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Full text of "Space, time and gravitation : an outline of the general relativity theory"

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SPACE TIME 

AND 

GRAVITATION 



CAMBRIDGE UNIVERSITY PRESS 

C. F. CLAY, MANAGER 
LONDON : FETTER LANE, E.C. 4 




NEW YORK : THE MACMILLAN CO. 

BOMBAY 1 

CALCUTTA > MACMILLAN AND CO., LTD. 

MADRAS J 

TORONTO : THE MACMILLAN CO. OP 

CANADA, LTD. 
TOKYO : MARUZEN-KABUSHIKI-KAISHA 



ALL RIGHTS RESERVED 



SPACE TIME 

AND 

GRAVITATION 

AN OUTLINE OF THE GENERAL 
RELATIVITY THEORY 



BY 

A. S. EDDINGTON, M.A., M.Sc., F.E.8. 

PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL 
PHILOSOPHY, CAMBRIDGE 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 
1920 



Mo. 



Perhaps to move 

His laughter at their quaint opinions wide 
Hereafter, when they come to model heaven 
And calculate the stars : how they will wield 
The mighty frame : how build, unbuild, contrive 

To save appearances. 

Paradise Lost. 



PREFACE 

BY his theory of relativity Albert Einstein has provoked a 
revolution of thought in physical science. 

The achievement consists essentially in this: Einstein has 
succeeded in separating far more completely than hitherto the 
share of the observer and the share of external nature in the 
things we see happen. The perception of an object by an observer 
depends on his own situation and circumstances; for example, 
distance will make it appear smaller and dimmer. We make 
allowance for this almost unconsciously in interpreting what we 
see. But it now appears that the allowance made for the motion 
of the observer has hitherto been too crude a fact overlooked 
because in practice all observers share nearly the same motion, 
that of the earth. Physical space and time are found to be 
closely bound up with this motion of the observer; and only an 
amorphous combination of the two is left inherent in the external 
world. When space and time are relegated to their proper source 
the observer the world of nature which remains appears 
strangely unfamiliar; but it is in reality simplified, and the 
underlying unity of the principal phenomena is now clearly 
revealed. The deductions from this new outlook have, with one 
doubtful exception, been confirmed when tested by experiment. 

It is my aim to give an account of this work without intro- 
ducing anything very technical in the way of mathematics, 
physics, or philosophy. The new view of space and time, so 
opposed to our habits of thought, must in any case demand 
unusual mental exercise. The results appear strange; and the 
incongruity is not without a humorous side. For the first nine 
chapters the task is one of interpreting a clear-cut theory, 
accepted in all its essentials by a large and growing school of 
physicists although perhaps not everyone would accept the 
author's views of its meaning. Chapters x and xi deal with 
very recent advances, with regard to which opinion is more 
fluid. As for the last chapter, containing the author's specula- 
tions on the meaning of nature, since it touches on the rudiments 
of a philosophical system, it is perhaps too sanguine to hope that 
it can ever be other than controversial. 



vi PREFACE 

A non-mathematical presentation has necessary limitations; 
and the reader who wishes to learn how certain exact results 
follow from Einstein's, or even Newton's, law of gravitation is 
bound to seek the reasons in a mathematical treatise. But this 
limitation of range is perhaps less serious than the limitation of 
intrinsic truth. There is a relativity of truth, as there is a 
relativity of space. 

"For is and IS-NOT though with Rule and Line 
And UP-AND-DOWN without, I could define." 

Alas ! It is not so simple. We abstract from the phenomena that 
which is peculiar to the position and motion of the observer; 
but can we abstract that which is peculiar to the limited imagina- 
tion of the human brain? We think we can, but only in the 
symbolism of mathematics. As the language of a poet rings with 
a truth that eludes the clumsy explanations of his commentators, 
so the geometry of relativity in its perfect harmony expresses a 
truth of form and type in nature, which my bowdlerised version 
misses. 

But the mind is not content to leave scientific Truth in a dry 
husk of mathematical symbols, and demands that it shall be 
alloyed with familiar images. The mathematician, who handles 
x so lightly, may fairly be asked to state, not indeed the in- 
scrutable meaning of x in nature, but the meaning which x 
conveys to him. 

Although primarily designed for readers without technical 
knowledge of the subject, it is hoped that the book may also 
appeal to those who have gone into the subject more deeply. 
A few notes have been added in the Appendix mainly to bridge 
the gap between this and more mathematical treatises, and t 
indicate the points of contact between the argument in the text 
and the parallel analytical investigation. 

It is impossible adequately to express my debt to con- 
temporary literature and discussion. The writings of Einstein, 
Minkowski, Hilbert, Lorentz, Weyl, Robb, and others, have 
provided the groundwork ; in the give and take of debate with 
friends and correspondents, the extensive ramifications have 
gradually appeared. A S E 

1 May, 1920. 



CONTENTS 

ECLIPSE INSTRUMENTS AT SOBRAL . Frontispiece 

PROLOGUE PAGE 

WHAT is GEOMETRY? ..... 1 

CHAPTER I 
THE FITZGERALD CONTRACTION . ^ . 17 

CHAPTER II 
RELATIVITY .. . . . , . 30 

CHAPTER III 
THE WORLD OF FOUR DIMENSIONS , . 45 

CHAPTER IV 
FIELDS or FORCE 63 

CHAPTER V 
KINDS OF SPACE . . . . * .77 

CHAPTER VI 

THE NEW LAW OF GRAVITATION AND THE 

OLD LAW . . . . . . 93 

CHAPTER VII 

WEIGHING LIGHT . . . . . 110 

CHAPTER VIII 

OTHER TESTS OF THE THEORY . . . 123 

CHAPTER IX 
MOMENTUM AND ENERGY .... 136 

CHAPTER X 
TOWARDS INFINITY . . . . . 152 

CHAPTER XI 
ELECTRICITY AND GRAVITATION . . 167 

CHAPTER XII 
ON THE NATURE OF THINGS . . ., 180 

APPENDIX 

MATHEMATICAL NOTES .-. . . . 202 

HISTORICAL NOTE 210 



PROLOGUE 
WHAT IS GEOMETRY? 

A conversation between 

An experimental PHYSICIST. 
A pure MATHEMATICIAN. 

A RELATIVIST, who advocates the newer conceptions of time and 
space in physics. 

Rel. There is a well-known proposition of Euclid which states 
that "Any two sides of a triangle are together greater than the 
third side." Can either of you tell me whether nowadays there 
is good reason to believe that this proposition is true? 

Math. For my part, I am quite unable to say whether the 
proposition is true or not. I can deduce it by trustworthy 
reasoning from certain other propositions or axioms, which are 
supposed to be still more elementary. If these axioms are true, 
the proposition is true ; if the axioms are not true, the proposition 
is not true universally. Whether the axioms are true or not 
I cannot say, and it is outside my province to consider. 

Phys. But is it not claimed that the truth of these axioms is 
self-evident? 

Math. They are by no means self-evident to me; and I think 
the claim has been generally abandoned. 

Phys. Yet since on these axioms you have been able to found 
a logical and self-consistent system of geometry, is not this 
indirect evidence that they are true? 

Math. No. Euclid's geometry is not the only self-consistent 
system of geometry. By choosing a different set of axioms I can, 
for example, arrive at Lobatchewsky's geometry, in which many 
of the propositions of Euclid are not in general true. From my 
point of view there is nothing to choose between these different 
geometries. 

Rel. How is it then that Euclid's geometry is so much the 
most important system? 

Math. I am scarcely prepared to admit that it is the most 
important. But for reasons which I do not profess to understand, 
my friend the Physicist is more interested in Euclidean geometry 

E.S. I 



2 PROLOGUE 

than in any other, and is continually setting us problems in it. 
Consequently we have tended to give an undue share of attention 
to the Euclidean system. There have, however, been great 
geometers like Riemann who have done something to restore 
a proper perspective. 

Eel. (to Physicist). Why are you specially interested in 
Euclidean geometry? Do you believe it to be the true geometry? 

Phys. Yes. Our experimental work proves it true. 

Rel. How, for example, do you prove that any two sides of 
a triangle are together greater than the third side? 

Phys. I can, of course, only prove it by taking a very large 
number of typical cases, and I am limited by the inevitable 
inaccuracies of experiment. My proofs are not so general or so 
perfect as those of the pure mathematician. But it is a recognised 
principle in physical science that it is permissible to generalise 
from a reasonably wide range of experiment; and this kind of 
proof satisfies me. 

Rel. It will satisfy me also. I need only trouble you with 
a special case. Here is a triangle ABC; how will you prove that 
AB + BC is greater than AC? 

Phys. I shall take a scale and measure the three sides. 

Rel. But we seem to be talking about different things. I was 
speaking of a proposition of geometry properties of space, not 
of matter. Your experimental proof only shows how a material 
scale behaves when you turn it into different positions. 

Phys. I might arrange to make the measures with an optical 
device. 

Rel. That is worse and worse. Now you are speaking of 
properties of light. 

Phys. I really cannot tell you anything about it, if you will 
not let me make measurements of any kind. Measurement is 
my only means of finding out about nature. I am not a meta- 
physicist. 

Rel. Let us then agree that by length and distance you always 
mean a quantity arrived at by measurements with material or 
optical appliances. You have studied experimentally the laws 
obeyed by these measured lengths, and have found the geometry 
to which they conform. We will call this geometry "Natural 
Geometry"; and it evidently has much greater importance for 



WHAT IS GEOMETRY? 3 

you than any other of the systems which the brain of the 
mathematician has invented. But we must remember that its 
subject matter involves the behaviour of material scales the 
properties of matter. Its laws are just as much laws of physics 
as, for example, the laws of electromagnetism. 

Phys. Do you mean to compare space to a kind of magnetic 
field? I scarcely understand. 

Rel. You say that you cannot explore the world without 
some kind of apparatus. If you explore with a scale, you find 
out the natural geometry ; if you explore with a magnetic needle, 
you find out the magnetic field. What we may call the field of 
extension, or space-field, is just as much a physical quality as 
the magnetic field. You can think of them both existing together 
in the aether, if you like. The laws of both must be determined 
by experiment. Of course, certain approximate laws of the space- 
field (Euclidean geometry) have been familiar to us from child- 
hood; but we must get rid of the idea that there is anything 
inevitable about these laws, and that it would be impossible to 
find in other parts of the universe space-fields where these laws 
do not apply. As to how far space really resembles a magnetic 
field, I do not wish to dogmatise; my point is that they present 
themselves to experimental investigation in very much the same 
way. 

Let us proceed to examine the laws of natural geometry. 
I have a tape-measure, and here is the triangle. AB = 39J in., 
BC = in., CA = 39| in. Why, your proposition does not hold ! 

Phys. You know very well what is wrong. You gave the 
tape-measure a big stretch when you measured AB. 

Rel. Why shouldn't I? 

Phys. Of course, a length must be measured with a rigid 
scale. 

Rel. That is an important addition to our definition of length. 
But what is a rigid scale? 

Phys. A scale which always keeps the same length. 

Rel. But we have just defined length as the quantity arrived 
at by measures with a rigid scale; so you will want another rigid 
scale to test whether the first one changes length; and a third 
to test the second ; and so ad inflnitum. You remind me of the 
incident of the clock and time-gun in Egypt. The man in charge 

i 2 



4 PROLOGUE 

of the time-gun fired it by the clock; and the man in charge of 
the clock set it right by the time-gun. No, you must not define 
length by means of a rigid scale, and define a rigid scale by 
means of length. 

Phys. I admit I am hazy about strict definitions. There is 
not time for everything; and there are so many interesting 
things to find out in physics, which take up my attention. Are 
you so sure that you are prepared with a logical definition of all 
the terms you use? 

Rel. Heaven forbid ! I am not naturally inclined to be 
rigorous about these things. Although I appreciate the value of 
the work of those who are digging at the foundations of science, 
my own interests are mainly in the upper structure. But some- 
times, if we wish to add another storey, it is necessary to deepen 
the foundations. I have a definite object in trying to arrive at 
the exact meaning of length. A strange theory is floating round, 
to which you may feel initial objections; and you probably 
would not wish to let your views go by default. And after all, 
when you claim to determine lengths to eight significant figures, 
you must have a pretty definite standard of right and wrong 
measurements. 

Phys. It is difficult to define what we mean by rigid; but in 
practice we can tell if a scale is likely to change length appreciably 
in different circumstances. 

Rel. No. Do not bring in the idea of change of length in 
describing the apparatus for defining length. Obviously the 
adopted standard of length cannot change length, whatever it 
is made of. If a metre is defined as the length of a certain bar, 
that bar can never be anything but a metre long; and if we 
assert that this bar changes length, it is clear that we must have 
changed our minds as to the definition of length. You recognised 
that my tape-measure was a defective standard that it was 
not rigid. That was not because it changed length, because, if 
it was the standard of length, it could not change length. It 
was lacking in some other quality. 

You know an approximately rigid scale when you see one. 
What you are comparing it with is not some non-measurable 
ideal of length, but some attainable, or at least approachable, 
ideal of material constitution. Ordinary scales have defects 



WHAT IS GEOMETRY? 5 

flexure, expansion with temperature, etc. which can be reduced 
by suitable precautions; and the limit, to which you approach 
as you reduce them, is your rigid scale. You can define these 
defects without appealing to any extraneous definition of length; 
for example, if you have two rods of the same material whose 
extremities are just in contact with one another, and when one 
of them is heated the extremities no longer can be adjusted to 
coincide, then the material has a temperature-coefficient of 
expansion. Thus you can compare experimentally the tempera- 
ture-coefficients of different metals and arrange them in 
diminishing sequence. In this sort of way you can specify the 
nature of your ideal rigid rod, before you introduce the term 
length. 

Phys. No doubt that is the way it should be defined. 

Eel. We must recognise then that all our knowledge of space 
rests on the behaviour of material measuring-scales free from 
certain definable defects of constitution. 

Phys. I am not sure that I agree. Surely there is a sense in 
which the statement AB = 2CD is true or false, even if we had 
no conception of a material measuring-rod. For instance, there 
is, so to speak, twice as much paper between A and B, as between 
C and D. 

Rel. Provided the paper is uniform. But then, what does 
uniformity of the paper mean? That the amount in given length 
is constant. We come back at once to the need of defining length. 

If you say instead that the amount of "space" between 
A and B is twice that between C and Z>, the same thing applies. 
You imagine the intervals filled with uniform space; but the 
uniformity simply means that the same amount of space corre- 
sponds to each inch of your rigid measuring-rod. You have 
arbitrarily used your rod to divide space into so-called equal 
lumps. It all comes back to the rigid rod. 

I think you were right at first when you said that you could 
not find out anything without measurement; and measurement 
involves some specified material appliance. 

Now you admit that your measures cannot go beyond a 
certain close approximation, and that you have not tried all 
possible conditions. Supposing that one corner of your triangle 
was in a very intense gravitational field far stronger than any 



6 PROLOGUE 

we have had experience of I have good ground for believing 
that under those conditions you might find the sum of two sides 
of a triangle, as measured with a rigid rod, appreciably less than 
the third side. In that case would you be prepared to give up 
Euclidean geometry? 

Phys. I think it would be risky to assume that the strong 
force of gravitation made no difference to the experiment. 

Rel. On my supposition it makes an important difference. 

Phys. I mean that we might have to make corrections to the 
measures, because the action of the strong force might possibly 
distort the measuring-rod. 

Rel. In a rigid rod we have eliminated any special response 
to strain. 

Phys. But this is rather different. The extension of the rod 
is determined by the positions taken up by the molecules under 
the forces to which they are subjected; and there might be a 
response to the gravitational force which all kinds of matter 
would share. This could scarcely be regarded as a defect; and 
our so-called rigid rod would not be free from it any more than 
any other kind of matter. 

Rel. True; but what do you expect to obtain by correcting 
the measures? You correct measures, when they are untrue to 
standard. Thus you correct the readings of a hydrogen-ther- 
mometer to obtain the readings of a perfect gas-thermometer, 
because the hydrogen molecules have finite size, and exert special 
attractions on one another, and you prefer to take as standard 
an ideal gas with infinitely small molecules. But in the present 
case, what is the standard you are aiming at when you propose 
to correct measures made with the rigid rod? 

Phys. I see the difficulty. I have no knowledge of space 
apart from my measures, and I have no better standard than 
the rigid rod. So it is difficult to see what the corrected measures 
would mean. And yet it would seem to me more natural to 
suppose that the failure of the proposition was due to the 
measures going wrong rather than to an alteration in the character 
of space. 

Rel. Is not that because you are still a bit of a metaphysicist? 
You keep some notion of a space which is superior to measure- 
ment, and are ready to throw over the measures rather than let 



WHAT IS GEOMETRY? 7 

this space be distorted. Even if there were reason for believing 
in such a space, what possible reason could there be for assuming 
it to be Euclidean? Your sole reason for believing space to be 
Euclidean is that hitherto your measures have made it appear so; 
if now measures of certain parts of space prefer non-Euclidean 
geometry, all reason for assuming Euclidean space disappears. 
Mathematically and conceptually Euclidean and non-Euclidean 
space are on the same footing; our preference for Euclidean 
space was based on measures, and must stand or fall by 
measures. 

Phys. Let me put it this way. I believe that I am trying to 
measure something called length, which has an absolute meaning 
in nature, and is of importance in connection with the laws of 
nature. This length obeys Euclidean geometry. I believe my 
measures with a rigid rod determine it accurately when no 
disturbance like gravitation is present; but in a gravitational 
field it is not unreasonable to expect that the uncorrected 
measures may not give it exactly. 

Rel. You have three hypotheses there: (1) there is an 
absolute thing in nature corresponding to length, (2) the 
geometry of these absolute lengths is Euclidean, and (3) practical 
measures determine this length accurately when there is no 
gravitational force. I see no necessity for these hypotheses, and 
propose to do without them. Hypotheses nonflngo. The second 
hypothesis seems to me particularly objectionable. You assume 
that this absolute thing in nature obeys the laws of Euclidean 
geometry. Surely it is contrary to scientific principles to lay 
down arbitrary laws for nature to obey ; we must find out her 
laws by experiment. In this case the only experimental evidence 
is that measured lengths (which by your own admission are not 
necessarily the same as this absolute thing) sometimes obey 
Euclidean geometry and sometimes do not. Again it would 
seem reasonable to doubt your third hypothesis beyond, say, 
the sixth decimal place; and that would play havoc with your 
more delicate measures. But where I fundamentally differ from 
you is the first hypothesis. Is there some absolute quantity in 
nature that we try to determine when we measure length? 
When we try to determine the number of molecules in a given 
piece of matter, we have to use indirect methods, and different 



8 PROLOGUE 

methods may give systematically different results; but no one 
doubts that there is a definite number of molecules, so that there 
is some meaning in saying that certain methods are theoretically 
good and others inaccurate. Counting appears to be an absolute 
operation. But it seems to me that other physical measures are 
on a different footing. Any physical quantity, such as length, 
mass, force, etc., which is not a pure number, can only be denned 
as the result arrived at by conducting a physical experiment 
according to specified rules. 

So I cannot conceive of any "length" in nature independent 
of a definition of the way of measuring length. And, if there is, 
we may disregard it in physics, because it is beyond the range 
of experiment. Of course, it is always possible that we may 
come across some quantity, not given directly by experiment, 
which plays a fundamental part in theory. If so, it will turn up 
in due course in our theoretical formulae. But it is no good 
assuming such a quantity, and laying down a priori laws for it 
to obey, on the off-chance of its proving useful. 

Phys. Then you will not let me blame the measuring-rod 
when the proposition fails? 

Rel. By all means put the responsibility on the measuring- 
rod. Natural geometry is the theory of the behaviour of material 
scales. Any proposition in natural geometry is an assertion as 
to the behaviour of rigid scales, which must accordingly take 
the blame or credit. But do not say that the rigid scale is 
wrong, because that implies a standard of right which does not 
exist. 

Phys. The space which you are speaking of must be a sort of 
abstraction of the extensional relations of matter. 

Rel. Exactly so. And when I ask you to believe that space 
can be non-Euclidean, or, in popular phrase, warped, I am not 
asking you for any violent effort of the imagination; I only 
mean that the extensional relations of matter obey somewhat 
modified laws. Whenever we investigate the properties of space 
experimentally, it is these extensional relations that we are 
finding. Therefore it seems logical to conclude that space as 
known to us must be the abstraction of these material relations, 
and not something more transcendental. The reformed methods 
of teaching geometry in schools would be utterly condemned, 



WHAT IS GEOMETRY? 9 

and it would be misleading to set schoolboys to verify propositions 
of geometry by measurement, if the space they are supposed to 
be studying had not this meaning. 

I suspect that you are doubtful whether this abstraction of 
extensional relations quite fulfils your general idea of space; and, 
as a necessity of thought, you require something beyond. I do 
not think I need disturb that impression, provided you realise 
that it is not the properties of this more transcendental thing 
we are speaking of when we describe geometry as Euclidean or 
non-Euclidean. 

Math. The view has been widely held that space is neither 
physical nor metaphysical, but conventional. Here is a passage 
from Poincare's Science and Hypothesis, which describes this 
alternative idea of space : 

" If Lobatchewsky's geometry is true, the parallax of a very 
distant star will be finite. If Riemann's is true, it will be negative. 
These are the results which seem within the reach of experiment, 
and it is hoped that astronomical observations may enable us 
to decide between the two geometries. But what we call a 
straight line in astronomy is simply the path of a ray of light. 
If, therefore, we were to discover negative parallaxes, or to 
prove that all parallaxes are higher than a certain limit, we 
should have a choice between two conclusions: we could give 
up Euclidean geometry, or modify the laws of optics, and 
suppose that light is not rigorously propagated in a straight 
line. It is needless to add that everyone would look upon this 
solution as the more advantageous. Euclidean geometry, 
therefore, has nothing to fear from fresh experiments." 

Eel. Poincare's brilliant exposition is a great help in under- 
standing the problem now confronting us. He brings out the 
interdependence between geometrical laws and physical laws, 
which we have to bear in mind continually. We can add on to 
one set of laws that which we subtract from the other set. 
I admit that space is conventional for that matter, the meaning 
of every word in the language is conventional. Moreover, we 
have actually arrived at the parting of the ways imagined by 
Poincare, though the crucial experiment is not precisely the 
one he mentions. But I deliberately adopt the alternative, 
which, he takes for granted, everyone would consider less 



10 PROLOGUE 

advantageous. I call the space thus chosen physical space, and 
its geometry natural geometry, thus admitting that other con- 
ventional meanings of space and geometry are possible. If it 
were only a question of the meaning of space a rather vague 
term these other possibilities might have some advantages. 
But the meaning assigned to length and distance has to go 
along with the meaning assigned to space. Now these are 
quantities which the physicist has been accustomed to measure 
with great accuracy; and they enter fundamentally into the 
whole of our experimental knowledge of the world. We have a 
knowledge of the so-called extent of the stellar universe, which, 
whatever it may amount to in terms of ultimate reality, is not 
a mere description of location in a conventional and arbitrary 
mathematical space. Are we to be robbed of the terms in which 
we are accustomed to describe that knowledge? 

