The advice was simply to listen during his 
first student year to many lectures on many subjects and to make 
a choice only at the end of the first year. So young Born went to 
the university at Breslau, listened to lectures on law, literature, 
biology, music, economics, astronomy. He liked the astronomy 
lectures the most. Perhaps not so much for the lectures them- 
selves as for the old Gothic building in which they were held. 
But he soon discovered that to understand astronomy one must 
know mathematics. He asked where the best mathematicians in 
the world were to be found and was told "Goettingen." So he 
went to Goettingen, where he finished his studies as a theoretical 
physicist, habilitated and finally became a professor. 

"At that time, before the war," he added, "I could have done 
whatever I wanted with my life since I did not even know what 
the struggle for existence meant. I believe I could have become a 
successful writer or a pianist. But I found the work in theoretical 
physics more pleasant and more exciting than anything else." 

Through our work I gained confidence in myself, a confidence 
that was strengthened by Bom's assurance that ours was one of 
the pleasantest collaborations he had ever known. Loyally he 
stressed my contributions in his lectures and pointed out my share 
in our collaboration. I was happy in the excitement of obtaining 
new results and in the conviction that I was working on essential 
problems, the importance of which I certainly exaggerated. Hav- 
ing new ideas, turning blankness into understanding, suddenly 
finding the right solution after weeks or months of painful doubt, 
creates perhaps the highest emotion man can experience. Every 
scientist knows this feeling of ecstasy even if his achievements 
are small. But this pure feeling of Eitreka Is mixed with overtones 
of very human, selfish emotions: "/ found it; / will have an im- 
portant paper; it will help me in my career." I was fully aware 
of the presence of these overtones in my ovv^n consciousness. 


Erwin Schrodinger developed some of the basic equations 
of modern atomic theory. This article considers a book in 
which Schrodinger discusses the repercussions of the quantum 

15 I am this Whole World: Erwin Schrodinger 

Jeremy Bernstein 

Chapter from Bernstein's book, A Comprehensible World: On Modern 
Science and its Origins, published in 1961. 

There is a parlor game often played by my colleagues 
in physics. It consists of trying to decide whether the 
physicists of the extraordinary generation that pro- 
duced the modem quantum theory, in the late 
twenties, were intrinsically more gifted than our pres- 
ent generation or whether they simply had the good 
fortune to be at the height of their creative powers 
(for physicists, with some notable exceptions, this lies 
between the ages of twenty-five and thirty-five at a 
time when there was a state of acute and total crisis in 
physics— a crisis brought about by the fact that existing 


physics simply did not account for what was known about 
the atom. In brief, if our generation had been alive at that 
time, could we have invented the quantum theory? 

It is a question that will never be answered. But there is 
no doubt that the group of men who did invent the theory 
was absolutely remarkable. Aside from Max Planck and 
Einstein (it was Planck who invented the notion of the 
quantum— the idea that energy was always emitted and 
absorbed in distinct units, or quanta, and not continu- 
ously, like water flowing from a tap— and it was Einstein 
who pointed out how Planck's idea could be extended and 
used to explain a variety of mysteries about matter and 
radiation that physicists were contending with) , who did 
their important work before 1925, the list includes Niels 
Bohr, who conceived the theory that the orbits of electrons 
around atoms were quantized (electrons, according to the 
Bohr theory, can move only in special elliptical paths— 
"Bohr orbits"— around the nucleus and not in any path, as 
the older physics would have predicted) ; Prince Louis de 
Broglie, a French aristocrat who conjectured in his doc- 
toral thesis that both light and matter had particle and 
wave aspects; Werner Heisenberg, who made the first 
breakthrough that led to the mathematical formulation of 
the quantum theory, from which the Bohr orbits can be 
derived, and whose "uncertainty relations" set the limita- 
tions on measurements of atomic systems; P. A. M. Dirac, 
who made basic contributions to the mathematics of the 
theory and who showed how it could be reconciled with 
Einstein's theory of relativity; Wolfgang Pauli, whose "ex- 
clusion principle" led to an explanation of why there is a 
periodic table of chemical elements; Max Bom and Pas- 
cual Jordan, who contributed to the interpretation of the 
theory; and, finally, Erwin Schrodinger, whose Schrodingcr 
Equation is in many ways the basic equation of the 
quantum theory, and is to the new physics what Newton's 


I am this Whole World: Erwin Schrodinger 

laws o£ motion were to the physics that went before it. 

While Heisenberg, Pauli, and Dirac were all in their 
early twenties when they did their work, de Broglie and 
Bohr were older, as was Schrodinger, who was born in 
Vienna in 1887. In 1926, he published the paper in which 
his equation was formulated. Oddly, just a few years be- 
fore, he had decided to give up physics altogether for 
philosophy. Philipp Frank, who had been a classmate of 
Schrodinger's in Vienna, once told me that just before 
Schrodinger began his work on the quantum theory he 
had been working on a psychological theory of color per- 
ception. Schrodinger himself writes in the preface of his 
last book. My View of the World (Cambridge) , published 
posthumously (he died in 1961), "In 1918, when I was 
thirty-one, I had good reason to expect a chair of theo- 
retical physics at Czemowitz. ... I was prepared to do 
a good job lecturing on theoretical physics . . . but for 
the rest, to devote myself to philosophy, being deeply 
imbued at the time with the writings of Spinoza, Schopen- 
hauer, Ernst Mach, Richard Semon, and Richard Aven- 
arius. My guardian angel intervened: Czemowitz soon no 
longer belonged to Austria. So nothing came of it. I had to 
stick to theoretical physics, and, to my astonishment, some- 
thing occasionally emerged from it." 

The early quantum theoreticians were a small group, 
mainly Europeans, who knew each other well. There was 
among them a sense of collaborating on one of the most 
important discoveries in the history of physics. In his 
Science and the Common Understanding, Robert Oppen- 
heimer wrote, "Our understanding of atomic physics, 
of what we call the quantum theory of atomic systems, 
had its origins at the turn of the century and its great 
synthesis and resolutions in the nineteen-twenties. It was a 
heroic time. It was not the doing of any one man; it in- 
volved the collaboration of scores of scientists from many 
different lands, though from first to last the deeply creative 


and subtle and critical spirit of Niels Bohr guided, re- 
strained, deepened, and finally transmuted the enterprise. 
It was a period of patient work in the laboratory, of crucial 
experiments and daring action, of many false starts and 
many untenable conjectures. It was a time of earnest corre- 
spondence and hurried conjectures, of debate, criticism, 
and brilliant mathematical improvisation. For those who 
participated, it was a time of creation; there was terror as 
well as exaltation in their new insight. It will probably not 
be recorded very completely as history. As history, its re- 
creation would call for an art as high as the story of 
Oedipus or the story of Cromwell, yet in a realm of action 
so remote from our common experience that it is unlikely 
to be known to any poet or any historian." 

However, as the outlines of the theory became clearer, a 
sharp division of opinion arose as to the ultimate signifi- 
cance of it. Indeed, de Broglie, Einstein, and Schrodinger 
came to feel that even though the theory illuminated vast 
stretches of physics and chemistry ("All of chemistry and 
most of physics," Dirac wrote) , there was fundamentally 
something unsatisfactory about it. The basic problem that 
troubled them was that the theory abandons causation of 
the kind that had been the goal of the classical physics of 
Newton and his successors: In the quantum theory, one 
cannot ask what one single electron in a single atom will 
do at a given time; the theory only describes the most 
probable behavior of an electron in a large collection of 
electrons. The theory is fundamentally statistical and deals 
solely with probabilities. The Schrodinger Equation en- 
ables one to work out the mathematical expressions for 
these probabilities and to determine how the probabilities 
will change in time, but according to the accepted inter- 
pretation it does not provide a step-by-step description of 
the motion of, say, a single electron in an atom, in the way 
that Newtonian mechanics projects the trajectory of a 
planet moving around the sun. 


I am this Whole World: Erwin Schrodinger 

To most physicists, these limitations are a fundamental 
limitation, in principle, on the type of information that 
can be gathered by carrying out measurements of atomic 
systems. These limitations, which were first analyzed by 
Heisenberg and Bohr, are summarized in the Heisenberg 
uncertainty relations, which state, generally speaking, that 
the very process of making most measurements of an 
atomic system disturbs the system's behavior so greatly that 
it is put into a state qualitatively different from the one it 
was in before the measurement. (For example, to measure 
the position of an electron in an atom, one must illumi- 
nate the electron with light of very short wave length. This 
light carries so much momentum that the process of illu- 
minating the electron knocks it clear out of the atom, so a 
second measurement of the position of the electron in the 
atom is impossible. "We murder to dissect," as Words- 
worth has said.) The observer— or, really, his measuring 
apparatus— has an essential influence on the observed. The 
physicists who have objected to the quantum theory feel 
that this limitation indicates the incompleteness of the 
theory and that there must exist a deeper explanation that 
would yield the same universal agreement with experi- 
ment that the quantum theory does but that would allow a 
completely deterministic description of atomic events. 
Naturally, the burden of finding such a theory rests upon 
those who feel that it must exist; so far, despite the re- 
peated eflForts of people like de Broglie, Einstein, and 
Schrodinger, no such theory has been forthcoming. 

Schrodinger, who was a brilliant writer of both scientific 
texts and popular scientific essays, summarized his distaste 
for the quantum theory in an essay entitled Are There 
Quantum Jumps? published in 1952: "I have been try- 
ing to produce a mood that makes one wonder what parts 
of contemporary science will still be of interest to more 
than historians two thousand years hence. There have 
been ingenious constructs of the human mind that gave an 


exceedingly accurate description of observed facts and have 
yet lost all interest except to historians. I am thinking of 
the theory of epicycles. [This theory was used, especially 
by the Alexandrian astronomer Ptolemy, to account for 
the extremely complicated planetary motions that had 
been observed; it postulated that they were compounded 
of innumerable simple circular motions. Reduced to the 
simplest terms, a planet was presumed to move in a small 
circle around a point that moved in a large circle around 
the earth. The theory was replaced by the assumption, 
conceived by Copernicus and Kepler, that the planets 
move in elliptical orbits around the sun.] I confess to the 
heretical view that their modern counterpart in physical 
theory are the quantum jumps." In his introduction to 
My View of the World, Schrodinger puts his belief even 
more strongly: "There is one complaint which I shall not 
escape. Not a word is said here of acausality, wave mechan- 
ics, indeterminacy relations, complementarity, an expand- 
ing universe, continuous creation, etc. Why doesn't he 
talk about what he knows instead of trespassing on the 
professional philosopher's preserves? Ne sutor supra crepi- 
dam. On this I can cheerfully justify myself: because I do 
not think that these things have as much connection as is 
currently supposed with a philosophical view of the 
world." There is a story that after Schrodinger lectured, in 
the twenties, at the Institute of Theoretical Physics, in 
Copenhagen, in which Bohr was teaching, on the implica- 
tions of his equation, a vigorous debate took place, in the 
course of which Schrodinger remarked that if he had 
known that the whole thing would be taken so seriously he 
never would have invented it in the first place. 

Schrodinger was too great a scientist not to recognize the 
significance of the all but universal success of the quantum 
theory— it accounts not only for "all of chemistry and most 
of physics" but even for astronomy; it can be used, for 
example, to make very precise computations of the energy 


I am this Whole World: Erwin Schrodinger 

generated in the nuclear reactions that go on in the sun 
and other stars. Indeed, Schrodinger's popular master- 
piece. What Is Life? deals with the impact of quantum 
ideas on biology and above all on the molecular processes 
that underlie the laws of heredity. The two striking fea- 
tures of the hereditary mechanism are its stability and its 
changeability— the existence of mutations, which allow for 
the evolution of a biological species. The characteristics 
that are inherited by a child from its mother and father are 
all contained in several large organic molecules— the genes. 
Genes are maintained at a fairly high temperature, 98° F., 
in the human body, which means that they are subject to 
constant thermal agitation. The question is how does this 
molecule retain its identity through generation after gen- 
eration. Schrodinger states the problem brilliantly: "Let 
me throw the truly amazing situation into relief once 
again. Several members of the Habsburg dynasty have a 
peculiar disfigurement of the lower lip ('Habsburger 
Lippe') . Its inheritance has been studied carefully and 
published, complete with historical portraits, by the Im- 
perial Academy of Vienna, under the auspices of the fam- 
ily. . . . Fixing our attention on the portraits of a member 
of the family in the sixteenth century and of his descend- 
ant, living in the nineteenth, we may safely assume that 
the material gene structure responsible for the abnormal 
feature has been carried on from generation to generation 
through the centuries, faithfully reproduced at every one 
of the not very numerous cell divisions that lie between. 
. . . The gene has been kept at a temperature around gS°F. 
during all that time. How are we to understand that it 
has remained unperturbed by the disordering tendency of 
the heat motion for centuries?" 

According to the quantum theory, the stability of any 
chemical molecule has a natural explanation. The mole- 
cule is in a definite energy state. To go from one state to 
another the molecule must absorb just the right amount of 


energy. If too little energy is supplied, the molecule will 
not make the transition. This situation differs completely 
from that envisaged by classical physics, in which the 
change of state can be achieved by absorbing any energy. It 
can be shown that the thermal agitations that go on in the 
human body do not in general supply enough energy to 
cause such a transition, but mutations can take place in 
those rare thermal processes in which enough energy is 
available to alter the gene. 

What Is Life? was published in 1944. Since then the 
field of molecular biology has become one of the most 
active and exciting in all science. A good deal of what 
Schrodinger said is now dated. But the book has had an 
enormous influence on physicists and biologists in that it 
hints how the two disciplines join together at their base. 
Schrodinger, who received the Nobel Prize jointly with 
Dirac, in 1933, succeeded Max Planck at the University of 
Berlin in 1927. When Hitler came to power, Schrodinger, 
although not a Jew, was deeply affected by the political 
climate. Philipp Frank has told me that Schrodinger at- 
tempted to intervene in a Storm Trooper raid on a Jewish 
ghetto and would have been beaten to death if one of the 
troopers, who had studied physics, had not recognized him 
as Germany's most recent Nobel Laureate and persuaded 
his colleagues to let him go. Shortly afterward, Schro- 
dinger went to England, then back to Austria, then to 
Belgium, when Austria fell, and finally to the Dublin In- 
stitute for Advanced Studies, where he remained until he 
returned to Vienna, in 1956. By the end of his life, he must 
have mastered as much general culture— scientific and non- 
scientific— as it is possible for any single person to absorb 
in this age of technical specialization. He read widely in 
several languages, and wrote perceptively about the rela- 
tion between science and the humanities and about Greek 
science, in which he was particularly interested. He even 
wrote poetry, which, I am told, was extremely romantic. 


I am this Whole World: Erwin Schrodinger 

(The pictures of Schrodinger as a young man give him a 
Byronic look.) What kind of personal metaphysics would 
such a man derive from his reading and experience? In 
My View of the World, he leaves a partial answer. 

My View of the World consists of two long essays— one 
written in 1925, just before the discovery of the Schro- 
dinger Equation, and one written in i960, just before his 
death. In both essays he reveals himself as a mystic deeply 
influenced by the philosophy of the Vedas. In 1925 he 
writes, "This life of yours which you are living is not 
merely a piece of the entire existence, but is in a certain 
sense the whole; only this whole is not so constituted that 
it can be surveyed in one single glance. This, as we know, 
is what the Brahmins express in that sacred, mystic for- 
mula which is yet really so simple and so clear: Tat tvam 
asi, this is you. Or, again, in such words as 'I am in the east 
and in the west. I am below and above, / am this whole 
world,' " and in the later essay he returns to this theme. 
He does not attempt to derive or justify his convictions 
with scientific argument. In fact, as he stresses in his pref- 
ace, he feels that modern science, his own work included, 
is not relevant to the search for the underlying metaphysi- 
cal and moral truths by which one lives. For him, they 
must be intuitively, almost mystically arrived at. He 
writes, "It is the vision of this truth (of which the indi- 
vidual is seldom conscious in his actions) which underlies 
all morally valuable activity. It brings a man of nobility 
not only to risk his life for an end which he recognizes or 
believes to be good but— in rare cases— to lay it down in 
full serenity, even when there is no prospect of saving his 
own person. It guides the hand of the well-doer— this per- 
haps even more rarely— when, without hope of future 
reward, he gives to relieve a stranger's suffering what he 
cannot spare without suffering himself." 

In i960, I had the chance to visit Schrodinger in 
Vienna. I was studying at the Boltzmann Institute for 


Theoretical Physics, whose director, Walter Thirring, is 
the son of Hans Thirring, a distinguished Austrian physi- 
cist, also a classmate of Schrodinger. Schrodinger had been 
very ill and he rarely appeared at the Institute. But he 
enjoyed maintaining his contact with physics and the 
young physicists who were v,rorking under Walter 
Thirring. Thirring took a small group of us to visit Schro- 
dinger. He lived in an old-fashioned Viennese apartment 
house, with a rickety elevator and dimly lit hallways. The 
Schrodinger living room-library was piled to the ceiling 
with books, and Schrodinger was in the process of writing 
the second of the two essays in My View of the World. 
Physically he was extremely frail, but his intellectual vigor 
was intact. He told us some of the lessons that modern 
scientists might learn from the Greeks. In particular, he 
stressed the recurrent theme of the writings of his later 
years— that modern science may be as far from revealing 
the underlying laws of the natural universe as was the 
science of ancient Greece. It was clear from watching and 
listening to him that the flame that illuminated his intel- 
lectual curiosity throughout his long life still burned 
brightly at the end of it. 


A master of the physics of the atom explains how he 
arrived at the modern theory of the atom. This lec- 
ture is not easy, but it is worth working through. 

16 The Fundamental Idea of Wave Mechanics 

Erwin Schrodinger 

Schrodinger's Nobel Prize lecture given in December 1933. 

On passing through an optical instrument, such as a telescope or a camera 
lens, a ray of light is subjected to a change in direction at each refracting or 
reflecting surface. The path of the rays can be constructed if we know the 
two simple laws which govern the changes in direction: the law of refrac- 
tion which was discovered by Snelhus a few hundred years ago, and the law 
of reflection with which Archimedes was famihar more than 2,000 years ago. 
As a simple example. Fig. i shows a ray A-B which is subjected to refraction 
at each of the four boundary surfaces of two lenses in accordance with the 
law of Snellius. 

Fig. I. 

Fcrmat defined the total path of a ray of hght from a much more general 
point of view. In different media, hght propagates with different velocities, 
and the radiation path gives the appearance as if the hght must arrive at its 
destination as quickly as possible. (Incidentally, it is permissible here to con- 
sider any two points along the ray as the starting- and end-points.) The least 
deviation from the path actually taken would mean a delay. This is the fa- 
mous Fermat principle of the shortest light time, which in a marvellous manner 
determines the entire fate of a ray of light by a single statement and also 
includes the more general case, when the nature of the medium varies not 
suddenly at individual surfaces, but gradually from place to place. The at- 
mosphere of the earth provides an example. The more deeply a ray of light 
penetrates into it from outside, the more slowly it progresses in an increas- 
ingly denser air. Although the differences in the speed of propagation are 


infinitesimal, Fermat's principle in these circumstances demands that the 
light ray should curve earthward (see Fig. 2), so that it remains a little longer 
in the higher « faster » layers and reaches its destination more quickly than 
by the shorter straight path (broken line in the figure; disregard the square, 

Fig. 2. 

WWW^W^ for the time being). I think, hardly any of you will have failed 
to observe that the sun when it is deep on the horizon appears to be not circular 
but flattened : its vertical diameter looks to be shortened. This is a result of 
the curvature of the rays. 

According to the wave theory of Hght, the hght rays, strictly speaking, 
have only fictitious significance. They are not the physical paths of some 
particles of light, but are a mathematical device, the so-called orthogonal 
trajectories of wave surfaces, imaginary guide lines as it were, which point in 
the direction normal to the wave surface in which the latter advances (cf. 
Fig. 3 which shows the simplest case of concentric spherical wave surfaces 
and accordingly rectilinear rays, whereas Fig. 4 illustrates the case of curved 

Fig- 3- 

Fig. 4. 


The Fundamental Idea of Wave Mechanics 

rays). It is surprising that a general principle as important as Fermat's relates 
directly to these mathematical guide lines, and not to the wave surfaces, and 
one might be inclined for this reason to consider it a mere mathematical 
curiosity. Far from it. It becomes properly understandable only from the 
point of view of wave theory and ceases to be a divine miracle. From the 
wave point of view, the so-called curvature of the light ray is far more readily 
understandable as a swerving of the wave surface, which must obviously oc- 
cur when neighbouring parts of a wave surface advance at different speeds; 
in exactly the same manner as a company of soldiers marching forward will 
carry out the order « right incline » by the men taking steps of varying lengths, 
the right-wing man the smallest, and the left-wing man the longest. In at- 
mospheric refraction of radiation for example (Fig. 2) the section of wave 
surface WW must necessarily swerve to the right towards W^W' because 
its left half is located in slightly higher, thinner air and thus advances more 
rapidly than the right part at lower point. (In passing, I wish to refer to one 
point at which the Sneilius' view fails. A horizontally emitted light ray should 
remain horizontal because the refraction index does not vary in the horizon- 
tal direction. In truth, a horizontal ray curves more strongly than any other, 
which is an obvious consequence of the theory of a swerving wave front.) 
On detailed examination the Fermat principle is found to be completely 
tantamount to the trivial and obvious statement that-given local distribution 
of light velocities -the wave front must swerve in the manner indicated. I 
cannot prove this here, but shall attempt to make it plausible. I would again 
ask you to visualize a rank of soldiers marching forward. To ensure that the 
line remains dressed, let the men be connected by a long rod which each 
holds firmly in his hand. No orders as to direction are given; the only order 
is : let each man march or run as fast as he can. If the nature of the ground 
varies slowly from place to place, it will be now the right wing, now the 
left that advances more quickly, and changes in direction will occur spon- 
taneously. After some time has elapsed, it will be seen that the entire path 
travelled is not rectilinear, but somehow curved. That this curved path is 
exactly that by which the destination attained at any moment could be at- 
tained most rapidly according to the nature of the terrain, is at least quite 
plausible, since each of the men did his best. It will also be seen that the swerv- 
ing also occurs invariably in the direction in which the terrain is worse, 
so that it will come to look in the end as if the men had intentionally « by- 
passed » a place where they would advance slowly. 

The Fermat principle thus appears to be the trivial quintessence of the wave 


theory. It was therefore a memorable occasion when Hamilton made the 
discovery that the true movement of mass points in a field of forces (e.g. of 
a planet on its orbit around the sun or of a stone thrown in the gravitational 
field of the earth) is also governed by a very similar general principle, 
which carries and has made famous the name of its discoverer since then. 
Admittedly, the Hamilton principle does not say exactly that the mass point 
chooses the quickest way, but it does say something so similar - the analogy 
with the principle of the shortest travelling time of light is so close, that one 
was faced with a puzzle. It seemed as if Nature had realized one and the 
same law twice by entirely different means: first in the case of light, by 
means of a fairly obvious play of rays ; and again in the case of the mass 
points, which was anything but obvious, unless somehow wave nature were 
to be attributed to them also. And this, it seemed impossible to do. Because 
the « mass points » on which the laws of mechanics had really been confirmed 
experimentally at that time were only the large, visible, sometimes very large 
bodies, the planets, for which a thing like « wave nature » appeared to be out 
of the question. 

The smallest, elementary components of matter which we today, much 
more specifically, call « mass points », were purely hypothetical at the time. 
It was only after the discovery of radioactivity that constant refinements of 
methods of measurement permitted the properties of these particles to be 
studied in detail, and now permit the paths of such particles to be photo- 
graphed and to be measured very exactly (stereophotogrammetrically) by 
the brilliant method of C. T.R.Wilson. As far as the measurements extend 
they confirm that the same mechanical laws are valid for particles as for large 
bodies, planets, etc. However, it was found that neither the molecule nor 
the individual atom can be considered as the « ultimate component »: but 
even the atom is a system of highly complex structure. Images are formed 
in our minds of the structure of atoms consisting 0/ particles, images which 
seem to have a certain similarity with the planetary system. It was ordy 
natural that the attempt should at first be made to consider as valid the same 
laws of motion that had proved themselves so amazingly satisfactory on a 
large scale. In other words, Hamilton's mechanics, which, as I said above, 
culminates in the Hamilton principle, were applied also to the « inner life » 
of the atom. That there is a very close analogy between Hamilton's principle 
and Fermat's optical principle had meanwhile become all but forgotten. If 
it was remembered, it was considered to be nothing more than a curious 
trait of the mathematical theory. 


The Fundamental idea of Wave Mechanics 

Now, it is very difficult, without further going into details, to convey a 
proper conception of the success or failure of these classical-mechanical im- 
ages of the atom. On the one hand, Hamilton's principle in particular proved 
to be the most faithful and reliable guide, which was simply indispensable; 
on the other hand one had to suffer, to do justice to the facts, the rough 
interference of entirely new incomprehensible postulates, of the so-called 
quantum conditions and quantum postulates. Strident disharmony in the 
symphony of classical mechanics-yet strangely familiar-played as it were 
on the same instrument. In mathematical terms we can formulate this as fol- 
lows : whereas the Hamilton principle merely postulates that a given integral 
must be a minimum, without the numerical value of the minimum being 
established by this postulate, it is now demanded that the numerical value 
of the minimum should be restricted to integral multiples of a universal natu- 
ral constant, Planck's quantum of action. This incidentally. The situation was 
fairly desperate. Had the old mechanics failed completely, it would not have 
been so bad. The way would then have been free to the development of a 
new system of mechanics. As it was, one was faced with the difficult task of 
saving the soul of the old system, whose inspiration clearly held sway in this 
microcosm, while at the same time flattering it as it were into accepting the 
quantum conditions not as gross interference but as issuing from its own 
innermost essence. 

The way out lay just in the possibility, already indicated above, of attrib- 
uting to the Hamilton principle, also, the operation of a wave mechanism 
on which the point-mechanical processes are essentially based, just as one 
had long become accustomed to doing in the case of phenomena relating to 
light and of the Fermat principle which governs them. Admittedly, the in- 
dividual path of a mass point loses its proper physical significance and be- 
comes as fictitious as the individual isolated ray of light. The essence of the 
theory, the minimum principle, however, remains not only intact, but reveals 
its true and simple meaning only under the wave-like aspect, as already ex- 
plained. Strictly speaking, the new theory is in fact not new, it is a completely 
organic development, one might almost be tempted to say a more elaborate 
exposition, of the old theory. 

How was it then that this new more « elaborate » exposition led to notably 
different results ; what enabled it, when applied to the atom, to obviate diffi- 
culties which the old theory could not solve? What enabled it to render gross 
interference acceptable or even to make it its own.!* 

Again, these matters can best be illustrated by analogy with optics. Quite 


properly, indeed, I previously called the Fermat principle the quintessence 
of the wave theory of light: nevertheless, it cannot render dispensible a more 
exact study of the wave process itself. The so-called refraction and inter- 
ference phenomena of light can only be understood if we trace the wave 
process in detail because what matters is not only the eventual destination of 
the wave, but also whether at a given moment it arrives there with a wave 
peak or a wave trough. In the older, coarser experimental arrangements, 
these phenomena occurred as small details only and escaped observation. 
Once they were noticed and were interpreted correctly, by means of waves, 
it was easy to devise experiments in which the wave nature of light fmds 
expression not only in small details, but on a very large scale in the entire 
character of the phenomenon. 

Allow me to illustrate this by two examples, first, the example of an op- 
tical instrument, such as telescope, microscope, etc. The object is to obtain a 
sharp image, i.e. it is desired that all rays issuing from a point should be re- 
united in a point, the so-called focus (cf. Fig. 5 a). It was at first beHeved that 
it was only geometrical-optical difficulties which prevented this : they are 
indeed considerable. Later it was found that even in the best designed instru- 

( .B 


The Fundamental Idea of Wave Mechanics 

nients focussing of the rays was considerably inferior than would be expected 
if each ray exactly obeyed the Fermat principle independently of the neigh- 
bouring rays. The light which issues from a point and is received by the 
instrument is reimited behind the instrument not in a single point any more, 
but is distributed over a small circular area, a so-called diffraction disc, which, 
otherwise, is in most cases a circle only because the apertures and lens con- 
tours are generally circular. For, the cause of the phenomenon which we call 
diffraction is that not all the spherical waves issuing from the object point can 
be accommodated by the instrument. The lens edges and any apertures 
merely cut out a part of the wave surfaces (cf Fig. 5b) and-if you will 
permit me to use a more suggestive expression-the injured margins resist 
rigid unification in a point and produce the somewhat blurred or vague 
image. The degree of blurring is closely associated with the wavelength of 
the light and is completely inevitable because of this deep-seated theoretical 
relationship. Hardly noticed at first, it governs and restricts the performance 
cf the modern microscope which has mastered all other errors of repro- 
duction. The images obtained of structures not much coarser or even still 
finer than the wavelengths of light are only remotely or not at all similar 
to the original. 

A second, even simpler example is the shadow of an opaque object cast 
on a screen by a small point light source. In order to construct the shape of 
the shadow, each light ray must be traced and it must be established whether 
or not the opaque object prevents it from reaching the screen. The margin 
of the shadow is formed by those light rays which only just brush past the 
edge of the body. Experience has shown that the shadow.margin is not ab- 
solutely sharp even with a point-shaped light source and a sharply defined 
shadow-casting object. The reason for this is the same as in the first example. 
The wave front is as it were bisected by the body (cf. Fig. 6) and the traces 
of this injury result in blurring of the margin of the shadow which would 
be incomprehensible if the individual light rays were independent entities 
advancing independently of one another without reference to their neigh- 

This phenomenon - which is also called diffraction -is not as a rule very 
noticeable with large bodies. But if the shadow-casting body is very small 
at least in one dimension, diffraction finds expression firstly in that no proper 
shadow is formed at all, and secondly - much more strikingly - in that the 
small body itself becomes as it were its own source of light and radiates light 
in all directions (preferentially to be sure, at small angles relative to the inci- 


Fig. 6. 

dent light). All of you are undoubtedly familiar with the so-called « motes 
of dust » in a hght beam falling into a dark room. Fine blades of grass and 
spiders' webs on the crest of a hill with the sun behind it, or the errant locks 
of hair of a man standing with the sun behind often hght up mysteriously 
by diffracted light, and the visibility of smoke and mist is based on it. It 
comes not really from the body itself, but from its immediate surroundings, 
an area in which it causes considerable interference with the incident wave 
fronts. It is interesting, and important for what follows, to observe that the 
area of interference always and in every direction has at least the extent of 
one or a few wavelengths, no matter how small the disturbing particle may 
be. Once again, therefore, we observe a close relationship between the phe- 
nomenon of diffraction and wavelength. This is perhaps best illustrated by 
reference to another wave process, i.e. sound. Because of the much (greater 
wavelength, which is of the order of centimetres and metres, shadow for- 
mation recedes in the case of sound, and diffraction plays a major, and prac- 
tically important, part: we can easily hear a man calling from behind a high 
wall or around the corner of a solid house, even if we cannot sec him. 

Let us return from optics to mechanics and explore the analogy to its 
fullest extent. In optics the oW system of mechanics corresponds to intellec- 


The Fundamental Idea of Wave Mechanics 

tually operating with isolated mutually independent light rays. The new 
undulatory mechanics corresponds to the wave theory of light. What is 
gained by changing from the old view to the new is that the diffraction 
phenomena can be accommodated or, better expressed, what is gained is 
something that is strictly analogous to the diffraction phenomena of light 
and which on the whole must be very unimportant, otherwise the old view 
of mechanics would not have given full satisfaction so long. It is, however, 
easy to surmise that the neglected phenomenon may in some circumstances 
make itself very much felt, will entirely dominate the mechanical process, 
and will face the old system with insoluble riddles, if the entire mechanical 
system is comparable in extent with the wavelengths of the « waves of matter ■» which 
play the same part in mechanical processes as that played by the hght waves 
in optical processes. 

This is the reason why in these minute systems, the atoms, the old view 
was bound to fail, which though remaining intact as a close approximation 
for gross mechanical processes, but is no longer adequate for the delicate 
interplay in areas of the order of magnitude of one or a few wavelengths. 
It was astounding to observe the manner in which all those strange addi- 
tional requirements developed spontaneously from the new undulatory 
view, whereas they had to be forced upon the old view to adapt them to 
the iimcr life of the atom and to provide some explanation of the observed 

Thus, the salient point of the whole matter is that the diameters of the 
atoms and the wavelength of the hypothetical material waves arc of approxi- 
mately the same order of magnitude. And now you arc bgund to ask wheth- 
er it must be considered mere chance that in our continued analysis of the 
structure of matter we should come upon the order of magnitude of the 
wavelength at this of all points, or whether this is to some extent compre- 
hensible. Further, you may ask, how we know that this is so, since the 
material waves are an entirely new requirement of this theory, unknown 
anywhere else. Or is it simply that this is an assumption which had to be 

The agreement between the orders of magnitude is no mere chance, nor 
is any special assumption about it necessary; it follows automatically from 
the theory in the following remarkable manner. That the heavy nucleus of 
the atom is very much smaller than the atom and may therefore be consid- 
ered as a point centre of attraction in the argument which follows may be 
considered as experimentally established by the experiments on the scattering 


of alpha rays done by Rutherford and Chadwick. Instead of the electrons we 
introduce hypothetical waves, whose wavelengths are left entirely open, 
because we know nothing about them yet. This leaves a letter, say a, in- 
dicating a still unknown figure, in our calculation. We are, however, used 
to this in such calculations and it docs not prevent us from calculating that 
the nucleus of the atom must produce a kind of diffraction phenomenon in 
these waves, similarly as a minute dust particle does in light waves. Analo- 
gously, it follows that there is a close relationship between the extent of the 
area of interference with which the nucleus surrounds itself and the wave- 
length, and that the two are of the same order of magnitude. What this is, 
we have had to leave open; but the most important step now follows: we 
identify the area of interference, the diffraction halo, with the atom; we assert that 
the atom in reality is merely the diffraction phenomenon of an electron nmve cap- 
tured as it were by the nucleus of the atom. It is no longer a matter of chance 
that the size of the atom and the wavelength are of the same order of magni- 
tude : it is a matter of course. We know the numerical value of neither, 
because we still have in our calculation the one unknown constant, which 
wc called a. There are two possible ways of determining it, which provide 
a mutual check on one another. First, wc can so select it that the manifesta- 
tions of life of the atom, above all the spectrum lines emitted, come out 
correctly quantitatively; these can after all be measured very accurately. 
Secondly, we can select a in a manner such that the diffraction halo acquires 
the size required for the atom. These two determinations of rt (of which the 
second is admittedly far more imprecise because «size of the atom» is no 
clearly defined term) are in co}}iplete agreement ii'ith one another. Thirdly, and 
lastly, wc can remark that the constant remaining unknown, physically 
speaking, docs not in fact have the dimension of a length, but of an action, 
i.e. energy X time. It is then an obvious step to substitute for it the numerical 
value of Planck's universal quantum of action, which is accurately known 
from the laws of heat radiation. It will be seen that we return, with the full, 
now considerable accuracy, to the frst (most accurate) determination. 

Quantitatively speaking, the theory therefore manages with a minimum 
of new assumptions. It contains a single available constant, to which a 
numerical value familiar from the older quantum theory must be given, 
first to attribute to the diffraction halos the right size so that they can be 
reasonably identified with the atoms, and secondly, to evaluate quantitative- 
ly and correctly all the manifestations of life of the atom, the light radiated 
by it, the ionization energy, etc. 


The Fundamental Idea of Wave Mechanics 

I have tried to place before you the fundamental idea of the wave theory 
of matter in the simplest possible form. I must admit now that in my desire 
not to tangle the ideas from the very beginning, I have painted the lily. Not 
as regards the high degree to which all sufficiently, carefully drawn conclu- 
sions are confirmed by experience, but with regard to the conceptual ease 
and simplicity with which the conclusions are reached. I am not speaking 
here of the mathematical difficulties, which always turn out to be trivial in 
the end, but of the conceptual difficulties. It is, of course, easy to say that we 
turn from the concept of a curved path to a system of wave surfaces normal 
to it. The wave surfaces, however, even if we consider only small parts of 
them (sec Fig. 7) include at least a narrow bundle of possible curved paths, 

Fig- 7- 

to all of which they stand in the same relationship. According to the old 
view, but not according to the new, one of them in each concrete individual 
case is distinguished from all the others which are « only possible », as that 
« really travelled ». Wc arc faced here with the full force of the logical oppo- 
sition between an 

either -or (point mechanics) 
and a 

both -and (wave mechanics) 

This would not matter much, if the old system were to be dropped entirely 
and to be replaced by the new. Unfortunately, this is not the case. From the 


point of view of wave mechanics, the infinite array of possible point paths 
would be merely fictitious, none of them would have the prerogative over 
the others of being that really travelled in an individual case. I have, how- 
ever, already mentioned that we have yet really observed such individual 
particle paths in some cases. The wave theory can represent this, either not 
at all or only very imperfectly. We find it confoundedly difficult to interpret 
the traces we sec as nothing more than narrow bundles of equally possible 
paths between which the wave surfaces establish cross-connections. Yet, 
these cross-connections are necessary for an understanding of the diffraction 
and interference phenomena which can be demonstrated for the same par- 
ticle with the same plausibility -and that on a large scale, not just as a conse- 
quence of the theoretical ideas about the interior of the atom, which we 
mentioned earlier. Conditions arc admittedly such that we can always man- 
age to make do in each concrete individual case without the two different 
aspects leading to different expectations as to the result of certain experi- 
ments. We cannot, however, manage to make do with such old, familiar, and 
seemingly indispensible terms as « real » or « only possible » ; we are never in 
a position to say what really is or what really happens, but we can only say 
what will be observed in any concrete individual case. Will we have to be 
permanently satisfied with this...? On principle, yes. On principle, there is 
nothing new in the postulate that in the end exact science should aim at 
nothing more than the description of what can really be observed. The ques- 
tion is only whether from now on we shall have to refrain from tying de- 
scription to a clear hypothesis about the real nature of the world. There are 
many who wish to pronounce such abdication even today. But I believe that 
this means making things a little too easy for oneself. 

I would define the present state of our knowledge as follows. The ray or 
the particle path corresponds to a longitudinal relationship of the propagation 
process (i.e. in the direction of propagation), the wave surface on the other 
hand to a transversal relationship (i.e. normal to it). Both relationships arc 
without doubt real; one is proved by photographed particle paths, the other 
by interference experiments. To combine both in a uniform system has 
proved impossible so far. Only in extreme cases does either the transversal, 
shell-shaped or the radial, longitudinal relationship predominate to such an 
extent that we think we can make do with the wave" theory alone or with 
the particle theory alone. 


In this short story by a well-known writer of science fiction, 
the moon explorers moke an unexpected discovery, and 
react in an all-too-human way. 

17 The Sentinel 

Arthur C. Clarke 

Chapter from his book, Expedition to Earth. 1953. 

The next time you see the full moon high in 
the south, look carefully at its right-hand edge and let 
your eye travel upward along the curve of the disk. 
Roimd about two o'clock you will notice a small, dark 
oval: anyone wdth normal eyesight can find it quite easily. 
It is the great walled plain, one of the finest on the Moon, 
known as the Mare Crisium — the Sea of Crises. Three 
hundred miles in diameter, and almost completely smr- 
rounded by a ring of magnificent mountains, it had never 
been explored until we entered it in the late summer of 

Our expedition was a large one. We had two heavy 
freighters which had flown oiu* supplies and equipment 
from the main limar base in the Mare Serenitatis, five hun- 
dred miles away. There were also three small rockets 
which were intended for short-range transport over re- 
gions which our surface vehicles couldn't cross. Luckily, 
most of the Mare Crisium is very flat. There are none of 
the great crevasses so common and so dangerous else- 
where, and very few craters or mountains of any size. As 
far as we could tell, our powerful caterpillar tractors 
would have no difficulty in taking us wherever we wished 
to go. 

I was geologist — or selenologist, if you want to be 
pedantic — in charge of the group exoloring the southern 
region of the Mare. We had crossed a himdred miles ot 
it in a week, skirting the foothills of the moimtains along 
the shore of what was once the ancient sea, some thou- 
sand miUion years before. When life was beginning on 
Earth, it was already dying here. The waters were re- 
treating down the flanks of those stupendous cliffs, re- 
treating into the empty heart of the Moon. Over the land 
which we were crossing, the tideless ocean had once 
been half a mile deep, and now the only trace of moisture 


was the hoarfrost one could sometimes find in caves wiucH 
the searing sunhght never penetrated. 

We had begun our journey early in the slow lunar 
dawn, and still had almost a week of Earth-time before 
nightfall. Half a dozen times a day we would leave our 
vehicle and go outside in the space-suits to hunt for in- 
teresting minerals, or to place markers for the guidance 
of future travelers. It was an uneventful routine. There 
is nothing hazardous or even particularly exciting about 
lunar exploration. We could hve comfortably for a month 
in our pressurized tractors, and if we ran into trouble we 
could always radio for help and sit tight until one of the 
spaceships came to our rescue. 

I said just now that there was nothing exciting about 
lunar exploration, but of course that isn't true. One could 
never grow tired of those incredible mountains, so much 
more rugged than the gentle hiUs of Earth. We never 
knew, as we rounded the capes and promontories of that 
vanished sea, what new splendors would be revealed to 
us. The whole southern curve of the Mare Crisium is a 
vast delta , where a score of rivers once found their 
way into the ocean, fed perhaps by the torrential rains 
that must have lashed the mountains in the brief vol- 
canic age when the Moon was young. Each of these 
ancient valleys was an invitation, challenging us to climb 
into the unknown uplands beyond. But we had a bun- 
dled miles still to cover, and could only look longingly 
at the heights which others must scale. 

We kept Earth-time aboard the tractor, and precisely 
at 22.00 hours the final radio message would be sent out 
to Base and we would close down for the day. Outside, 

the rocks would still be burning beneath the almost ver- 
tical sun, but to us it was night until we awoke again 
eight hours later. Then one of us would prepare break- 
fast, there would be a great buzzing of electric razors, 
and someone would switch on the short-wave radio from 
Earth. Indeed, when the smell of frying sausages began 
to fiU the cabin, it was sometimes hard to beheve that we 
were not back on our own world — everything was so 
normal and homely, apart from the feeling of decreased 
weight and the unnatural slowness with which objects 


The Sentinel 

It was my turn to prepare breakfast in the comer of 
the main cabin that served as a galley. I can remember 
that moment quite vividly after all these years, for the 
radio had just played one of my favorite melodies, the 
old Welsh air, "David of the White Rock." Our driver 
was already outside in his space-suit, inspecting our cater- 
pillar treads. My assistant, Louis Gamett, was up for- 
ward in the control position, making some belated entries 
in yesterday's log. 