The law of Boyle states that the pressure of a gas is propor- 
tional to its density. It is found by experiment that this law is 
only approximately true. A certain mathematical simplicity 
would be gained by conventionally redefining pressure in such 
a way that Boyle's law would be rigorously obeyed. But it 
would be high-handed to appropriate the word pressure in this 
way, unless it had been ascertained that the physicist had no 
further use for it in its original meaning. 

Phys. I have one other objection. Apart from measures, we 
have a general perception of space, and the space we perceive 
is at least approximately Euclidean. 

Rel. Our perceptions are crude measures. It is true that our 
perception of space is very largely a matter of optical measures 
with the eyes. If in a strong gravitational field optical and 
mechanical measures diverged, we should have to make up our 
minds which was the preferable standard, and afterwards abide 
by it. So far as we can ascertain, however, they agree in all 
circumstances, and no such difficulty arises. So, if physical 
measures give us a non-Euclidean space, the space of perception 
will be non-Euclidean. If you were transplanted into an ex- 
tremely intense gravitational field, you would directly perceive 
the non-Euclidean properties of space. 

Phys. Non-Euclidean space seems contrary to reason. 

Math. It is not contrary to reason, but contrary to common 



WHAT IS GEOMETRY? 11 

experience, which is a very different thing, since experience is 
very limited. 

Phys. I cannot imagine myself perceiving non-Euclidean space ! 

Math. Look at the reflection of the room in a polished door- 
knob, and imagine yourself one of the actors in what you see 
going on there. 

Rel. I have another point to raise. The distance between 
two points is to be the length measured with a rigid scale. Let 
us mark the two points by particles of matter, because we must 
somehow identify them by reference to material objects. For 
simplicity we shall suppose that the two particles have no 
relative motion, so that the distance whatever it is remains 
constant. Now you will probably agree that there is no such 
thing as absolute motion; consequently there is no standard 
condition of the scale which we can call "at rest." We may 
measure with the scale moving in any way we choose, and if 
results for different motions disagree, there is no criterion for 
selecting the true one. Further, if the particles are sliding past 
the scale, it makes all the difference what instants we choose 
for making the two readings. 

Phys. You can avoid that by denning distance as the measure- 
ment made with a scale which has the same velocity as the two 
points. Then they will always be in contact with two particular 
divisions of the scale. 

Rel. A very sound definition; but unfortunately it does not 
agree with the meaning of distance in general use. When the 
relativist wishes to refer to this length, he calls it the proper- 
length; in non-relativity physics it does not seem to have been 
used at all. You see it is not convenient to send your apparatus 
hurling through the laboratory after a pair of a particles, for 
example. And you could scarcely measure the length of a wave 
of light by this convention*. So the physicist refers his lengths 
to apparatus at rest on the earth ; and the mathematician starts 
with the words "Choose unaccelerated rectangular axes 0#, Oy, 
Oz, ..." and assumes that the measuring-scales are at rest 
relatively to these axes. So when the term length is used some 
arbitrary standard motion of the measuring apparatus must 
always be implied. 

* The proper- length of a light- wave is actually infinite. 



12 PROLOGUE 

Phys. Then if you have fixed your standard motion of the 
measuring-rod, there will be no ambiguity if you take the 
readings of both particles at the same moment. 

Rel. What is the same moment at different places? The 
conception of simultaneity in different places is a difficult one. 
Is there a particular instant in the progress of time on another 
world, Arcturus, which is the same as the present instant on the 
Earth? 

Phys. I think so, if there is any connecting link. We can 
observe an event, say a change of brightness, on Arcturus, and, 
allowing for the time taken by light to travel the distance, 
determine the corresponding instant on the earth. 

Rel. But then you must know the speed of the earth through 
the aether. It may have shortened the light-time by going some 
way to meet the light coming from Arcturus. 

Phys. Is not that a small matter? 

Rel. At a very modest reckoning the motion of the earth in 
the interval might alter the light-time by several days. Actually, 
however, any speed of the earth through the aether up to the 
velocity of light is admissible, without affecting anything observ- 
able. At least, nothing has been discovered which contradicts 
this. So the error may be months or years. 

Phys. What you have shown is that we have not sufficient 
knowledge to determine in practice which are simultaneous 
events on the Earth and Arcturus. It does not follow that there 
is no definite simultaneity. 

Rel. That is true, but it is at least possible that the reason 
why we are unable to determine simultaneity in practice (or, 
what comes to pretty much the same thing, our motion through 
the aether) in spite of many brilliant attempts, is that there is 
no such thing as absolute simultaneity of distant events. It is 
better therefore not to base our physics on this notion of absolute 
simultaneity, which may turn out not to exist, and is in any 
case out of reach at present. 

But what all this comes to is that time as well as space is 
implied in all our measures. The fundamental measurement is 
not the interval between two points of space, but between two 
points of space associated with instants of time. 

Our natural geometry is incomplete at present. We must 



WHAT IS GEOMETRY? 13 

supplement it by bringing in time as well as space. We shall 
need a perfect clock as well as a rigid scale for our measures. 
It may be difficult to choose an ideal standard clock : but what- 
ever definition we decide on must be a physical definition. We 
must not dodge it by saying that a perfect clock is one which 
keeps perfect time. Perhaps the best theoretical clock would be 
a pulse of light travelling in vacuum to and fro between mirrors 
at the ends of a rigid scale. The instants of arrival at one end 
would define equal intervals of time. 

Phys. I think your unit of time would change according to 
the motion of your "clock" through the aether. 

Rel. Then you are comparing it with some notion of absolute 
time. I have no notion of time except as the result of measure- 
ment with some kind of clock. (Our immediate perception of 
the flight of time is presumably associated with molecular 
processes in the brain which play the part of a material clock.) 
If you know a better clock, let us adopt it; but, having once 
fixed on our ideal clock there can be no appeal from its judg- 
ments. You must remember too that if you wish to measure 
a second at one place, you must keep your clock fixed at what 
you consider to be one place; so its motion is defined. The 
necessity of defining the motion of the clock emphasises that 
one cannot consider time apart from space; there is one geometry 
comprising both. 

Phys. Is it right to call this study geometry. Geometry deals 
with space alone. 

Math. I have no objection. It is only necessary to consider 
time as a fourth dimension. Your complete natural geometry 
will be a geometry of four dimensions. 

Phys. Have we then found the long-sought fourth dimension? 

Math. It depends what kind of a fourth dimension you were 
seeking. Probably not in the sense you intend. For me it only 
means adding a fourth variable, t, to my three space-variables 
x, y, z. It is no concern of mine what these variables really 
represent. You give me a few fundamental laws that they 
satisfy, and I proceed to deduce other consequences that may 
be of interest to you. The four variables may for all I know be 
the pressure, density, temperature and entropy of a gas; that 
is of no importance to me. But you would not say that a gas 



14 PROLOGUE 

had four dimensions because four mathematical variables were 
used to describe it. Your use of the term "dimensions" is 
probably more restricted than mine. 

Phys. I know that it is often a help to represent pressure 
and volume as height and width on paper; and so geometry 
may have applications to the theory of gases. But is it not going 
rather far to say that geometry can deal directly with these 
things and is not necessarily concerned with lengths in 
space? 

Math. No. Geometry is nowadays largely analytical, so that 
in form as well as in effect, it deals with variables of an unknown 
nature. It is true that I can often see results more easily by 
taking my x and y as lengths on a sheet of paper. Perhaps it 
would be helpful in seeing other results if I took them as pressure 
and density in a steam-engine; but a steam-engine is not so 
handy as a pencil. It is literally true that I do not want to 
know the significance of the variables x 9 y, z,t that I am discussing. 
That is lucky for the Relativist, because although he has defined 
carefully how they are to be measured, he has certainly not 
conveyed to me any notion of how I am to picture them, if my 
picture of absolute space is an illusion. 

Phys. Yours is a strange sub j ect . You told us at the beginning 
that you are not concerned as to whether your propositions are 
true, and now you tell us you do not even care to know what 
you are talking about. 

Math. That is an excellent description of Pure Mathematics, 
which has already been given by an eminent mathematician*. 

Rel. I think there is a real sense in which time is a fourth 
dimension as distinct from a fourth variable. The term 
dimension seems to be associated with relations of order. 
I believe that the order of events in nature is one indissoluble 
four-dimensional order. We may split it arbitrarily into space 
and time, just as we can split the order of space into length, 

* " Pure mathematics consists entirely of such asseverations as that, if such 
and such a proposition is true of anything, then such and such a proposition 
is true of that thing. It is essential not to discuss whether the first proposition 
is really true, and not to mention what the anything is of which it is supposed 

to be true Thus mathematics may be denned as the subject in which we 

never know what we are talking about, nor whether what we are saying is true." 

BEBTBAND RTJSSELL. 



WHAT IS GEOMETRY? 15 

breadth and thickness. But space without time is as incomplete 
as a surface without thickness. 

Math. Do you argue that the real world behind the pheno- 
mena is four-dimensional? 

Rel. I think that in the real world there must be a set of 
entities related to one another in a four-dimensional order, and 
that these are the basis of the perceptual world so far as it is 
yet explored by physics. But it is possible to pick out a four- 
dimensional set of entities from a basal world of five dimensions, 
or even of three dimensions. The straight lines in three- 
dimensional space form a four-dimensional set of entities, i.e. 
they have a fourfold order. So one cannot predict the ultimate 
number of dimensions in the world if indeed the expression 
dimensions is applicable. 

Phys. What would a philosopher think of these conceptions? 
Or is he solely concerned with a metaphysical space and time 
which is not within reach of measurement. 

Rel. In so far as he is a psychologist our results must concern 
him. Perception is a kind of crude physical measurement; and 
perceptual space and time is the same as the measured space 
and time, which is the subject-matter of natural geometry. In 
other respects he may not be so immediately concerned. 
Physicists and philosophers have long agreed that motion 
through absolute space can have no meaning; but in physics 
the question is whether motion through aether has any meaning. 
I consider that it has no meaning; but that answer, though it 
brings philosophy and physics into closer relation, has no bearing 
on the philosophic question of absolute motion. I think, 
however, we are entitled to expect a benevolent interest from 
philosophers, in that we are giving to their ideas a perhaps 
unexpected practical application. 

Let me now try to sum up my conclusions from this conversa- 
tion. We have been trying to give a precise meaning to the 
term space, so that we may be able to determine exactly the 
properties of the space we live in. There is no means of deter- 
mining the properties of our space by a priori reasoning, because 
there are many possible kinds of space to choose from, no one 
of which can be considered more likely than any other. For 



16 PROLOGUE 

more than 2000 years we have believed in a Euclidean space, 
because certain experiments favoured it ; but there is now reason 
to believe that these same experiments when pushed to greater 
accuracy decide in favour of a slightly different space (in the 
neighbourhood of massive bodies). The relativist sees no reason 
to change the rules of the game because the result does not 
agree with previous anticipations. Accordingly when he speaks 
of space, he means the space revealed by measurement, whatever 
its geometry. He points out that this is the space with which 
physics is concerned ; and, moreover, it is the space of everyday 
perception. If his right to appropriate the term space in this 
way is challenged, he would urge that this is the sense in which 
the term has always been used in physics hitherto; it is only 
recently that conservative physicists, frightened by the revolu- 
tionary consequences of modern experiments, have begun to 
play with the idea of a pre-existing space whose properties 
cannot be ascertained by experiment a metaphysical space, to 
which they arbitrarily assign Euclidean properties, although it 
is obvious that its geometry can never be ascertained by experi- 
ment. But the relativist, in defining space as measured space, 
clearly recognises that all measurement involves the use of 
material apparatus ; the resulting geometry is specifically a study 
of the extensional relations of matter. He declines to consider 
anything more transcendental. 

My second point is that since natural geometry is the study 
of extensional relations of natural objects, and since it is found 
that their space-order cannot be discussed without reference to 
their time-order as well, it has become necessary to extend our 
geometry to four dimensions in order to include time. 



CHAPTEE I 
THE FITZGERALD CONTRACTION 

In order to reach the Truth, it is necessary, once in one's life, to put every 
thing in doubt so far as possible. DESCARTES. 

WILL it take longer to swim to a point 100 yards up-stream 
and back, or to a point 100 yards across-stream and back? 

In the first case there is a long toil up against the current, 
and then a quick return helped by the current, which is all too 
short to compensate. In the second case the current also hinders, 
because part of the effort is devoted to overcoming the drift 
down-stream. But no swimmer will hesitate to say that the 
hindrance is the greater in the first case. 

Let us take a numerical example. Suppose the swimmer's 
speed is 50 yards a minute in still water, and the current is 
30 yards a minute. Thus the speed against the current is 20, 
and with the current 80 yards a minute. The up journey then 
takes 5 minutes and the down journey 1 J minutes. Total time, 
6 J minutes. 

Going across-stream the swimmer must aim at a point E above 
the point B where he wishes to arrive, so 
that OE represents his distance travelled 
in still water, and EB the amount he has J, 1 

drifted down. These must be in the ratio E 

50 to 30, and we then know from the right- 
angled triangle QBE that OB will corre- 
spond to 40. Since OB is 100 yards, OE 
is 125 yards, and the time taken is 2J 
minutes. Another 2| minutes will be 
needed for the return journey. Total time, 



5 minutes. Ito ' L 

In still water the time would have been 4 minutes. 
The up-and-down swim is thus longer than the transverse 

swim in the ratio 6J : 5 minutes. Or we may write the ratio 

1 



. S. 



18 THE FITZGERALD CONTRACTION [CH. 

which shows how the result depends on the ratio of the speed 
of the current to the speed of the swimmer, viz. f g. . 

A very famous experiment on these lines was tried in America 
in the year 1887. The swimmer was a wave of light, which we 
know swims through the aether with a speed of 186,330 miles 
a second. The aether was flowing through the laboratory like 
a river past its banks. The light- wave was divided, by partial 
reflection at a thinly silvered surface, into two parts, one of 
which was set to perform the up-and-down stream journey and 
the other the across-stream journey. When the two waves 
reached their proper turning-points they were sent back to the 
starting-point by mirrors. To judge the result of the race, there 
was an optical device for studying interference fringes ; because 
the recomposition of the two waves after the journey would 
reveal if one had been delayed more than the other, so that, for 
example, the crest of one instead of fitting on to the crest of 
the other coincided with its trough. 

To the surprise of Michelson and Morley, who conducted the 
experiment, the result was a dead-heat. It is true that the 
direction of the current of aether was not known they hoped 
to find it out by the experiment. That, however, was got over 
by trying a number of different orientations. Also it was 
possible that there might actually be no current at a particular 
moment. But the earth has a velocity of 18 J miles a second, 
continually changing direction as it goes round the sun ; so that 
at some time during the year the motion of a terrestrial labora- 
tory through the aether must be at least 18 J miles a second. 
The experiment should have detected the delay by a much 
smaller current; in a repetition of it by Morley and Miller 
in 1905, a current of 2 miles a second would have been 
sufficient. 

If we have two competitors, one of whom is known to be 
slower than the other, and yet they both arrive at the winning- 
post at the same time, it is clear that they cannot have travelled 
equal courses. To test this, the whole apparatus was rotated 
through a right angle, so that what had been the up-and-down 
course became the transverse course, and vice versa. Our two 
competitors interchanged courses, but still the result was a 
dead-heat. 



i] THE FITZGERALD CONTRACTION 19 

The surprising character of this result can be appreciated by 
contrasting it with a similar experiment on sound-waves. 
Sound consists of waves in air or other material, as light con- 
sists of waves in aether. It would be possible to make a precisely 
similar experiment on sound, with a current of air past the 
apparatus instead of a current of aether. In that case the greater 
delay of the wave along the direction of the current would 
certainly show itself experimentally. Why does light seem to 
behave differently? 

The straightforward interpretation of this remarkable result 
is that each course undergoes an automatic contraction when it 
is swung from the transverse to the longitudinal position, so 
that whichever arm of the apparatus is placed up-stream it 
straightway becomes the shorter. The course is marked out in 
the rigid material apparatus, and we have to suppose that the 
length of any part of the apparatus changes as it is turned in 
different directions with respect to the aether-current. It is 
found that the kind of material metal, stone or wood makes 
no difference to the experiment. The contraction must be the 
same for all kinds of matter; the expected delay depends only 
on the ratio of the speed of the aether current to the speed of 
light, and the contraction which compensates it must be equally 
definite. 

This explanation was proposed by FitzGerald, and at first 
sight it seems a strange and arbitrary hypothesis. But it has 
been rendered very plausible by subsequent theoretical researches 
of Larmor and Lorentz. Under ordinary circumstances the form 
and size of a solid body is maintained by the forces of cohesion 
between its particles. What is the nature of cohesion? We guess 
that it is made up of electric forces between the molecules. But 
the aether is the medium in which electric force has its seat; 
hence it will not be a matter of indifference to these forces how 
the electric medium is flowing with respect to the molecules. 
When the flow changes there will be a readjustment of cohesive 
forces, and we must expect the body to take a new shape and 
size. 

The theory of Larmor and Lorentz enables us to trace in 
detail the readjustment. Taking the accepted formulae of 
electromagnetic theory, they showed that the new form of 

2 2 



20 THE FITZGERALD CONTRACTION [CH. 

equilibrium would be contracted in just such a way and by 
just such an amount as FitzGerald's explanation requires*. 

The contraction in most cases is extremely minute. We have 
seen that when the ratio of the speed of the current to that 
of the swimmer is f, a contraction in the ratio ^/(l (f) 2 ) 
is needed to compensate for the delay. The earth's orbital 
velocity is 10 } 00 of the velocity of light, so that it will give a 
contraction of \/(l - ( y ^^) 2 ), or 1 part in 200,000,000. This 
would mean that the earth's diameter in the direction of its 
motion is shortened by 2 J inches. 

The Michelson-Morley experiment has thus failed to detect 
our motion through the aether, because the effect looked for- 
the delay of one of the light waves is exactly compensated by 
an automatic contraction of the matter forming the apparatus. 
Other ingenious experiments have been tried, electrical and 
optical experiments of a more technical nature. They likewise 
have failed, because there is always an automatic compensation 
somewhere. We now believe there is something in the nature 
of things which inevitably makes these compensations, so that 
it will never be possible to determine our motion through the 
aether. Whether we are at rest in it, or whether we are rushing 
- through it with a speed not much less than that of light, will 
make no difference to anything that can possibly be observed. 

This may seem a rash generalization from the few experiments 
actually performed ; more particularly, since we can only experi- 
ment with the small range of velocity caused by the earth's 
orbital motion. With a larger range residual differences might 
be disclosed. But there is another reason for believing that the 
compensation is not merely approximate but exact. The com- 
pensation has been traced theoretically to its source in the 
well-known laws of electromagnetic force ; and here it is mathe- 
matically exact. Thus the generalization is justified, at least in 
so far as the observed phenomena depend on electromagnetic 
causes, and in so far as the universally accepted laws of electro- 
magnetism are accurate. 

The generalization here laid down is called the restricted 
*X Principle of Relativity: It is impossible by any experiment to 
detect uniform motion relative to the aether. 
* Appendix, Note 1. 



i] THE FITZGERALD CONTRACTION 21 

There are other natural forces which have not as yet been 
recognised as coming within the electromagnetic scheme 
gravitation, for example and for these other tests are required. 
Indeed we were scarcely justified in stating above that the 
diameter of the earth would contract 2j inches, because the 
figure of the earth is determined mainly by gravitation, whereas 
the Michelson-Morley experiment relates to bodies held together 
by cohesion. There is fair evidence of a rather technical kind 
that the compensation exists also for phenomena in which 
gravitation is concerned ; and we shall assume that the principle 
covers all the forces of nature. 

Suppose for a moment it were not so, and that it were possible 
to determine a kind of absolute motion of the earth by experi- 
ments or observations involving gravitation. Would this throw 
light on our motion through the aether? I think not. It would 
show that there is some standard of rest with respect to which 
the law of gravitation takes a symmetrical and simple form; 
presumably this standard corresponds to some gravitational 
medium, and the motion determined would be motion with 
respect to that medium. Similarly if the motion were revealed 
by vital or psychical phenomena, it would be motion relative 
to some vital or psychical medium. The aether, denned as the 
seat of electric forces, must be revealed, if at all, by electric 
phenomena. 

It is well to remember that there is reasonable justification 
for adopting the principle of relativity even if the evidence is 
insufficient to prove it. In Newtonian dynamics the phenomena 
are independent of uniform motion of the system; no explanation 
is asked for, because it is difficult to see any reason why there 
should be an effect. If in other phenomena the principle fails, 
then we must seek for an explanation of its failure and no 
doubt a plausible explanation can be devised; but so long as 
experiment gives no indication of a failure, it is idle to anticipate 
such a complication. Clearly physics cannot concern itself with 
all the possible complexities which may exist in nature, but have 
not hitherto betrayed themselves in any experiment. 

The principle of relativity has implications of a most revolu- 
tionary kind. Let us consider what is perhaps an exaggerated 
case or perhaps the actual case, for we cannot tell. Let the 



22 THE FITZGERALD CONTRACTION [CH. 

reader suppose that he is travelling through the aether at 
161,000 miles a second vertically upwards; if he likes to make 
the positive assertion that this is his velocity, no one will be 
able to find any evidence to contradict him. For this speed the 
FitzGerald contraction is just J, so that every object contracts 
to half its original length when turned into the vertical position. 

As you lie in bed, you are, say, 6 feet long. Now stand upright; 
you are 3 feet. You are incredulous? Well, let us prove it! 
Take a yard-measure; when turned vertically it must undergo 
the FitzGerald contraction, and become only half a yard. If you 
measure yourself with it, you will find you are just two half- 
yards. "But I can see that the yard-measure does not change 
length when I turn it." What you perceive is an image of the 
rod on the retina of your eye; you imagine that the image 
occupies the same space in both positions; but your retina has 
contracted in the vertical direction without your knowing it, so 
that your visual estimates of vertical length are double what 
they should be. And so on with every test you can devise. 
Because everything is altered in the same way, nothing appears 
to be altered at all. 