As I stood by the frying pan waiting, like any terres- 
trial housewife, for the sausages to brown, I let my gaze 
wander idly over the mountain walls which covered the 
whole of the southern horizon, marching out of sight to 
east and west below the curve of the Moon. They seemed 
only a mile or two from the tractor, but I knew that the 
nearest was twenty miles away. On the Moon, of course, 
there is no loss of detail with distance — none of that al- 
most imperceptible haziness which softens and sometimes 
transfigures all far-off things on Earth. 

Those mountains were ten thousand feet high, and 
they cHmbed steeply out of the plain as if ages ago some 
subterranean eruption had smashed them skyward 
through the molten crust. The base of even the nearest 
was hidden from sight by the steeply curving surface of 
the plain, for the Moon is a very Httle world, and from 
where I was standing the horizon was only two miles 

I lifted my eyes toward the peaks which no man had 
ever climbed, the peaks which, before the coming of 
terrestrial life, had watched the retreating oceans sink 
sullenly into their graves, taking with them the hope and 
the morning promise of a world. The sunhght was beat- 
ing against those ramparts with a glare that hurt the eyes, 
yet only a Uttle way above them the stars were shining 
steadily in a sky blacker than a winter midnight on Earth. 

I was turning away when my eye caught a metallic 
glitter high on the ridge of a great promontory thrust- 
ing out into the sea thirty miles to the west. It was a di- 
mensionless point of light, as if a star had been clawed 
from the sky by one of those cruel peaks, and I imagined 
that some smooth rock surface was catching the sunhght 
and heUographing it straight into my eyes. Such things 


were not uncommon. When the Moon is in her second 
quarter, observers on Earth can sometimes see the great 
ranges in the Oceanus Procellarum burning with a blue- 
white iridescence as the sunhght flashes from their slopes 
and leaps again from world to world. But I was curious 
to know what land of rock could be shining so brightly 
up there, and I climbed into the observation turret and 
swimg our four-inch telescope round to the west. 

I could see just enough to tantalize me. Clear and 
sharp in the field of vision, the mountain peaks seemed 
only half a mile away, but whatever was catching the 
sunlight was still too small to be resolved. Yet it seemed 
to have an elusive symmetry, and the summit upon which 
it rested was curiously flat. I stared for a long time at that 
ghttering enigma, straining my eyes into space, until 
presently a smell of burning from the galley told me that 
our breakfast sausages had made their quarter-million 
mfle journey in vain. 

All that morning we argued our way across the Mare 
Crisium while the western mountains reared higher in 
the sky. Even when we were out prospecting in the space- 
suits, the discussion would continue over the radio. It 
was absolutely certain, my companions argued, that there 
had never been any form of inteUigent life on the Moon. 
The only Hving things that had ever existed there were a 
few primitive plants and their shghtly less degenerate 
ancestors. I knew that as well as anyone, but there are 
times when a scientist must not be afraid to make a fool 
of himself. 

"Listen," I said at last, "I'm going up there, if only for 
my own peace of mind. That mountain's less than twelve 
thousand feet high — that's only two thousand under 
Earth gravity — and I can make the trip in twenty hours 
at the outside. I've always wanted to go up into those 
hills, anyway, and this gives me an excellent excuse." 

"If you don't break your neck," said Gamett, "you'll be 
the laughing-stock of the expedition when we get back 
to Base. That mountain will probably be called Wilson's 
Folly from now on." 

"I won't break my neck," I said firmly. "Who was the 
first man to climb Pico and Helicon?" 


The Sentinel 

"But weren't you rather younger in those days?" asked 
Louis gently. 

"That," I said with great dignity, "is as good a reason 
as any for going." 

We went to bed early that night, after driving the 
tractor to within half a mile of the promontory. Gamett 
was coming with me in the morning; he was a good 
climber, and had often been with me on such exploits 
before. Our driver was only too glad to be left in charge 
of the machine. 

At first sight, those cliffs seemed completely unscale- 
able, but to anyone with a good head for heights, climb- 
ing is easy on a world where all weights are only a sixth 
of their normal value. The real danger in lunar mountain- 
eering lies in overconfidence; a six-hundred-foot drop 
on the Moon can kill you just as thoroughly as a hundred- 
foot fall on Earth. 

We made our first halt on a wide ledge about four 
thousand feet above the plain. Climbing had not been 
very diflBcult, but my Hmbs were stiff with the unac- 
customed effort, and I was glad of the rest. We could 
still see the tractor as a tiny metal insect far down at the 
foot of the cliff, and we reported our progress to the 
driver before starting on the next ascent. 

Inside our suits it was comfortably cool, for the re- 
frigeration units were fighting the fierce sun and carrying 
away the body-heat of our exertions. We seldom spoke 
to each other, except to pass climbing instructions and 
to discuss our best plan of ascent. I do not know what 
Gamett was thinking, probably that this was the craziest 
goose-chase he had ever embarked upon. I more than half 
agreed with him, but the joy of climbing, the knowledge 
that no man had ever gone this way before and the ex- 
hilaration of the steadily widening landscape gave me 
all the reward I needed. 

I don't think I was particularly excited when I saw in 
front of us the wall of rock I had first inspected through 
the telescope from thirty miles away. It would level off 
about fifty feet above our heads, and there on the plateau 
would be the thing that had lured me over these barren 
wastes. It was, almost certainly, nothing more than a 


boulder splintered ages ago by a falling meteor, and with 
its cleavage planes still fresh and bright in this incorrupt- 
ible, unchanging silence. 

There were no hand-holds on the rock face, and we 
had to use a grapnel. My tired arms semed to gain new 
strength as I swxmg the three-pronged metal anchor 
round my head and sent it sailing up toward the stars. 
The first time it broke loose and came falling slowly back 
when we pulled the rope. On the third attempt, the 
prongs gripped firmly and our combined weights could 
not shift it. 

Gamett looked at me anxiously. I could tell that he 
wanted to go first, but I smiled back at him through the 
glass of my helmet and shook my head. Slowly, taking 
my time, I began the final ascent. 

Even with my space-suit, I weighed only forty pounds 
here, so I pulled myself up hand over hand without 
bothering to use my feet. At the rim I paused and waved 
to my companion, then I scrambled over the edge and 
stood upright, staring ahead of me. 

You must understand that until this very moment I 
had been almost completely convinced that there could 
be nothing strange or unusual for me to find here. Al- 
most, but not quite; it was that haunting doubt that had 
driven me forward. Well, it was a doubt no longer, but 
the haunting had scarcely begim. 

I was standing on a plateau perhaps a hundred feet 
across. It had once been smooth — too smooth to be nat- 
ural — but falling meteors had pitted and scored its sur- 
face through immeasurable eons. It had been leveled 
to support a glittering, roughly pyramidal structure, twice 
as high as a man, that was set in the rock like a gigantic, 
many-faceted jewel. 

Probably no emotion at all filled my mind in those first 
few seconds. Then I felt a great lifting of my heart, and a 
strange, inexpressible joy. For I loved the Moon, and now 
I knew that the creeping moss of Aristarchus and Eratos- 
thenes was not the only life she had brought forth in her 
youth. The old, discredited draam of the first explorers 
was true. There had, after all, been a lunar civilization — 
and I was the first to find it. That I had come perhaps a 


The Sentinel 

hundred million years too late did not distress me; it was 
enough to have come at all. 

My mind was beginning to function normally, to ana- 
lyze and to ask questions. Was this a building, a shrine — 
or something for which my language had no name? If a 
building, then why was it erected in so uniquely inac- 
cessible a spot? I wondered if it might be a temple, and 
I could picture the adepts of some strange priesthood 
calling on their gods to preserve them as the hfe of the 
Moon ebbed with the dying oceans, and calling on their 
gods in vain. 

I took a dozen steps forward to examine the thing more 
closely, but some sense of caution kept me from going 
too near. I knew a little of archaeology, and tried to guess 
the cultural level of the civilization that must have 
smoothed this mountain and raised the glittering mirror 
sm-faces that still dazzled my eyes. 

The Egyptians could have done it, I thought, if their 
workmen had possessed whatever strange materials these 
far more ancient architects had used. Because of the 
thing's smallness, it did not occur to me that I might be 
looking at the handiwork of a race more advanced than 
my own. The idea that the Moon had possessed intelli- 
gence at all was still almost too tremendous to grasp, 
and my pride would not let me take the final, humiliating 

And then I noticed something that set the scalp 
crawling at the back of my neck — something so trivial 
and so innocent that many would never have noticed it 
at all. I have said that the plateau was scarred by 
meteors; it was also coated inches-deep with the cosmic 
dust that is always filtering down upon the surface of 
any world where there are no winds to disturb it. Yet 
the dust and the meteor scratches ended quite abruptly 
in a wide circle enclosing the httle pyramid, as though an 
invisible wall was protecting it from the ravages of time 
and the slow but ceaseless bombardment from space. 

There was someone shouting in my earphones, and I 
realized that Gamett had been calling me for some time. 
I walked unsteadily to the edge of the cliff and signaled 
him to join me, not trusting myself to speak. Then I went 
back toward that circle in tht; dust. I picked up a frag- 


ment of splintered rock and tossed it gently toward the 
shining enigma. If the pebble had vanished at that in- 
visible barrier I should not have been surprised, but it 
seemed to hit a smooth, hemispherical surface and slide 
gently to the ground. 

I knew then that I was looking at nothing that could 
be matched in the antiquity of my own race. This was 
not a building, but a machine, protecting itself with 
forces that had challenged Eternity. Those forces, what- 
ever they might be, were still operating, and perhaps I 
had already come too close. I thought of all the radiations 
man had trapped and tamed in the past century. For all 
I knew, I might be as irrevocably doomed as if I had 
stepped into the deadly, silent aura of an unshielded 
atomic pile. 

I remember turning then toward Garnett, who had 
joined me and was now standing motionless at my side. 
He seemed quite oblivious to me, so I did not disturb him 
but walked to the edge of the cliflF in an effort to marshal 
my thoughts. There below me lay the Mare Crisium — 
Sea of Crises, indeed — strange and weird to most men, 
but reassuringly familiar to me. I lifted my eyes toward 
the crescent Earth, lying in her cradle of stars, and I 
wondered what her clouds had covered when these un- 
known builders had finished their work. Was it the steam- 
ing jimgle of the Carboniferous, the bleak shoreline over 
which the first amphibians must crawl to conquer the land 
— or, earher still, the long loneliness before the coming of 

Do not ask me why I did not guess the truth sooner — 
the truth that seems so obvious now. In the first excite- 
ment of my discovery, I had assumed without question 
that this crystalline apparition had been built by some 
race belonging to the Moon's remote past, but suddenly, 
and with overwhelming force, the belief came to me 
that it was as alien to the Moon as I myself. 

In twenty years we had found no trace of life but a few 
degenerate plants. No lunar civilization, whatever its 
doom, could have left but a single token of its existence. 

I looked at the shining pyramid again, and the more 
remote it seemed from anything that had to do with the 
Moon. And suddenly I felt myself shaking with a foolish, 
hysterical laughter, brought on by excitement and over- 


The Sentinel 

exertion: for I had imagined that the httle pyramid was 
speaking to me and was saying: "Sorry, I'm a stranger 
here myself." 

It has taken us twenty years to crack that invisible 
shield and to reach the machine inside those crystal walls. 
What we could not understand, we broke at last with 
the savage might of atomic power and now I have seen 
the fragments of the lovely, glittering thing I foimd up 
there on the mountain. 

They are meaningless. The mechanisms — if indeed 
they are mechanisms — of the pyramid belong to a tech- 
nology that lies far beyond our horizon, perhaps to the 
technology of para-physical forces. 

The mystery haunts us all the more now that the other 
planets have been reached and we know that only Earth 
has ever been the home of inteUigent life in our Universe. 
Nor could any lost civilization of our own world have 
built that machine, for the thickness of the meteoric dust 
on the plateau has enabled us to measure its age. It was 
set there upon its mountain before life had emerged from 
the seas of Earth. 

When our world was half its present age, something 
from the stars swept through the Solar System, left this 
token of its passage, and went again upon its way. Until 
we destroyed it, that machine was still fulfilling the pur- 
pose of its builders; and as to that purpose, here is my 

Nearly a hundred thousand million stars are turning in 
the circle of the Milky Way, and long ago other races on 
the worlds of other suns must have scaled and passed 
the heights that we have reached. Think of such civiliza- 
tions, far back in time against the fading afterglow of 
Creation, masters of a universe so young that life as yet 
had come only to a handful of worlds. Theirs would have 
been a loneliness we cannot imagine, the loneliness of 
gods looking out across infinity and finding none to share 
their thoughts. 

They must have searched the star-clusters as we have 
searched the planets. Everywhere there would be worlds, 
but they would be empty or peopled with crawling, mind- 
less things. Such was our own Earth, the smoke of the 
great volcanoes still staining the sides, when that first ship 
of the peoples of the dawn came shding in from the abyss 


beyond Pluto. It passed the frozen outer worlds, know- 
ing that life could play no part in their destinies. It came 
to rest among the inner planets, warming themselves 
around the fire of the Sun and waiting for their stories to 

Those wanderers must have looked on Earth, circling 
safely in the narrow zone between fire and ice, and must 
have guessed that it was the favorite of the Sun's chil- 
dren. Here, in the distant future, would be intelligence; 
but there were coimtless stars before them still, and they 
might never come this way again. 

So they left a sentinel, one of miUions they have scat- 
tered throughout the Universe, watching over all worlds 
with the promise of life. It was a beacon that down the 
ages has been patiently signaling the fact that no one had 
discovered it. 

Perhaps you understand now why that crystal pyra- 
mid was set upon the Moon instead of on the EartL Its 
builders were not concerned with races still struggling 
up from savagery. They would be interested in our civi- 
lization only if we proved our fitness to survive — by cross- 
ing space and so escaping from the Earth, our cradle. 
That is the challenge that all inteUigent races must meet, 
sooner or later. It is a double challenge, for it depends ia 
turn upon the conquest of atomic energy and the last 
choice between life and death. 

Once we had passed that crisis, it was only a matter of 
time before we found the pyramid and forced it open. 
Now its signals have ceased, and those whose duty it is 
will be turning their minds upon Earth. Perhaps they 
wish to help our infant civilization. But they must be 
very, very old, and the old are often insanely jealous of 
the young. 

I can never look now at the Milky Way without won- 
dering from which of those banked clouds of stars the 
emissaries are coming. If you will pardon so commonplace 
a simile, we have set off the fire-alarm and have nothing 
to do but to wait. 

I do not think we will have to wait for long. 


A distinguished mathematical physicist, the nephew 
of the great Irish playwright John Millington Synge, 
uses an amusing allegory to discuss the nature of 
scientific knowledge. 

18 The Sea-Captain's Box 

John L. Synge 

Excerpt from his book. Science: Sense and Nonsense, 
published in 1951. 

Long ago there lived a retired sea-captain who Hked to go 
to auctions where he bought all sorts of queer things, much 
to the annoyance of his wife. One day he brought home a 
box with strange hieroglyphics painted all over it and set it 
down in a place of honour on the table where he kept his 

As far as could be seen, there was no way of opening the 
box. This aroused the curiosity of the sea-captain and he 
started carefully to scrape off the rust and grime with which 
the box was covered. To his great delight he found a small 
shaft or axle protruding from one side of the box, as shown 
in Fig. I 

He discovered that he could turn this shaft with a pair of 
pliers, but nothing seemed to happen when he did so. 
Certainly the box did not open. 'Perhaps I haven't turned 
the shaft far enough,' he said to himself, 'or perhaps I'm 
turning it the wrong way.' 

He realized then that he had lost track of the amount by 
which he had turned the shaft, and rebuked himself severely 
for not keeping a log. He must be more systematic. 

There was a tiny arrow on the end of the shaft, and when 
the shaft was turned so that this arrow was vertical, it would 
go no further to the left. That he called 'the zero position'. 
Then he set to work and fixed a knob on the end of the shaft 


with a pointer attached and a graduated scale running 
round the shaft so that he could take readings with the 
pointer when he turned the shaft (see Fig. 2). He marked 
off the scale in units, tenths of units and hundredths of 
units, but he could not draw any finer divisions. 

He got out one of the old log books he had brought back 
from the sea and wrote the words 'Log of my box' at the top 
of a blank page. He ruled two columns very neatly and 
wrote at the head of the first column 'Date of observa- 
tion' and at the head of the second column 'Reading of 

Then he turned the knob, looked at the calendar and the 
pointer, and made this entry: 

Date of observation Reading of pointer 

3 March 1453, morning, 

cloudy, wind fresh S.E. 2 00 

by E. 

There was an auction in the neighbourhood that day. 
The sea-captain came home from it in the evening and made 
another entry: 

3 March 1453, evening, 2" 00 

fair, wind slight S.E. by S. 

'We'll never reach port at this rate,' said the sea-captain 
to himself 'Man the capstan!' Then he took the knob and 
turned the pointer to another position, which he noted in his 
log; but the box did not open. He turned the knob to 
various positions, noting them all, but still the box did not 

By this time he was pretty disgusted and half resolved to 
throw the box away, but he was afraid his wife would laugh 
at him. He opened his clasp knife and attacked the box in a 
fury, but succeeded only in knocking off a few flakes of rust 
and breaking his knife. But he was excited to see that he had 


The Sea-Captain's Box 

Fig. I . The Box and the Shaft 

Fig. 2. The Pointer and the Scale 

Fig. 3. Protus and Deutus 


exposed a second shaft! He quickly went to work and fitted 
this shaft with a knob, pointer and graduated scale, so that 
it looked as in Fig. 3. 

Then he turned over a fresh page in his log and ruled three 
columns. The first he headed as before 'Date of observation'. 
Then he hesitated. He must not get the two pointers mixed 
up — he must give them names — what would he call them? 
Castor and Pollux? Scylla and Charybdis? Port and star- 

The sea-captain was a long time making up his mind. An 
unlucky name might send a good ship to the bottom on 
her maiden voyage. He rejected for reasons of domestic 
peace the idea of naming the pointers after girl friends of 
his youth or even after Greek goddesses. He must choose 
names which would apply to his pointers only and to nothing 
else, and the only thing to do was to make up names. He 
finally decided on protus for the one he had discovered first 
and DEUTUS for the one he had discovered second. The 
grammarians might not think much of these names, but the 
mixture of Greek and Latin sounds had a pleasant ring and 
should make them safe from confusion with anything else. 
So he now prepared three columns in his log like this: 

Date of observation protus deutus 

The sea-captain's wife thought that he bought things at 
auctions merely to satisfy a childish yearning to possess 
curious pieces of rubbish, but that was not the real reason. 
Actually, he was a very avaricious man, and he was con- 
vinced that sooner or later he would find a hoard of gold in 
some trunk or box picked up for next to nothing at an 
auction. That is the reason for the gleam in his eyes as he 
now grasps the two knobs on the box and prepares to turn 
them. Surely the box will open now! 


The Sea-Captain's Box 

But the box does not open. Instead, the sea-captain 
jumps back, shaking in every Kmb and with his hair on end. 
'Shiver my timbers!' he cries. 'There's a witch in the fo'c'sle!' 

For, as he had tried to turn the knobs, there seemed to be 
human hands inside the box resisting his efforts. 

Then cautiously, as if afraid of getting burned, he stretches 
out his hand to Protus and turns it gently. No resistance. 
But he draws back his hand in alarm. When he turned 
Protus, Deutus turned at the same time! 

The sea-captain is no coward. In his time he has fought 
pirates in the Levant and dived last from the bridge of his 

ship sinking under him in the Bay of Biscay. But this is a 
different matter. There is magic in this box, and his con- 
science is troubled by his secret avarice for gold. Muttering 
a prayer and an incantation he picked up in an Eastern port, 
he takes up his pen in a shaky hand and with the other starts 
to manipulate Protus, writing down the figures as he does so. 
He is so excited that he forgets to record the date and the 

Here are his readings: 


00 GOO 

1 00 2*00 
2*00 2*83 

3-00 3-46 

4*00 400 

The box does not open, but he does not care. The lust for 
gold has been replaced by scientific curiosity. His sporting 
instinct is roused. 'Good old Protus!' he cries. 'You made a 
poor start but you're gaining. Two to one on Protus!' 

He turns Protus further and gets these readings : 


5-00 447 

6-00 4-90 


'Protus wins!' roars the sea-captain, springing to his feet 
and nearly knocking the table over. His wife puts her head 
round the door. 'What's all the noise about?' Then she 
sneers: 'Still playing with that silly old box! A man of your 

As the days pass, the sea-captain plays the game of Protus 
versus Deutus over and over again. Protus always makes a 
bad start and Protus always wins. It gets boring and he 
begins to dream a little. He forgets that Protus and Deutus 
were names he made up to distinguish one pointer from the 
other. They take on reality and he begins to think of them as 
two ships. Protus must be a heavy ship and Deutus a little 
sloop, very quick at the get-away but not able to hold the 
pace against the sail-spread of Protus. 