It is possible to devise electrical and optical tests; in that 
case the argument is more complicated, because we must con- 
sider the effect of the rapid current of aether on the electric 
forces and on waves of light. But the final conclusion is always 
the same; the tests will reveal nothing. Here is one illustration. 
To avoid distortion of the retina, lie on your back on the floor, 
and watch in a suitably inclined mirror someone turn the rod 
from the horizontal to the vertical position. You will, of course, 
see no change of length, and it is not possible to blame the 
retina this time. But is the appearance in the mirror a faithful 
reproduction of what is actually occurring? In a plane mirror 
at rest the appearance is correct ; the rays of light come off the 
mirror at the same angle as they fall on to it, like billiard balls 
rebounding from an elastic cushion. But if the cushion is in 
rapid motion the angle of the billiard-ball will be altered ; and 
similarly the rapid motion of the mirror through the aether 
alters the law of reflection. Precise calculation shows that the 
moving mirror will distort the image, so as to conceal exactly 
the changes of length which occur. 



i] THE FITZGERALD CONTRACTION 23 

The mathematician does not need to go through all the 
possible tests in detail; he knows that the complete compensa- 
tion is inherent in the fundamental laws of nature, and so must 
occur in every case. So if any suggestion is made of a device 
for detecting these effects, he starts at once to look for the 
fallacy which must surely be there. Our motion through the 
aether may be very much less than the value here adopted, and 
the changes of length may be very small ; but the essential point 
is that they escape notice, not because they are small (if they 
are small), but because from their very nature they are unde- 
tectable. 

There is a remarkable reciprocity about the effects of motion 
on length, which can best be illustrated by another example. 
Suppose that by development in the powers of aviation, a man 
flies past us at the rate of 161,000 miles a second. We shall 
suppose that he is in a comfortable travelling conveyance in 
which he can move about, and act normally and that his length 
is in the direction of the flight. If we could catch an instantaneous 
glimpse as he passed, we should see a figure about three feet 
high, but with the breadth and girth of a normal human being. 
And the strange thing is that he would be sublimely unconscious 
of his own undignified appearance. If he looks in a mirror in 
his conveyance, he sees his usual proportions; this is because of 
the contraction of his retina, or the distortion by the moving 
mirror, as already explained. But when he looks down on us, 
he sees a strange race of men who have apparently gone through 
some flattening-out process; one man looks barely 10 inches 
across the shoulders, another standing at right angles is almost 
" length and breadth, without thickness." As they turn about 
they change appearance like the figures seen in the old-fashioned 
convex-mirrors. If the reader has watched a cricket-match 
through a pair of prismatic binoculars, he will have seen this 
effect exactly. 

It is the reciprocity of these appearances that each party 
should think the other has contracted that is so difficult to 
realise. Here is a paradox beyond even the imagination of 
Dean Swift. Gulliver regarded the Lilliputians as a race of 
dwarfs; and the Lilliputians regarded Gulliver as a giant. That 
is natural. If the Lilliputians had appeared dwarfs to Gulliver, 



24 THE FITZGERALD CONTRACTION [CH. 

and Gulliver had appeared a dwarf to the Lilliputians but no ! 
that is too absurd for fiction, and is an idea only to be found in 
the sober pages of science. 

This reciprocity is easily seen to be a necessary consequence 
of the Principle of Relativity. The aviator must detect a Fitz- 
Gerald contraction of objects moving rapidly relatively to him, 
just as we detect the contraction of objects moving relatively to us, 
and as an observer at rest in the aether detects the contraction 
of objects moving relatively to the aether. Any other result 
would indicate an observable effect due to his own motion 
through the aether. 

Which is right? Are we or the aviator? Or are both the 
victims of illusion? It is not illusion in the ordinary sense, 
because the impressions of both would be confirmed by every 
physical test or scientific calculation suggested. No one knows 
which is right. No one will ever know, because we can never 
find out which, if either, is truly at rest in the aether. 

It is not only in space but in time that these strange variations 
occur. If we observed the aviator carefully we should infer that 
he was unusually slow in his movements; and events in the 
conveyance moving with him would be similarly retarded as 
though time had forgotten to go on. His cigar lasts twice as 
long as one of ours. I said "infer" deliberately; we should see 
a still more extravagant slowing down of time; but that is easily 
explained, because the aviator is rapidly increasing his distance 
from us and the light-impressions take longer and longer to 
reach us. The more moderate retardation referred to remains 
after we have allowed for the time of transmission of light. 

But here again reciprocity comes in, because in the aviator's 
opinion it is we who are travelling at 161,000 miles a second 
past him; and when he has made all allowances, he finds that 
it is we who are sluggish. Our cigar lasts twice as long as his. 

Let us examine more closely how the two views are to be 
reconciled. Suppose we both light similar cigars at the instant 
he passes us. At the end of 30 minutes our cigar is finished. 
This signal, borne on the waves of light, hurries out at the rate 
of 186,000 miles a second to overtake the aviator travelling at 
161,000 miles a second, who has had 30 minutes start. It will 
take nearly 194 minutes to overtake him, giving a total time of 



I] 



THE FITZGERALD CONTRACTION 



25 



224 minutes after lighting the cigar. His watch like everything 
else about him (including his cigar) is going at half-speed; so 
it records only 112 minutes elapsed when our signal arrives. 
The aviator knows, of course, that this is not the true time when 
our cigar was finished, and that he must correct for the time of 
transmission of the light-signal. He sets himself this problem 
that man has travelled away from me at 161,000 miles a second 
for an unknown time x minutes ; he has then sent a signal which 
travels the same distance back at 186,000 miles a second; the 
total time is 112 minutes; problem, find x. Answer, x 60 
minutes. He therefore judges that our cigar lasted 60 minutes, 
or twice as long as his own. His cigar lasted 30 minutes by his 
watch (because the same retardation affects both watch and 
cigar); and that was in our opinion twice as long as ours, because 
his watch was going at half-speed. . 

Here is the full time-table. 



Stationary 
watch 



min. 
30 
60 

112 
120 

224 , 



Stationary Observer 

Lights cigar 
Finishes cigar 

Inferred time aviator's 
cigar finished 

Receives signal aviator's 
cigar finished 



Aviator 

Lights cigar 

Finishes cigar 



Aviator's 
watch 

min. 
15 

30 

56 , 



Inferred time stationary 60 

cigar finished 
Receives signal stationary 112 

cigar finished 



This is analysed from our point of view, not the aviator's; 
because it makes out that he was wrong in his inference and we 
were right. But no one can tell which was really right. 

The argument will repay a careful examination, and it will 
be recognised that the chief cause of the paradox is that we 
assume that we are at rest in the aether, whereas the aviator 
assumes that he is at rest. Consequently whereas in our opinion 
the light-signal is overtaking him at merely the difference 
between 186,000 and 161,000 miles a second, he considers that 
it is coming to him through the relatively stationary aether at 
the normal speed of light. It must be remembered that each 
observer is furnished with complete experimental evidence in 
support of his own assumption. If we suggest to the aviator 



26 THE FITZGERALD CONTRACTION [CH. 

that owing to his high velocity the relative speed of the wave 
overtaking him can only be 25,000 miles a second, he will reply 
" I have determined the velocity of the wave relatively to me 
by timing it as it passes two points in my conveyance; and it 
turns out to be 186,000 miles a second. So I know my correction 
for light-time is right*." His clocks and scales are all behaving 
in an extraordinary way from our point of view, so it is not 
surprising that he should arrive at a measure of the velocity of 
the overtaking wave which differs from ours; but there is no 
way of convincing him that our reckoning is preferable. 

Although not a very practical problem, it is of interest to 
inquire what happens when the aviator's speed is still further 
increased and approximates to the velocity of light. Lengths 
in the direction of flight become smaller and smaller, until for 
the speed of light they shrink to zero. The aviator and the 
objects accompanying him shrink to two dimensions. We are 
saved the difficulty of imagining how the processes of life can 
go on in two dimensions, because nothing goes on. Time is 
arrested altogether. This is the description according to the 
terrestrial observer. The aviator himself detects nothing un- 
usual; he does not perceive that he has stopped moving. He is 
merely waiting for the next instant to come before making the 
next movement; and the mere fact that time is arrested means 
that he does not perceive that the next instant is a long time 
coming. 

It is a favourite device for bringing home the vast distances 
of the stars to imagine a voyage through space with the velocity 
of light. The youthful adventurer steps on to his magic carpet 
loaded with provisions for a century. He reaches his journey's 
end, say Arcturus, a decrepit centenarian. This is wrong. It is 
quite true that the journey would last something like a hundred 
years by terrestrial chronology ; but the adventurer would arrive 
at his destination no more aged than when he started, and he 
would not have had time to think of eating. So long as he travels 
with the speed of light he has immortality and eternal youth. 

* We need not stop to prove this directly. If the aviator could detect any- 
thing in his measurements inconsistent with the hypothesis that he was at rest 
in the aether (e.g. a difference of velocity of overtaking waves of light and 
waves meeting him) it would contradict the restricted principle of relativity. 



i] THE FITZGERALD CONTRACTION 27 

If in some way his motion were reversed so that he returned to 
the earth again, he would find that centuries had elapsed here, 
whilst he himself did not feel a day older for him the voyage 
had lasted only an instant*. 

Our reason for discussing at length the effects of these 
improbably high velocities is simply in order that we may speak 
of the results in terms of common experience; otherwise it 
would be necessary to use the terms of refined technical measure- 
ment. The relativist is sometimes suspected of an inordinate 
fondness for paradox; but that is rather a misunderstanding of 
his argument. The paradoxes exist when the new experimental 
discoveries are woven into the scheme of physics hitherto 
current, and the relativist is ready enough to point this out. 
But the conclusion he draws is that a revised scheme of physics 
is needed in which the new experimental results will find a natural 
place without paradox. 

To sum up on any planet moving with a great velocity 
through the aether, extraordinary changes of length of objects 
are continually occurring as they move about, and there is a 
slowing down of all natural processes as though time were 
retarded. These things cannot be perceived by anyone on the 
planet; but similar effects would be detected by any observer 
having a great velocity relative to the planet (who makes all 
allowances for the effect of the motion on the observations, but 
takes if for granted that he himself is at rest in the aethert). 
There is complete reciprocity so that each of two observers in 
relative motion will find the same strange phenomena occurring 

* Since the earth is moving relatively to our adventurer with the velocity 
of light, we might be tempted to argue that from this point of view the terrestrial 
observer would have perpetual youth whilst the voyager grew older. Evidently, 
if they met again, they could disprove one or other of the two arguments. But 
in order to meet again the velocity of one of them must be reversed by super- 
natural means or by an intense gravitational force so that the conditions are 
not symmetrical and reciprocity does not apply. The argument given in the 
text appears to be the correct one. 

t The last clause is perhaps unnecessary. The correction applied for light 
transmission will naturally be based on the observer's own experimental deter- 
mination of the velocity of light. According to experiment the velocity of light 
relatively to him is apparently the same in all directions, and he will apply 
the corrections accordingly. This is equivalent to assuming that he is at rest 
in the aether; but he need not, and probably would not, make the assumption 
explicitly. 



28 THE FITZGERALD CONTRACTION [CH. 

to the other; and there is nothing to help us to decide which is 
right. 

I think that no one can contemplate these results without 
feeling that the whole strangeness must arise from something 
perverse and inappropriate in our ordinary point of view. 
Changes go on on a planet, all nicely balanced by adjustments 
of natural forces, in such a way that no one on the planet can 
possibly detect what is taking place. Can we seriously imagine 
that there is anything in the reality behind the phenomena, 
which reflects these changes? Is it not more probable that we 
ourselves introduce the complexity, because our method of 
description is not well-adapted to give a simple and natural 
statement of what is really occurring? 

The search for a more appropriate apparatus of description 
leads us to the standpoint of relativity described in the next 
chapter. I draw a distinction between the principle and the 
standpoint of relativity. The principle of relativity is a state- 
ment of experimental fact, which may be right or wrong; the 
first part of it the restricted principle has already been 
enunciated. Its consequences can be deduced by mathematical 
reasoning, as in the case of any other scientific generalization. 
It postulates no particular mechanism of nature, and no particular 
view as to the meaning of time and space, though it may suggest 
theories on the subject. The only question is whether it is 
experimentally true or not. 

The standpoint of relativity is of a different character. It 
asserts first that certain unproved hypotheses as to time and 
space have insensibly crept into current physical theories, and 
that these are the source of the difficulties described above. 
Now the most dangerous hypotheses are those which are tacit 
and unconscious. So the standpoint of relativity proposes 
tentatively to do without these hypotheses (not making any 
others in their place); and it discovers that they are quite 
unnecessary and are not supported by any known fact. This in 
itself appears to be sufficient justification for the standpoint. 
Even if at some future time facts should be discovered which 
confirm the rejected hypotheses, the relativist is not wrong in 
reserving them until they are required. 

It is not our policy to take shelter in impregnable positions; 



i] THE FITZGERALD CONTRACTION 29 

and we shall not hesitate to draw reasonable conclusions as well 
as absolutely proved conclusions from the knowledge available. 
But to those who think that the relativity theory is a passing 
phase of scientific thought, which may be reversed in the light 
of future experimental discoveries, we would point out that, 
though like other theories it may be developed and corrected, 
there is a certain minimum statement possible which represents 
irreversible progress. Certain hypotheses enter into all physical 
descriptions and theories hitherto current, dating back in some 
cases for 2000 years, in other cases for 200 years. It can now 
be proved that these hypotheses have nothing to do with any 
phenomena yet observed, and do not afford explanations of any 
known fact. This is surely a discovery of the greatest importance 
quite apart from any question as to whether the hypotheses 
are actually wrong. 

I am not satisfied with the view so often expressed that the 
sole aim of scientific theory is "economy of thought." I cannot 
reject the hope that theory is by slow stages leading us nearer 
to the truth of things. But unless science is to degenerate into 
idle guessing, the test of value of any theory must be whether 
it expresses with as little redundancy as possible the facts 
which it is intended to cover. Accidental truth of a conclusion 
is no compensation for erroneous deduction. 

The relativity standpoint is then a discarding of certain 
hypotheses, which are uncalled for by any known facts, and 
stand in the way of an understanding of the simplicity of nature. 



CHAPTER II 
RELATIVITY 

The views of time and space, which I have to set forth, have their foundation 
in experimental physics. Therein is their strength. Their tendency is revolu- 
tionary. From henceforth space in itself and time in itself sink to mere shadows, 
and only a kind of union of the two preserves an independent existence. 

H. MINKOWSKI (1908). 

THERE are two parties to every observation the observed and 
the observer. 

What we see depends not only on the object looked at, but 
on our own circumstances position, motion, or more personal 
idiosyncracies. Sometimes by instinctive habit, sometimes by 
design, we attempt to eliminate our own share in the observa- 
tion, and so form a general picture of the world outside us, 
which shall be common to all observers. A small speck on the 
horizon of the sea is interpreted as a giant steamer. From the 
window of our railway carriage we see a cow glide past at fifty 
miles an hour, and remark that the creature is enjoying a rest. 
We see the starry heavens revolve round the earth, but decide 
that it is really the earth that is revolving, and so picture the 
state of the universe in a way which would be acceptable to an 
astronomer on any other planet. 

The first step in throwing our knowledge into a common 
stock must be the elimination of the various individual stand- 
points and the reduction to some specified standard observer. 
The picture of the world so obtained is none the less relative. 
We have not eliminated the observer's share; we have only 
fixed it definitely. 

To obtain a conception of the world from the point of view 
of no one in particular is a much more difficult task. The 
position of the observer can be eliminated ; we are able to grasp 
the conception of a chair as an object in nature looked at all 
round, and not from any particular angle or distance. We can 
think of it without mentally assigning ourselves some position 
with respect to it. This is a remarkable faculty, which has 
evidently been greatly assisted by the perception of solid relief 



CH. n] RELATIVITY 31 

with our two eyes. But the motion of the observer is not 
eliminated so simply. We had thought that it was accomplished; 
but the discovery in the last chapter that observers with 
different motions use different space- and time-reckoning shows 
that the matter is more complicated than was supposed. It may 
well require a complete change in our apparatus of description, 
because all the familiar terms of physics refer primarily to the 
relations of the world to an observer in some specified circum- 
stances. 

Whether we are able to go still further and obtain a knowledge 
of the world, which not merely does not particularise the 
observer, but does not postulate an observer at all; whether if 
such knowledge could be obtained, it would convey any in- 
telligible meaning; and whether it could be of any conceivable 
interest to anybody if it could be understood these questions 
need not detain us now. The answers are not necessarily 
negative, but they lie outside the normal scope of physics. 

The circumstances of an observer which affect his observations 
are his position, motion and gauge of magnitude. More personal 
idiosyncracies disappear if, instead of relying on his crude 
senses, he employs scientific measuring apparatus. But scientific 
apparatus has position, motion and size, so that these are still 
involved in the results of any observation. There is no essential 
distinction between scientific measures and the measures of the 
senses. In either case our acquaintance with the external world 
comes to us through material channels; the observer's body can 
be regarded as part of his laboratory equipment, and, so far as 
we know, it obeys the same laws. We therefore group together 
perceptions and scientific measures, and in speaking of "a 
particular observer" we include all his measuring appliances. 

Position, motion, magnitude-scale these factors have a pro- 
found influence on the aspect of the world to us. Can we form 
a picture of the world which shall be a synthesis of what is seen 
by observers in all sorts of positions, having all sorts of velocities, 
and all sorts of sizes. As already stated we have accomplished 
the synthesis of positions. We have two eyes, which have 
dinned into our minds from babyhood that the world has to be 
looked at from more than one position. Our brains have so far 
responded as to give us the idea of solid relief, which enables us 



32 RELATIVITY [CH. 

to appreciate the three-dimensional world in a vivid way that 
would be scarcely possible if we were only acquainted with 
strictly two-dimensional pictures. We not merely deduce the 
three-dimensional world; we see it. But we have no such aid 
in synthesising different motions. Perhaps if we had been 
endowed with two eyes moving with different velocities our 
brains would have developed the necessary faculty; we should 
have perceived a kind of relief in a fourth dimension so as to 
combine into one picture the aspect of things seen with different 
motions. Finally, if we had had two eyes of different sizes, we 
might have evolved a faculty for combining the points of view 
of the mammoth and the microbe. 

It will be seen that we are not fully equipped by our senses 
for forming an impersonal picture of the world. And it is 
because the deficiency is manifest that we do not hesitate to 
advocate a conception of the world which transcends the images 
familiar to the senses. Such a world can perhaps be grasped, 
but not pictured by the brain. It would be unreasonable to 
limit our thought of nature to what can be comprised in sense- 
pictures. As Lodge has said, our senses were developed by the 
struggle for existence, not for the purpose of philosophising on 
the world. 

Let us compare two well-known books, which might be 
described as elementary treatises on relativity, Alice in Wonder- 
land and Gulliver's Travels. Alice was continually changing size, 
sometimes growing, sometimes on the point of vanishing alto- 
gether. Gulliver remained the same size, but on one occasion 
he encountered a race of men of minute size with everything in 
proportion, and on another voyage a land where everything was 
gigantic. It does not require much reflection to see that both 
authors are describing the same phenomenon a relative change 
of scale of observer and observed. Lewis Carroll took what is 
probably the ordinary scientific view, that the observer had 
changed, rather than that a simultaneous change had occurred 
to all her surroundings. But it would never have appeared like 
that to Alice; she could not have "stepped outside and looked 
at herself," picturing herself as a giant filling the room. She 
would have said that the room had unaccountably shrunk. 
Dean Swift took the truer view of the human mind when he 



n] RELATIVITY 33 

made Gulliver attribute his own changes to the things around 
him ; it never occurred to Gulliver that his own size had altered ; 
and, if he had thought of the explanation, he could scarcely 
have accustomed himself to that way of thinking. But both 
points of view are legitimate. The size of a thing can only be 
imagined as relative to something else; and there is no means of 
assigning the change to one end of the relation rather than the 
other. 

We have seen in the theory of the Michelson-Morley experi- 
ment that, according to current physical views, our standard of 
size the rigid measuring-rod must change according to the 
circumstances of its motion; and the aviator's adventures 
illustrated a similar change in the standard of duration of time. 
Certain rather puzzling irregularities have been discovered in 
the apparent motions of the Sun, Mercury, Venus and the Moon ; 
but there is a strong family resemblance between these, which 
leads us to believe that the real phenomenon is a failure of the 
time-keeping of our standard clock, the Earth. Instances could 
be multiplied where a change of the observer or his standards 
produces or conceals changes in the world around him. 

The object of the relativity theory, however, is not to attempt 
the hopeless task of apportioning responsibility between the 
observer and the external world, but to emphasise that in our 
ordinary description and in our scientific description of natural 
phenomena the two factors are indissolubly united. All the 
familiar terms of physics length, duration of time, motion, 
force, mass, energy, and so on refer primarily to this relative 
knowledge of the world; and it remains to be seen whether any 
of them can be retained in a description of the world which is 
not relative to a particular observer. 

Our first task is a description of the world independent of 
the motion of the observer. The question of the elimination of 
his gauge of magnitude belongs to a later development of the 
theory discussed in Chapter xi. Let us draw a square ABCD on 
a sheet of paper, making the sides equal, to the best of our 
knowledge. We have seen that an aviator flying at 161,000 
miles a second in the direction AB, would judge that the sides 
AB, DC had contracted to half their length, so that for him 
the figure would be an oblong. If it were turned through a right 
E. s. * 



34 RELATIVITY [CH. 

angle AB and DC would expand and the other two sides con- 
tract in his judgment. For us, the lengths of AB and AC are 
equal; for him, one length is twice the other. Clearly length 
cannot be a property inherent in our drawing; it needs the 
specification of some observer. 

We have seen further that duration of time also requires that 
an observer should be specified. The stationary observer and 
the aviator disagreed as to whose cigar lasted the longer time. 

Thus length and duration are not things inherent in the 
external world; they are relations of things in the external 
world to some specified observer. If we grasp this all the mystery 
disappears from the phenomena described in Chapter i. When 
the rod in the Michelson-Morley experiment is turned through 
a right angle it contracts ; that naturally gives the impression 
that something has happened to the rod itself. Nothing whatever 
has happened to the rod the object in the external world. 
Its length has altered, but length is not an intrinsic property of 
the rod, since it is quite indeterminate until some observer is 
specified. Turning the rod through a right angle has altered the 
relation to the observer (implied in the discussion of the experi- 
ment); but the rod itself, or the relation of a molecule at one 
end to a molecule at the other, is unchanged. Measurement of 
length and duration is a comparison with partitions of space 
and time drawn by the observer concerned, with the help of 
apparatus which shares his motion. Nature is not concerned 
with these partitions; it has, as we shall see later, a geometry 
of its own which is of a different type. 

Current physics has hitherto assumed that all observers are 
not to be regarded as on the same footing, and that there is 
some absolute observer whose judgments of length and duration 
are to be treated with respect, because nature pays attention to 
his space-time partitions. He is supposed to be at rest in the 
aether, and the aether materialises his space-partitions so that 
they have a real significance in the external world. This is 
sheer hypothesis, and we shall find it is unsupported by any 
facts. Evidently our proper course is to pursue our investiga- 
tions, and call in this hypothetical observer only if we find there 
is something which he can help to explain. 

We have been leading up from the older physics to the new 



n] RELATIVITY 35 

outlook of relativity, and the reader may feel some doubt as to 
whether the strange phenomena of contraction and time- 
retardation, that were described in the last chapter, are to be 
taken seriously, or are part of a reductio ad absurdum argument. 
The answer is that we believe that the phenomena do occur as 
described ; only the description (like that of all observed pheno- 
mena) concerns the relations of the external world to some 
observer, and not the external world itself. The startling 
character of the phenomena arises from the natural but fallacious 
inference that they involve intrinsic changes in the objects 
themselves. 

We have been considering chiefly the observer's end of the 
observation; we must now turn to the other end the thing 
observed. Although length and duration have no exact counter- 
parts in the external world, it is clear that there is a certain 
ordering of things and events outside us which we must now 
find more appropriate terms to describe. The order of events is 
a four-fold order; we can arrange them as right-and-left, back- 
wards-and-forwards, up-and-down, sooner-and-later. An indi- 
vidual may at first consider these as four independent orders, 
but he will soon attempt to combine some of them. It is 
recognised at once that there is no essential distinction between 
right-and-left and backwards-and-forwards. The observer has 
merely to turn through a right angle and the two are inter- 
changed. If he turns through a smaller angle, he has first to 
combine them, and then to redivide them in a different way. 
Clearly it would be a nuisance to continually combine and re- 
divide; so we get accustomed to the thought of leaving them 
combined in a two-fold or two-dimensional order. The amalga- 
mation of up-and-down is less simple. There are obvious reasons 
for considering this dimension of the world as fundamentally 
distinct from the other two. Yet it would have been a great 
stumbling-block to science if the mind had refused to combine 
space into a three-dimensional whole. The combination has not 
concealed the real distinction of horizontal and vertical, but has 
enabled us to understand more clearly its nature for what 
phenomena it is relevant, and for what irrelevant. We can 
understand how an observer in another country redivides the 
combination into a different vertical and horizontal. We must 

32 



36 RELATIVITY [CH. 

now go further and amalgamate the fourth order, sooner-and- 
later. This is still harder for the mind. It does not imply that 
there is no distinction between space and time; but it gives a 
fresh unbiassed start by which to determine what the nature of 
the distinction is. 

The idea of putting together space and time, so that time is 
regarded as a fourth dimension, is not new. But until recently 
it was regarded as merely a picturesque way of looking at things 
without any deep significance. We can put together time and 
temperature in a thermometer chart, or pressure and volume 
on an indicator-diagram. It is quite non-committal. But our 
theory is going to lead much further than that. We can lay 
two dimensional surfaces sheets of paper on one another till 
we build up a three-dimensional block; but there is a difference 
between a block which is a pile of sheets and a solid block of 
paper. The solid block is the true analogy for the four-dimen- 
sional combination of space- time ; it does not separate naturally 
into a particular set of three-dimensional spaces piled in time- 
order. It can be redivided into such a pile ; but it can be redivided 
in any direction we please. 

Just as the observer by changing his orientation makes a new 
division of the two-dimensional plane into right-and-left, back- 
wards-and-forwards just as the observer by changing his 
longitude makes a new division of three-dimensional space into 
vertical and horizontal so the observer by changing his motion 
makes a new division of the four-dimensional order into time 
and space. 

This will be justified in detail later; it indicates that observers 
with different motions will have different time and space- 
reckoning a conclusion we have already reached from another 
point of view. 

Although different observers separate the four orders differ- 
ently, they all agree that the order of events is four-fold; and 
it appears that this undivided four-fold order is the same for 
all observers. We therefore believe that it is inherent in the 
external world; it is in fact the synthesis, which we have been 
seeking, of the appearances seen by observers having all sorts of 
positions and all sorts of (uniform) motions. It is therefore to 
be regarded as a conception of the real world not relative to any 
particularly circumstanced observer. 



ii] RELATIVITY 37 

The term " real world " is used in the ordinary sense of physics, 
without any intention of prejudging philosophical questions as 
to reality. It has the same degree of reality as was formerly 
attributed to the three-dimensional world of scientific theory or 
everyday conception, which by the advance of knowledge it 
replaces. As I have already indicated, it is merely the accident 
that we are not furnished with a pair of eyes in rapid relative 
motion, which has allowed our brains to neglect to develop a 
faculty for visualising this four-dimensional world as directly 
as we visualise its three-dimensional section. 

It is now easy to see that length and duration must be the 
components of a single entity in the four-dimensional world of 
space-time. Just as we resolve a structure into plan and eleva- 
tion, so we resolve extension in the four-dimensional world into 
length and duration. The structure has a size and shape 
independent of our choice of vertical. Similarly with things in 
space-time. Whereas length and duration are relative, the 
single " extension " of which they are components has an absolute 
significance in nature, independent of the particular decomposi- 
tion into space and time separately adopted by the observer. 
\ Consider two events; for example, the stroke of one o'clock 
and the stroke of two o'clock by Big Ben. These occupy two 
points in space-time, and there is a definite separation between 
them. An observer at Westminster considers that they occur at 
the same place, and that they are separated by an hour in time; 
thus he resolves their four-dimensional separation into zero 
distance in space and one hour distance in time. An observer 
on the sun considers that they do not occur at the same place; 
they are separated by about 70,000 miles, that being the distance 
travelled by the earth in its orbital motion with respect to the 
sun. It is clear that he is not resolving in quite the same direc- 
tions as the terrestrial observer, since he finds the space-com- 
ponent to be 70,000 miles instead of zero. But if he alters one 
component he must necessarily alter the other; so he will make 
the time-component differ slightly from an hour. By analogy 
with resolution into components in three-dimensions, we should 
expect him to make it less than an hour having, as it were, 
borrowed from time to make space ; but as a matter of fact he 
makes it longer. This is because space-time has a different 



38 RELATIVITY [CH. 

geometry, which will be described later. Our present point is 
that there is but one separation of two events in four dimensions, 
which can be resolved in any number of ways into the com- 
ponents length and duration. 

We see further how motion must be purely relative. Take 
two events A and B in the history of one particle. We can choose 
any direction as the time-direction; let us choose it along AB. 
Then A and B are separated only in time and not in space, so 
the particle is at rest. If we choose a slightly inclined time- 
direction, the separation AB will have a component in space; 
the two events then do not occur at the same place, that is to 
say, the particle has moved. The negation of absolute motion 
is thus associated with the possibility of choosing the time- 
direction in any way we please. What determines the separation 
of space and time for any particular observer can now be seen. 
Let the observer place himself so that he is, to the best of his 
knowledge, at rest. If he is a normal human being, he will 
seat himself in an arm-chair; if he is an astronomer, he will 
place himself on the sun or at the centre of the stellar universe. 
Then all the events happening directly to him will in his opinion 
occur at the same place. Their separation will have no space- 
component, and they will accordingly be ranged solely in the 
time-direction. This chain of events, marking his track through 
the four-dimensional world, will be his time-direction. Each 
observer bases his separation of space and time on his own track 
through the world. 

Since any separation of space and time is admissible, it is 
possible for the astronomer to base his space and time on the 
track of a solar observer instead of that of a terrestrial observer; 
but it must be remembered that in practice the space and time 
of the solar observer have to be inferred indirectly from those 
of the terrestrial observer; and, if the corrections are made 
according to the crude methods hitherto employed, they may 
be inferred wrongly (if extreme accuracy is needed). 

The most formidable objection to this relativist view of the 
world is the aether difficulty. We have seen that uniform motion 
through the aether cannot be detected by experiment, and 
therefore it is entirely in accordance with experiment that such 
motion should have no counterpart in the four-dimensional 



n] RELATIVITY 39 

world. Nevertheless, it would almost seem that such motion 
must logically exist, if the aether exists; and, even at the 
expense of formal simplicity, it ought to be exhibited in any 
theory which pretends to give a complete account of what is 
going on in nature. If a substantial aether analogous to a 
material ocean exists, it must rigidify, as it were, a definite 
space; and whether the observer or whether nature pays any 
attention to that space or not, a fundamental separation of 
space and time must be there. Some would cut the knot by 
denying the aether altogether. We do not consider that desirable, 
or, so far as we can see, possible ; but we do deny that the aether 
need have such properties as to separate space and time in the 
way supposed. It seems an abuse of language to speak of a 
division existing, when nothing has ever been found to pay any 
attention to the division. 

Mathematicians of the nineteenth century devoted much time 
to theories of elastic solid and other material aethers. Waves of 
light were supposed to be actual oscillations of this substance ; 
it was thought to have the familiar properties of rigidity and 
density; it was sometimes even assigned a place in the table of 
the elements. The real death-blow to this materialistic concep- 
tion of the aether was given when attempts were made to explain 
matter as some state in the aether. For if matter is vortex- 
motion or beknottedness in aether, the aether cannot be matter 
some state in itself. If any property of matter comes to be 
regarded as a thing to be explained by a theory of its structure, 
clearly that property need not be attributed to the aether. 
If physics evolves a theory of matter which explains some 
property, it stultifies itself when it postulates that the same 
property exists unexplained in the primitive basis of matter. 

Moreover the aether has ceased to take any very active part 
in physical theory and has, as it were, gone into reserve. A 
modern writer on electromagnetic theory will generally start 
with the postulate of an aether pervading all space ; he will then 
explain that at any point in it there is an electromagnetic vector 
whose intensity can be measured; henceforth his sole dealings 
are with this vector, and probably nothing more will be heard 
of the aether itself. In a vague way it is supposed that this 
vector represents some condition of the aether, and we need not 



40 RELATIVITY [CH. 

dispute that without some such background the vector would 
scarcely be intelligible but the aether is now only a background 
and not an active participant in the theory. 

There is accordingly no reason to transfer to this vague back- 
ground of aether the properties of a material ocean. Its properties 
must be determined by experiment, not by analogy. In particular 
there is no reason to suppose that it can partition out space in 
a definite way, as a material ocean would do. We have seen in 
the Prologue that natural geometry depends on laws of matter ; 
therefore it need not apply to the aether. Permanent identity 
of particles is a property of matter, which Lord Kelvin sought 
to explain in his vortex-ring hypothesis. This abandoned 
hypothesis at least teaches us that permanence should not be 
regarded as axiomatic, but may be the result of elaborate con- 
stitution. There need not be anything corresponding to 
permanent identity in the constituent portions of the aether; 
we cannot lay our finger at one spot and say ''this piece of 
aether was a few seconds ago over there." Without any con- 
tinuity of identity of the aether motion through the aether 
becomes meaningless; and it seems likely that this is the true 
reason why no experiment ever reveals it. 

This modern theory of the relativity of all uniform motion is 
essentially a return to the original Newtonian view, temporarily 
disturbed by the introduction of aether problems ; for in Newton's 
dynamics uniform motion of the whole system has not and no 
one would expect it to have any effect. But there are consider- 
able difficulties in the limitation to uniform motion. Newton 
himself seems to have appreciated the difficulty; but the experi- 
mental evidence appeared to him to be against any extension 
of the principle. Accordingly Newton's laws of mechanics are 
not of the general type in which it is unnecessary to particularise 
the observer; they hold only for observers with a special kind 
of motion which is described as "unaccelerated." The only 
definition of this epithet that can be given is that an "un- 
accelerated" observer is one for whom Newton's laws of motion 
Ihold. On this theory, the phenomena are not indifferent to an 
^acceleration or non-uniform motion of the whole system. Yet 
an absolute non-uniform motion through space is just as im- 
possible to imagine as an absolute uniform motion. The partial 



n] RELATIVITY 41 

relativity of phenomena makes the difficulty all the greater. 
If we deny a fundamental medium with continuous identity of 
its parts, motion uniform or non-uniform should have no 
significance; if we admit such a medium, motion uniform or 
non-uniform should be detectable; but it is much more difficult 
to devise a plan of the world according to which uniform motion 
has no significance and non-uniform motion is significant. 

It is through experiment that we have been led back to the 
principle of relativity for uniform motion. In seeking some kind 
of extension of this principle to accelerated motion, we are led 
by the feeling that, having got so far, it is difficult and arbitrary 
to stop at this point. We now try to conceive a system of nature 
for which all kinds of motion of the observer are indifferent. 
It will be a completion of our synthesis of what is perceived by 
observers having all kinds of motions with respect to one 
another, removing the restriction to uniform motion. The 
experimental tests must follow after the consequences of this 
generalisation have been deduced. 

The task of formulating such a theory long appeared impossible. 
It was pointed out by Newton that, whereas there is no criterion 
for detecting whether a body is at rest or in uniform motion, it 
is easy to detect whether it is in rotation. For example the 
bulge of the earth's equator is a sign that the earth is rotating, 
since a plastic body at rest would be spherical. 

This problem of rotation affords a hint as to the cause of the 
incomplete relativity of Newtonian mechanics. The laws of 
motion are formulated with respect to an unaccelerated observer, 
and do not apply to a frame of reference rotating with the earth. 
Yet mathematicians frequently do use a rotating frame. Some 
modification of the laws is then necessary ; and the modification 
is made 'by introducing a centrifugal force not regarded as a 
real force like gravitation, but as a mathematical fiction em- 
ployed to correct for the improper choice of a frame of reference. 
The bulge of the earth's equator may be attributed indifferently 1 
to the earth's rotation or to the outward pull of the centrifugal I 
force introduced when the earth is regarded as non-rotating. 

Now it is generally assumed that the centrifugal force is 
something sui generis, which could always be distinguished 
experimentally from any other natural phenomenon. If then 



42 RELATIVITY [CH. 

on choosing a frame of reference we find that a centrifugal force 
is detected, we can at once infer that the frame of reference is 
a "wrong" one; rotating and non-rotating frames can be dis- 
tinguished by experiment, and rotation is thus strictly absolute. 
But this assumes that the observed effects of centrifugal force 
cannot be produced in any other way than by rotation of the 
observer's frame of reference. If once it is admitted that centri- 
fugal force may not be completely distinguishable by experiment 
from another kind of force gravitation perceived even by 
Newton's unaccelerated observer, the argument ceases to apply. 
We can never determine exactly how much of the observed field 
of force is centrifugal force and how much is gravitation; and 
we cannot find experimentally any definite standard that is to 
be considered absolutely non-rotating. 

The question then, whether there exists a distinction between 
"right" frames of reference and "wrong" frames, turns on 
whether the*use of a "wrong" frame produces effects experi- 
mentally distinguishable from any natural effects which can be 
perceived when a "right" frame is used. If there is no such 
difference, all frames may be regarded as on the same footing 
and equally right. In that case we can have a complete relativity 
of natural phenomena. Since the effect of departing from 
Newton's standard frame is the introduction of a field of force, 
this generalised relativity theory must be largely occupied with 
the nature of fields of force. 

The precise meaning of the statement that all frames of 
reference are on the same footing is rather difficult to grasp. 
We believe that there are absolute things in the world not only 
matter, but certain characteristics in empty space or aether. 
In the atmosphere a frame of reference which moves with the 
air is differentiated from other frames moving in a different 
manner; this is because, besides discharging the normal functions 
of a frame of reference, the air-frame embodies certain of the 
absolute properties of the matter existing in the region. 
Similarly, if in empty space we choose a frame of reference 
which more or less follows the lines of the absolute structure in 
the region, the frame will usurp some of the absolute qualities 
of that structure. What we mean by the equivalence of all 
frames is that they are not differentiated by any qualities 



n] RELATIVITY 43 

formerly supposed to be intrinsic in the frames themselves 
rest, rectangularity, acceleration independent of the absolute 
structure of the world that is referred to them. Accordingly the 
objection to attributing absolute properties to Newton's frame 
of reference is not that it is impossible for a frame of reference 
to acquire absolute properties, but that the Newtonian frame 
has been laid down on the basis of relative knowledge without 
any attempt to follow the lines of absolute structure. 

Force, as known to us observationally, is like the other 
quantities of physics, a relation. The force, measured with a 
spring-balance, for example, depends on the acceleration of the 
observer holding the balance; and the term may, like length 
and duration, have no exact counterpart in a description of 
nature independent of the observer. Newton's view assumes 
that there is such a counterpart, an active cause in nature 
which is identical with the force perceived by his standard 
unaccelerated observer. Although any other observer perceives 
this force with additions of his own, it is implied that the 
original force in nature and the observer's additions can in some 
way be separated without ambiguity. There is no experimental 
foundation for this separation, and the relativity view is that 
a field of force can, like length and duration, be nothing but 
a link between nature and the observer. There is, of course, 
something at the far end of the link, just as we found an 
extension in four dimensions at the far end of the relations 
denoted by length and duration. We shall have to study the 
nature of this unknown whose relation to us appears as force. 
Meanwhile we shall realise that the alteration of perception of 
force by non-uniform motion of the observer, as well as the 
alteration of the perception of length by his uniform motion, is 
what might be expected from the nature of these quantities as 
relations solely. 

We proceed now to a more detailed study of the four-dimen- 
sional world, of the things which occur in it, and of the laws by 
which they are regulated. It is necessary to dive into this 
absolute world to seek the truth about nature; but the physicist's 
object is always to obtain knowledge which can be applied to 
the relative and familiar aspect of the world. The absolute 
world is of so different a nature, that the relative world, with 



44 RELATIVITY [CH. n 

which we are acquainted, seems almost like a dream. But if 
indeed we are dreaming, our concern is with the baseless fabric 
of our vision. We do not suggest that physicists ought to 
translate their results into terms of four-dimensional space for 
the empty satisfaction of working in the realm of reality. It is 
rather the opposite. They explore the new field and bring back 
their spoils a few simple generalisations to apply them to the 
practical world of three-dimensions. Some guiding light will be 
given to the attempts to build a scheme of things entire. For 
the rest, physics will continue undisturbedly to explore the 
relative world, and to employ the terms applicable to relative 
knowledge, but with a fuller appreciation of its relativity. 



CHAPTER III 
THE WORLD OF FOUR DIMENSIONS 

Here is a portrait of a man at eight years old, another at fifteen, another at 
seventeen, another at twenty-three, and so on. All these are evidently sections, 
as it were, Three -Dimensional representations of his Four-Dimensional being, 
which is a fixed and unalterable thing. H. G. WELLS, The Time Machine. 

THE distinction between horizontal and vertical is not an 
illusion; and the man who thinks it can be disregarded is likely 
to come to an untimely end. Yet we cannot arrive at a com- 
prehensive view of nature unless we combine horizontal and 
vertical dimensions into a three-dimensional space. By doing 
this we obtain a better idea of what the distinction of horizontal 
and vertical really is in those cases where it is relevant, e.g. the 
phenomena of motion of a projectile c We recognise also that 
vertical is not a universally differentiated direction in space, as 
the flat-earth philosophers might have imagined. 

Similarly by combining the time-ordering and space-ordering 
of the events of nature into a single order of four dimensions, 
we shall not only obtain greater simplicity for the phenomena 
in which the separation of time and space is irrelevant, but we 
shall understand better the nature of the differentiation when it 
is relevant. 

^ A point in this space-time, that is to say a given instant at 
a given place, is called an "event." An event in its customary 
meaning would be the physical happening which occurs at and 
identifies a particular place and time. However, we shall use 
the word in both senses, because it is scarcely possible to think 
of a point in space-time without imagining some identifying 
occurrence. 

In the ordinary geometry of two or three dimensions, the 
distance between two points is something which can be measured, 
usually with a rigid scale; it is supposed to be the same for all 
observers, and there is no need to specify horizontal and vertical 
directions or a particular system of coordinates. In four- 
dimensional space^time there is likewise a certain extension or 



46 THE WORLD OF FOUR DIMENSIONS [CH. 

generalised distance between two events, of which the distance 
in space and the separation in time are particular components. 
This extension in space and time combined is called the 
"interval" between the two events; it is the same for all 
observers, however they resolve it into space and time separately. 
We may think of the interval as something intrinsic in external 
nature an absolute relation of the two events, which postulates 
no particular observer. Its practical measurement is suggested 
by analogy with the distance of two points in space. 

In two dimensions on a plane, two points P lt P 2 (Fig. 2) can 

be specified by their rectangular 
coordinates (x 1} t/j) and (x 2 , y 2 ), 
when arbitrary axes have been se- 
lected. In the figure, OX t = #j , 
, etc. We have 




XiXf + 

(X 2 - X 



Xi x 2 x S o that if s is the distance between 

FIG. 2. P 1 and P 2 

s 2 = (x 2 - xtf + (y 2 - ytf. 
The extension to three dimensions is, as we should expect, 

* 2 = (x 2 - xj* + (y 2 - t/J 2 + (z 2 - zj*. 

Introducing the times of the events t lt t 2 , we should naturally 
expect that the interval in the four-dimensional world would 
be given by 

* 2 = (x 2 - ^) 2 + (y 2 - ytf + ( - %) 2 + (*, - <i) 2 . 
An important point arises here. It was, of course, assumed 
that the same scale was used for measuring x and y and z. But 
how are we to use the same scale for measuring tt We cannot 
use a scale at all; some kind of clock is needed. The most 
natural connection between the measure of time and length is 
given by the fact that light travels 300,000 kilometres in 1 
second. For the four-dimensional world we shall accordingly 
regard 1 second as the equivalent of 300,000 kilometres, and 
measure lengths and times in seconds or kilometres indiscrimi- 
nately ; in other words we make the velocity of light the unit of 



m] THE WORLD OF FOUR DIMENSIONS 47 

velocity. It is not essential to do this, but it greatly simplifies 
the discussion. 

Secondly, the formulae here given for s 2 are the characteristic 
formulae of Euclidean geometry. So far as three-dimensional 
space is concerned the applicability of Euclidean geometry is 
very closely confirmed by experiment. But space-time is not 
Euclidean; it does, however, conform (at least approximately) 
to a very simple modification of Euclidean geometry indicated 
by the corrected formula 

s* = (, - *,) + (y t - y^ + (, - *i) 2 - (* - <!)* 

There is only a sign altered; but that minus sign is the secret 
of the differences of the manifestations of time and space in 
nature. 