But he pulls himself together. The lust for gold is now 
completely gone and the sea-captain starts to ask himself 

What is there really inside the box? He toys again with 
the idea that there may be a witch inside the box, but reason 
tells him that witches don't behave like that. No witch 
would reproduce the same readings over and over again. 

Since Deutus moves whenever you move Protus, there 
must be some connection between them. Ha! Blocks and 
tackle, that's what it must be! Very small ivory pulley- 
blocks and silk threads! 

So the sea-captain stumps down to the dock and gets one 
of his friends to put his ship at his disposal. He tries all sorts 
of ways of connecting two windlasses so that their motions 
will reproduce the motions of Protus and Deutus, but it will 
not work. He can easily make one windlass turn faster than 
the other, but he can never arrange matters so that one wind- 
lass makes a bad start and then overtakes the other. He 
returns home dejected. He is as wise as before about the 
contents of the mysterious box. 


The Sea-Captain's Box 

He reads over his log again and notices that he has always 
set Protus to an integer value. What would happen if he 
moved Protus through half a unit to 0*50? He is about to set 
Protus to 0*50 when his pride explodes in an oath. 'Sacred 
catfish!' he cries. 'What am I? A knob-twiddler and pointer- 
reader? No. I am a man — a man endowed with the gift 
of reason. I shall think it out for myself!' 

Then he ponders: 'When Protus goes from o*oo to I'oo, 
Deutus goes from O'oo to 2'00. That means that Deutus goes 
twice as fast as Protus, at least at the start of the race. So 
when Protus goes from 000 to 0-50, Deutus will go from 
0.00 to 1. 00. That's obvious!' And he writes in the log 


050 I'OO (theoretical) 

By adding that word 'theoretical' the sea-captain shows 
himself to be a cautious, conscientious man, distinguishing 
what he has deduced from his 'theory' from what he observes 
directly. (A noble precedent, often sadly neglected, but 
much harder to follow than one might suppose at first 

Was the sea-captain right? No. When he actually turned 
Protus, he had to record the readings as follows: 


0-50 I '41 (observed) 

What do you think of the sea-captain's 'theory'? Not bad 
for 1453, but any modern schoolboy could tell him how to do 
better. He should have taken a sheet of squared paper and 
plotted a graph, Protus versus Deutus, marking first the 
points corresponding to the observations made and then 
drawing a smooth curve through them. Then he could have 
read off from the curve the 'theoretical' Deutus-reading 
corresponding to the Protus-reading 050. That might have 


saved him from making a fool of himself, provided that 
nature does not make jumps. That is an assumption always 
made in the absence of evidence to the contrary, and (as we 
shall see later) it might have been made here. 

But a graph is not completely satisfactory. It is hard to 
tell another person in a letter the precise shape of the graph; 
you have to enclose a copy of the graph, and the making of 
copies of a grapli is a nuisance unless you use photography. 
A mathematical formula is always regarded as a much more 
convenient and satisfactory way of describing a natural law. 
The sea-captain had never heard of graphs or photography, 
but the other idea slowly evolved in his mind. Let us con- 
tinue the story. 

After thinking the matter over for several years, the sea- 
captain walked down to the pier one evening and stuck up 
a notice which read as follows: 


The people of the sea-port were of course very proud of 
the sea-captain, and they crowded cheering round the 
notice-board. But there was one young man who did not 
cheer. He had just returned from the University of Paris 
and took all scientific matters very seriously. This young 
man now pressed through the crowd until he reached the sea- 
captain, and, taking him by the lapel of his coat, said 
earnestly 'This notice, what does it mean?' 

The sea-captain had been celebrating his discovery and 
was a little unsteady on his feet. He stared belligerently at 
the young man. 'Deutus is twice the square root of Protus,' 
he said. 'That's what it means. Can't you read?' 

'And who is Deutus?' said the young man. 'And who is 
this creature Protus that has a square root?' 

'You don't know Protus and Deutus?' cried the sea-captain. 


The Sea-Captain's Box 

'Why, everyone knows Protus and Deutus! Come up to my 
house and meet them over a glass of grog!' 

So they went up to the sea-captain's house and he intro- 
duced the young man to Protus and Deutus. 'That's Protus 
on the left,' said he, 'and Deutus on the right.' Then he 
leaned over and whispered confidentially in the young man's 
ear: 'Protus carries more sail, but Deutus is quicker on the 

The young man looked at the sea-captain coldly. 'You 
mean,' he said, 'that Protus is a word which stands for the 
number indicated by the left-hand pointer and Deutus is a 
word which stands for the number indicated by the right- 
hand pointer. When you say that Protus is twice the square 
root of Deutus, you mean that one of these numbers is twice 
the square root of the other. In Paris we do not use words 
like Protus and Deutus for numbers. We use letters. We 
would write your result 

D = 2 V P~ 

But is it really true?' 

'Of course it's true,' said the sea-captain, 'and we don't 
need all your French fancy-talk to prove it. Here, read my 
ship's log.' He opened the log and showed the young man 
the readings which you have read on p. 65. 

'Let us see,' said the young man. 'These things are not so 
obvious. Let us do a little calculation. The square root of 
zero is zero, and twice zero is zero, so the first line is right.' 

He was about to put a check mark opposite the first line 
when the sea-captain roared 'Keep your hands off my log! 
Time enough to start writing when you find a mistake, 
which you won't. You can't teach a master mariner how to 

'To proceed,' went on the young man, 'in the second line 
P is one; the square root of one is one, and twice one is two. 


Quite correct,' He put out his hand to make a check mark, 
but withdrew it hastily. 

'In the next line,' he continued, 'P is two. The square 
root of two is an irrational number and cannot be represented 
by a terminating decimal. The third line is wrong, in the 
sense that the law D = 2 \/ P is not satisfied by these numbers. 

The sea-captain was taken aback. 'What's that?' he said. 
'An irrational number? I've sailed the seven seas, but never 
did I meet up with an irrational number. Take your 
irrational numbers back where they come from, and don't 
try to teach me about Protus and Deutus!' 

'I can put it another way,' said the young man, 'If you 
square both sides of your equation, and then interchange the 
sides of the equation, you get 

4P = T>\ 

Now we shall put in the figures from the third line of your 
log. P is 2*00 and D = 2*83. Four times P is therefore eight. 
Now we calculate the square of 2*83; it comes out to be 
80089. "^o Y^^ assert — or do you? — that 

8 = 8-0089. 

Surely you cannot mean that?' 

The sea-captain scratched his head. 'That's not the way 
I figured it,' he said, 'Let's see now. Protus is 2 00. What 
is the square root of 2*00? Why, it's 1-4142. If you double 
that you get 2 8284, and that is 2*83 to the nearest second 
decimal place. You can't trip me up, my boy. The law is 
satisfied all right,' 

'Honest sir,' said the young man, smoothing his Parisian 
hair-cut, 'do you tell me that 

28284 = 283?' 


The Sea-Captain's Box 

'Yes,' said the sea-captain stoutly, 'it is. Those numbers 
are equal to two decimal places.' 

The young man jumped to his feet in anger. 'What a 
waste of my time!' he cried. 'It is a lying notice you have 
posted on the pier! Go down and add to it those words which 
will make it true.' 

'And what words might those be?' asked the sea-captain 

'Write that Deutus is twice the square root of Protus to 
two decimal places.'' 

'I will not,' replied the sea-captain stubbornly. 'Every- 
body knows that Protus and Deutus have only two decimal 
places and they don't need to be told. Keep your irrational 
numbers and other French fiddle-faddle away from Protus 
and Deutus. Commonsense is enough for them. But,' he 
added, 'you're a nice young fellow for a land-lubber, so sit 
ye down and we'll have a glass of grog together.' 

So the young man sat down for a glass of grog and as the 
evening wore on the two became more and more friendly 
and open-hearted with one another. Finally, speaking at 
once, they both broke out with the question: 'What is inside 
the box?' 

The sea-captain told the young man how he had first 
thought that there was gold in the box, how then he had 
thought that there must be a witch, and now for the life of 
him he could think of nothing but that there were two ships, 
Protus with a great sail-spread and Deutus smaller and 
quicker on the get-away. 'But,' he added, 'it bothers me how 
you could fit ships in such a little box, with a sea for them to 
sail on and a wind to sail by. And how is it that they always 
sail the same, with Protus slow at first and Deutus quick on 
the get-away?' 

Not having followed the sea, the young man paid little 
attention to the idea of the two ships. Then suddenly he 


stood up and stared at the box. He had now drunk several 
glasses of grog, so he stood with difficulty and leaned heavily 
on the table. 

'I see it,' he said. 'Yes, I see it!' 

'What do you see?' asked the sea-captain. 'Protus with her 
tops'ls set?' And he too stared at the box. 

'I see no ships,' said the young man, speaking slowly at 
first and then more and more rapidly. 'I see a world of 
mathematics. I see two variable numbers, P and D, taking 
all values rational and irrational from zero to infinity. What 
fools we were to talk of two decimal places! The law is exactl 
D = 2 \/ P. It is true for all values, rational and irrational. 
Protus is a number and Deutus is a number, and if you can- 
not measure them to more than two decimal places, that is 
your infirmity, not theirs. Go,' he cried to the sea-captain, 
'go to the silversmith and make him contrive for you more 
cunning scales so that they may be read more accurately. 
I will go to Paris and procure some optic glasses wherewith 
to read the scales. Then you will see that I am right. The 
law D = 2 \/ P is an exact mathematical law and you will 
verify it with readings that go to four or five or six decimal 

The sea-captain yawned. 'The silversmith is now abed,' 
he said, 'and with the wind now holding you cannot sail for 
France. It may be that this grog has been too much for your 
young stomach. Lie down on the couch there and sleep it 

But before long the silversmith made the cunning scales 
and the young man brought the optic glasses from Paris; to 
the great surprise of the sea-captain, the young man was 
right — the law was satisfied to two more decimal places. 
Beyond that they could not go, although the young man 
married the sea-captain's daughter and worked with his 


The Sea-Captain's Box 

father-in-law on the box for many years. The sea-captain 
died thinking of Protus and Deutus racing in a stiff breeze 
and bequeathed the box to the young man, who in course of 
time grew old and died too. The box was handed down 
from generation to generation as a family heirloom, and it 
was a point of honour with each generation to try to add a 
decimal place to the readings and see whether the law D = 
2 \/ P remained true. Generation after generation found 
that it did remain true, and finally the idea that there might 
be any doubt about it faded. 

No one has ever succeeded in getting inside the box, and 
there is a mixed tradition as to what its contents are. Gold 
and witches were ruled out long ago, but still some members 
of the family see Protus and Deutus sailing with foaming 
wakes where others see two variable numbers capable of 
taking all positive values, rational and irrational. 

An allegory must not be pushed too far, and so one 
hesitates to say what has happened to the sea-captain's box 
in these days of relativity and quantum mechanics. You 
might say that if you look very hard at Protus, your mere 
inspection disturbs him, and when you feel quite certain you 
have pinned him down to a definite reading Deutus is danc- 
ing all over the place. Or perhaps you might say that the 
two pointers do not move continuously but only in definite 
small jumps. 

However, the whole picture is blurred by the discovery of 
a vast number of shafts, connected to one another by many 
complicated laws which the sea-captain would find it im- 
possible to visualize in terms of nautical manoeuvres. 

But the essential feature of the allegory remains — the 
unopened and unopenable box, and the question: 'What is 
really inside it?' Is it the world of mathematics, or can it 
be explained in terms of ships and shoes and sealing wax? 


The answer must surely be a subjective matter; if you ask 
for an 'explanation', you cannot be satisfied unless the 
explanation you get rings a bell somewhere inside you. If 
you are a mathematician, you will respond to a mathe- 
matical explanation, but if you are not, then probably you 
will want an explanation which establishes analogies between 
the deep laws of nature and simple facts of ordinary life. 

Up to the year 1900, roughly, such homely explanations 
were available. It is true that they never told the whole 
story (that inevitably involved mathematics), but they pro- 
vided crusts for the teeth of the mind to bite on. The earth 
pursues its orbit round the sun on account of the pull of 
gravity; then think of an apple with a string through it which 
you whirl round your head. Light travels from the sun to 
the earth in ether- waves; then think of the ripples on the 
surface of a pond when you throw a stone into it. 

Modern physics tends to decry 'explanations' of this sort — 
not out of any malevolent desire to hide secrets, but because 
the simple analogies prove too deceptive and inadequate. In 
fact there are those who deny that physicists have the 
responsibility of giving explanations. This modern attitude 
has been expressed compactly by Professor Dirac: 'The only 
object of theoretical physics is to calculate results that can be 
compared with experiment, and it is quite unnecessary that 
any satisfying description of the whole course of the pheno- 
mena should be given. '^ 

A new creed! Something to weigh and consider and 
contrast with the old creed implicit in science for centuries. 

* Dirac, P. A. M., Quantum Mechanics, Clarendon Press (Oxford, 1930), p. 7. 


This article, based on lectures of Edward M. Purcell, 
distinguishes between sound proposals and unworkable 
fantasies about space travel. 

19 Space Travel: Problems of Physics and Engineering 

Harvard Project Physics Staff 

Traveling through empty space . After centuries of gazing curiously at 
stars, moon, and planets from the sanctuary of his own planet with its 
blanket of lifegiving atmosphere, man has learned to send instruments 
to some of the nearer celestial objects; and he will no doubt soon try 
to make such a trip himself. 

Starting with Johannes Kepler's Somnium , a flood of fanciful stories 
dealt with journeys to the moon, often in balloons equipped with all 
the luxuries of a modern ocean liner. These stories, of course, ig- 
nored something that had already been known for almost a century, name- 
ly, that the earth's atmosphere must be only a thin shell of gas, held 
in place by gravity, and that beyond it must lie a nearly perfect 
vacuum. In this vacuum of outer space there is no friction to retard 
the motion of a space ship, and this is a great advantage. But the 
forces of gravity from the sun and other bodies will not always take 
a vehicle where we want it to go, and we must be able to produce oc- 
casional bursts of thrust to change its course from time to time. Thus, 
quite aside from how we may launch such a space vehicle, we must equip 
it with an engine that can exert a thrust in empty space. 

The only way to obtain a thrust in a completely empty space is to use 
recoil forces like those actina on a gun when it fires a projectile. 
Indeed, Newton's third law says that to obtain a thrusting force on the 
space vehicle an equal and opposite force must be exerted on something 
else, and in empty space this "something else" can only be a matter that 
comes from the space vehicle itself, a matter that we are willing to 
leave behind us. Only by throwing out a part of its own mass can a 
vehicle achieve recoil forces to change its own velocity — or at least 
the velocity of the part of it that remains intact. 

A rocket is a recoil engine of this type. It carries its ov>?n oxygen 
(or other oxidizer) with which to burn its fuel, and the mass of the 
burned fuel and oxygen is ejected from the rear and left behind. The 
rocket is much like a continuously firing gun that constantly sprays 
out an enormous number of very tinv bullets. The recoil from these 
"bullets" is precisely the thrusting force on the body of the rocket. 

Obviously there is a limit to the length of time that such a process 
can continue, for the mass remaining in the space ship qets smaller all 
the time, except when the engine is turned off entirely. In this chap- 
ter we will examine this limitation and see what it implies about space 
travel. To be definite, we shall usually speak about rocket engines, 
but it will be clear that what we have to say applies to any recoil 
engine whether it is run by chemical power, nuclear power, or any other 
source of power. All such engines, to produce a thrust in empty space, 
must eject some of the mass that has been carried alonq. 

The rocket equation . It turns out, as we shall see, that the only prop- 
erty of a rocket engine that seriously limits its performance is the 
"exhaust velocity" of the burned fuel gases, i.e. the velocity of the 
exhaust material as seen from the rocket. This exhaust velocity, which 
we denote by Vgj^, is determined by the energy released inside the com- 
bustion chamber and hence by the fuel (and oxidizer) used by the rocket. 
The same "kick" backward is given to the exhaust-gas molecule whether 
or not the rocket is already moving. Therefore, to a man standing on 
the rocket using a specific combustion process, the gases rushing out 
the exhaust will always appear to have the same velocity relative to 
the rocket, whatever the motion of the rocket itself with respect to 
another body. 


Imagine you are watching a rocket coasting along at constant velocity, 
far away from any other massive bodies. Suppose that the engine is ig- 
nited briefly and ejects a small mass Am of burned gases. The situation 
is sketched in Fig. 1, where we have denoted the initial mass and veloc- 
ity of the vehicle by m and v respectively. The velocity v may be mea- 
sured with respect to any (unaccelerated) coordinate system, for example, 
another space ship coasting alongside the first, or the sun-centered 
coordinate system that we commonly use to analyze the motions of the 
planets. (The actual value of v will cancel out of our final results. 
Why is this expected?) After the burst of power, the rocket will move 
away from us at velocity v + Av, having a mass m - Am; and the "cloud" 
of exhaust gases, of mass Am, will be moving away from us at a velocity 
equal to the exhaust velocity diminished by the forward velocity of the 
rocket, Vg^ ~ '^• 

Since no external forces are acting on the system, we know that 
momentum must be conserved. In Fig. 1(a) , before the burst of power, 
the momentum is mv; right afterwards, in Fig. 1(b) , it is (m - Am) (v + Av) 
- (Am) (Vgj^ - v) . These momenta must be the same: 

(m - Am) (v + av) - (Am) ("^ v ~ '^^ = mv . 

Multiplying out the terms on the left-hand side, we find that all terms 
containing v cancel out (as they must) , and the result can be written in 

the form, 

{Am)v + (Am) (Av) = m(Av) . 

If we consider a sufficiently small burst of thrust, we can make Av as 
small as we wish compared to Vex » ^^^ the second term on the left-hand 
side of this equation can be made completely negligible compared to the 
first term. Then we can write (for very small bursts of thrust) : 

Am _ AV -, , 

m ~ V ' 

Notice that this relation does not depend in any way on the length of 
time during which the change av occurs. The fuel Am may be burned very 
rapidly or very slowly. As long as the exhaust gases emerge with veloc- 
ity Vex relative to the rocket, the resulting momentum changes will be 
the same, and will lead to the same relation Eq . (1) , whenever the changes 
are sufficiently small. Notice also that this result depends only on 
the conservation of momentum; we have used no other law in deriving it. 

Now, a moderately large burst of power can be divided conceptually 
into a great many consecutive small bursts, and Eq. (1) shows that each 
small increase in velocity requires ejecting a given fraction of the re- 
maining mass of the rocket. The rules of this "inverted compound-interest 
payment" are examined in the appendix to this chapter. There we find 
(Eq. A6) that any velocity change v^ / large or small, requires reducing 
the mass of the rocket as follows: 

-(V /v ) 
c ex 
m = m e 

- (v /v ) 
m/m = e . ( 2 ) 

Here mg is the mass before the change, and m is the mass after the 
change. The quantity e is a certain number whose value is 


Space Travel: Problems of Physics and Engineering 




Fig. 1. Analysis of the performance of a 
rocket. Note that the "backwards" velocity of the 
spent fuel, namely v - v, might actually be 
negative as seen by an external observer. This 
would happen if v is larger than v , in which case 
the exhaust "cloud" is seen to move off to the right, 
too, although at a speed less than that of the rocket. 



= 10 



One use of Eq. (2) is in computing the final velocity vf of a rocket 
that has initial mass mo, initial speed v©/ final mass mf, and exhaust 
velocity Vgx • The result is 

"^f -(^f/^ex^ 

fir = ^ 


as shown graphically in Fig. 2. 

Fig. 2 


^/ INT/ML MtlSi 


Eq . (2) is the rocket equation . Unless a table of powers of e happens 
to be handy, the most convenient way to write this equation is the fol- 

(m) 10 

(0.4343 V /v ) 
c ex 


log^o ^"^o^"^^ 

.4343 (V /v ) 
c' ex 


This relation is based only on the conservation of momentum and on the 
concept of a constant exhaust velocity Vex (constant with respect to 
the body of the rocket) for the spent part of the fuel. (But the rela- 
tion is idealized in the sense that we have not taken into account any 
accelerations due to gravity.) 

As an example, suppose that we wish to give a rocket a final velocity 
equal to twice the exhaust velocity of its engines, starting with the 
rocket at rest. Then Vc = 2vex > ^'^'^ ^^ have: 

= (m)10 


= 7.39 (m) 

That is, the original takeoff mass m must be over 7 times the final 
mass. In other words, about 87 percent of the initial mass must be 
expelled to achieve a velocity of 2vgy. The useful payload must be 
somewhat less than the remaining 13 percent of the takeoff mass, because 
the rocket casing, its fuel tanks, and the like will constitute much of 
this remaining mass. 


Space Travel: Problems of Physics and Engineering 

Practical rockets . The rocket equation shows that the most important 
feature of a rocket is Vgx » the velocity with which the spent fuel gases 
are expelled. When chemical fuels are used, there is a limit to how 
large this exhaust velocity can be. We can see this by applying the law 
of energy conservation to the interior of the rocket. 