This change of sign is often found puzzling at the start. We 
could not define s by the expression originally proposed (with 
the positive sign), because the expression does not define any- 
thing objective. Using the space and time of one observer, one 
value is obtained; for another observer, another value is 
obtained. But if s is defined by the 'expression now given, it is 
found that the same result is obtained by all observers*. The 
quantity s is thus something which concerns solely the two 
events chosen ; we give it a name the interval between the two 
events. In ordinary space the distance between two points is 
the corresponding property, which concerns only the two points 
and not the extraneous coordinate system of location which is 
used. Hence interval, as here defined, is the analogue of dis- 
tance ; and the analogy is strengthened by the evident resemblance 
of the formula for s in both cases. Moreover, when the difference 
of time vanishes, the interval reduces to the distance. But the 
discrepancy of sign introduces certain important differences. 
These differences are summed up in the statement that the 
geometry of space is Euclidean, but the geometry of space-time 
is semi-Euclidean or " hyperbolic." The association of a geometry 
with any continuum always implies the existence of some 
uniquely measurable quantity like interval or distance; in 
ordinary space, geometry without the idea of distance would be 
meaningless. 

* Appendix, Note 2. 



48 THE WORLD OF FOUR DIMENSIONS [CH. 

For the moment the difficulty of thinking in terms of an 
unfamiliar geometry may be evaded by a dodge. Instead of 
real time t, consider imaginary time r; that is to say, let 



t= r - 1. 

Then (* 2 - tj* = - (r 2 - Tl ) 2 , 

so that 

s 2 = (x 2 - xtf + (2/2 - 2/i) 2 + (* 2 - *i) a + (T, - T X ) 2 . 
Everything is now symmetrical and there is no distinction 
between r and the other variables. The continuum formed of 
space and imaginary time is completely isotropic for all measure- 
ments; no direction can be picked out in it as fundamentally 
distinct from any other. 

The observer's separation of this continuum into space and 
time consists in slicing it in some direction, viz. that perpen- 
dicular to the path along which he is himself travelling. The 
section gives three-dimensional space at some moment, and the 
perpendicular dimension is (imaginary) time. Clearly the slice 
may be taken in any direction; there is no question of a true 
separation and a fictitious separation. There is no conspiracy 
of the forces of nature to conceal our absolute motion because, 
looked at from this broader point of view, there is nothing to 
conceal. The observer is at liberty to orient his rectangular axes 
of x 9 y, z and T arbitrarily, just as in three-dimensions he can 
orient his axes of x, y, z arbitrarily. 

It can be shown that the different space and time used by 
the aviator in Chapter I correspond to an orientation of the 
time-axis along his own course in the four-dimensional world, 
whereas the ordinary time and space are given when the time- 
axis is oriented along the course of a terrestrial observer. The 
FitzGerald contraction and the change of time-measurement 
are given exactly by the usual formulae for rotation of rect- 
angular axes*. 

It is not very profitable to speculate on the implication of the 
mysterious factor V 1, which seems to have the property of 
turning time into space. It can scarcely be regarded as more 
than an analytical device. To follow out the theory of the four- 

* Appendix, Note 3. 



m] THE WORLD OF FOUR DIMENSIONS 49 

dimensional world in more detail, it is necessary to return to 
real time, and face the difficulties of a strange geometry. 

Consider a particular observer, S, and represent time according 
to his reckoning by distance up the page parallel to OT. One 
dimension of his space will be represented by horizontal distance 
parallel to OX; another will stand out at right angles from the 
page; and the reader must imagine the third as best he can. 
Fortunately it will be sufficient for us to consider only the one 
dimension of space OX and deal with the phenomena of "line- 




FIG. 3. 

land," i.e. we limit ourselves to motion to and fro in one straight 
line in space. 

The two lines U'OU, V'OV, at 45 to the axes, represent the 
tracks of points which progress 1 unit horizontally (in space) 
for 1 unit vertically (in time) ; thus they represent points moving 
with unit velocity. We have chosen the velocity of light as unit 
velocity; hence U'OU, V'OV will be the tracks of pulses of light 
in opposite directions along the straight line. 

Any event P within the sector UOV is indubitably after the 
event O, whatever system of time-reckoning is adopted. For it 
would be possible for a material particle to travel from O to P 9 
the necessary velocity being less than that of light; and no 



50 THE WORLD OF FOUR DIMENSIONS [CH. 

rational observer would venture to state that the particle had 
completed its journey before it had begun it. It would, in fact, 
be possible for an observer travelling along NP to receive a 
light-signal or wireless telegram announcing the event 0, just 
as he reached N, since ON is the track of such a message ; and 
then after the time NP he would have direct experience of the 
event P. To have actual evidence of the occurrence of one 
event before experiencing the second is a clear proof of their 
absolute order in nature, which should convince not merely 
the observer concerned but any other observer with whom he 
can communicate. 

Similarly events in the sector U'OV are indubitably before 
the event 0. 

With regard to an event P' in the sector UOV or YOU' we 
v/ cannot assert that it is absolutely before or after O. According 
to the time-reckoning of our chosen observer S, P' is after 0, 
because it lies above the line OX; but there is nothing absolute 
about this. The track OP' corresponds to a velocity greater 
than that of light, so that we know of no particle or physical 
impulse which could follow the track. An observer experiencing 
the event P' could not get news of the event by any known 
means until after P' had happened. The order of the two events 
can therefore only be inferred by estimating the delay of the 
message and this estimate will depend on the observer's mode 
of reckoning space and time. 

Space-time is thus divided into three zones with respect to 
the event O. U'OV belongs to the indubitable past. UOV is 
the indubitable future. UOV and YOU' are (absolutely) neither 
past nor future, but simply "elsewhere." It may be remarked 
that, as we have no means of identifying points in space as "the 
same point," and as the events O and P might quite well happen 
to the same particle of matter, the events are not necessarily to 
be regarded as in different places, though the observer S will 
judge them so; but the events O and P' cannot happen to the 
same particle, and no observer could regard them as happening 
at the same place. The main interest of this analysis is that it 
shows that the arbitrariness of time-direction is not inconsistent 
with the existence of regions of absolute past and future. 

Although there is an absolute past and future, there is between 



m] THE WORLD OF FOUR DIMENSIONS 51 

them an extended neutral zone; and simultaneity of events at 
different places has no absolute meaning. For our selected 
observer all events along OX are simultaneous with one another; 
for another observer the line of events simultaneous with 
would lie in a different direction. The denial of absolute 
simultaneity is a natural complement to the denial of absolute 
motion. The latter asserts that we cannot find out what is the 
same place at two different times; the former that we cannot 
find out what is the same time at two different places. It is 
curious that the philosophical denial of absolute motion is 
readily accepted, whilst the denial of absolute simultaneity 
appears to many people revolutionary. 

The division into past and future (a feature of time-order 
which has no analogy in space-order) is closely associated with 
our ideas of causation and free-will. In a perfectly determinate 
scheme the past and future may be regarded as lying mapped 
out as much available to present exploration as the distant 
parts of space. Events do not happen; they are just there, and 
we come across them. " The formality of taking place" is merely 
the indication that the observer has on his voyage of exploration 
passed into the absolute future of the event in question ; and it 
has no important significance. We can be aware of an eclipse 
in the year 1999, very much as we are aware of an unseen 
companion to Algol. Our knowledge of things where we are not, 
and of things when we are not, is essentially the same an 
inference (sometimes a mistaken inference) from brain impres- 
sions, including memory, here and now. 

So, if events are determinate, there is nothing to prevent a 
person from being aware of an event before it happens ; and an 
event may cause other events previous to it. Thus the eclipse 
of the Sun in May 1919 caused observers to embark in March. 
It may be said that it was not the eclipse, but the calculations 
of the eclipse, which caused the embarkation; but I do not 
think any such distinction is possible, having regard to the 
indirect character of our acquaintance with all events except 
those at the precise point of space where we stand. A detached 
observer contemplating our world would see some events 
apparently causing events in their future, others apparently 
causing events in their past the truth being that all are linked 



52 THE WORLD OF FOUR DIMENSIONS [CH. 

by determinate laws, the so-called causal events being merely 
conspicuous foci from which the links radiate. 

The recognition of an absolute past and future seems to 
depend on the possibility of events which are not governed by 
a determinate scheme. If, say, the event O is an ultimatum, 
and the person describing the path NP is a ruler of the country 
affected, then it may be manifest to all observers that it is his 
knowledge of the actual occurrence of the event O which has 
caused him to create the event P. P must then be in the absolute 




FIG. 4. 

future of 0, and, as we have seen, must lie in the sector UOV. 
But the inference is only permissible, if the event P could be 
determined by the event 0, and was not predetermined by 
causes anterior to both if it was possible for it to happen or 
not, consistently with the laws of nature. Since physics does 
not attempt to cover indeterminate events of this kind, the 
distinction of absolute past and future is not directly important 
for physics ; but it is of interest to show that the theory of four- 
dimensional space-time provides an absolute past and future, in 
accordance with common requirements, although this can 
usually be ignored in applications to physics. 



m] THE WORLD OF FOUR DIMENSIONS 53 

Consider now all the events which are at an interval of one 
unit from 0, according to the definition of the interval s 

s 2 = - (* a - xtf - (y 2 - ytf - (* a - *!) 2 + (t 2 - *x) 2 ...(l). 
We have changed the sign of s 2 , because usually (though not 
always) the original s 2 would have come out negative. In 
Euclidean space points distant a unit interval lie on a circle: 
but, owing to the change in geometry due to the altered sign 
of (t 2 Zi) 2 , they now lie on a rectangular hyperbola with two 
branches KLM, K'L'M'. Since the interval is an absolute 
quantity, all observers will agree that these points are at unit 
interval from O. 

Now make the following construction: draw a straight line 
OFTj , to meet the hyperbola in F; draw the tangent FG at F, 
meeting the light-line U'OU in G; complete the parallelogram 
OFGH; produce OH to X^ We now assert that an observer 
S l who chooses OT for his time-direction will regard OX as 
his space direction and will consider OF and OH to be the units 
of time and space. 

The two observers make their partitions of space and time 
in different ways, as illustrated in Figs. 5 and 6, where in each 
case the partitions are at unit distance (in space and time) 
according to the observers' own reckoning. The same" diagram 
of events in the world will serve for both observers ; S 1 merely 
removes *S"s partitions and overlays his own, locating the events 
in his space and time accordingly. It will be seen at once that 
the lines of unit velocity progress of one unit of space for one 
unit of time agree, so that the velocity of a pulse of light is 
unity for both observers. It can be shown from the properties 
of the hyperbola that the locus of points at any interval s from 
O, given by equation (1), viz. 

* 2 = (t - t Q )* -(x- x.)\ 

is the same locus (a hyperbola) for both systems of reckoning 
a? and t. The two observers will always agree on the measures 
of intervals, though they will disagree about lengths, durations, 
and the velocities of everything except light. This rather com- 
plex transformation is mathematically equivalent to the simple 
rotation of the axes required when imaginary time is used. 
It must not be supposed that there is any natural distinction 



54 THE WORLD OF FOUR DIMENSIONS [CH. 

corresponding to the difference between the square-partitions 
of observer S and the diamond-shaped partitions of observer S l . 
We might say that S l transplants the space-time world un- 
changed from Fig. 5 to Fig. 6, and then distorts it until the 
diamonds shown become squares ; or we might equally well start 
with this distorted space-time, partitioned by S into squares, 
and then S's partitions would be represented by diamonds. 
It cannot be said that either observer's space-time is distorted 
absolutely, but one is distorted relatively to the other. It is the 
relation of order which is intrinsic in nature, and is the same 
both for the squares and diamonds ; shape is put into nature by 
the observer when he has chosen his partitions. 




Xi 



o 



X O 



FIG. 5. 



FIG. 6. 



We can now deduce the FitzGerald contraction. Consider 
a rod of unit length at rest relatively to the observer S. The 
two extremities are at rest in his space, and consequently remain 
on the same space-partitions ; hence their tracks in four dimen- 
sions PP', QQ' (Fig. 7) are entirely in the time-direction. The 
real rod in nature is the four-dimensional object shown in section 
as P'PQQ'. Overlay the same figure with S^s space and time 
partitions, shown by the dotted lines. Taking a section at any 
one "time," the instantaneous rod is PjQx, viz. the section of 
P'PQQ' by S^s time-line. Although on paper P 1 Q 1 is actually 
longer than PQ, it is seen that it is a little shorter than one of 
S^s space-partitions; and accordingly Sj_ judges that it is less 



Ill] 



THE WORLD OF FOUR DIMENSIONS 



55 



than one unit long it has contracted on account of its motion 
relative to him. 

Similarly RR' - SS' is a rod of unit length at rest relatively 
to Si . Overlaying S's partitions we see that it occupies -R^ at 



Q' 





1 


.-,' 




/,* 


/ 


^ 


' f 


/ 


**" 


1 


I 




1 


^ + 


j 


**"*! 


r Qi 


^' 


p, / 




/ 






/ 






/ 

rj 




n 




a particular instant for S; and this is less than one of S's 
partitions. Thus S judges it to have contracted on account of 
its motion relative to him. 

In the same way we can illustrate the problem of the duration 
of the cigar; each observer 
believed the other's cigar to 
last the longer time. Taking 
LM (Fig. 8) to represent the ** 
duration of $'s cigar (two 
units), we see that in S^s 
reckoning it reaches over a 
little more than two time- 
partitions. Moreover it has not 
kept to one space-partition, 
i.e. it has moved. Similarly L'N' is the duration of S^s cigar 
(two time-units for him) ; and it lasts a little beyond two unit- 
partitions in >S"s time-reckoning. (Note, in comparing the two 
diagrams, L', M', N' are the same points as L, M, N.) 

If in Fig. 4 we had taken the line OT^ very near to OU, our 



FIG. 8. 



56 THE WORLD OF FOUR DIMENSIONS [CH. 

diamonds would have been very elongated, and the unit- 
divisions OF, OH very large. This kind of partition would be 
made by an observer whose course through the world is OT l} 
and who is accordingly travelling with a velocity approaching 
that of light relative to S. In the limit, when the velocity 
reaches that of light, both space-unit and time-unit become 
infinite, so that in the natural units for an observer travelling 
with the speed of light, all the events in the finite experience of 
S take place "in no time" and the size of every object is zero. 
This applies, however, only to the two dimensions x and t; the 
space-partitions parallel to the plane of the paper are not 
affected by this motion along x. Consequently for an observer 
travelling with the speed of light all ordinary objects become 
two-dimensional, preserving their lateral dimensions, but in- 
finitely thin longitudinally. The fact that events take place "in 
no time" is usually explained by saying that the inertia of any 
particle moving with the velocity of light becomes infinite so 
that all molecular processes in the observer must stop; many 
things may happen in $'s world in a twinkling of an eye of 
Si's eye. 

However successful the theory of a four-dimensional world 
may be, it is difficult to ignore a voice inside us which whispers 
"At the back of your mind, you know that a fourth dimension 
is all nonsense." I fancy that that voice must often have had a 
busy time in the past history of physics. What nonsense to 
say that this solid table on which I am writing is a collection 
of electrons moving with prodigious speeds in empty spaces, 
which relatively to electronic dimensions are as wide as the 
spaces between the planets in the solar system ! What nonsense 
to say that the thin air is trying to crush my body with a load 
of 14 Ibs. to the square inch! What nonsense that the star- 
cluster, which I see through the telescope obviously there now, 
is a glimpse into a past age 50,000 years ago! Let us not be 
beguiled by this voice. It is discredited. 

But the statement that time is a fourth dimension may 
suggest unnecessary difficulties which a more precise definition 
avoids. It is in the external world that the four dimensions are 
united not in the relations of the external world to the 
individual which constitute his direct acquaintance with space 



m] THE WORLD OF FOUR DIMENSIONS 57 

and time. Just in that process of relation to an individual, the 
order falls apart into the distinct manifestations of space and time. 
An individual is a four-dimensional object of greatly elongated 
form; in ordinary language we say that he has considerable exten- 
sion in time and insignificant extension in space. Practically he 
is represented by a line his track through the world. When the 
world is related to such an individual, his own asymmetry is 
introduced into the relation; and that order of events which is 
parallel with his track, that is to say with himself, appears in 
his experience to be differentiated from all other orders of events. 
Probably the best known exposition of the fourth dimension 
is that given in E. Abbott's popular book Flatland. It may be of 
interest to see how far the four-dimensional world of space-time 
conforms with his anticipations. He lays stress on three points. 

(1) As a four-dimensional body moves, its section by the 
three-dimensional world may vary; thus a rigid body can alter 
size and shape. 

(2) It should be possible for a body to enter a completely 
closed room, by travelling into it in the direction of the fourth 
dimension, just as we can bring our pencil down on to any point 
within a square without crossing its sides. 

(3) It should be possible to see the inside of a solid, just as 
we can see the inside of a square by viewing it from a point 
outside its plane. 

The first phenomenon is manifested by the FitzGerald con- 
traction. 

If quantity of matter is to be identified with its mass, the 
second phenomenon does not happen. It could easily be con- 
ceived of as happening, but it is provided against by a special 
law of nature the conservation of mass. It could happen, 
but it does not happen. 

The third phenomenon does not happen for two reasons. 
A natural body extends in time as well as in space, and is 
therefore four-dimensional; but for the analogy to hold, the 
object must have one dimension less than the world, like the 
square seen from the third dimension. If the solid suddenly 
went out of existence so as to present a plane section towards 
time, we should still fail to see the interior of it ; because light- 
tracks in four-dimensions are restricted to certain lines like 



58 THE WORLD OF FOUR DIMENSIONS [CH. 

UOV, U'OV in Fig. 3, whereas in three-dimensions light can 
traverse any straight line. This could be remedied by interposing 
some kind of dispersive medium, so that light of some wave- 
length could be found travelling with every velocity and following 
every track in space-time ; then, looking at a solid which suddenly 
went out of existence, we should receive at the same moment 
light-impressions from every particle in its interior (supposing 
them self-luminous). We actually should see the inside of it. 

How our poor eyes are to disentangle this overwhelming 
experience is quite another question. 

The interval is a quantity so fundamental for us that we may 
consider its measurement in some detail. Suppose we have a 
scale AB divided into kilometres, say, and at each division is 
placed a clock also registering kilometres. (It will be remembered 
that time can be measured in seconds or kilometres indifferently.) 



1 


' 2 


1 3l 


4 


J 



FIG. 9. 

When the clocks are correctly set and viewed from A the sum 
of the readings of any clock and the division beside it is the 
same for all, since the scale-reading gives the correction for the 
time taken by light, travelling with unit velocity, to reach A. 
This is shown in Fig. 9 where the clock-readings are given as 
though they were being viewed from A. 

Now lay the scale in line with the two events ; note the clock 
and scale-readings t lt x lt of the first event, and the corresponding 
readings t 2 , x 2 , of the second event. Then by the formula 
already given 

* = (*. ~ *l) 2 - (*2 - *l) 2 ' 

But suppose we took a different standard of rest, and set the 
scale moving uniformly in the direction AB. Then the divisions 
would have advanced to meet the second event, and (x 2 #1) 
would be smaller. This is compensated, because t 2 ^ also 
becomes altered. A is now advancing to meet the light coming 
from any of the clocks along the rod; the light arrives too 



in] THE WORLD OF FOUR DIMENSIONS 59 

quickly, and in the initial adjustment described above the clock 
must be set back a little. The clock-reading of the event is thus 
smaller. There are other small corrections arising from the 
FitzGerald contraction, etc. ; and the net result is that, it does 
not matter what uniform motion is given to the scale, the final 
result for s is always the same. 

In elementary mechanics we are taught that velocities can be 
compounded by adding. If l?'s velocity relative to A (as observed 
by either of them) is 100 km. per sec., and C's velocity relative 
to B is 100 km. per sec. in the same direction, then C's velocity 
relative to A should be 200 km. per sec. This is not quite 
accurate; the true answer is 199-999978 km. per sec. The dis- 
crepancy is not difficult to explain. The two velocities and their 
resultant are not all reckoned with respect to the same partitions 
of space and time. When B measures C's velocity relative to 
him he uses his own space and time, and it must be corrected 
to reduce to A's space and time units, before it can be added 
on to a velocity measured by A. 

If we continue the chain, introducing D whose velocity 
relative to C, and measured by C, is 100 km. per sec., and so on 
ad infinitum, we never obtain an infinite velocity with respect 
to A, but gradually approach the limiting velocity of 300,000 
km. per sec., the speed of light. This speed has the remarkable 
property of being absolute, whereas every other speed is relative. 
If a speed of 100 km. per sec. or of 100,000 km. per sec. is 
mentioned, we have to ask speed relative to what? But if 
a speed of 300,000 km. per sec. is mentioned, there is no need 
to ask the question; the answer is relative to any and every 
piece of matter. A ft particle shot off from radium can move at 
more than 200,000 km. per sec. ; but the speed of light relative 
to an observer travelling with it is still 300,000 km. per sec. It 
reminds us of the mathematicians' transfinite number Aleph; 
you can subtract any number you like from it, and it still 
remains the same. 

The velocity of light plays a conspicuous part in the relativity 
theory, and it is of importance to understand what is the 
property associated with it which makes it fundamental. The 
fact that the velocity of light is the same for all observers is a 
consequence rather than a cause of its pre-eminent character. 




60 THE WORLD OF FOUR DIMENSIONS [CH. 

Our first introduction of it, for the purpose of coordinating 
units of length and time, was merely conventional with a view 
to simplifying the algebraic expressions. Subsequently, con- 
siderable use has been made of the fact that nothing is known 
in physics which travels with greater speed, so that in practice 
our determinations of simultaneity depend on signals trans- 
mitted with this speed. If some new kind of ray with a higher 
speed were discovered, it would perhaps tend to displace light- 
signals and light- velocity in this part of the work, time-reckoning 
being modified to correspond; on the other hand, this would 
lead to greater complexity in the formulae, because the Fitz- 
Gerald contraction which affects space-measurement depends 
on light- velocity. But the chief importance of the velocity of 
light is that no material body can exceed this velocity. This 
gives a general physical distinction between paths which are time- 
like and space-like, respectively those which can be traversed 
by matter, and those which cannot. The material structure of 
the four-dimensional world is fibrous, with the threads all running 
along time-like tracks; it is a tangled warp without a woof. 
Hence, even if the discovery of a new ray led us to modify the 
reckoning of time and space, it would still be necessary in the 
study of material systems to preserve the present absolute 
distinction of time-like and space-like intervals, under a new 
name if necessary. 

It may be asked whether it is possible for anything to have 
a speed greater than the velocity of light. Certainly matter 
cannot attain a greater speed; but there might be other things 
in nature which could. "Mr Speaker," said Sir Boyle Roche, 
" not being a bird, I could not be in two places at the same time." 
Any entity with a speed greater than light would have the 
peculiarity of Sir Boyle Roche's bird. It can scarcely be said to 
be a self-contradictory property to be in two places at the same 
time any more than for an object to be at two times in the same 
place. The perplexities of the quantum theory of energy some- 
times seem to suggest that the possibility ought not to be 
overlooked ; but, on the whole, the evidence seems to be against 
the existence of anything moving with a speed beyond that of 
light. 

The standpoint of relativity and the principle of relativity 



in] THE WORLD OF FOUR DIMENSIONS 61 

are quite independent of any views as to the constitution of 
matter or light. Hitherto our only reference to electrical theory 
has been in connection with Larmor and Lorentz's explanation 
of the FitzGerald contraction; but now from the discussion of 
the four-dimensional world, we have found a more general 
explanation of the change of length. The case for the electrical 
theory of matter is actually weakened, because many experi- I 
mental effects formerly thought to depend on the peculiar J 
properties of electrical forces are now found to be perfectly j 
general consequences of the relativity of observational know- j 
ledge. 

Whilst the evidence for the electrical theory of matter is not 
so conclusive, as at one time appeared, the theory may be 
accepted without serious misgivings. To postulate two entities, 
matter and electric charges, when one will suffice is an arbitrary 
hypothesis, unjustifiable in our present state of knowledge. The 
great contribution of the electrical theory to this subject is a 
precise explanation of the property of inertia. It was shown 
theoretically by J. J. Thomson that if a charged conductor is 
to be moved or stopped, additional effort will be necessary 
simply on account of the charge. The conductor has to carry 
its electric field with it, and force is needed to set the field 
moving. This property is called inertia, and it is measured by 
mass. If, keeping the charge constant, the size of the conductor 
is diminished, this inertia increases. Since the smallest separable 
particles of matter are found by experiment to be very minute 
and to carry charges, the suggestion arises that these charges 
may be responsible for the whole of the inertia detected in 
matter. The explanation is sufficient; and there seems no reason 
to doubt that all inertia is of this electrical kind. 

When the calculations are extended to charges moving with 
high velocities, it is found that the electrical inertia is not 
strictly constant but depends on the speed; in all cases the 
variation is summed up in the statement that the inertia is 
simply proportional to the total energy of the electromagnetic 
field. We can say if we like that the mass of a charged particle 
at rest belongs to its electrostatic energy; when the charge is 
set in motion, kinetic energy is added, and this kinetic energy 
also has mass. Hence it appears that mass (inertia) and energy 



62 THE WORLD OF FOUR DIMENSIONS [CH.III 

are essentially the same thing, or, at the most, two aspects of 
the same thing. It must be remembered that on this view the 
greater part of the mass of matter is due to concealed energy, 
which is not as yet releasable. 

The question whether electrical energy not bound to electric 
charges has mass, is answered in the affirmative in the case of 
light. Light has mass. Presumably also gravitational energy 
has mass; or, if not, mass will be created when, as often happens, 
gravitational energy is converted into kinetic energy. The mass 
of the whole (negative) gravitational energy of the earth is of 
the order minus a billion tons. 

The theoretical increase of the mass of an electron with speed 
has been confirmed experimentally, the agreement with calcula- 
tion being perfect if the electron undergoes the FitzGerald 
contraction by its motion. This has been held to indicate that 
the electron cannot have any inertia other than that due to the 
electromagnetic field carried with it. But the conclusion (though 
probable enough) is not a fair inference; because these results, 
obtained by special calculation for electrical inertia, are found 
to be predicted by the theory of relativity for any kind of 
inertia. This will be shown in Chapter ix. The factor giving 
the increase of mass with speed is the same as that which affects 
length and time. Thus if a rod moves at such a speed that its 
length is halved, its mass will be doubled. Its density will be 
increased four-fold, since it is both heavier and less in volume. 

We have thought it necessary to include this brief summary 
of the electrical theory of matter and mass, because, although 
it is not required by the relativity theory, it is so universally 
accepted in physics that we can scarcely ignore it. Later on we 
shall reach in a more general way the identification of mass with 
energy and the variation of mass with speed; but, since the 
experimental measurement of inertia involves the study of a 
body in non-uniform motion, it is not possible to enter on a 
satisfactory discussion of mass until the more general theory of 
relativity for non-uniform motion has been developed. 



CHAPTER IV 
FIELDS OF FORCE 

For whenever bodies fall through water and thin air, they must quicken their 
descents in proportion to their weights, because the body of water and subtle 
nature of air cannot retard everything in equal degree, but more readily give 
way overpowered by the heavier; on the other hand empty void cannot offer 
resistance to anything in any direction at any time, but must, as its nature 
craves, continually give way; and for this reason all things must be moved and 
borne along with equal velocities though of unequal weights through the 
unresisting void. LUCRETIUS, De Natura Rerum, 

THE primary conception offeree is associated with the muscular 
sensation felt when we make an effort to cause or prevent the 
motion of matter. Similar effects on the motion of matter can 
be caused by non-living agency, and these also are regarded as 
due to forces. As is well known, the scientific measure of a force 
is the momentum that it communicates to a body in given time. 
There is nothing very abstract about a force transmitted by 
material contact ; modern physics shows that the momentum is 
communicated by a process of molecular bombardment. We can 
visualise the mechanism, and see the molecules carrying the 
motion in small parcels across the boundary into the body that 
is being acted on. Force is no mysterious agency; it is merely 
a convenient summary of this flow of motion, which we can 
trace continuously if we take the trouble. It is true that the 
difficulties are only set back a stage, and the exact mode by 
which the momentum is redistributed during a molecular 
collision is not yet understood; but, so far as it goes, this analysis 
gives a clear idea of the transmission of motion by ordinary 
forces. 

But even in elementary mechanics an important natural force 
appears, which does not seem to operate in this manner. Gravita- 
tion is not resolvable into a succession of molecular blows. 
A massive body, such as the earth, seems to be surrounded by 
a field of latent force, ready, if another body enters the field, to 
become active, and transmit motion. One usually thinks of this 
influence as existing in the space round the earth even when 



64 FIELDS OF FORCE [CH. 

there is no test-body to be affected, and in a rather vague way 
it is suspected to be some state of strain or other condition of 
an unperceived medium. 

Although gravitation has been recognised for thousands of 
years, and its laws were formulated with sufficient accuracy for 
almost all purposes more than 200 years ago, it cannot be said 
that much progress has been made in explaining the nature or 
mechanism of this influence. It is said that more than 200 
theories of gravitation have been put forward; but the most 
plausible of these have all had the defect that they lead nowhere 
and admit of no experimental test. Many of them would nowa- 
days be dismissed as too materialistic for our taste filling space 
with the hum of machinery a procedure curiously popular in 
the nineteenth century. Few would survive the recent discovery 
that gravitation acts not only on the molecules of matter, but 
on the undulations of light. 

The nature of gravitation has seemed very mysterious, yet it 
is a remarkable fact that in a limited region it is possible to 
create an artificial field of force which imitates a natural 
gravitational field so exactly that, so far as experiments have 
yet gone, no one can tell the difference. Those who seek for an 
explanation of gravitation naturally aim to find a model which 
will reproduce its effects; but no one before Einstein seems to 
have thought of finding the clue in these artificial fields, familiar 
as they are. 

When a lift starts to move upwards the occupants feel a 
characteristic sensation, which is actually identical with a 
sensation of increased weight. The feeling disappears as soon 
as the motion becomes uniform; it is associated only with the 
change of motion of the lift, that is to say, the acceleration. 
Increased weight is not only a matter of sensation ; it is shown 
by any physical experiments that can be performed. The usual 
laboratory determination of the value of gravity by Atwood's 
machine would, if carried out inside the accelerated lift, give 
a higher value. A spring-balance would record higher weights. 
Projectiles would follow the usual laws of motion but with a 
higher value of gravity. In fact, the upward acceleration of 
the lift is in its mechanical effects exactly similar to an additional 
gravitational field superimposed on that normally present. 



iv] FIELDS OF FORCE 65 

Perhaps the equivalence is most easily seen when we produce 
in this manner an artificial field which just neutralises the earth's 
field of gravitation. Jules Verne's book Round the Moon tells 
the story of three men in a projectile shot from a cannon into 
space. The author enlarges on their amusing experiences when 
their weight vanished altogether at the neutral point, where the 
attraction of the earth and moon balance one another. As a 
matter of fact they would not have had any feeling of weight 
at any time during their journey after they left the earth's 
atmosphere. The projectile was responding freely to the pull of 
gravity, and so were its occupants. When an occupant let go 
of a plate, the plate could not "fall" any more than it was 
doing already, and so it must remain poised. 

It will be seen that the sensation of weight is not felt when 
we are free to respond to the force of gravitation; it is only 
felt when something interferes to prevent our falling. It is 
primarily the floor or the chair which causes the sensation of 
weight by checking the fall. It seems literally true to say that 
we never feel the force of the earth's gravitation ; what we do 
feel is the bombardment of the soles of our boots by the molecules 
of the ground, and the consequent impulses spreading upwards 
through the body. This point is of some importance, since the 
idea of the force of gravitation as something which can be felt, 
predisposes us to a materialistic view of its nature. 

Another example of an artificial field of force is the centrifugal 
force of the earth's rotation. In most books of Physical Con- 
stants will be found a table of the values of "g," the acceleration 
due to gravity, at different latitudes. But the numbers given 
do not relate to gravity alone; they are the resultant of gravity 
and the centrifugal force of the earth's rotation. These are so 
much alike in their effects that for practical purposes physicists 
have not thought it worth while to distinguish them. 

Similar artificial fields are produced when an aeroplane 
changes its course or speed; and one of the difficulties of naviga- 
tion is the impossibility of discriminating between these and the 
true gravitation of the earth with which they combine. One 
usually finds that the practical aviator requires little persuasion 
of the relativity of force. 

To find a unifying idea as to the origin of these artificial 
B,S, 5 



66 FIELDS OF FORCE [CH. 

fields of force, we must return to the four-dimensional world of 
space-time. The observer is progressing along a certain track 
in this world. Now his course need not necessarily be straight. 
It must be remembered that straight in the four-dimensional 
world means something more than straight in space; it implies 
also uniform velocity, since the velocity determines the inclina- 
tion of the track to the time-axis. 

The observer in the accelerated lift travels upwards in a 
straight line, say 1 foot in the first second, 4 feet in two seconds, 
9 feet in three seconds, and so on. If we plot these points as 
x and t on a diagram we obtain a curved track. Presently the 
speed of the lift becomes uniform and the track in the diagram 
becomes straight. So long as the track is curved (accelerated 
motion) a field of force is perceived; it disappears when the 
track becomes straight (uniform motion). 

Again the observer on the earth is carried round in a circle 
once a day by the earth's rotation ; allowing for steady progress 
through time, the track in four dimensions is a spiral. For an 
observer at the north pole the track is straight, and there the 
centrifugal force is zero. 

Clearly the artificial field of force is associated with curvature 
of track, and we can lay down the following rule : 
i Whenever the observer's track through the four-dimensional 

world is curved he perceives an artificial field of force. 

The field of force is not only perceived by the observer in his 
sensations, but reveals itself in his physical measures. It should 
be understood, however, that the curvature of track must not 
have been otherwise allowed for. Naturally if the observer in 
the lift recognises that his measures are affected by his own 
acceleration and applies the appropriate corrections, the artificial 
force will be removed by the process. It only exists if he is 
unaware of, or does not choose to consider, his acceleration. 

The centrifugal force is often called " unreal." From the point 
of view of an observer who does not rotate with the earth, there 
is no centrifugal force; it only arises for the terrestrial observer 
who is too lazy to make other allowance for the effects of the 
earth's rotation. It is commonly thought that this "unreality" 
quite differentiates it from a "real" force like gravity; but if 
we try to find the grounds of this distinction they evade us. 



iv] FIELDS OF FORCE 67 

The centrifugal force is made to disappear if we choose a suitable 
standard observer not rotating with the earth; the gravitational 
force was made to disappear when we chose as standard observer 
an occupant of Jules Verne's falling projectile. If the possibility 
of annulling a field of force by choosing a suitable standard 
observer is a test of unreality, then gravitation is equally unreal 
with centrifugal force. 

It may be urged that we have not stated the case quite 
fairly. When we choose the non-rotating observer the centrifugal 
force disappears completely and everywhere. When we choose 
the occupant of the falling projectile, gravitation disappears in 
his immediate neighbourhood; but he would notice that, 
although unsupported objects round him experienced no accelera- 
tion relative to him, objects on the other side of the earth would 
fall towards him. So far from getting rid of the field of force, 
he has merely removed it from his own surroundings, and piled 
it up elsewhere. Thus gravitation is removable locally, but 
centrifugal force can be removed everywhere. The fallacy of 
this argument is that it speaks as though gravitation and 
centrifugal force were distinguishable experimentally. It pre- 
supposes the distinction that we are challenging. Looking simply 
at the resultant of gravitation and centrifugal force, which is all 
that can be observed, neither observer can get rid of the resultant 
force at all parts of space. Each has to be content with leaving 
a residuum. The non-rotating observer claims that he has got 
rid of all the unreal part, leaving a remainder (the usual gravita- 
tional field) which he regards as really existing. We see no 
justification for this claim, which might equally well be made 
by Jules Verne's observer. 

It is not denied that the separation of centrifugal and gravita- 
tional force generally adopted has many advantages for 
mathematical calculation. If it were not so, it could not have 
endured so long. But it is a mathematical separation only, 
without physical basis ; and it often happens that the separation 
of a mathematical expression into two terms of distinct nature, 
though useful for elementary work, becomes vitiated for more 
accurate work by the occurrence of minute cross-terms which 
have to be taken into account. 

Newtonian mechanics proceeds on the supposition that there 

52 



68 FIELDS OF FORCE [CH. 

is some super-observer. If he feels a field of force, then that 
force really exists. Lesser beings, such as the occupants of the 
falling projectile, have other ideas, but they are the victims of 
illusion. It is to this super-observer that the mathematician 
appeals when he starts a dynamical investigation with the words 
"Take unaccelerated rectangular axes, Ox, Oy, Oz ...." Un- 
accelerated rectangular axes are the measuring-appliances of the 
super-observer. 

It is quite possible that there might be a super-observer, 
whose views have a natural right to be regarded as the truest, 
or at least the simplest. A society of learned fishes would pro- 
bably agree that phenomena were best described from the point 
of view of a fish at rest in the ocean. But relativity mechanics 
finds that there is no evidence that the circumstances of any 
observer can be such as to make his views pre-eminent. All are 
on an equality. Consider an observer A in a projectile falling 
freely to the earth, and an observer B in space out of range of 
any gravitational attraction. Neither A nor B feel any field of 
force in their neighbourhood. Yet in Newtonian mechanics an 
artificial distinction is drawn between their circumstances; B is 
in no field of force at all, but A is really in a field of force, only 
its effects are neutralised by his acceleration. But what is this 
acceleration of ^4? Primarily it is an acceleration relative to the 
earth; but then that can equally well be described as an accelera- 
tion of the earth relative to A, and it is not fair to regard it as 
something located with A. Its importance in Newtonian 
philosophy is that it is an acceleration relative to what we have 
called the super-observer. This potentate has drawn planes and 
lines partitioning space, as space appears to him. I fear that 
the time has come for his abdication. 

Suppose the whole system of the stars were falling freely 
under the uniform gravitation of some vast external mass, like 
a drop of rain falling to the ground. Would this make any 
difference to phenomena? None at all. There would be a 
gravitational field; but the consequent acceleration of the 
observer and his landmarks would produce a field of force 
annulling it. Who then shall say what is absolute acceleration? 

We shall accordingly give up the attempt to separate artificial 
fields of force and natural gravitational fields; and call the whole 



iv] FIELDS OF FORCE 69 

measured field of force the gravitational field, generalising the 
expression. This field is not absolute, but always requires that 
some observer should be specified. 

It may avoid some mystification if we state at once that there 
are certain intricacies in the gravitational influence radiating 
from heavy matter which are distinctive. A theory which did 
not admit this would run counter to common sense. What our 
argument has shown is that the characteristic symptom in a 
region in the neighbourhood of matter is not the field of force; 
it must be something more intricate. In due course we shall 
have to explain the nature of this more complex effect of matter 
on the condition of the world. 

Our previous rule, that the observer perceives an artificial 
field of force when he deviates from a straight track, must now 
be superseded. We need rather a rule determining when he 
perceives a field of force of any kind. Indeed the original rule 
has become meaningless, because a straight track is no longer 
an absolute conception. Uniform motion in a straight line is 
not the same for an observer rotating with the earth as for a 
non-rotating observer who takes into account the sinuosity of 
the rotation. We have decided that these two observers are on 
the same footing and their judgments merit the same respect. 
A straight-line in space-time is accordingly not an absolute 
conception, but is only defined relative to some observer. 

Now we have seen that so long as the observer and his 
measuring-appliances are unconstrained (falling freely) the field 
of force immediately round him vanishes. It is only when he is 
deflected from his proper track that he finds himself in the 
midst of a field of force. Leaving on one side the question of 
the motion of electrically charged bodies, which must be reserved 
for more profound treatment, the observer can only leave his 
proper track if he is being disturbed by material impacts, e.g. 
the molecules of the ground bombarding the soles of his boots. 
We may say then that a body does not leave its natural track 
without visible cause ; and any field of force round an observer 
is the result of his leaving his natural track by such cause. 
There is nothing mysterious about this field of force; it is merely 
the reflection in the phenomena of the observer's disturbance; 
just as the flight of the houses and hedgerows past our railway- 
carriage is the reflection of our motion with the train. 



70 FIELDS OF FORCE [CH. 

Our attention is thus directed to the natural tracks of un- 
constrained bodies, which appear to be marked out in some 
absolute way in the four-dimensional world. There is no 
question of an observer here; the body takes the same course 
in the world whoever is watching it. Different observers will 
describe the track as straight, parabolical, or sinuous, but it is 
the same absolute locus. 

Now we cannot pretend to predict without reference to 
experiment the laws determining the nature of these tracks; 
but we can examine whether our knowledge of the four-dimen- 
sional world is already sufficient to specify definite tracks of this 
kind, or whether it will be necessary to introduce new hypothetical 
factors. It will be found that it is already sufficient. So far we 
have had to deal with only one quantity which is independent 
of the observer and has therefore an absolute significance in the 
world, namely the interval between two events in space and time. 
Let us choose two fairly distant events P x and P 2 . These can 
be joined by a variety of tracks, and the interval-length from 
P! to P 2 along any track can be measured. In order to make 
sure that the interval-length is actually being measured along 
the selected track, the method is to take a large number of 
intermediate points on the track, measure the interval corre- 
sponding to each subdivision, and take the sum. It is virtually 
the same process as measuring the length of a twisty road on 
a map with a piece of cotton. The interval-length along a 
particular track is thus something which can be measured 
absolutely, since all observers agree as to the measurement of 
the interval for each subdivision. It follows that all observers 
will agree as to which track (if any) is the shortest track between 
the two points, judged in terms of interval-length. 

This gives a means of defining certain tracks in space-time as 
having an absolute significance, and we proceed tentatively to 
identify them with the natural tracks taken by freely moving 
particles. 

In one respect we have been caught napping. Dr A. A. Robb 
has pointed out the curious fact that it is not the shortest track, 
but the longest track, which is unique*. There are any number 

* It is here assumed that P 2 is in the future of P x so that it is possible for 
a particle to travel from P to P 2 . If P x and P 2 are situated like O and P' in 
Fig. 3, the interval-length is imaginary, and the shortest track is unique. 



iv] FIELDS OF FORCE 71 

of tracks from P 1 to P 2 of zero interval-length ; there is just one 
which has maximum length. This is because of the peculiar 
geometry which the minus sign of (t 2 t-^) 2 introduces. For 
instance, it will be seen from equation (1), p. 53, that when 

(, - ,) + (y, - ytf + to - %) 2 = (*, - y 2 , 

that is to say when the resultant distance travelled in space is 
equal to the distance travelled in time, then s is zero. This 
happens when the velocity is unity the velocity of light. To 
get from P x to P 2 by a path of no interval-length, we must 
simply keep on travelling with the velocity of light, cruising 
round if necessary, until the moment comes to turn up at P 2 . 
On the other hand there is evidently an upper limit to the interval- 
length of the track, because each portion of s is always less than 
the corresponding portion of (t 2 ^), and s can never exceed 

* a -*i- 

There is a physical interpretation of interval-length along the 

path of a particle which helps to give a more tangible idea of 
its meaning. It is the time as perceived by an observer, or 
measured by a clock, carried on the particle. This is called the 
proper-time ; and, of course, it will not in general agree with the 
time-reckoning of the independent onlooker who is supposed to 
be watching the whole proceedings. To prove this, we notice 
from equation (1) that if x 2 = x lf y 2 = y lt and z z = z lt then 
s = t 2 /! . The condition # 2 = x , etc. means that the particle 
must remain stationary relative to the observer who is measuring 
x, y, 2, t. To secure this we mount our observer on the particle 
and then the interval-length s will be t 2 t lt which is the time 
elapsed according to his clock. 

We can use proper-time as generally equivalent to interval- 
length; but it must be admitted that the term is not very 
logical unless the track in question is a natural track. For any 
other track, the drawback to denning the interval-length as the 
time measured by a clock which follows the track, is that no 
clock could follow the track without violating the laws of nature. 
We may force it into the track by continually hitting it; but 
that treatment may not be good for its time-keeping qualities. 
The original definition by equation (1) is the more general 
definition. 



72 FIELDS OF FORCE [CH. 

We are now able to state formally our proposed law of motion 
Every particle moves so as to take the track of greatest interval- 
length between two events, except in so far as it is disturbed by 
impacts of other particles or electrical forces. 

This cannot be construed into a truism like Newton's first 
law of motion. The reservation is not an undefined agency like 
force, whose meaning can be extended to cover any breakdown 
of the law. We reserve only direct material impacts and electro- 
magnetic causes, the latter being outside our present field of 
discussion. 

Consider, for example, two events in space-time, viz. the 
position of the earth at the present moment, and its position a 
hundred years ago. Call these events P 2 and P x . In the interim 
the earth (being undisturbed by impacts) has moved so as to 
take the longest track from Pj to P 2 or, if we prefer, so as to 
take the longest possible proper-time over the journey. In the 
weird geometry of the part of space-time through which it 
passes (a geometry which is no doubt associated in some way 
with our perception of the existence of a massive body, the sun) 
this longest track is a spiral a circle in space, drawn out into 
a spiral by continuous displacement in time. Any other course 
would have had shorter interval-length. 

In this way the study of fields of force is reduced to a study 
of geometry. To a certain extent this is a retrograde step ; we 
adopt Kepler's description of the sun's gravitational field instead 
of Newton's. The field of force is completely described if the 
tracks through space and time of particles projected in every 
possible way are prescribed. But we go back in order to go 
forward in a new direction. To express this unmanageable mass 
of detail in a unified way, a world-geometry is found in which 
the tracks of greatest length are the actual tracks of the particles. 
It only remains to express the laws of this geometry in a concise 
form. The change from a mechanical to a geometrical theory of 
fields of force is not so fundamental a change as might be 
supposed. If we are now reducing mechanics to a branch of 
natural geometry, we have to remember that natural geometry 
is equally a branch of mechanics, since it is concerned with the 
behaviour of material measuring-appliances. 