Consider what happens when a given mass m of fuel and oxidizer are 
combined, with the fuel burning in the oxidizer. Let the total energy 
produced by this chemical reaction be E. Obviously, the ratio E/m, 
which is the energy per unit mass of fuel and oxidizer, will be a con- 
stant that depends only on the chemical nature of the fuel and the 
oxidizer. After the materials have reacted, the total mass m is ejected 
from the rocket with velocity Vgx / ^i^d the kinetic energy of the ejected 
mass is just kmiv^y^)'^. Since this energy comes from burning the fuel, 
it can be no greater than the chemical energy liberated, namely E: 

'^m(v )^ < E . 
ex - 

Dividing by km and taking square roots, we find: 

^ex ^2 (E/m) . (6) 

These relations are not simple equalities because much of the released 
energy will be wasted, primarily as internal (random motion) heat energy 

in the still-hot exhaust gases. 

Chemists have measured the "heats of reaction" (which determines E/m) 
for almost all chemical reactions. For example, for typical hydrocarbons 
such as fuel oil, gasoline, kerosene, and the like, they have found that 
about 1.1 X 10'* kcal are given off for each kilogram of fuel burned. 
When we add the mass of oxygen required (about 3.4 kg per kg of fuel) 
and convert to mechanical units, we find that E/m for all of these fuels 
is very nearly 10^ j/kg. Therefore, according to Eq. (6), 

v < /20^ X 10^ m/sec = 4.5 km/sec 

for hydrocarbon fuels burned in oxygen. This, of course, is the largest 
value that could possibly be obtained, even if the exhaust gases emerged 
ice-cold. In actual practice, many current rockets using kerosene and 
liquid oxygen (called LOX) obtain roughly: 

V =2.5 km/sec. (7) 


Even liquid hydrogen and liquid fluorine will yield exhaust velocities 
only about 20 percent greater than this in practice.* Consequently 
whenever the speed of the rocket has to be substantially more than this 
value of Vex — and we shall see in the next section that this is indeed 
so even for orbital flights — the useful payload is in practice only a 
small fraction of the original mass, by Eq . (2). 

In view of this limitation on the fundamental quantity Vgj^ for chem- 
ical rockets, a number of proposals and experimental models have been 
made for nonchemical rockets where Vgx "^ay not have these limitations. 
To date, none of these has offered any real advantage, although they 
may do so in the future. The difficulty is that today the auxiliary 
apparatus for ion-beam engines, nuclear reactors, and the like, always 
contains too much mass relative to the mass allowance needed for any 
significant payload. Eventually, of course, we might be able to do much 
better with nonchemical engines. 

* Specific impulse is a term often used by rocket engineers who use the 
symbol I for it. It is essentially impulse per unit weight of fuel and 
equals the exhaust velocity divided by the acceleration of gravity at 
the earth's surface: I = v^^/q . Typical practical values are therefore 
about 250 sec. 



Artificial satellites . Now let us see what velocities we need to perform 
the simplest task of space engineering, namely placing an artificial sat- 
ellite in orbit above the surface of the earth. Since the radius of the 
earth is about 4000 miles, the force of gravity on a satellite moving per- 
haps a few hundred miles above the earth's surface will be not very dif- 
ferent from that on the surface. Thus, the satellite will experience an 
acceleration of approximately g toward the center of the earth. As we 
saw in Chapter 5 if it is travelling in a circular orbit with speed v, 
its centripetal acceleration must be v^/r where R is the radius of the 
orbit. For these two facts to be consistent, 

V-/R = g or V = /Rg . 

Since the satellite is assumed to be fairly close to the earth, the 
radius of its orbit R will be about the same as the radius of the earth, 
or about 6400 km. Substituting this value, along with g = 9.8 m/sec^ 
= 0.0098 km/sec^ , into our formula, we obtain 

V = 8 km/sec (close orbit) . (8) 

This is the approximate speed an object must have if it is to remain 
in orbit. Eq. (7) displays the rocket-exhaust velocities achieved when 
chemical engines are used. Are these velocities adequate? From Eqs. (7) 
and (8) , we have 

V /v = 8/2.5 = 3.2 . 
c ex 

Substituting this value into the rocket equation (4) or (5), we find: 

m^ = (m) lO-"--^^ = 24.5(m) . 

That is, the takeoff mass mo must be almost 25 times the mass m of the 
satellite and all other non-fuel structures; thus only about 4 percent 
of the initial mass can actually go into orbit (even ignoring the problem 
of lifting it to orbit altitude, which we shall examine shortly). 

But the situation is even worse than these numbers may seem to imply 
at first. The ''other nonfuel structures" — the rocket's casing, frame- 
work, fuel tanks, fuel pumps, and the like — have much more mass than the 
payload, the satellite. In fact even with the best of modern structural 
materials and techniques, there is so far no rocket mechanism with a mass 
less than about 1/10 of the mass of the fuel it can carry (rather than 
1/25). According to our foregoing result, a rocket of this sort could 
not be put into orbit at all. 

The way out of these difficulties is to use the technique of staging , 
which essentially amounts to putting a small rocket onto a larger rocket 
(and this combination onto a third, still larger rocket, and so on as 
necessary) . The fundamental rocket equation is not circumvented by this 
strategem; it remains valid. Eut heavy casings and fuel tanks can be 
thrown away as soon as their fuel is used up, and the remaining fuel in 
the remaining rocket then need only accelerate the remaining mass, which 
can be much smaller. In this way, the remaining fuel is used more 
efficiently toward the end of the process, and the ideal limit expressed 
by the rocket equation can be more nearly approached. It cannot be ex- 
ceeded, for that would violate the conservation of momentum, upon which 
the rocket equation is based. 

There is one further matter that we should look into. We have ne- 
glected to compute the work we must do to lift the payload up into its 
orbit against the downward force of gravity. (Anyone who has watched 


Space Travel: Problems of Physics and Engineering 

pictures of a big rocket taking off has seen how, at the start, thrust 
must be increased until the rocket's own weight on the launching pad is 
balanced and the net acceleration upward can begin.) This work, how- 
ever, is not terribly large, relatively speaking, for a close-in orbit, 
as we can easily show. In obtaining Eq . (8), we derived the relation 
v^ = Rg for the orbital velocity. If we multiply this equation by km, 
we find that the orbital kinetic energy is 'jmv^ = '^mgR, The potential 
energy change in lifting the mass m to height h above the surface of 
the earth is mgh. Since h is only a few hundred miles while R is 4000 
miles or more, the work (mgh) required to raise the satellite will be 
only about 1/10 to 1/5 of the work {'imgR) required to give it orbiting 
speed in a close-in orbit. (Naturally, this is not true for a very 
large orbit with a height of, say, 4000 miles or more above the earth's 
surface . ) 

Interplanetary travel . To send instruments to other planets, we must 
first free them from the gravitational attraction of the earth. This 
requires that the payload be given a velocity sufficient to prevent it 
from, returning close to the earth of its own accord. The smallest such 
velocity is called the escape velocity . A vehicle with this velocity 
will just barely escape, and its final velocity will be nearly zero 
relative to the earth. 

As might be expected, the escape velocity is not enormously greater 
than orbital velocity, and in the appendix to this chapter, we show that 
it is about: 

V (for escape) = 11.2 km/sec, (9) 

as compared to v (for close orbit) = 8 km/sec. (8) 

Even this moderately greater (than orbital) velocity for escape requires 
a rather large increase in the ratio of takeoff mass to payload mass. 
With Vgx equal to 2.5 km/sec as in Eq. (7), we have Vc/vex = 11.2/2.5 
= 4.48, and the rocket equation (Eq. 4) yields: 

m = (m) lO""-^^ = 89 (m) • 

So, despite the seemingly modest change in velocity (11.2 km/sec 
instead of 8 km/sec) , freeing a payload from the earth with chemically 
fueled rockets (even in stages) requires about 3^ times as much fuel 
as required for placing the same payload into a close-in orbit. 

Once essentially free of the earth, a body will still be under the 
direct influence of the sun's gravitational forces. Here it is neces- 
sary to recall that the earth already has a rather large orbital veloc- 
ity around the sun, and that any body launched from the earth will con- 
tinue to have that orbital velocity if it has been merely freed from 
the earth with no additional accelerations. This velocity is about 
30 km/sec and clearly represents a very substantial bonus for interplane- 
tary travel. Even so, the Mariner 4 probe to Mars, for example, actually 
required a takeoff mass 400 times as large as the mass of the probe itself, 
The rocket was an Atlas-Agena with an initial mass of about 200,000 lbs. 
and a payload of 500 lbs. It was designed to cover the 3 x 10^ mile trip 
in the solar system in about 7 months (this works out at about 16 
miles/sec) . 


Rocket o 


Fig. 3 

Travel to a star ? When we think of sending a payload to examine a star, 
we find once more that the necessary velocity is the crucial factor, but 
the origin of the needed velocity is different. The velocity required 
to escape from the solar system is about 4 5 km/sec, but even the nearest 
stars are enormously far away, and the payload must travel much faster 
than this if it is to complete its journey within a century. 

The distances to the nearest stars have been measured by observing 
the shift in their apparent positions in, say, summer and winter as 
the earth moves from one side of its orbit to the other. Even with this 
very large baseline (186 million miles), the apparent shift in direction — 
the parallax — is extremely small, and the corresponding distances are 
found to be several million million miles, i.e., several trillion miles. 
Such large distances are more conveniently expressed in light years, a 
light year being the distance that light will travel in one year. A 
simple multiplication shows that one light year is about 10^^ km. 

The two nearest stars are in the constellation Centaurus. The nearest 
one, Proxima Centauri, is 4.2 light years away but is very dim and emits 
only about 10"** times as much light as our sun. The next-to-nearest star. 
Alpha Centauri, is 4.3 light years away and is actually a double star, 
consisting of two stars similar to our sun and separated by about the 
distance between the sun and Jupiter. The brighter of the two emits 
energy at about the same rate as our sun, and the other at about 1/5 
that rate. 

While none of these particular stars seems likely to have habitable 
planets comparable to our own, it might be very interesting to send in- 
struments in close to one of them and take pictures of it. To see just 
what problems such a project might entail, let us examine this simplest 
of all interstellar journeys a little more closely. 

The first question to be answered is how long we would be willing to 
wait for the results of the journey. Although an unmanned instrument 
package need not return to the earth within a man's lifespan, it never- 
theless seems that we would be unlikely to plan today for a very expensive 
project whose results would be known later than, say, a century from now. 


Space Travel: Problems of Physics and Engineering 

If the payload is to travel 4.2 light years during 100 years, its 
speed must be 0.042 times the speed of light (3 x lo^ km/sec). This 
speed Vc is 12.6 x 10 ^ km/sec. Let us optimistically assume that we 
can soon design rockets with exhaust velocities twice as high as the 
ones we now have, even though it is difficult to see now how this could 
be done with chemical fuels. Thus, we assume Vgx = 5 km/sec. Then we 
have Vc/Vex = 2.52 x 10 3, which we substitute into the rocket equation. 
(Both speeds are small enough so that we can use this nonrelativistic 
equation; actually the relativistic one gives slightly more pessimistic 
results . ) 

When we make this substitution in Eq. (4) , we find a result that can 
only be described as ridiculous: 

m^ = (m) 10l°5'' . 

To see just how impossibly large this mass ratio is, we might note that 
the total number of atoms in the entire solar system has been estimated 
to be less than 10^^. There is not enough chemical fuel in the entire 
solar system to send even one atom on such a journey! In fact, we are 
short of having enough fuel for even that trivial task by a factor of 
over 101°°°! 

These numbers are so large that the mind can not really form an ade- 
quate picture of their hugeness. To reduce them, let us throw caution 
to the winds and allow a much longer time for the journey, for example, 
5000 years or 50 centuries — a terribly long wait. Retracing the arith- 
metic we find that we then obtain 

219 21 

m^ = (m) lO''-''^ = 8 X 10''-^ (m) . 

Even this more familiar sort of number is still absurdly large. The 
mass of the entire earth is only 6 x lo^^ tons, less than enough (even 
if it were all good fuel — and to be so used!) to send a one-ton payload 
on a journey of 5000 years to the nearest star. 

There is only one sensible conclusion: interstellar travel is impos - 
sible if chemical fuels are used for propulsion. 

Future star travel? Perhaps one of the conceivable nonchemical rockets 
might someday offer an escape from this pessimistic conclusion. To look 
at this possibility, let us return to our simplest of interstellar 
journeys, a trip to the nearest star in 100 years. As we saw, we need 
a velocity v^ of 12.6 x lo^ km/sec for such a journey. (With this veloc- 
ity, the payload arrives at Alpha Centauri after 100 years; it must con- 
tain either a very powerful radio transmitter, or enough fuel to return 
in another 100 years or so.) 

The various "plasma" engines and "magnetohydrodynamic" engines that 
have been proposed are essentially electric "guns" that shoot out ionized 
gases. It is difficult to set limiting numbers on the best possible per- 
formance from such engines, partly because the exhaust gases are usually 
accelerated by some separate source of power. Certainly, they can be no 
better than nuclear engines, which we shall examine later. It is probably 
fair to say that exhaust velocities much larger than 1/300 the velocity 
of light could not be expected when very large masses of ionized gas must 
be expelled. 


If we adopt this estimate, then a value of Vgx of 1000 km/sec is about 

the best that could ever be expected from such non-nuclear engines. With 

this value we obtain the ratio Vq/Vqx - 12.6, and by inserting this into 
the rocket equation, Eq . (4), we get the result, 

m = (m) 10^''^^ = 3 X 10^ (m) . 

Thus, a 3-ton payload would require at least a million tons of "fuel" 
(material to be expelled as ionized gas) . If the payload is to contain 
a sufficiently powerful radio transmitter, it is likely to weigh at least 
3 tons. To form some picture of wl ^.t a million tons of material might 
look like, we may note that a million tons of water would cover a football 
field to a depth of 200 yards. 

Abandoning the radio transmitter and waiting another 100 years for the 
payload to return would be no way to avoid this large mass of "fuel," be- 
cause the effective payload on the outward journey would then have to in- 
clude all the "fuel" for reversing the velocity for the return trip. This 
essentially squares the mass ratio, making mo/m equal to 10^', which is 
far worse: even only one pound of true payload then requires 50 million 
tons of takeoff mass. 

These results are not quite so ridiculous as the ones we obtained when 
we tried to use chemical fuels, but they clearly show that ion-beam en- 
gines will not be very practical for interstellar travel unless they can 
consistently give an exhaust velocity significantly greater than 1/300 
the velocity of light. 

Nuclear fission yields about 8.2 x 10^ ^ joules per kilogram of fission- 
able material. According to Eq. (6) , this will result in a maximum ex- 
haust velocity of the products of fission of 12.8 x lo^ m/sec , or 12.8 ^ 10- 
km/sec, about 1/23 the velocity of light.* 

These exhaust velocities at last begin to approach what we need for 
the simplest of interstellar journeys. For the 100-year, one-way trip to 
Alpha Centauri , the necessary v^ is just about exactly equal to the Vgj^ 
that we might hope to obtain for nuclear fission products, and the rocket 
equation then gives mo/m = 2.7 to 3. This, in itself, is so clearly 
practical that we might begin to consider making the elapsed time some- 
what shorter or journeying further to a few of the slightly more distant 
stars. Note, however, that a 20-year, one-way trip to Alpha Centauri 
would still require mo/m = 200 approximately. 

But present day engineering is a long way from being able to put a 
small nuclear reactor on a rocket to provide these exhaust velocities 
for fission products. Today's nuclear reactors involve so much additional 
mass besides their fuel that they would be even less useful than engines 
working with chemical fuels — and the latter are hopeless for interstellar 
journeys, as we have seen. It was only by ignoring these auxiliary dif- 
ficulties that we have made nuclear power appear to be the answer for 
interstellar travel. What is^ likely, however, is the development of 
nuclear reactors that do not emit the relatively heavy fission products, 
but that provide heat to a supply of hydrogen that is pumped over the 
reactor, heated by it, and ejecte d a t correspondingly higher speed (see 
Eq. (6) ; v^^ is proportional to /ITin . 

* The best possible nuclear fusion reaction, converting 4 hydrogen 

nuclei into a helium nucleus , gTves about 1/8 the velocity of light. 
But non-explosive "slow" fusion reactors are far from being available 
on the earth, not to speak of the availability of a portable model 
for use in rockets! 


Space Travel: Problems of Physics and Engineering 

If we are ever going to send instruments, let alone men, to even the 
nearest stars, we must first develop an almost ideal nuclear rocket (or 
an ion-beam rocket virtually equivalent to it) . Even then, the simplest 
such trip will require many decades. 

The perfect rocket . If we agree to ignore questions of engineering know- 
how, is there any absolute limit to how effective any rocket could possibly 
be? There is indeed such a limit and it is imposed by the facts of physics; 
physical energy cannot leave the rocket at an exhaust velocity greater than 
c, the velocity of light. And when any energy (say of amount E) is lost 
by the rocket, it also loses a (rest) mass of m = E/c^ , This is true 
whether the energy E is carried off in the exhaust of some gas or in the 
form of a beam of light that escapes from the back of the rocket. This 
last possibility is suggested by certain reactions between elementary 
particles, reactions known as annihilations . VJhen an electron (e~) and 
a positron (e"*") react sufficiently strongly, both particles disappear and 
in their place appear two gamma rays; the latter are photons, like light 
or x-ray photons, that travel at the speed of light and together carry 
all of the energy represented by the masses of the vanished electron and 
positron. The reaction suggests that one may call the electron a particle 
of matter and the positron a particle of anti-matter . 

This annihilation of positrons with electrons was the first reaction 
of this kind that was observed; but in the late 19 50 's, anti-protons and 
anti-neutrons were also discovered, and each was observed to annihilate 
with its ordinary counterpart, the usual proton or neutron respectively, 
producing two energetic gamma rays in each case. Thus, it became clear 
that a whole system of anti-matter — anti-hydrogen, anti-helium, and so 
on — could be constructed from the elementary anti-particles. We do not 
yet know how to do this to any significant extent, but we know of no 
physical law that would f orbit it. 

Since we have already agreed to ignore practical manufacturing problems 
in this discussion, let us assume that large amounts of anti-matter might 
be made available. What could we do with such a material if we had it? 
It would not be an inexpensive supply, because to manufacture it would 
require at least as much energy as it would later give back. But it 
would represent a very efficient way of storing energy. Indeed, anti- 
matter, plus ordinary matter to "burn" it with, would have the smallest 
ratio of stored energy to total mass that is physically possible, namely 
E/m = c^ . Moreover, because the released (photon) energy will depart at 
the speed of light, such a "fuel" would constitute the best possible 
rocket fuel (provided we could find a way of making the photons travel 
backwards from the rocket) . 

Naturally, we must use relativistic mechanics to derive the equations 
for such an exotic rocket. We shall not do so here, but will merely 
quote the result: if the exhaust velocity equals the velocity of light, 

m /c + V 

_o = / £. (10) 

m V c - V 

where all the symbols have the same meanings as before. This is the 
mass equation for a perfect rocket. 

[Note, by the way, that a man on the rocket sees the exhaust energy 
leaving the rocket at the velocity of light; at the same time a man on the 
earth, say, will see the rocket traveling at the velocity of light rela- 
tive to the earth. This is one of those paradoxes (seeming contradictions) 
of relativity that cannot be reconciled with our ordinary experience.! 


Would such a "perfect" rocket make it easier for us to travel to the 
stars? One answer is: "A little, perhaps, but not much." Even this 
small degree of optimism is justifiable only if we may ignore a number 
of serious practical difficulties in addition to that of creating the 
necessary anti-matter for fuel. 

Let us analyze a "typical" journey, preferably a rather simple one. 
As stated before, the nearest stars are about 4 light years away, but an 
ideal nuclear rocket would suffice for such a trip, so let us consider a 
slightly longer journey. Within a distance of 12 to 13 light years from 
the earth there are about 20 stars. (Of these, only Alpha Centauri is 
closely similar to our sun; two others emit about 1/3 as much energy 
as does the sun and one other emits about 5 times as much. The remain- 
ing ones are either very much brighter or very much dimmer than the sun.) 

Accordingly, let us consider a round trip from the earth to a star 
12 light years away and back. Since we would have to wait 24 years for 
light rays to make the round trip, the top speed of the rocket must be 
close to the speed of light if the rocket is to return to the base on 
earth during our lifetime. But we would not want the rocket to fly past 
its distant goal at nearly the speed of light, and it will take about 
as long to slow the rocket down as it did to speed it up in the first 
place. Thus the velocity of the rocket would have to vary approximately 
as shown in Fig. 4. 

To avoid imposing unduly large forces on the men inside the rocket, 
we must keep the accelerations and decelerations small; at an average 
acceleration of 1 g, one can calculate that about a year will be re- 
quired to reach full speed, and another year to stop. To keep the 
total time for the journey reasonably small, we shall choose a top 
speed of 0.8c, that is, only 20% less than the speed of light. 

Journeys of this type involve, therefore, four separate steps: accel- 
eration, deceleration, reacceleration, and a final deceleration. The 
mass equation applies to each one, but we must remember that, during each 
step, we must accelerate (or decelerate) all of the fuel mass that will 
be needed for all the succeeding steps. For one step of the journey in 
Fig. 4, the mass equation Eq. (1) yields 

'c + 0.8c /I + 0.1 

= 3 . 