Reference has been made to weird geometry. There is no 



iv] FIELDS OF FORCE 73 

help for it, if the longest track can be a spiral like that known 
to be described by the earth. Non-Euclidean geometry is 
necessary. In Euclidean geometry the shortest track is always 
a straight line ; and the slight modification of Euclidean geometry 
described in Chapter in is found to give a straight line as the 
longest track. The status of non-Euclidean geometry has already 
been thrashed out in the Prologue; and there seems to be no 
reason whatever for preferring Euclid's geometry unless observa- 
tions decide in its favour. Equation (1), p. 53, is the expression 
of the Euclidean (or semi-Euclidean) geometry we have hitherto 
adopted ; we shall have to modify it, if we adopt non-Euclidean 
geometry. 

But the point arises that the geometry arrived at in Chapter in 
was not arbitrary. It was the synthesis of measures made with 
clocks and scales, by observers with all kinds of uniform motion 
relative to one another; we cannot modify it arbitrarily to fit 
the behaviour of moving particles like the earth. Now, if the 
worst came to the worst, and we could not reconcile a geometry 
based on measures with clocks and scales and a geometry based 
on the natural tracks of moving particles if we had to select 
one or the other and keep to it I think we ought to prefer to 
use the geometry based on the tracks of moving particles. The 
free motion of a particle is an example of the simplest possible 
kind of phenomenon; it is unanalysable; whereas, what the 
readings of any kind of clock record, what the extension of a 
material rod denotes, may evidently be complicated phenomena 
involving the secrets of molecular constitution. Each geometry 
would be right in its own sphere; but the geometry of moving 
particles would be the more fundamental study. But it turns 
out that there is probably no need to make the choice; clocks, 
scales, moving particles, light-pulses, give the same geometry. 
This might perhaps be expected since a clock must comprise 
moving particles of some kind. 

A formula, such as equation (1), based on experiment can 
only be verified to a certain degree of approximation. Within 
certain limits it will be possible to introduce modifications. Now 
it turns out that the free motion of a particle is a much more 
sensitive way of exploring space-time, than any practicable 
measures with scales and clocks. If then we employ our accurate 



74 FIELDS OF FORCE [CH. 

knowledge of the motion of particles to correct the formula, we 
shall find that the changes introduced are so small that they are 
inappreciable in any practical measures with scales and clocks. 
There is only one case where a possible detection of the modifica- 
tion is indicated ; this refers to the behaviour of a clock on the 
surface of the sun, but the experiment is one of great difficulty 
and no conclusive answer has been given. We conclude then 
that the geometry of space and time based on the motions of 
particles is accordant with the geometry based on the cruder 
observations with clocks and scales; but if subsequent experi- 
ment should reveal a discrepancy, we shall adhere to the moving 
particle on account of its greater simplicity. 

The proposed modification can be regarded from another 
point of view. Equation (1) is the synthesis of the experiences 
of all observers in uniform motion. But uniform motion means 
that their four-dimensional tracks are straight lines. We must 
suppose that the observers were moving in their natural tracks ; 
for, if not, they experienced fields of force, and presumably 
allowed for these in their calculations, so that reduction was 
made to the natural tracks. If then equation (1) shows that 
the natural tracks are straight lines, we are merely getting out 
of the equation that which we originally put into it. 

The formula needs generalising in another way. Suppose there 
is a region of space-time where, for some observer, the natural 
tracks are all straight lines and equation (1) holds rigorously. 
For another (accelerated) observer the tracks will be curved, 
and the equation will not hold. At the best it is of a form which 
can only hold good for specially selected observers. 

Although it has become necessary to throw our formula into 
the melting-pot, that does not create any difficulty in measuring 
the interval. Without going into technical details, it may be 
pointed out that the innovations arise solely from the intro- 
duction of gravitational fields of force into our scheme. When 
there is no force, the tracks of all particles are straight lines as 
our previous geometry requires. In any small region we can 
choose an observer (falling freely) for whom the force vanishes, 
and accordingly the original formula holds good. Thus it is only 
necessary to modify our rule for determining the interval by 
two provisos (1) that the interval measured must be small, 



iv] FIELDS OF FORCE 75 

(2) that the scales and clocks used for measuring it must be 
falling freely. The second proviso is natural, because, if we do 
not leave our apparatus to fall freely, we must allow for the 
strain that it undergoes. The first is not a serious disadvantage, 
because a larger interval can be split up into a number of small 
intervals and the parts measured separately. In mathematical 
problems the same device is met with under the name of integra- 
tion. To emphasise that the formula is strictly true only for 
infinitesimal intervals, it is written with a new notation 

ds z = - dx* - dy* - dz* + dt* (2) 

where dx stands for the small difference x 2 x 1 , etc. 

The condition that the measuring appliances must not be 
subjected to a field of force is illustrated by Ehrenfest's para- 
dox. Consider a wheel revolving rapidly. Each portion of the 
circumference is moving in the direction of its length, and 
might be expected to undergo the FitzGerald contraction due 
to its velocity ; each portion of a radius is moving transversely 
and would therefore have no longitudinal contraction. It looks 
as though the rim of the wheel should contract and the spokes 
remain the same length, when the wheel is set revolving. The 
conclusion is absurd, for a revolving wheel has no tendency to 
buckle which would be the only way of reconciling these 
conditions. The point which the argument has overlooked is 
that the results here appealed to apply to unconstrained bodies, 
which have no acceleration relative to the natural tracks in 
space. Each portion of the rim of the wheel has a radial accelera- 
tion, and this affects its extensional properties. When accelera- 
tions as well as velocities occur a more far-reaching theory is 
needed to determine the changes of length. 

To sum up the interval between two (near) events is some- 
thing quantitative which has an absolute significance in nature. 
The track between two (distant) events which has the longest 
interval-length must therefore have an absolute significance. 
Such tracks are called geodesies. Geodesies can be traced practi- 
cally, because they are the tracks of particles undisturbed by 
material impacts. By the practical tracing of these geodesies 
we have the best means of studying the character of the natural 
geometry of the world. An auxiliary method is by scales and 



76 FIELDS OF FORCE [CH. iv 

clocks, which, it is believed, when unconstrained, measure a 
small interval according to formula (2). 

The identity of the two methods of exploring the geometry 
of the world is connected with a principle which must now be 
enunciated definitely. We have said that no experiments have 
been able to detect a difference between a gravitational field 
and an artificial field of force such- as the centrifugal force. This 
is not quite the same thing as saying that it has been proved 
that there is no difference. It is well to be explicit when a 
positive generalisation is made from negative experimental 
evidence. The generalisation which it is proposed to adopt is 
known as the Principle of Equivalence. 

A gravitational field of force is precisely equivalent to an artificial 
field of force, so that in any small region it is impossible by any 
conceivable experiment to distinguish between them. 

In other words, force is purely relative. 



CHAPTER V 
KINDS OF SPACE 

The danger of asserting dogmatically that an axiom based on the experience 
of a limited region holds universally will now be to some extent apparent to 
the reader. It may lead us to entirely overlook, or when suggested at once 
reject, a possible explanation of phenomena. The hypothesis that space is not 
homaloidal [flat], and again that its geometrical character may change with 
the time, may or may not be destined to play a great part in the physics of the 
future; yet we cannot refuse to consider them as possible explanations of 
physical phenomena, because they may be opposed to the popular dogmatic 
belief in the universality of certain geometrical axioms a belief which has 
risen from centuries of indiscriminating worship of the genius of Euclid. 

W. K. CLIFFORD (and K. PEARSON), Common Sense of the Exact Sciences. 

ON any surface it requires two independent numbers or "co- 
ordinates" to specify the position of a point. For this reason 
a surface, whether flat or curved, is called a two-dimensional 
space. Points in three-dimensional space require three, and in 
four-dimensional space-time four numbers or coordinates. 

To locate a point on a surface by two numbers, we divide the 
surface into meshes by any two systems of lines which cross one 
another. Attaching consecutive numbers to the lines, or better 
to the channels between them, one number from each system 
will identify a particular mesh ; and if the subdivision is sufficiently 
fine any point can be specified in this way with all the accuracy 
needed. This method is used, for example, in the Post Office 
Directory of London for giving the location of streets on the 
map. The point (4, 2) will be a point in the mesh where channel 
No. 4 of the first system crosses channel No. 2 of the second. 
If this indication is not sufficiently accurate, we must divide 
channel No. 4 into ten parts numbered 4-0, 4-1, etc. The sub- 
division must be continued until the meshes are so small that 
all points in one mesh can be considered identical within the 
limits of experimental detection. 

The diagrams, Figs. 10, 11, 12, illustrate three of the many 
kinds of mesh-systems commonly used on a flat surface. 

If we speak of the properties of the triangle formed by the 
points (1, 2), (3, 0), (4, 4), we shall be at once asked, What mesh- 



78 



KINDS OF SPACE 



[CH. 



system are you using? No one can form a picture of the triangle 
until that information has been given. But if we speak of the 
properties of a triangle whose sides are of lengths 2, 3, 4 inches, 
anyone with a graduated scale can draw the triangle, and follow 
our discussion of its properties. The distance between two points 
can be stated without referring to any mesh-system. For this 
reason, if we use a mesh-system, it is important to find formulae 
connecting the absolute distance with the particular system that 
is being used. 

In the more complicated kinds of mesh-systems it makes a 
great simplification if we content ourselves with the formulae for 
very short distances. The mathematician then finds no difficulty 
in extending the results to long distances by the process called 
integration. We write ds for the distance between two points 





1 1 / 1 1 1 



FIG. 10. 



FIG. 11. 



FIG. 12. 



close together, x and x 2 for the two numbers specifying the 
location of one of them, dx l and dx 2 for the small differences of 
these numbers in passing from the first point to the second. 
But in using one of the particular mesh-systems illustrated in 
the diagrams, we usually replace x lt x 2 by particular symbols 
sanctioned by custom, viz. (x lt x 2 ) becomes (x, y), (r, 6), (, 77) 
for Figs. 10, 11, 12, respectively. 

The formulae, found by geometry, are: 

For rectangular coordinates (x, y), Fig. 10, 
ds 2 = dx 2 + dy\ 

For polar coordinates (r, 0), Fig. 11, 
ds* = dr* + rW 8 . 

For oblique coordinates (f , 77), Fig. 12, 



where K is the cosine of the angle between the lines of partition. 



v] KINDS OF SPACE 79 

As an example of a mesh-system on a curved surface, we may 
take the lines of latitude and longitude on a sphere. 

For latitude and longitude (/?, A) 

ds 2 = d^ + cos 2 $d\ 2 . 

These expressions form a test, and in fact the only possible 
test, of the kind of coordinates we are using. It may perhaps 
seem inconceivable that an observer should for an instant be in 
doubt whether he was using the mesh-system of Fig. 10 or 
Fig. 11. He sees at a glance that Fig. 11 is not what he would 
call a rectangular mesh-system. But in that glance, he makes 
measures with his eye, that is to say he determines ds for pairs 
of points, and he notices how these values are related to the 
number of intervening channels. In fact he is testing which 
formula for ds will fit. For centuries man was in doubt whether 
the earth was flat or round whether he was using plane rect- 
angular coordinates or some kind of spherical coordinates. In 
some cases an observer adopts his mesh-system blindly and long 
afterwards discovers by accurate measures that ds does not fit 
the formula he assumed that his mesh-system is not exactly of 
the nature he supposed it was. In other cases he deliberately 
sets himself to plan out a mesh-system of a particular variety, 
say rectangular coordinates ; he constructs right angles and rules 
parallel lines; but these constructions are all measurements of 
the way the ^-channels and ^-channels ought to go, and the 
rules of construction reduce to a formula connecting his measures 
ds with x and y. 

The use of special symbols for the coordinates, varying 
according to the kind of mesh-system used, thus anticipates a 
knowledge which is really derived from the form of the formulae. 
In order not to give away the secret prematurely, it will be 
better to use the symbols x l , x 2 in all cases. The four kinds of 
coordinates already considered then give respectively the re- 
lations, 

ds 2 = dxj 2 + dcc 2 2 (rectangular), 

ds* = dx^ + xfdxf (polar), 

ds 2 = dx^ ZKdXidx^ + dx 2 2 (oblique), 

ds 2 = dxj 2 + cos 2 x^dx^ (latitude and longitude). 

If we have any mesh-system and want to know its nature, we 



80 KINDS OF SPACE [CH. 

must make a number of measures of the length ds between 
adjacent points (x lt x 2 ) and (x -f dx lf x 2 + dx 2 ) and test which 
formula fits. If, for example, we then find that ds 2 is always 
equal to dx-f + x^dx, we know that our mesh-system is like 
that in Fig. 11, x l and x 2 being the numbers usually denoted by 
the polar coordinates r, 6. The statement that polar coordinates 
are being used is unnecessary, because it adds nothing to our 
knowledge which is not already contained in the formula. It is 
merely a matter of giving a name ; but, of course, the name calls 
to our minds a number of familiar properties which otherwise 
might not occur to us. 

For instance, it is characteristic of the polar coordinate system 
that there is only one point for which x t (or r) is equal to 0, 
whereas in the other systems x = gives a line of points. This 
is at once apparent from the formula; for if we have two points 
for which x l = and x + dx l = 0, respectively, then 

dxj 2 + x^dxf = 0. 

The distance ds between the two points vanishes, and accordingly 
they must be the same point. 

The examples given can all be summed up in one general 
expression 

ds* = gndxj 2 + 2g l2 dz 1 dx 2 + g 22 dx 2 2 , 

where g u , g l2 , g 22 may be constants or functions of x and cc 2 . 
For instance, in the fourth example their values are 1, 0, cos 2 x^. 
It is found that all possible mesh-systems lead to values of ds 2 
which can be included in an expression of this general form; so 
that mesh-systems are distinguished by three functions of 
position g llt 12, 22 which can be determined by making physical 
measurements. These three quantities are sometimes called 
potentials. 

We now come to a point of far-reaching importance. The 
formula for ds 2 teaches us not only the character of the mesh- 
system, but the nature of our two-dimensional space, which is 
independent of any mesh-system. If ds 2 satisfies any one of the 
first three formulae, then the space is like a flat surface; if it 
satisfies the last formula, then the space is a surface curved like 
a sphere. Try how you will, you cannot draw a mesh-system on 
.a flat (Euclidean) surface which agrees with the fourth formula. 



v] KINDS OF SPACE 81 

If a being limited to a two-dimensional world finds that his 
measures agree with the first formula, he can make them agree 
with the second or third formulae by drawing the meshes 
differently. But to obtain the fourth formula he must be trans- 
lated to a different world altogether. 

We thus see that there are different kinds of two-dimensional 
space, betrayed by different metrical properties. They are 
naturally visualised as different surfaces in Euclidean space of 
three dimensions. This picture is helpful in some ways, but 
perhaps misleading in others. The metrical relations on a plane 
sheet of paper are not altered when the paper is rolled into a 
cylinder the measures being, of course, confined to the two- 
dimensional world represented by the paper, and not allowed to 
take a short cut through space. The formulae apply equally 
well to a plane surface or a cylindrical surface; and in so far as 
our picture draws a distinction between a plane and a cylinder, 
it is misleading. But they do not apply to a sphere, because 
a plane sheet of paper cannot be wrapped round a sphere. 
A genuinely two-dimensional being could not be cognisant of 
the difference between a cylinder* and a plane; but a sphere 
would appear as a different kind of space, and he would recognise 
the difference by measurement. 

Of course there are many kinds of mesh-systems, and many 
kinds of two-dimensional spaces, besides those illustrated in the 
four examples. Clearly it is not going to be a simple matter to 
discriminate the different kinds of spaces by the values of the 
g's. There is no characteristic, visible to cursory inspection, 
which suggests why the first three formulae should all belong to 
the same kind of space, and the fourth to a different one. 
Mathematical investigation has discovered what is the common 
link between the first three formulae. The g n , g 12 , g 22 satisfy in 
all three cases a certain differential equationf; and whenever 
this differential equation is satisfied, the same kind of space 
occurs. 

No doubt it seems a very clumsy way of approaching these 
intrinsic differences of kinds of space to introduce potentials 

* One should perhaps rather say a roll, to avoid any question of joining the 
two edges. 

t Appendix, Note 4. 
E.S. 6 



82 KINDS OF SPACE [CH. 

which specifically refer to a particular mesh-system, although 
the mesh-system can have nothing to do with the matter. It is 
worrying not to be able to express the differences of space in a 
purer form without mixing them up with irrelevant differences 
of potential. But we have neither the vocabulary nor the 
imagination for a description of absolute properties as such. 
All physical knowledge is relative to space and time partitions; 
and to gain an understanding of the absolute it is necessary to 
approach it through the relative. The absolute may be denned 
as a relative which is always the same no matter what it is 
relative to *. Although we think of it as self-existing, we cannot 
give it a place in our knowledge without setting up some dummy 
to relate it to. And similarly the absolute differences of space 
always appear as related to some mesh-system, although the 
mesh-system is only a dummy and has nothing to do with the 
problem. 

The results for two dimensions can be generalised, and applied 
to four-dimensional space-time. Distance must be replaced by 
interval, which it will be remembered, is an absolute quantity, 
and therefore independent of the mesh-system used. Partitioning 
space-time by any system of meshes, a mesh being given by the 
crossing of four channels, we must specify a point in space-time 
by four coordinate numbers, as lf x 2 , #3, #4- By analogy the 
general formula will be 

+ 22 d*2 2 + 3 3<^3 2 + &4 ^ 4 2 + 2g lt da} l dsc t 



# 4 ........................... (3). 

The only difference is that there are now ten g's, or potentials, 
instead of three, to summarise the metrical properties of the 
mesh-system. It is convenient in specifying special values of 
the potentials to arrange them in the standard form 

11 812 Sis i4 

22 23 24 

33 34 



* Cf. p. 31, where a distinction was drawn between knowledge which does 
not particularise the observer and knowledge which does not postulate an 
observer at all. 



v] KINDS OF SPACE 83 

The space-time already discussed at length in Chapter ni 
corresponded to the formula (2), p. 75, 

ds*= - dx*- dy 2 - dz* + dp. 

Here (#, y, z, t) are the conventional symbols for (x 1> # 2 , x z , # 4 ) 
when this special mesh-system is used, viz. rectangular coordi- 
nates and time. Comparing with (3) the potentials have the 
special values 

- 1 

- 1 00 

- 1 

+ 1 

These are called the "Galilean values." If the potentials have 
these values everywhere, space-time may be called "flat," 
because the geometry is that of a plane surface drawn in 
Euclidean space of five dimensions. Recollecting what we found 
for two dimensions, we shall realise that a quite different set 
of values of the potentials may also belong to flat space-time, 
because the meshes may be drawn in different ways. We must 
clearly understand that 

(1) The only way of discovering what kind of space-time is 
being dealt with is from the values of the potentials, which are 
determined practically by measurements of intervals, 

(2) Different values of the potentials do not necessarily 
indicate different kinds of space-time, 

(3) There is some complicated mathematical property 
common to all values of the potentials which belong to the 
same space-time, which is not shared by those which belong to 
a different kind of space-time. This property is expressed by 
a set of differential equations. 

It can now be deduced that the space-time in which we live 
is not quite flat. If it were, a mesh-system could be drawn for 
which the g's have the Galilean values, and the geometry with 
respect to these partitions of space and time would be that 
discussed in Chapter in. For that geometry the geodesies, giving 
the natural tracks of particles, are straight lines. 

Thus in flat space-time the law of motion is that (with 
suitably chosen coordinates) every particle moves uniformly in 
a straight line except when it is disturbed by the impacts of 

62 



84 KINDS OF SPACE [CH. 

other particles. Clearly this is not true of our world ; for example, 
the planets do not move in straight lines although they do not 
suffer any impacts. It is true that if we confine attention to a 
small region like the interior of Jules Verne's projectile, all the 
tracks become straight lines for an appropriate observer, or, 
as we generally say, he detects no field of force. It needs a 
large region to bring out the differences of geometry. That is 
not surprising, because we cannot expect to tell whether a 
surface is flat or curved unless we consider a reasonably large 
portion of it. 

According to Newtonian ideas, at a great distance from all 
matter beyond the reach of any gravitation, particles would all 
move uniformly in straight lines. Thus at a great distance from 
all matter space-time tends to become perfectly flat. This can 
only be checked by experiment to a certain degree of accuracy, 
and there is some doubt as to whether it is rigorously true. We 
shall leave this afterthought to Chapter x, meanwhile assuming 
with Newton that space-time far enough away from everything 
is flat, although near matter it is curved. It is this puckering 
near matter which accounts for its gravitational effects. 

Just as we picture different kinds of two-dimensional space 
as differently curved surfaces in our ordinary space of three- 
dimensions, so we are now picturing different kinds of four- 
dimensional space-time as differently curved surfaces in a 
Euclidean space of five dimensions. This is a picture only *. 
The fifth dimension is neither space nor time nor anything that 
can be perceived ; so far as we know, it is nonsense. I should not 
describe it as a mathematical fiction, because it is of no great 
advantage in a mathematical treatment. It is even liable to 
mislead because it draws distinctions, like the distinction be- 
tween a plane and a roll, which have no meaning. It is, like 
the notion of a field of force acting in space and time, merely 
introduced to bolster up Euclidean geometry, when Euclidean 
geometry has been found inappropriate. The real difference 
between the various kinds of space-time is that they have 

* A fifth dimension suffices for illustrating the property here considered; 
but for an exact representation of the geometry of the world, Euclidean space 
of ten dimensions is required. We may well ask whether there is merit in 
Euclidean geometry sufficient to justify going to such extremes. 



v] KINDS OF SPACE 85 

different kinds of geometry, involving different properties of the 
g's. It is no explanation to say that this is because the surfaces 
are differently curved in a real Euclidean space of five dimensions. 
We should naturally ask for an explanation why the space of 
five dimensions is Euclidean ; and presumably the answer would 
be, because it is a plane in a real Euclidean space of six dimen- 
sions, and so on ad infinitwn. 

The value of the picture to us is that it enables us to describe 
important properties with common terms like "pucker" and 
"curvature" instead of technical terms like "differential 
invariant." We have, however, to be on our guard, because 
analogies based on three-dimensional space do not always apply 
immediately to many-dimensional space. The writer has keen 
recollections of a period of much perplexity, when he had not 
realised that a four-dimensional space with "no curvature" is 
not the same as a "flat" space! Three-dimensional geometry 
does not prepare us for these surprises. 