0.8c / 1-0.8 

But if m represents the true pay load, this result applies only to the 
final deceleration. For example, the mass at the beginning of this final 
step must be mo = 3, and this must be the "payload" for the next-to-last 
step, the acceleration for the return trip. Thus, the return trip must 
begin with a total mass of 3mo = (3^m) . It is easy to show in the same 
way that the two steps of the outward leg of the journey will introduce 
two more factors of 3. Thus, if mQQ denotes the take-off mass when the 
rocket leaves the earth (and m denotes the true payload, as before) , 
we find: 

-°° = 3^ = 81 . 


That is, each ton of payload requires 81 tons of combined take-off mass. 
A 10-ton payload would require almost a thousand tons of fuel for the 
journey we have considered — and half of this fuel must be anti-matter. 
Obviously, we would have to learn how to manufacture anti-matter in very 
large amounts indeed. 


Space Travel: Problems of Physics and Engineering 


KT-veKYeND, v'O 

v-«0.8e -- 

o firr yejgy staXT 






(, U6HT y£Mi DfS-r^^JCC 

Fig. 4 A modest interstellar journey 


With these assumptions about the trip, it is possible to show that 
the journey we have discussed would take 32 years as measured on the 
earth. But because of relativistic time-dilation for the inhabitants 
of the moving systems, it turns out that the crew of the rocket would 
age by only 20 years. That is, as measured by the crew, the journey 
would require only 20 years. 

The perfect rocket has further difficulties that we have not yet 
mentioned. First, the energy flux of gamma rays from such a rocket, 
with a 10-ton payload, can be shown to be 2.4 x 10 15 watts, a power 
that is equivalent to a 1-kilo bomb once every 1.7 seconds! And all 
of this energy flux is in the form of very penetrating, deadly gamma 
rays. The payload would have to be shielded very well indeed from 
even the slightest leakage of all this energy — to say nothing of the 
difficulties of shielding the earth and its inhabitants as the rocket 
takes off. Figure 5 indicates how the rocket might look in principle. 

Secondly, a glance at Fig. 5 reveals another very serious difficulty. 
Anti-matter would act as a "universal solvent," reacting readily with 
any ordinary matter that it contacts. Then, in what can we store it? 
Within our present knowledge, this problem has no solution. 

Thus, we have found that a perfect rocket probably cannot be built, 
and that, even if it could be built, it would not extend the range of 
possible space travel very much beyond the meager capabilities of an 
ideal nuclear rocket. Even the nuclear rocket is presently a long way 
from being practical. For the time being, of course, there are many 
exciting possibilities for exploring our own solar system with the 
chemically fueled rockets we already know how to build. The dreams of 
space travel are coming true, but only on a "local" basis. 

Communicating through space . This final section is closely based on, 
and copiously cites from, E. M, Purcell's article "Radioastronomy and 
Communication through Space." Brookhaven National Lecture series 
#BNL 658 -{T-214); we wish to thank Dr. Purcell and the BNL for per- 
mission to use this material. 

Now we shall discuss a very different aspect of space engineering, 
namely, sending signals, rather than physical hardware, across the huge 
distances of space. The signals that we know how to send most efficiently 
are coded radio waves, but our discussion will also apply to the light 
beam from a laser or to any other type of electromagnetic radiation 
if the necessary engineering "know-how" can be developed. Radio signals 
suitable for communicating over a distance of a few hundred miles require 
relatively little energy, but a large amount of energy is needed in com- 
municating across the vast reaches of space. 

The simplest possible radio signal is just the presence or absence of 
a radio wave — or equally well, the presence or absence of a small shift 
in its frequency (so-called "frequency-shift keying") . Correspondingly, 
the simplest possible sign that can be written on a piece of paper is the 
presence or absence of a black dot in some agreed-upon location. News- 
paper photographs are arrays of such dots. Television pictures are built 
up in much the same way. 

The simplest possible signal, then, expresses a two-fold ("binary") 
choice, a simple "yes or no," a "something or nothing" signal. More 
complicated codes can always be broken down into such signals. For 
example, a Morse code dot might be called a "yes" and the space between 


Space Travel: Problems of Physics and Engineering 

( y P^yiMO (10 TONS) 

-LON& BOOM ■^ 

navy shiuo 


TO PMTecr PunoAo 

Unu 717 5£PW»Tt flATT£M W/> 

Af^i-nttTr£i( in/ioeoF $om 

so Fm NONaiSTe.NT 


Fig. 5 A perfect rocket? 


two dots a "no"; then the dash becomes two successive "yesses," and the 
longer space between two letters is represented by two successive "noes," 
and so on. 

This way of analyzing signals was first suggested by the American 
radio engineer R. V. L. Hartley in 1928, and it was further developed 
by C. E. Shannon at Bell Telephone Laboratories in 1948. Shannon called 
the simplest yes-no signal a bit (for "binary digit") , and he first de- 
veloped much of the analysis that we shall be using in this section. 
This analysis is a part of "information theory." 

For space communication, the important fact is that each bit (each 
yes-no signal) requires a very small amount of energy. Just as space 
is filled with very faint light rays from the stars, it is filled also 
with a background of weak radio waves of all types. If we are to de- 
tect a signal from outer space against this "noise," we must receive 
enough energy to be sure that the supposed signal is not just one of 
the random mutterings of space itself. Near our solar system, a re- 
ceived signal energy of at least 10"^^ joule per bit is required. This 
requirement is essentially independent of the radio frequency or the 
manner in which the signal is coded in the radio wave, and presumably 
it remains about the same in many parts of empty space. 

As an example, let us consider the task of the Mariner IV space probe, 
namely to send good television pictures of Mars back to the earth. Since 
such a picture contains an array of about 1000-by-lOOO dots, one picture 
can be transmitted by a signal consisting of about 10^ bits. The signal 
can be detected if, on reaching the earth, it delivers (to our receiving 
antenna) 10^ x 10"^^ joule = 10"^^ joule for each picture that is to be 

But what the transmitter emits must be much more energy than what we 
intercept and receive at a distance. A simple radio antenna sends the 
energy outward more or less equally in all directions. A properly de- 
signed complex antenna can concentrate most of the energy into a narrow 
beam, but such an antenna must be large (compared to the wavelength of 
the radio waves) , and it must be very accurately shaped. Not only is 
this difficult to do, but once it is done, the antenna must be pointed 
toward the receiver, accurately enough to be sure that the receiver 
lies inside the radio beam, and this pointing operation in turn re- 
quires additional machinery and sensors that must be equally accurate. 
Thus, a space probe such as Mariner must contain either a rather large 
radio transmitter or else a smaller transmitter and a lot of complex, 
rather heavy machinery. 

The best compromise amongst all the possibilities will depend on 
the purpose of the space probe and on the status of various engineering 
arts at the time the probe is designed. But we can obtain a rough idea 
of the weight of the necessary equipment by analyzing the situation when 
a simple antenna is used. 

Fig. 6 summarizes the situation. Notice that the receiving antenna 
on the earth can be quite large, and we shall assume that it has a diam- 
eter of 100 m (about 100 yards) . Only the radio energy that happens to 
strike the receiving antenna will be useful. Thus, the fraction of the 
energy that is useful will be given by the ratio of the area of the re- 
ceiving antenna to the area of a sphere whose radius is equal to the 
distance from Mars to the earth, about 10^ km = 10^ ^ m (see Fig. 6). 
The ratio of these areas is 

^(50)^ _ ^ „ ,„-20 

11 2 


= 6 X 10 


Space Travel: Problems of Physics and Engineering 

A.KDie eM«R«iy 

Fig. 6 

Sending television pictures from Mars to the 
(The diagram is not to scale!) 

We have seen that the received energy must be at least 10"^^ joule per 
picture. The energy that must be transmitted for each picture, however, 
must be 

3 = 16 X 10 joules per picture . 

6 X 10"^" 

Although this amounts to only about 0.005 kw-hr, a rather small amount 
of energy by our normal standards, it does represent something of a bur- 
den to a space probe. To compare it with something familiar, we might 
note that the average automobile battery could store only enough energy 
for sending about 100 such pictures. Actually, this is a very optimistic 
estimate because we have computed it by using the minimum possible energy 
per bit of "information," namely 10"^^ joules per bit. If we are going to 
go to all the trouble of sending a probe to Mars, we would want the signal 
that it sends back to be quite strong, not just barely detectable, lest we 
miss it entirely. Thus, it would be more realistic to say that an auto- 
mobile battery can store enough energy to send about 10 television pic- 
tures from Mars to the earth. 

Since such a battery would weigh about 35 lb, and since the ratio of 
take-off mass to payload mass was about 400 for Mariner IV, the energy 
storage for 10 television pictures of Mars would add about 7 tons to the 
take-off mass of such a probe, if a nondirection antenna were used to 
send the pictures back to the earth. Actually, Mariner IV used a rather 
highly directional "dish" antenna, but note that the antenna and its 
pointing equipment must have weighed less than 35 lb if it was to econo- 
mize on take-off weight. 


Although these energies and masses are perhaps surprisingly large when 
we consider that they all arose from the very small number of joules per 

bit (10- 

see p. 16) , they are nevertheless small compared to the masses 

and energies that would be necessary to send physical hardware back from 
Mars. For example, even a small canister of exposed photographic film 
might weigh 1 lb, but we would have to send along with it enough fuel to 
start it on its return journey, namely about 400 lb of fuel. This would 
add 400 x 400 lb or no less than 80 tons to the original take-off mass when 
the probe leaves the earth — and we have completely ignored the extra equip- 
ment that would be needed to ensure both a proper return orbit and a safe 
re-entry through the earth's atmosphere. 

When we consider the very much greater distances to the nearer stars, 
the economy of sending signals rather than hardware becomes even more 
marked. We have seen that nothing short of an ideal nuclear rocket can 
send a physical payload to the nearest star, and that even then the trip 
would require several tens of years. On the other hand, if we consider 
distances as great as 12 light years (containing 20 to 30 stars) , it is 
possible to show that, with 300-ft antennas at the transmitter and re- 
ceiver, a ten-word telegram can be sent with about a kilowatt-hour of 
radiated energy (Fig. 7). This is less than one dollar's worth of 
energy at current prices . 

Of course, the trouble is that there is no body at the other end to 
communicate to. Or is there? In the remainder of this section, we 
shall discuss the question of communicating with other people out there — 
if there are any. 

3Do' OtSH 


Teiw6«iB/*t'> kjC HA«« TO 

/lAbiATr /«\e>ooT '/ uoK-m 

Fig. 7 from E. M. Purcell, "Radioastronomy and Communication 

through Space" [BNL lecture series #BNL 658 (T-214) ] 1960 , p. 9, 


Space Travel: Problems of Physics and Engineering 

Let us look at just our own galaxy. There are some 10^' stars in the 
galaxy. Double stars are by no means uncommon, and in fact, there appear 
to be almost as many double stars as single stars. Astronomers take 
this as a hint that planetary systems around stars may not be very un- 
common either. Moreover, a large number of stars are not rapidly spin- 
ning. One good way for a star to lose most of its spin is by interacting 
with its planets; that is what probably happened in our own solar system. 
So the chances that there are hundreds of millions of planetary systems 
among the hundred billion stars in our galaxy seem good. One can elab- 
orate on this, but we shall not try to estimate the probability that a 
planet occurs at a suitable distance from a star, that it has an atmo- 
sphere in which life is possible, that life developed, and so on. Very 
soon in such speculation, the word "probability" loses any practical 
meaning. On the other hand, one can scarcely escape the impression that 
it would be rather remarkable if only one planet in a billion (to speak 
only of our own galaxy) had become the home of intelligent life. 

Since we can communicate so easily over such vast distances, it ought 
to be easy to establish communication with a society (if we may use that 
word) in a remote spot. It would be even easier for them to initiate 
communication with us if they were technologically ahead of us. Should 
we try to listen for such communications, or should we broadcast a mes- 
sage and hope that someone will hear it? If you think about this a 
little, you will probably agree that we want to listen before we trans- 
mit. The historic time scale of our galaxy is very long, whereas wire- 
less telegraphy on Earth is only 50 years old, and really sensitive re- 
ceivers are much more recent. If we bank on people who are able to 
receive our signals but have not surpassed us technologically, that is, 
people who are not more than 20 years behind us but still not ahead, we 
are exploring a very thin slice of history. On the other hand, if we 
listen instead of transmitting, we might hear messages from people any - 
where who are ahead of us and happen to have the urge to send out signals. 
Also, being technologically more advanced than we are, they can presumably 
transmit much better than we can. So it would not be sensible for us to 
transmit until we have listened for a long time. 

If you want to transmit to someone — and you and he cannot agree on 
what radio frequency to use — the task is nearly hopeless. To search 
the entire radio spectrum for a feeble signal entails a vast waste of 
time. It is like trying to meet someone in New York when you have been 
unable to communicate and agree on a meeting place. Still, you know you 
want to meet him and he wants to meet you. Where do you end up? There 
are only a few likely places: at the clock of Grand Central Station, in 
the lobby of the Metropolitan Museum, and so on. Here, there is only one 
Grand Central Station, namely the 1420-megacycle/sec frequency emitted 
by hydrogen, which is the most prominent radio frequency in the whole 
galaxy (by a factor of at least 1000) . There is no question as to which 
frequency to use if you want the other fellow to hear: you pick out the 
frequency that he knows. Conversely, he will pick out the frequency that 
he knows we know, and that must surely be 1420-megacycle/sec frequency. 

Let us assume rhat his transmitter can radiate a megawatt of power 
within a 1-cycle/sec bandwidth. This is something that we could do our- 
selves if we wished to; it is just a modest stretch of the present state 
of the art. If we receive with a 300-ft dish-antenna and he transmits 
with a similar one, we should be able to recognize his signal even if 
it comes from several hundred light years away. With the new maser re- 
ceivers, which are now being used in radioastronomy , 500 light years 
ought to be easy. But even a sphere only 100 light years in radius 
contains about 400 stars of roughly the same brightness as the sun. And 


the voliome accessible to coiranunication increases as the cube of the range, 
We have previously argued that it is hopelessly difficult to travel even 
a few light years, and we now see that it is in principle quite easy to 
coinmunicate over a few hundreds of light years. The ratio of the volumes 
is about one million. (Fig . 5T 

• cor^rnifim6 £00 sntAJ 
V UHs Tue S"/* 






Fig. 8 (From Purcell, 0£. cit . ) 

There are other interesting questions. When we get a signal, how do 
we know it is real and not just some accident of cosmic static? This 
might be called the problem of the axe head: an archeologist finds a 
lump of stone that looks vaguely like an axe head; how does he know it 
is an axe head and not an oddly shaped lump of stone? Actually, the 
archeologist is usually very sure. An arrowhead can look rather like 
an elliptical pebble, and still there is no doubt that it is an arrow- 
head. Our axe head problem can be solved in many ways. Perhaps the 
neatest suggestion for devising a message having the unmistakable hall- 
mark of intelligent beings is the suggestion made by G. Cocconi and P. 
Morrison. They would have the sender transmit a few prime numbers, i.e., 
1, 3, 5, 7, 11, 13, 17 . . . . There are no magnetic storms that send 
messages like this. 

What can we talk about with our remote friends? We have a lot in 
common. To start with, we have mathematics in common, and physics, 
chemistry, and astronomy. We have the galaxy in which we are near 
neighbors. So we can open our discourse on common ground before we move 
into the more exciting exploration of what is not common experience. Of 
course, the conversation has the peculiar feature of a very long built- 
in delay. The answer comes back decades later. But it gives one's 
children something to look forward to. 


Space Travel: Problems of Physics and Engineering 

Appendix A 

Appendix A . The rocket equation 

In Eq, (1), we showed that, during very small changes of velocity Av, 
the following relation is required by the conservation of momentum: 

^ = ^ . (Al) 

m V 


Now we want to extend this relation to arbitrarily large changes of ve- 

A large change of velocity can be conceptually divided into a great 
many steps with a small change in each. Let us choose these in such a 
manner that all of them involve the same fractional change in the mass 
of the rocket. For example, we may choose 

m n 

where n is a large number that we will leave unspecified for the moment, 
but it is to be the same for each small step. 

Then if m is the original mass of the rocket and m^ is its mass after 
the first small step of velocity change, we will have: 

mi = (1 - — ) m 
^ no 

After the second step, the mass will become: 

m2 = (1 - i) mi = (1 - i)2 m^ . 

After the third step, it will be: 

ma = (1 - i) 3 m^ . 
^ no 

and it is easy to see that after k of our very small changes in velocity, 
the mass of the rocket will be 

'"k = ^1 - y "^o • (^3) 

Now, what will be the change in the rocket's velocity during these 
k steps of acceleration? By substituting Eq. (A2) into Eq. (Al) , we 
find that during each step the velocity change will be: 

Av = — v 

n ex 

Since these are all the same, the total change in velocity during k 
steps will be just k(Av). If we denote this total change in the rocket's 
velocity by v , we have: 

v = — V 
c n ex 


Now solve this relation for k; 

k = n (v /v ) 
c' ex 

And substitute into Eq. (A3) : 

, n (v /v ) 

m, = m ( 1 - — ) 
k o n 

If we write m in place of m, with the understanding that m now repre- 
sents the rocket's mass after its velocity has changed by v , and if 
we use the multiplication rule for exponents, we can write our result in 
the following form: 


T (V /v ) 
, n I c' ex 


We have eliminated k from our relations, by expressing it in terms of 
the velocity change v . Can we eliminate n? In a sense, we cannot, but 
we can replace it by a less arbitrary quantity. 

As we noted earlier, the simple relation Eq. (Al) is valid only for 
very small bursts of thrust. The smaller the burst, the more accurate 
Eq. (Al) becomes. In view of Eq. (A2) , then our relations will all be- 
come more and more accurate as we choose n larger and larger. Obviously, 
the best thing to do is to choose n so very large that the quantity in 
square brackets in Eq. (A4) approaches a steady value and no longer 
changes significantly. Better still, we should take the limit of the 
square brackets as n "approaches infinity." 

Perhaps it is not obvious that this limit exists in the sense that it 
is a well-defined number, but this fact can be shown by methods that we 
cannot pursue in this book. To agree with standard mathematical nota- 
tion, we shall define a number e by the relation: 

— = limit (as n 


1 - i 


The number e has been evaluated to very many decimal places, but in physics 
we seldom need more than a few places: e = 2.718 is usually quite suffi- 
cient. Another way of stating the value is often more convenient: 

e = ioO-'*3'*3 . 

Now, if we let n approach infinity in Eq. 
definition (A5) , we obtain the result: 

(A4) and substitute the 


(V /v ) 
c ex 

-(V /v ) 
c ex 


This final relation can be rewritten in many ways. Eq. (2) of this chap- 
ter is the same as Eq. (A6) ; and Eqs . (4) and (5) are other forms ob- 
tained by solving Eq. (A6) for m and substituting a numerical value for 


Space Travel: Problems of Physics and Engineering 

Appendix B 

Appendix B . Escape velocity 

If a body is projected away from the earth with sufficient velocity, 
it will never return. The smallest such velocity is called the escape 
velocity, and we shall derive it in this section from the law of conser- 
vation of energy. 

The initial kinetic energy of a body of mass m that has been projected 
out from the earth with velocity v is equal to Jjmv^ . If this is just 
equal to the work that must be done against the earth's gravitational 
force on the body as it travels away, then the body will slow down 
greatly when it gets very far away, but it will never entirely stop, as 
it would if its initial kinetic energy were less than the work that must 
be done against the gravitational attraction. 

Thus, our main task is to evaluate the work that is done against the 
earth's gravitational force by a body that moves from the earth's surface 
to a very large distance away. But to simplify the language of our argu- 
ments, we shall evaluate the work done on^ the body b^ the earth's gravi- 
tational field. 

Newton's law of gravitation states that the force on a body of mass m 
due to the earth (mass M) is 

F = G 5LJ1 • (Bl) 

where G is Newton's gravitational constant and R is the distance from the 
body to the center of the earth. When the body moves a small distance 
AR further away from the earth, the work done on it by the gravitational 
force will be 

AW =-F(AR) = (GmM) ^^2 (B2) 

where the minus sign arises because the force opposes the increase in R. 

Now we must add up all the AW's for all the AR's as the body moves 
from the earth's surface to a very great distance. In Eq. (B2) , the 
quantity (GmM) is a simple constant, but l/R^ changes continually as the 
body moves away, and we must find some way to express the ratio -(AR)/r2 
as a change in some other quantity. One way to find this desired quan- 
tity is to guess at it and then try to prove that the guess is correct. 
From the fact that -(AR)/r2 has the units of a reciprocal length, we 
might guess that it could equal A(l/R). The change in 1/R, as R itself 
changes by AR, will be: 

, (h = J^ 1 = -^R 

^r' R+AR R R(R+AR) 

This is almost the result we were seeking, and now we note that we are 
free to make the individual steps AR as small as we like. Thus, we can 
make -(AR)/r2 equal to A (1/R) to any accuracy that we may wish to choose. 
In the limit as the steps are made smaller and smaller, the relation be- 
comes exact, although we cannot go into the proof of this here. 


Accordingly, we can rewrite Eq. (B2) as follows: 

AW = (GmM) A(i) 

This equation states that the steps AW in the total work done are just 
equal to the constant (GmM) times the corresponding changes in the quan- 
tity 1/R. The sum of all the AW's, therefore, will be equal to the total 
change in the quantity GmM/R. If the body moves far enough from the 
earth, we may take the final value of this quantity as zero (because R 
"approaches infinity") , and the initial value was GmM/R , where R is the 
radius of the earth. The total net change is the final value minus the 
initial one: 

w = - 


W = - i^^ . (B3) 

We can simplify this result and eliminate the factor GM by observing 
that, when R = R , Eq. (Bl) will give the gravitational force on the body 
when it is at tne earth's surface and that this force must be simply mg. 

GM r. f ^ c ^ 

m = F (at surface) = mg. 

R 2 

Thus, GM = gR ^ f and when this is substituted into Eq. (B3) , we obtain: 

W = - m g Rg. (B4) 

The work done b^^ the body against the gravitational attraction of the 
earth will be just the negative of this quantity, and we have already ob- 
served that, if V is equal to the escape velocity, this work must equal 
the initial kinetic energy of the body: 

m g R = H mv^ . 

Multiplying through by 2/m and taking the square root of both sides of 
this equation, we obtain the final formula for the escape velocity: 

V (escape) = \/2 g R . (B5) 

Notice that this is independent of the mass of the body. Inserting the 
numerical values R = 6400 km, g = 0.0098 km/sec^ , we arrive at the vali 
we have been seeking: 

V (escape) = 11.2 km/sec. 


One of the foremost theoretical physicists discusses informolly 
in this talk the process of discovering physical theories. 

20 Looking for a New Law 

Richard P. Feynman 

Excerpt from his book. The Character of Physical Law, published 
in 1965. 

In general we look for a new law by the following process. 
First we guess it. Then we compute the consequences of the 
guess to see what would be implied if this law that we guessed 
is right. Then we compare the result of the computation to 
nature, with experiment or experience, compare it directly 
with observation, to see if it works. If it disagrees with ex- 
periment it is wrong. In that simple statement is the key to 
science. It does not make any difference how beautiful your 
guess is. It does not make any difference how smart you are, 
who made the guess, or what his name is - if it disagrees 
with experiment it is wrong. That is all there is to it. It is 
true that one has to check a little to make sure that it is 
wrong, because whoever did the experiment may have re- 
ported incorrectly, or there may have been some feature in 
the experiment that was not noticed, some dirt or something; 
or the man who computed the consequences, even though it 
may have been the one who made the guesses, could have 
made some mistake in the analysis. These are obvious re- 
marks, so when I say if it disagrees with experiment it is 
wrong, I mean after the experiment has been checked, the 


calculations have been checked, and the thing has been 
rubbed back and forth a few times to make sure that the 
consequences are logical consequences from the guess, and 
that in fact it disagrees with a very carefully checked experi- 

This will give you a somewhat wrong impression of 
science. It suggests that we keep on guessing possibihties 
and comparing them with experiment, and this is to put 
experiment into a rather weak position. In fact experimen- 
ters have a certain individual character. They hke to do 
experiments even if nobody has guessed yet, and they very 
often do their experiments in a region in which people know 
the theorist has not made any guesses. For instance, we may 
know a great many laws, but do not know whether they 
really work at high energy, because it is just a good guess that 
they work at high energy. Experimenters have tried experi- 
ments at higher energy, and in fact every once in a while 
experiment produces trouble; that is, it produces a dis- 
covery that one of the things we thought right is wrong. In 
this way experiment can produce unexpected results, and 
that starts us guessing again. One instance of an unexpec- 
ted result is the mu meson and its neutrino, which was not 
guessed by anybody at all before it was discovered, and even 
today nobody yet has any method of guessing by which 
this would be a natural result. 

You can see, of course, that with this method we can 
attempt to disprove any definite theory. If we have a definite 
theory, a real guess, from which we can conveniently com- 
pute consequences which can be compared with experiment, 
then in principle we can get rid of any theory. There is 
always the possibility of proving any definite theory wrong" 
but notice that we can never prove it right. Suppose that 
you invent a good guess, calculate the consequences, and 
discover every time that the consequences you have calcula- 
ted agree with experiment. The theory is then right? No, it 
is simply not proved wrong. In the future you could com- 
pute a wider range of consequences, there could be a wider 
range of experiments, and you might then discover that the 


Looking for a New Law 

thing is wrong. That is why laws Hke Newton's laws for the 
motion of planets last such a long time. He guessed the law 
of gravitation, calculated all kinds of consequences for the 
system and so on, compared them with experiment - and it 
took several hundred years before the slight error of the 
motion of Mercury was observed. During all that time the 
theory had not been proved wrong, and could be taken 
temporarily to be right. But it could never be proved right, 
because tomorrow's experiment might succeed in proving 
wrong what you thought was right. We never are definitely 
right, we can only be sure we are wrong. However, it is 
rather remarkable how we can have some ideas which will 
last so long. 

One of the ways of stopping science would be only to do 
experiments in the region where you know the law. But 
experimenters search most diligently, and with the greatest 
effort, in exactly those places where it seems most likely that 
we can prove our theories wrong. In other words we are 
trying to prove ourselves wrong as quickly as possible, be- 
cause only in that way can we find progress. For example, 
today among ordinary low energy phenomena we do not 
know where to look for trouble, we think everything is all 
right, and so there is no particular big programme looking 
for trouble in nuclear reactions, or in super-conductivity. 
In these lectures I am concentrating on discovering funda- 
mental laws. The whole range of physics, which is interest- 
ing, includes also an understanding at another level of these 
phenomena like super-conductivity and nuclear reactions, in 
terms of the fundamental laws. But I am talking now about 
discovering trouble, something wrong with the fundamental 
laws, and since among low energy phenomena nobody 
knows where to look, all the experiments today in this field 
of finding out a new law, are of high energy. 

Another thing I must point out is that you cannot prove 
a vague theory wrong. If the guess that you make is poorly 
expressed and rather vague, and the method that you use 
for figuring out the consequences is a little vague - you are 
not sure, and you say, 'I think everything's right because it's 


all due to so and so, and such and such do this and that more 
or less, and I can sort of explain how this works . . .', then 
you see that this theory is good, because it cannot be 
proved wrong! Also if the process of computing the con- 
sequences is indefinite, then with a httle skill any experi- 
mental results can be made to look like the expected 
consequences. You are probably famihar with that in other 
fields. 'A' hates his mother. The reason is, of course, because 
she did not caress him or love him enough when he was a 
child. But if you investigate you find out that as a matter of 
fact she did love him very much, and everything was all 
right. Well then, it was because she was over-indulgent when 
he was a child! By having a vague theory it is possible to 
get either result. The cure for this one is the following. If it 
were possible to state exactly, ahead of time, how much love 
is not enough, and how much love is over-indulgent, then 
there would be a perfectly legitimate theory against which 
you could make tests. It is usually said when this is pointed 
out, 'When you are deahng with psychological matters 
things can't be defined so precisely'. Yes, but then you 
cannot claim to know anything about it. 

You will be horrified to hear that we have examples in 
physics of exactly the same kind. We have these approximate 
symmetries, which work something like this. You have an 
approximate symmetry, so you calculate a set of conse- 
quences supposing it to be perfect. When compared with 
experiment, it does not agree. Of course - the symmetry 
you are supposed to expect is approximate, so if the agree- 
ment is pretty good you say, 'Nice!', while if the agreement 
is very poor you say, 'Well, this particular thing must be 
especially sensitive to the failure of the symmetry'. Now you 
may laugh, but we have to make progress in that way. When 
a subject is first new, and these particles are new to us, this 
jockeying around, this 'feeling' way of guessing at the 
results, is the beginning of any science. The same thing is 
true of the symmetry proposition in physics as is true of 
psychology, so do not laugh too hard. It is necessary in the 
beginning to be very careful. It is easy to fall into the deep 


Looking for a New Law 

end by this kind of vague theory. It is hard to prove it 
wrong, and it takes a certain skill and experience not to walk 
off the plank in the game. 

In this process of guessing, computing consequences, and 
comparing with experiment, we can get stuck at various 
stages. We may get stuck in the guessing stage, when we have 
no ideas. Or we may get stuck in the computing stage. For 
example, Yukawa* guessed an idea for the nuclear forces in 
1934, but nobody could compute the consequences because 
the mathematics was too difficult, and so they could not 
compare his idea with experiment. The theories remained 
for a long time, until we discovered all these extra particles 
which were not contemplated by Yukawa, and therefore it 
is undoubtedly not as simple as the way Yukawa did it. 
Another place where you can get stuck is at the experimen- 
tal end. For example, the quantum theory of gravitation is 
going very slowly, if at all, because all the experiments that 
you can do never involve quantum mechanics and gravita- 
tion at the same time. The gravity force is too weak com- 
pared with the electrical force. 

Because I am a theoretical physicist, and more delighted 
with this end of the problem, I want now to concentrate 
on how you make the guesses. 

As I said before, it is not of any importance where the 
guess comes from ; it is only important that it should agree 
with experiment, and that it should be as definite as pos- 
sible. 'Then', you say, 'that is very simple. You set up a 
machine, a great computing machine, which has a random 
wheel in it that makes a succession of guesses, and each time 
it guesses a hypothesis about how nature should work it 
computes immediately the consequences, and makes a com- 
parison with a Ust of experimental results it has at the other 
end'. In other words, guessing is a dumb man's job. Actually 
it is quite the opposite, and I will try to explain why. 

The first problem is how to start. You say, 'Well I'd 
start off with all the known principles'. But all the principles 

♦Hideki Yukawa, Japanese physicist. Director of Research Institute for 
Fundamental Physics at Kyoto. Nobel Prize 1949. 


that are known are inconsistent with each other, so some- 
thing has to be removed. We get a lot of letters from people 
insisting that we ought to makes holes in our guesses. You 
see, you make a hole, to make room for a new guess. Some- 
body says, 'You know, you people always say that space is 
continuous. How do you know when you get to a small 
enough dimension that there really are enough points in 
between, that it isn't just a lot of dots separated by Httle 
distances?' Or they say, 'You know those quantum mechani- 
cal amplitudes you told me about, they're so complicated 
and absurd, what makes you think those are right? Maybe 
they aren't right'. Such remarks are obvious and are per- 
fectly clear to anybody who is working on this problem. It 
does not do any good to point this out. The problem is not 
only what might be wrong but what, precisely, might be sub- 
stituted in place of it. In the case of the continuous space, 
suppose the precise proposition is that space really consists 
of a series of dots, and that the space between them does not 
mean anything, and that the dots are in a cubic array. Then 
we can prove immediately that this is wrong. It does not 
work. The problem is not just to say something might be 
wrong, but to replace it by something - and. that is not so 
easy. As soon as any really definite idea is substituted it 
becomes almost immediately apparent that it does not work. 
The second difficulty is that there is an infinite number of 
possibilities of these simple types. It is something like this. 
You are sitting working very hard, you have worked for a 
long time trying to open a safe. Then some Joe comes along 
who knows nothing about what you are doing, except that 
you are trying to open the safe. He says 'Why don't you 
try the combination 10:20:30?' Because you are busy, you 
have tried a lot of things, maybe you have already tried 
10:20:30. Maybe you know already that the middle number 
is 32 not 20. Maybe you know as a matter of fact that it is 
a five digit combination. ... So please do not send me any 
letters trying to tell me how the thing is going to work. 
I read them - I always read them to make sure that I have 
not already thought of what is suggested - but it takes too 


Looking for a New Law 

long to answer them, because they are usually in the class 
'try 10:20:30'. As usual, nature's imagination far surpasses 
our own, as we have seen from the other theories which are 
subtle and deep. To get such a subtle and deep guess is not 
so easy. One must be really clever to guess, and it is not 
possible to do it bUndly by machine. 

I want to discuss now the art of guessing nature's laws. 
It is an art. How is it done ? One way you might suggest is 
to look at history to see how the other guys did it. So we 
look at history. 

We must start with Newton. He had a situation where he 
had incomplete knowledge, and he was able to guess the 
laws by putting together ideas which were all relatively close 
to experiment; there was not a great distance between the 
observations and the tests. That was the first way, but today 
it does not work so well. 

The next guy who did something great was Maxwell, who 
obtained the laws of electricity and magnetism. What he 
did was this. He put together all the laws of electricity, due 
to Faraday and other people who came before him, and he 
looked at them and reaUzed that they were mathematically 
inconsistent. In order to straighten it out he had to add one 
term to an equation. He did this by inventing for himself a 
model of idler wheels and gears and so on in space. He found 
what the new law was - but nobody paid much attention 
because they did not believe in the idler wheels. We do not 
beheve in the idler wheels today, but the equations that he 
obtained were correct. So the logic may be wrong but the 
answer right. 

In the case of relativity the discovery was completely 
different. There was an accumulation of paradoxes; the 
known laws gave inconsistent results. This was a new kind 
of thinking, a thinking in terms of discussing the possible 
symmetries of laws. It was especially difficult, because for 
the first time it was reaUzed how long something hke New- 
ton's laws could seem right, and still ultimately be wrong. 
Also it was difficult to accept that ordinary ideas of time 
and space, which seemed so instinctive, could be wrong. 


Quantum mechanics was discovered in two independent 
ways - which is a lesson. There again, and even more so, an 
enormous number of paradoxes were discovered experi- 
mentally, things that absolutely could not be explained in 
any way by what was known. It was not that the knowledge 
was incomplete, but that the knowledge was too complete. 
Your prediction was that this should happen - it did not. 
The two different routes were one by Schrodinger,* who 
guessed the equation, the other by Heisenberg, who argued 
that you must analyse what is measurable. These two dif- 
ferent philosophical methods led to the same discovery in 
the end. 

More recently, the discovery of the laws of the weak 
decay I spoke of, when a neutron disintegrates into a proton, 
an electron and an anti-neutrino - which are still only partly 
known - add up to a somewhat different situation. This time 
it was a case of incomplete knowledge, and only the equation 
was guessed. The special difficulty this time was that the 
experiments were all wrong. How can you guess the right 
answer if, when you calculate the result, it disagrees with 
experiment? You need courage to say the experiments must 
be wrong. I will explain where that courage comes from later. 

Today we have no paradoxes - maybe. We have this in- 
finity that comes in when we put all the laws together, but 
the people sweeping the dirt under the rug are so clever that 
one sometimes thinks this is not a serious paradox. Again, 
the fact that we have found all these particles does not tell 
us anything except that our knowledge is incomplete. I am 
sure that history does not repeat itself in physics, as you can 
tell from looking at the examples I have given. The reason 
is this. Any schemes - such as 'think of symmetry laws', or 
'put the information in mathematical form', or 'guess 
equations' - are known to everybody now, and they are all 
tried all the time. When you are stuck, the answer cannot 
be one of these, because you will have tried these right away. 

*Erwin Schrodinger, Austrian theoretical physicist. Won Nobel Prize 
for Physics 1933 with Paul Dirac. 


Looking for a New Law 

There must be another way next time. Each time we get into 
this log-jam of too much trouble, too many problems, it is 
because the methods that we are using are just like the ones 
we have used before. The next scheme, the new discovery, 
is going to be made in a completely different way. So his- 
tory does not help us much. 

I should Uke to say a httle about Heisenberg's idea that 
you should not talk about what you cannot measure, be- 
cause many people talk about this idea without really under- 
standing it. You can interpret this in the sense that the 
constructs or inventions that you make must be of such a 
kind that the consequences that you compute are comparable 
with experiment - that is, that you do not compute a con- 
sequence hke 'a moo must be three goos', when nobody 
knows what a moo or a goo is. Obviously that is no good. 
But if the consequences can be compared to experiment, 
then that is all that is necessary. It does not matter that moos 
and goos cannot appear in the guess. You can have as much 
junk in the guess as you hke, provided that the consequences 
can be compared with experiment. This is not always fully 
appreciated. People often complain of the unwarranted ex- 
tension of the ideas of particles and paths, etc., into the 
atomic realm. Not so at all; there is nothing unwarranted 
about the extension. We must, and we should, and we always 
do, extend as far as we can beyond what we already know, 
beyond those ideas that we have already obtained. Dan- 
gerous ? Yes. Uncertain ? Yes. But it is the only way to make 
progress. Although it is uncertain, it is necessary to make 
science useful. Science is only useful if it tells you about 
some experiment that has not been done; it is no good if it 
only tells you what just went on. It is necessary to extend the 
ideas beyond where they have been tested. For example, in 
the law of gravitation, which was developed to understand 
the motion of planets, it would have been no use if Newton 
had simply said, T now understand the planets', and had 
not felt able to try to compare it with the earth's pull on the 
moon, and for later men to say 'Maybe what holds the 
galaxies together is gravitation'. We must try that. You 


could say, 'When you get to the size of the galaxies, since 
you know nothing about it, anything can happen'. I know, 
but there is no science in accepting this type of limitation. 
There is no ultimate understanding of the galaxies. On the 
other hand, if you assume that the entire behaviour is due 
only to known laws, this assumption is very limited and 
definite and easily broken by experiment. What we are 
looking for is just such hypotheses, very definite and easy 
to compare with experiment. The fact is that the way the 
galaxies behave so far does not seem to be against the 

I can give you another example, even more interesting 
and important. Probably the most powerful single assump- 
tion that contributes most to the progress of biology is the 
assumption that everything animals do the atoms can do, 
that the things that are seen in the biological world are the 
results of the behaviour of physical and chemical pheno- 
mena, with no 'extra something'. You could always say, 
'When you come to living things, anything can happen'. 
If you accept that you will never understand living things. 
It is very hard to believe that the wiggling of the tentacle of 
the octopus is nothing but some fooling around of atoms 
according to the known physical laws. But when it is investi- 
gated with this hypothesis one is able to make guesses quite 
accurately about how it works. In this way one makes great 
progress in understanding. So far the tentacle has not been 
cut off - it has not been found that this idea is wrong. 

It is not unscientific to make a guess, although many 
people who are not in science think it is. Some years ago I 
had a conversation with a layman about flying saucers - be- 
cause I am scientific I know all about flying saucers! I said 
'I don't think there are flying saucers'. So my antagonist 
said, 'Is it impossible that there are flying saucers? Can you 
prove that it's impossible?' 'No', I said, 'I can't prove it's 
impossible. It's just very unlikely'. At that he said, 'You are 
very unscientific. If you can't prove it impossible then how 
can you say that it's unlikely?' But that is the way that is 
scientific. It is scientific only to say what is more likely and 


Looking for a New Law 

what less likely, and not to be proving all the time the pos- 
sible and impossible. To define what I mean, I might have 
said to him, 'Listen, I mean that from my knowledge of the 
world that I see around me, I think that it is much more 
likely that the reports of flying saucers are the results of the 
known irrational characteristics of terrestrial intelligence 
than of the unknown rational eff"orts of extra-terrestrial 
intelUgence'. It is just more hkely, that is all. It is a good 
guess. And we always try to guess the most Hkely explana- 
tion, keeping in the back of the mind the fact that if it does 
not work we must discuss the other possibiUties. 

How can we guess what to keep and what to throw away ? 
We have all these nice principles and known facts, but we 
are in some kind of trouble : either we get the infinities, or 
we do not get enough of a description - we are missing some 
parts. Sometimes that means that we have to throw away 
some idea; at least in the past it has always turned out that 
some deeply held idea had to be thrown away. The question 
is, what to throw away and what to keep. If you throw it all 
away that is going a httle far, and then you have not much 
to work with. After all, the conservation of energy looks 
good, and it is nice, and I do not want to throw it away. To 
guess what to keep and what to throw away takes con- 
siderable skill. Actually it is probably merely a matter of 
luck, but it looks as if it takes considerable skill. 

Probability amplitudes are very strange, and the first 
thing you think is that the strange new ideas are clearly 
cock-eyed. Yet everything that can be deduced from the 
ideas of the existence of quantum mechanical probability 
amplitudes, strange though they are, do work, throughout 
the long list of strange particles, one hundred per cent. 
Therefore I do not believe that when we find out the inner 
guts of the composition of the world we shall find these 
ideas are wrong. I think this part is right, but I am only 
guessing: I am telling you how I guess. 

On the other hand, I believe that the theory that space is 
continuous is wrong, because we get these infinities and other 
difficulties, and we are left with questions on what deter- 


mines the size of all the particles. I rather suspect that the 
simple ideas of geometry, extended down into infinitely 
small space, are wrong. Here, of course, I am only making a 
hole, and not telling you what to substitute. If I did, I should 
finish this lecture with a new law. 

Some people have used the inconsistency of all the prin- 
ciples to say that there is only one possible consistent world, 
that if we put all the principles together, and calculate very 
exactly, we shall not only be able to deduce the principles, 
but we shall also discover that these are the only principles 
that could possibly exist if the thing is still to remain con- 
sistent. That seems to me a big order. I beUeve that sounds 
hke wagging the dog by the tail. I beUeve that it has to be 
given that certain things exist - not all the 50-odd particles, 
but a few httle things like electrons, etc. - and then with all 
the principles the great complexities that come out are prob- 
ably a definite consequence. I do not think that you can get 
the whole thing from arguments about consistencies. 

Another problem we have is the meaning of the partial 
symmetries. These symmetries, like the statement that 
neutrons and protons are nearly the same but are not the 
same for electricity, or the fact that the law of reflection 
symmetry is perfect except for one kind of reaction, are very 
annoying. The thing is almost symmetrical but not com- 
pletely. Now two schools of thought exist. One will say that 
it is really simple, that they are really symmetrical but that 
there is a little complication which knocks it a bit cock-eyed. 
Then there is another school of thought, which has only one 
representative, myself, which says no, the thing may be com- 
plicated and become simple only through the complications. 
The Greeks believed that the orbits of the planets were 
circles. Actually they are ellipses. They are not quite sym- 
metrical, but they are very close to circles. The question is, 
why are they very close to circles? Why are they nearly 
symmetrical ? Because of a long complicated effect of tidal 
friction - a very complicated idea. It is possible that nature in 
her heart is completely unsymmetrical in these things, but 
in the complexities of reahty it gets to look approximately 


Looking for a New Law 

as if it is symmetrical, and the ellipses look almost like 
circles. That is another possibihty; but nobody knows, it is 
just guesswork. 

Suppose you have two theories, A and B, which look 
completely different psychologically, with different ideas in 
them and so on, but that all the consequences that are com- 
puted from each are exactly the same, and both agree with 
experiment. The two theories, although they sound different 
at the beginning, have all consequences the same, which is 
usually easy to prove mathematically by showing that the 
logic from A and B will always give corresponding con- 
sequences. Suppose we have two such theories, how are we 
going to decide which one is right? There is no way by 
science, because they both agree with experiment to the 
same extent. So two theories, although they may have deeply 
different ideas behind them, may be mathematically identi- 
cal, and then there is no scientific way to distinguish them. 

However, for psychological reasons, in order to guess new 
theories, these two things may be very far from equivalent, 
because one gives a man different ideas from the other. By 
putting the theory in a certain kind of framework you get 
an idea of what to change. There will be something, for 
instance, in theory A that talks about something, and you 
will say, Til change that idea in here'. But to find out what 
the corresponding thing is that you are going to change in 
B may be very complicated - it may not be a simple idea at 
all. In other words, although they are identical before they 
are changed, there are certain ways of changing one which 
looks natural which will not look natural in the other. There- 
fore psychologically we must keep all the theories in our 
heads, and every theoretical physicist who is any good 
knows six or seven different theoretical representations for 
exactly the same physics. He knows that they are all equiva- 
lent, and that nobody is ever going to be able to decide 
which one is right at that level, but he keeps them in his 
head, hoping that they will give him different ideas for 

That reminds me of another point, that the philosophy or 


ideas around a theory may change enormously when there 
are very tiny changes in the theory. For instance, Newton's 
ideas about space and time agreed with experiment very well, 
but in order to get the correct motion of the orbit of Mer- 
cury, which was a tiny, tiny difference, the difference in the 
character of the theory needed was enormous. The reason 
is that Newton's laws were so simple and so perfect, and 
they produced definite results. In order to get something 
that would produce a slightly different result it had to be 
completely different. In stating a new law you cannot make 
imperfections on a perfect thing; you have to have another 
perfect thing. So the differences in philosophical ideas be- 
tween Newton's and Einstein's theories of gravitation are 

What are these philosophies ? They are really tricky ways 
to compute consequences quickly. A philosophy, which is 
sometimes called an understanding of the law, is simply a 
way that a person holds the laws in his mind in order to 
guess quickly at consequences. Some people have said, and 
it is true in cases like Maxwell's equations, 'Never mind the 
philosophy, never mind anything of this kind, just guess the 
equations. The problem is only to compute the answers so 
that they agree with experiment, and it is not necessary to 
have a philosophy, or argument, or words, about the equa- 
tion'. That is good in the sense that if you only guess the 
equation you are not prejudicing yourself, and you will 
guess better. On the other hand, maybe the philosophy helps 
you to guess. It is very hard to say. 

For those people who insist that the only thing that is 
important is that the theory agrees with experiment, I would 
like to imagine a discussion between a Mayan astronomer 
and his student. The Mayans were able to calculate with 
great precision predictions, for example, for eclipses and for 
the position of the moon in the sky, the position of Venus, 
etc. It was all done by arithmetic. They counted a certain 
number and subtracted some numbers, and so on. There 
was no discussion of what the moon was. There was no 
discussion even of the idea that it went around. They just 


Looking for a New Law 

calculated the time when there would be an eclipse, or when 
the moon would rise at the full, and so on. Suppose that a 
young man went to the astronomer and said, 'I have an 
idea. Maybe those things are going around, and there are 
balls of something like rocks out there, and we could cal- 
culate how they move in a completely different way from 
just calculating what time they appear in the sky'. 'Yes', says 
the astronomer, 'and how accurately can you predict 
ecUpses?' He says, 'I haven't developed the thing very far 
yet'. Then says the astronomer, 'Well, we can calculate 
ecHpses more accurately than you can with your model, so 
you must not pay any attention to your idea because ob- 
viously the mathematical scheme is better'. There is a very 
strong tendency, when someone comes up with an idea and 
says, 'Let's suppose that the world is this way', for people 
to say to him, 'What would you get for the answer to such 
and such a problem?' And he says, 'I haven't developed it 
far enough'. And they say, 'Well, we have already developed 
it much further, and we can get the answers very accurately'. 
So it is a problem whether or not to worry about philoso- 
phies behind ideas. 

Another way of working, of course, is to guess new prin- 
ciples. In Einstein's theory of gravitation he guessed, on top 
of all the other principles, the principle that corresponded to 
the idea that the forces are always proportional to the masses. 
He guessed the principle that if you are in an accelerating 
car you cannot distinguish that from being in a gravitational 
field, and by adding that principle to all the other principles, 
he was able to deduce the correct laws of gravitation. 

That outUnes a number of possible ways of guessing. I 
would now like to come to some other points about the 
final result. First of all, when we are all finished, and we 
have a mathematical theory by which we can compute con- 
sequences, what can we do ? It really is an amazing thing. 
In order to figure out what an atom is going to do in a given 
situation we make up rules with marks on paper, carry them 
into a machine which has switches that open and close in 
some complicated way, and the result will tell us what the 


atom is going to do ! If the way that these switches open and 
close were some kind of model of the atom, if we thought 
that the atom had switches in it, then I would say that I 
understood more or less what is going on. I find it quite 
amazing that it is possible to predict what will happen by 
mathematics, which is simply following rules which really 
have nothing to do with what is going on in the original 
thing. The closing and opening of switches in a computer 
is quite different from what is happening in nature. 

One of the most important things in this 'guess - compute 
consequences - compare with experiment' business is to 
know when you are right. It is possible to know when you 
are right way ahead of checking all the consequences. You 
can recognize truth by its beauty and simplicity. It is always 
easy when you have made a guess, and done two or three 
little calculations to make sure that it is not obviously 
wrong, to know that it is right. When you get it right, it is 
obvious that it is right - at least if you have any experience 
- because usually what happens is that more comes out 
than goes in. Your guess is, in fact, that something is very 
simple. If you cannot see immediately that it is wrong, and 
it is simpler than it was before, then it is right. The in- 
experienced, and crackpots, and people like that, make 
guesses that are simple, but you can immediately see that 
they are wrong, so that does not count. Others, the inex- 
perienced students, make guesses that are very complicated, 
and it sort of looks as if it is all right, but I know it is not 
true because the truth always turns out to be simpler than 
you thought. What we need is imagination, but imagination 
in a terrible strait-jacket. We have to find a new view of the 
world that has to agree with everything that is known, but 
disagree in its predictions somewhere, otherwise it is not 
interesting. And in that disagreement it must agree with 
nature. If you can find any other view of the world which 
agrees over the entire range where things have already been 
observed, but disagrees somewhere else, you have made a 
great discovery. It is very nearly impossible, but not quite, 
to find any theory which agrees with experiments over the 


Looking for a New Law 

entire range in which all theories have been checked, and 
yet gives different consequences in some other range, even 
a theory whose different consequences do not turn out to 
agree with nature. A new idea is extremely difficult to think 
of. It takes a fantastic imagination. 

What of the future of this adventure ? What will happen 
ultimately ? We are going along guessing the laws ; how many 
laws are we going to have to guess ? I do not know. Some of 
my colleagues say that this fundamental aspect of our 
science will go on; but I think there will certainly not be 
perpetual novelty, say for a thousand years. This thing can- 
not keep on going so that we are always going to discover 
more and more new laws. If we do, it will become boring 
that there are so many levels one underneath the other. It 
seems to me that what can happen in the future is either that 
all the laws become known - that is, if you had enough laws 
you could compute consequences and they would always 
agree with experiment, which would be the end of the hne - 
or it may happen that the experiments get harder and harder 
to make, more and more expensive, so you get 99-9 per cent 
of the phenomena, but there is always some phenomenon 
which has just been discovered, which is very hard to 
measure, and which disagrees ; and as soon as you have the 
explanation of that one there is always another one, and 
it gets slower and slower and more and more uninteresting. 
That is another way it may end. But I think it has to end in 
one way or another. 

We are very lucky to live in an age in which we are still 
making discoveries. It is like the discovery of America - 
you only discover it once. The age in which we hve is the age 
in which we are discovering the fundamental laws of nature, 
and that day will never come again. It is very exciting, it is 
marvellous, but this excitement will have to go. Of course in 
the future there will be other interests. There will be the 
interest of the connection of one level of phenomena to 
another - phenomena in biology and so on, or, if you are 
talking about exploration, exploring other planets, but there 
will not still be the same things that we are doing now. 


Another thing that will happen is that ultimately, if it 
turns out that all is known, or it gets very dull, the vigorous 
philosophy and the careful attention to all these things that 
I have been talking about will gradually disappear. The 
philosophers who are always on the outside making stupid 
remarks will be able to close in, because we cannot push 
them away by saying, 'If you were right we would be able 
to guess all the rest of the laws', because v/hen the laws are 
all there they will have an explanation for them. For in- 
stance, there are always explanations about why the world 
is three-dimensional. Well, there is only one world, and it is 
hard to tell if that explanation is right or not, so that if 
everything were known there would be some explanation 
about why those were the right laws. But that explanation 
would be in a frame that we cannot criticize by arguing that 
that type of reasoning will not permit us to go further. 
There will be a degeneration of ideas, just like the degenera- 
tion that great explorers feel is occurring when tourists 
begin moving in on a territory. 

In this age people are experiencing a dehght, the tremen- 
dous delight that you get when you guess how nature will 
work in a new situation never seen before. From experi- 
ments and information in a certain range you can guess what 
is going to happen in a region where no one has ever ex- 
plored before. It is a little different from regular exploration 
in that there are enough clues on the land discovered to 
guess what the land that has not been discovered is going 
to look like. These guesses, incidentally, are often very 
different from what you have already seen - they take a lot 
of thought. 

What is it about nature that lets this happen, that it is 
possible to guess from one part what the rest is going to do? 
That is an unscientific question: I do not know how to 
answer it, and therefore I am going to give an unscientific 
answer. I think it is because nature has a simplicity and 
therefore a great beauty. 



A Portfolio of Computer-mode Drawings 

D CD I 1 


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D DD DO crn I ll l ll lll I 

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1 D □ D DD n [ir~ ^=^ ^ — 

□ - □ 

A Computer Drawing, Darel Eschbach, Jr. 


A Computer Drawing , Calcomp Test Pattern 


spires of Contribution, Lloyd Sumner 


Devil's Staircase, A Computer Drawing Lloyd Sumner 


Yesterday & Forever, A Computer Drawing Lloyd Sumner 


Krystollos, by CalComp 


The Snail , by CalComp 


Authors and Artists 


Jeremy Bernstein, born in Rochester, New York in 
1929, is Professor of Physics at Stevens Institute 
of Technology in New Jersey. He was educated at 
Columbia Grammar School in New York City and 
received a bachelor's and master's degree in mathe- 
matics, and a doctorate in physics from Harvard 
University. He has doT\» research at the Harvard 
Cyclotron Laboratory, the Institute for Advanced 
Study at Princeton, Los Alamos, at the Brook- 
haven National Laboratories, and is frequently a 
visiting physicist at CERN (Conseil European 
pour la Recherche Nucleaire) in Geneva. Bernstein 
is the author of The Analytical Engine: Computers, 
Past, Present and Future, Ascent, on account of 
mountaineering in the Alps, and has written book 
reviews and profile articles for the magazine. The 
New Yorker. 


Arthur C. Clark, British scientist and writer, is a 
Fellow of the Royal Astronomical Society. During 
World War II he served as technical officer in 
charge of the first aircraft ground-controlled ap- 
proach project. He has won the Kalinga Prize, 
given by UNESCO for the popularization of science. 
The feasibility of many of the current space devel- 
opments was perceived and outlined by Clarke in 
the 1930's. His science fiction novels include 
Childhoods End and The City ond the Stars. 


Charles Galton Darwin, British physicist, and grand* 
son of the founder of the theory of evolution, was 
born in Cambridge, England in 1887, and died in 
1962. He was educated at Trinity College, Cam- 
bridge University, and held positions at Manchester 
University, Cambridge University, and Edinburgh 
University. In 1938 he became director of the 
Notional Physical Laboratory. Charles Galton 
Darwin was the author of The New Conceptions of 
Matter (1931), The Next Million Years (1952), and 
he wrote many papers on theoretical physics. 


Paul Adrien Maurice Dirac is one of the major 
figures in modern mathematics and theoretical 
physics. He received the Nobel Prize in 1933 
for his contribution to quantum mechanics. Diroc 
was born in 1902 in Bristol and received his 
bachelor's degree in engineering from Bristol 

University. Later he became a research student 
in mathematics at St. John's College, Cambridge, 
and received his Ph.D. in 1926. He is now 
Lucosion Professor of Mathematics at Cambridge, 


Albert Einstein, considered to be the most creative 
physical scientist since Newton, was nevertheless 
humble and sometimes rather shy man. He was 
born in Ulm, Germany, in 1879. He seemed to learn 
so slowly that his parents feared that he might be 
retarded. After graduating from the Polytechnic 
Institute in Zurich, he became a junior official at 
the Patent Office at Berne. At the age of twenty- 
six, and quite unknown, he published three revolu- 
tionary papers in theoretical physics in 1905. The 
first paper extended Max Planck's ideas of quanti- 
zation of energy, and established the quantum 
theory of radiation. For this work he received the 
Nobel Prize for 1921. The second paper gave a 
mathematical theory of Brownian motion, yielding 
a calculation of the size of a molecule. His third 
paper founded the special theory of relativity. 
Einstein's later work centered on the general 
theory of relativity. His work hod a profound in- 
fluence not only on physics, but olso on philo- 
sophy. An eloquent and widely beloved man, 
Einstein took an active part in liberal and anti-war 
movements. Fleeing from Nozi Germany, he settled 
in the United States in 1933 at the Institute for Ad- 
vanced Study in Princeton. He died in 1955. 


George Gomow, a theoretical physicist from Russia, 
received his Ph.D. in physics at the University of 
Leningrad. At Leningrad he became professor after 
being o Carlsberg fellow and a university fellow ot 
the University of Copenhagen and a Rockefeller 
fellow at Cambridge University. He came to the 
United States in 1933 to teach at the George Wash- 
ington University and later at the University of 
Colorado. His popularizations of physics are much 


Victor Guillemin, Jr., an American physicist, was 
born in Milwaukee in 1896. He was educated ot the 
University of Wisconsin, Harvard, and the Univer- 
sity of Munich. He taught at Harvard from 1930 to 


1935, wos research associote at the Fatigue 
Laboratories from 1935-41, senior physicist ot 
the United States Army Air Force in Dayton, Ohio, 
from 1941 to 1948, and professor of biophysics at 
th* University of Illinois from 1948 to 1959. He 
is outhor of The Story of Quontum Mechanics 
(1968). His research interests are in atomic ond 
molecular structure, and biological and aero- 
medical sciences. 


Banesh Hoffman, born in Richmond, Englond in 
1906, ottended Oxford and Princeton. He has been 
a member of the Institute of Advanced Study, elec- 
trical engineer at the Federal Telephone and Radio 
Laboratories, researcher at King's College, London, 
and a consuitont for Westinghouse Electric Corpora- 
tion's science talent search tests. He has won the 
distinguished teacher award at Queen's College, 
where he is Professor of Mathematics. During the 
1966-67 year he was on the staff of Harvard 
Project Physics. 


Leopold Infeld, a co-worker with Albert Einstein 
in general relativity theory, was born in 1898 in 
Poland. After studying ot the Cracow and Berlin 
Universities, he bacame a Rockefeller Fellow at 
Cambridge where he worked with Max Born in 
ele ctromagnetic theory, and then a member of the 
Institute for Advanced Study at Princeton. For 
eleven years he was Professor of Applied Mathe- 
matics at the University of Toronto. He then re- 
turned to Poland and become Professor of Physics 
at the University of Warsaw and until his death on 
16 January 1968 he was director of the Theoretical 
Physics Institute at the university. A member of 
the presidium of the Polish Academy of Science, 
Infeld conducted research in theoretical physics, 
especially relativity and quantum theories. In- 
feld was the author of The New Field Theory ^ The 
World in Modern Science , Quest , Albert Einstein , 
and with Einstein The Evolution of Physics . 


Martin J. Klein was born in New York City and 
attended Columbia University and Massachusetts 
Institute of Technology. He has been a Notional 
Research Fellow at the Dublin Institute for Ad- 
vanced Studies and a Guggenheim Fellow at the 
University of Leyden, Holland. He has taught at 
MIT and Case Institute and is now Professor at 
Yale University. His main interest is in the his- 
tory of relativity and quantum mechanics. 


E. M. Purcell, Professor of Physics at Harvard Uni- 
versity, was born in 1912 in Toylorville, Illinois. 
He was educated at Purdue University ond ot Har- 
vard. During World War II he worked as a researcher 
at the Radiation Laboratory, and he has been a 
member of the Science Advisory Board for the 
United States Air Force and of the President's 
Science Advisory Committee. For his work in nu- 
cleor magnetism, E. M. Purcell was awarded the 
1952 Nobel Prize in Physics. He has worked on 
microwove phenomena and radio-frequency spectro- 
scopy, and has also written physics textbooks. 


Eric M. Rogers, Professor of Physics at Princeton 
University, was born in Bickley, England in 1902. 
He received his education at Cambridge and later 
was demonstrator at the Cavendish Laboratory. 
Since 1963 he has been the organizer in physics 
for the Nuffield Foundation Science Teaching 
Project. He is the author of the textbook, Physics 
for the Inquiring Mind. 


Erwin Schrodinger (1887-1961) was born in Vienna 
and became successor of Max Planck as professor 
of physics at the University of Berlin. His work 
provided some of the basic equations of the quan- 
tum theory. Jointly with Paul A. M. Diroc he was 
awarded the Nobel Prize in physics in 1933 for the 
discovery of new productive forms of atomic theory. 
Originally he hod planned to be a philosopher, and 
he wrote widely-read books concerning the relation 
between science and the humanities, as well as 
some poetry. 


Cyril Stanley Smith, Professor of Physics at Massa- 
chusetts Institute of Technology, was born in Bir- 
mingham, England, in 1903. In 1926 he received his 
doctor of science from MIT. He has done research 
in physical metollurgy at MIT, the American Brass 
Company, and during World War II, the Los Alamos 
Laboratory. For his work there he received the 
United States Medal of Merit in 1946. Professor 
Smith has served on the General Advisory Commit- 
tee to the Atomic Energy Commission and on the 
President's Scientific Advisory Committee. His 
interest reaches deeply into history of science and 
technology; he is also jn art collector. 


Authors and Artists 


Charles Percy Snow, Baron of Leicester, was born 
in 1905 and educated at University College, 
Leicester, and at Christ's College, Cambridge. 
Although well known as a novelist, especially 
dealing with the lives and problems of professional 
men, he has held such diverse positions as chief 
of scientific personnel for the Ministry of Labour, 
Civil Service Commiss ioner, and a Director of the 
English Electric Co., Ltd. His writings have been 
widely acclaimed; among his novels are The 
Search, The New Men, and Corridors of Power. His 
nonfiction books on science and its consequences 
include The Two Cultures ond The Scientific Revo- 
lution, and Science and Government. 


J. L. Synge was born in Ireland in 1897. He has 
taught at universities in Ireland, Canada, and the 
United States, and is currently Professor of Mathe- 
matics at the Institute for Advanced Studies in 
Dublin. He is the President of the Royal Irish 
Academy. Synge has written papers on Riemannian 
geometry, relativity, hydrodynamics, and elasticity, 
has been author or co-author of Geometrical Optics 
and Principles of Mechanics, and has coedited the 
Mathematical Papers of Sir W. R. Hamilton. 


Sir Joseph John Thomson (1856-1940) was born 
near Manchester, England. At fourteen he entered 
a college in Manchester, at twenty he entered 
Cambridge on a scholarship, and at twenty-seven 
became professor of physics at Cambridge. It was 
Thomson whose work ushered in the period of sul^ 
atomic research when he showed conclusively that 
"cathode rays" consisted of electrons. With this a* 
a building block he constructed the "Thomson" 
model of the atom — a sphere of positive electricity 
in which were embedded negatively charged elec- 
trons. In 1906 J. J. Thomson was awarded the 
Nobel Prize, and in 1908 he was knighted. During 
Thomson's period as Director of the Cavendish 
Laboratory at Cambridge, eight Nobel Prizes were 
won by his colleagues. With this start England re- 
mained the leader in subatomic experimental 
physics for almost forty years.