Picturing the space-time in the gravitational field round the 
earth as a pucker, we notice that we cannot locate the pucker 
at a point; it is "somewhere round" the point. At any special 
point the pucker can be pressed out flat, and the irregularity 
runs off somewhere else. That is what the inhabitants of Jules 
Verne's projectile did; they flattened out the pucker inside the 
projectile so that they could not detect any field of force there; 
but this only made things worse somewhere else, and they 
would find an increased field of force (relative to them) on the 
other side of the earth. 

What determines the existence of the pucker is not the values 
of the g's at any point, or, what comes to the same thing, the 
field of force there. It is the way these values link on to those 
at other points the gradient of the g's, and more particularly 
the gradient of the gradient. Or, as has already been said, the 
kind of space-time is fixed by differential equations. 

Thus, although a gravitational field of force is not an absolute 
thing, and can be imitated or annulled at any point by an 
acceleration of the observer or a change of his mesh-system, 
nevertheless the presence of a heavy particle does modify the 
world around it in an absolute way which cannot be imitated 
artificially. Gravitational force is relative; but there is this 



86 KINDS OF SPACE [CH. 

more complex character of gravitational influence which is 
absolute. 

The question must now be put, Can every possible kind of 
space-time occur in an empty region in nature? Suppose we 
give the ten potentials perfectly arbitrary values at every point ; 
that will specify the geometry of some mathematically possible 
space-time. But could that kind of space-time actually occur 
by any arrangement of the matter round the region? 

The answer is that only certain kinds of space-time can occur 
in an empty region in nature. The law which determines what 
kinds can occur is the law of gravitation. 

It is indeed clear that, since we have reduced the theory of 
fields of force to a theory of the geometry of the world, if there 
is any law governing fields of force (including the gravitational 
field), that law must be of the nature of a restriction on the 
possible geometries of the world. 

The choice of g's in any special problem is thus arrived at by 
a three-fold sorting out: (1) many sets of values can be dismissed 
because they can never occur in nature, (2) others, while possible, 
do not relate to the kind of space-time present in the problem 
considered, (3) of those which remain, one set of values relates 
to the particular mesh-system that has been chosen. We have 
now to find the law governing the first discrimination. What is 
the criterion that decides what values of the g's give a kind of 
space-time possible in nature? 

In solving this problem Einstein had only two clues to guide 
him. 

(1) Since it is a question of whether the kind of space-time is 
possible, the criterion must refer to those properties of the g's 
which distinguish different kinds of space-time, not to those 
which distinguish different kinds of mesh-system in the same 
space-time. The formulae must therefore not be altered in any 
way, if we change the mesh-system. 

(2) We know that flat space-time can occur in nature (at 
great distances from all gravitating matter). Hence the criterion 
must be satisfied by any values of the g's belonging to flat 
space-time. 

It is remarkable that these slender clues are sufficient to 
indicate almost uniquely a particular law. Afterwards the 



v] KINDS OF SPACE 87 

further test must be applied whether the law is confirmed by 
observation. 

The irrelevance of the mesh-system to the laws of nature is 
sometimes expressed in a slightly different way. There is one 
type of observation which, we can scarcely doubt, must be 
independent of any possible circumstances of the observer, 
namely a complete coincidence in space and time. The track of 
a particle through four-dimensional space-time is called its 
world-line. Now, the world-lines of two particles either intersect 
or they do not intersect; the standpoint of the observer is not 
involved. In so far as our knowledge of nature is a knowledge 
of intersections of world-lines, it is absolute knowledge inde- 
pendent of the observer. If we examine the nature of our 
observations, distinguishing what is actually seen from what is 
merely inferred, we find that, at least in all exact measurements, 
our knowledge is primarily built up of intersections of world- 
lines of two or more entities, that is to say their coincidences. 
For example, an electrician states that he has observed a current 
of 5 milliamperes. This is his inference: his actual observation 
was a coincidence of the image of a wire in his galvanometer 
with a division of a scale. A meteorologist finds that the tem- 
perature of the air is 75; his observation was the coincidence of 
the top of the mercury-thread with division 75 on the scale of 
his thermometer. It would be extremely clumsy to describe the 
results of the simplest physical experiment entirely in terms of 
coincidence. The absolute observation is, whether or not the 
coincidence exists, not when or where or under what circum- 
stances the coincidence exists ; unless we are to resort to relative 
knowledge, the place, time and other circumstances must in 
their turn be described by reference to other coincidences. But 
it seems clear that if we could draw all the world-lines so as to 
show all the intersections in their proper order, but otherwise 
arbitrary, this would contain a complete history of the world, 
and nothing within reach of observation would be omitted. 

Let us draw such a picture, and imagine it embedded in a 
jelly. If we deform the jelly in any way, the intersections will 
still occur in the same order along each world-line and no 
additional intersections will be created. The deformed jelly will 
represent a history of the world, just as accurate as the one 



88 KINDS OF SPACE [CH. 

originally drawn; there can be no criterion for distinguishing 
which is the best representation. 

Suppose now we introduce space and time-partitions, which 
we might do by drawing rectangular meshes in both jellies. 
We have now two ways of locating the world-lines and events 
in space and time, both on the same absolute footing. But 
clearly it makes no difference in the result of the location whether 
we first deform the jelly and then introduce regular meshes, or 
whether we introduce irregular meshes in the undeformed jelly. 
And so all mesh-systems are on the same footing. 

This account of our observational knowledge of nature shows 
that there is no shape inherent in the absolute world, so that 
when we insert a mesh-system, it has no shape initially, and a 
rectangular mesh-system is intrinsically no different from any 
other mesh-system. 

Returning to our two clues, condition (1) makes an extra- 
ordinarily clean sweep of laws that might be suggested ; among 
them Newton's law is swept away. The mode of rejection can 
be seen by an example; it will be sufficient to consider two 
dimensions. If in one mesh-system (#, y) 

ds 2 = gu&v* + 2g lz dxdy + fe^ 2 , 
and in another system (x' 9 y') 

ds* = n W* + 2g 12 ' dx'dy' + g^dy'\ 

the same law must be satisfied if the unaccented letters are 
throughout replaced by accented letters. Suppose the law 
11 = 22 i s suggested. Change the mesh-system by spacing the 
2/-lines twice as far apart, that is to say take y' = \y> with 



2g 12 dxdy 

4g 12 dx'dy' 

so that g u r = g u , g 22 = 4g 22 . 

And if g u is equal to 2 2 11' cannot be equal to 22 '. 

After a few trials the reader will begin to be surprised that 
any possible law could survive the test. It seems so easy to 
defeat any formula that is set up by a simple change of mesh- 
system. Certainly it is unlikely that anyone would hit on such 
a law by trial. But there are such laws, composed of exceedingly 
complicated mathematical expressions. The theory of these is 



v] KINDS OF SPACE 89 

called the "theory of tensors," and had already been worked 
out by the pure mathematicians Riemann, Christoffel, Ricci, 
Levi-Civita who, it may be presumed, never dreamt of a physical 
application for it. 

One law of this kind is the condition for flat space-time, 
which is generally written in the simple, but not very illuminating, 
form 



The quantity on the left is called the Riemann-Christoffel 
tensor, and it is written out in a less abbreviated form in the 
Appendix*. It must be explained that the letters /i,, v, a, p 
indicate gaps, which are to be filled up by any of the numbers 
1, 2, 3, 4, chosen at pleasure. (When the expression is written 
out at length, the gaps are in the suffixes of the #'s and g's.) 
Filling the gaps in different ways, a large number of expressions, 
B l ul , I?i 23> B\ 32 , etc., are obtained. The equation (4) states that 
all of these are zero. There are 4 4 , or 256, of these expressions 
altogether, but many of them are repetitions. Only 20 of the 
equations are really necessary; the others merely say the same 
thing over again. 

It is clear that the law (4) is not the law of gravitation for 
which we are seeking, because it is much too drastic. If it were 
a law of nature, then only flat space-time could exist in nature, 
and there would be no such thing as gravitation. It is not the 
general condition, but a special case when all attracting 
matter is infinitely remote. 

But in finding a general condition, it may be a great help to 
know a special case. Would it do to select a certain number of 
the 20 equations to be satisfied generally, leaving the rest to 
be satisfied only in the special case? Unfortunately the equations 
hang together; and, unless we take them all, it is found that 
the condition is not independent of the mesh-system. But there 
happens to be one way of building up out of the 20 conditions 
a less stringent set of conditions independent of the mesh- 
system. Let 

**11 = #111 + #112 + #113 + #114> 

and, generally 

GW = flj,x + Bf^ + BJ^ + BJ_ 4 , 

* Note 5. 



90 KINDS OF SPACE [CH. 

then the conditions 

fl^-O (5), 

will satisfy our requirements for a general law of nature. 

This law is independent of the mesh-system, though this can 
only be proved by elaborate mathematical analysis. Evidently, 
when all the B's vanish, equation (5) is satisfied; so, when flat 
space-time occurs, this law of nature is not violated. Further 
it is not so stringent as the condition for flatness, and admits 
of the occurrence of a limited variety of non-Euclidean geome- 
tries. Rejecting duplicates, it comprises 10 equations; but four 
of these can be derived from the other six, so that it gives 
six conditions, which happens to be the number required for a 
law of gravitation*. 

The suggestion is thus reached that 

<v = o 

may be the general law of gravitation. Whether it is so or not 
can only be settled by experiment. In particular, it must in 
ordinary cases reduce to something so near the Newtonian law, 
that the remarkable confirmation of the latter by observation 
is accounted for. Further it is necessary to examine whether 
there are any exceptional cases in which the difference between 
it and Newton's law can be tested. We shall see that these 
tests are satisfied. 

What would have been the position if this suggested law had 
failed? We might continue the search for other laws satisfying 
the two conditions laid down; but these would certainly be far 
more complicated mathematically. I believe too that they would 
not help much, because practically they would be indistinguish- 
able from the simpler law here suggested though this has not 
been demonstrated rigorously. The other alternative is that 
there is something causing force in nature not comprised in the 

* Isolate a region of empty space- time; and suppose that everywhere outside 
the region the potentials are known. It should then be possible by the law of 
gravitation to determine the nature of space-time in the region. Ten differential 
equations together with the boundary-values would suffice to determine the 
ten potentials throughout the region; but that would determine not only the 
kind of space-time but the mesh-system, whereas the partitions of the mesh- 
system can be continued across the region in any arbitrary way. The four 
sets of partitions give a four-fold arbitrariness; and to admit of this, the number 
of equations required is reduced to six. 



v] KINDS OF SPACE 91 

geometrical scheme hitherto considered, so that force is not 
purely relative, and Newton's super-observer exists. 

Perhaps the best survey of the meaning of our theory can be 
obtained from the standpoint of a ten-dimensional Euclidean 
continuum, in which space-time is conceived as a particular 
four-dimensional surface. It has to be remarked that in ten 
dimensions there are gradations intermediate between a flat 
surface and a fully curved surface, which we shall speak of as 
curved in the "first degree" or "second degree*." The dis- 
tinction is something like that of curves in ordinary space, 
which may be curved like a circle, or twisted like a helix ; but the 
analogy is not very close. The full " curvature" of a surface is a 
single quantity called G, built up out of the various terms G MI/ in 
somewhat the same way as these are built up out of B P ^ V<T . 
The following conclusions can be stated. 

If B vv = (20 conditions) 

space-time is flat. This is the state of the world at an infinite 
distance from all matter and all forms of energy. 

If G^ = (6 conditions) 

space-time is curved in the first degree. This is the state of the 
world in an empty region not containing matter, light or 
electromagnetic fields, but in the neighbourhood of these forms 
of energy. 

If G = (1 condition) 

space-time is curved in the second degree. This is the state of 
the world in a region not containing matter or electrons (bound 
energy), but containing light or electromagnetic fields (free 
energy). 

If G is not zero 

space-time is fully curved. This is the state of the world in a 
region containing continuous matter. 

According to current physical theory continuous matter does 
not exist, so that strictly speaking the last case never arises. 
Matter is built of electrons or other nuclei. The regions lying 
between the electrons are not fully curved, whilst the regions 
inside the electrons must be cut out of space-time altogether. 
We cannot imagine ourselves exploring the inside of an electron 

* This is not a recognised nomenclature. 



92 KINDS OF SPACE [CH.V 

with moving particles, light-waves, or material clocks and 
measuring-rods; hence, without further definition, any geometry 
of the interior, or any statement about space and time in the 
interior, is meaningless. But in common life, and frequently in 
physics, we are not concerned with this microscopic structure of 
matter. We need to know, not the actual values of the g's at 
a point, but their average values through a region, small from 
the ordinary standpoint but large compared with the molecular 
structure of matter. In this macroscopic treatment molecular 
matter is replaced by continuous matter, and uncurved space- 
time studded with holes is replaced by an equivalent fully 
curved space-time without holes. 

It is natural that our senses should have developed faculties 
for perceiving some of these intrinsic distinctions of the possible 
states of the world around us. I prefer to think of matter and 
energy, not as agents causing the degrees of curvature of the 
world, but as parts of our perceptions of the existence of the 
curvature. 

It will be seen that the law of gravitation can be summed up 
in the statement that in an empty region space-time can be 
curved only in the first degree. 



CHAPTER VI 

THE NEW LAW OF GRAVITATION AND 
THE OLD LAW 

I don't know what I may seem to the world, but, as to myself, I seem to have 
been only as a boy playing on the sea-shore, and diverting myself in now and 
then finding a smoother pebble or a prettier shell than ordinary, whilst the 
great ocean of truth lay all undiscovered before me. SIR ISAAC NEWTON. 

WAS there any reason to feel dissatisfied with Newton's law of 
gravitation? 

Observationally it had been subjected to the most stringent 
tests, and had come to be regarded as the perfect model of an 
exact law of nature. The cases, where a possible failure could 
be alleged, were almost insignificant. There are certain unex- 
plained irregularities in the moon's motion; but astronomers 
generally looked and must still look in other directions for 
the cause of these discrepancies. One failure only had led to 
a serious questioning of the law; this was the discordance of 
motion of the perihelion of Mercury. How small was this dis- 
crepancy may be judged from the fact that, to meet it, it was 
proposed to amend square of the distance to the 2-00000016 
power of the distance. Further it seemed possible, though 
unlikely, that the matter causing the zodiacal light might be of 
sufficient mass to be responsible for this effect. 

The most serious objection against the Newtonian law as an 
exact law was that it had become ambiguous. The law refers 
to the product of the masses of the two bodies ; but the mass 
depends on the velocity a fact unknown in Newton's day. 
Are we to take the variable mass, or the mass reduced to rest? 
Perhaps a learned judge, interpreting Newton's statement like 
a last will and testament, could give a decision; but that is 
scarcely the way to settle an important point in scientific 
theory. 

Further distance, also referred to in the law, is something 
relative to an observer. Are we to take the observer travelling 
with the sun or with the other body concerned, or at rest in the 
aether or in some gravitational medium? 



94 THE NEW LAW OF GRAVITATION fen. 

Finally is the force of gravitation propagated instantaneously, 
or with the velocity* of light, or some other velocity? Until 
comparatively recently it was thought that conclusive proof 
had been given that the speed of gravitation must be far higher 
than that of light. The argument was something like this. If 
the Sun attracts Jupiter towards its present position S, and 
Jupiter attracts the Sun towards its present position J, the two 
forces are in the same line and balance. But if the Sun attracts 
Jupiter towards its previous position S r , and Jupiter attracts 
the Sun towards its previous position J', when the force of 
attraction started out to cross the gulf, then the two forces 




PIG. 13. 

give a couple. This couple will tend to increase the angular 
momentum of the system, and, acting cumulatively, will soon 
cause an appreciable change of period, disagreeing with observa- 
tion if the speed is at all comparable with that of light. The 
argument is fallacious, because the effect of propagation will not 
necessarily be that S is attracted in the direction towards J'. 
Indeed it is found that if S and J are two electric charges, S will 
be attracted very approximately towards J (not J') in spite of 
the electric influence being propagated with the velocity of 
light*. In the theory given in this book, gravitation is propa- 
gated with the speed of light, and there is no discordance with 
observation. 

It is often urged that Newton's law of gravitation is much 
* Appendix, Note 6. 



vi] AND THE OLD LAW 95 

simpler than Einstein's new law. That depends on the point of 
view; and from the point of view of the four-dimensional world 
Newton's law is far more complicated. Moreover, it will be seen 
that if the ambiguities are to be cleared up, the statement of 
Newton's law must be greatly expanded. 

Some attempts have been made to expand Newton's law on 
the basis of the restricted principle of relativity (p. 20) alone. 
This was insufficient to determine a definite amendment. Using 
the principle of equivalence, or relativity of force, we have 
arrived at a definite law proposed in the last chapter. Probably 
the question has arisen in the reader's mind, why should it be 
called the law of gravitation? It may be plausible as a law of 
nature; but what has the degree of curvature of space-time to 
do with attractive forces, whether real or apparent? 

A race of flat-fish once lived in an ocean in which there were 
only two dimensions. It was noticed that in general fishes swam 
in straight lines, unless there was something obviously interfering 
with their free courses. This seemed a very natural behaviour. 
But there was a certain region where all the fish seemed to be 
bewitched; some passed through the region but changed the 
direction of their swim, others swam round and round inde- 
finitely. One fish invented a theory of vortices, and said that 
there were whirlpools in that region which carried everything 
round in curves. By-and-by a far better theory was proposed; 
it was said that the fishes were all attracted towards a particu- 
larly large fish a sun-fish which was lying asleep in the middle 
of the region ; and that was what caused the deviation of their 
paths. The theory might not have sounded particularly plausible 
at first; but it was confirmed with marvellous exactitude by all 
kinds of experimental tests. All fish were found to possess this 
attractive power in proportion to their sizes ; the law of attraction 
was extremely simple, and yet it was found to explain all the 
motions with an accuracy never approached before in any 
scientific investigations. Some fish grumbled that they did not 
see how there could be such an influence at a distance; but it 
was generally agreed that the influence was communicated 
through the ocean and might be better understood when more 
was known about the nature of water. Accordingly, nearly 
every fish who wanted to explain the attraction started by 



96 THE NEW LAW OF GRAVITATION [CH. 

proposing some kind of mechanism for transmitting it through 
the water. 

But there was one fish who thought of quite another plan. 
He was impressed by the fact that whether the fish were big 
or little they always took the same course, although it would 
naturally take a bigger force to deflect the bigger fish. He there- 
fore concentrated attention on the courses rather than on the 
forces. And then he arrived at a striking explanation of the 
whole thing. There was a mound in the world round about 
where the sun-fish lay. Flat-fish could not appreciate it directly 
because they were two-dimensional; but whenever a fish went 
swimming over the slopes of the mound, although he did his 
best to swim straight on, he got turned round a bit. (If a traveller 
goes over the left slope of a mountain, he must consciously 
keep bearing away to the left if he wishes to keep to his original 
direction relative to the points of the compass.) This was the 
secret of the mysterious attraction, or bending of the paths, 
which was experienced in the region. 

The parable is not perfect, because it refers to a hummock in 
space alone, whereas we have to deal with hummocks in space- 
time. But it illustrates how a curvature of the world we live 
in may give an illusion of attractive force, and indeed c%n only 
be discovered through some such effect. How this works out in 
detail must now be considered. 

In the form G^ v = 0, Einstein's law expresses conditions to be 
satisfied in a gravitational field produced by any arbitrary 
distribution of attracting matter. An analogous form of Newton's 
law was given by Laplace in his celebrated expression V 2 F = 0. 
A more illuminating form of the law is obtained if, instead of 
putting the question what kinds of space-time can exist under 
the most general conditions in an empty region, we ask what 
kind of space-time exists in the region round a single attracting 
particle? We separate out the effect of a single particle, just as 
Newton did. We can further simplify matters by introducing 
some definite mesh-system, which, of course, must be of a type 
which is not inconsistent with the kind of space-time found. 

We need only consider space of two dimensions sufficient 
for the so-called plane orbit of a planet time being added as 
the third dimension. The remaining dimension of space can 



vi] AND THE OLD LAW 97 

always be added, if desired, by conditions of symmetry. The 
result of long algebraic calculations * is that, round a particle 

ds 2 =- - dr 2 - r 2 d6 2 + -ydt 2 ....(6) 

2m 
where y = 1 . 

The quantity m is the gravitational mass of the particle 
but we are not supposed to know that at present, r and 6 are 
polar coordinates, the mesh-system being as in Fig. 11 ; or rather 
they are the nearest thing to polar coordinates that can be 
found in space which is not truly flat. 

The fact is that this expression for ds 2 is found in the first 
place simply as a particular solution of Einstein's equations of 
the gravitational field; it is a variety of hummock (apparently 
the simplest variety) which is not curved beyond the first degree. 
There could be such a state of the world under suitable circum- 
stances. To find out what those circumstances are. we have to 
trace some of the consequences, find out how any particle 
moves when ds 2 is of this form, and then examine whether we 
know of any case in which these consequences are found 
observationally. It is only after having ascertained that this 
form of ds 2 does correspond to the leading observed effects 
attributable to a particle of mass m at the origin that we have 
the right to identify this particular solution with the one we 
hoped to find. 

It will be a sufficient illustration of this procedure, if we 
indicate how the position of the matter causing this particular 
solution is located. Wherever the formula (6) holds good there 
can be no matter, because the law which applies to empty space 
is satisfied. But if we try to approach the origin (r = 0), a 
curious thing happens. Suppose we take a measuring-rod, and, 
laying it radially, start marking off equal lengths with it along 
a radius, gradually approaching the origin. Keeping the time 
t constant, and dO being zero for radial measurements, the 
formula (6) reduces to 

ds 2 = - - dr 2 

y 

or dr 2 = yds 2 . 

* Appendix, Note 7. 
E. s. i 



98 THE NEW LAW OF GRAVITATION [CH. 

We start with r large. By-and-by we approach the point 
where r = 2m. But here, from its definition, y is equal to 0. 
So that, however large the measured interval ds may be, dr = 0. 
We can go on shifting the measuring-rod through its own length 
time after time, but dr is zero; that is to say, we do not reduce 
r. There is a magic circle which no measurement can bring us 
inside. It is not unnatural that we should picture something 
obstructing our closer approach, and say that a particle of 
matter is filling up the interior. 

The fact is that so long as we keep to space-time curved only 
in the first degree, we can never round off the summit of the 
hummock. It must end in an infinite chimney. In place of the 
chimney, however, we round it off with a small region of greater 
curvature. This region cannot be empty because the law applying 
to empty space does not hold. We describe it therefore as con- 
taining matter a procedure which practically amounts to a 
definition of matter. Those familiar with hydrodynamics may 
be reminded of the problem of the irrotational rotation of a 
fluid; the conditions cannot be satisfied at the origin, and it is 
necessary to cut out a region which is filled by a vortex- 
filament. 

A word must also be said as to the coordinates r and t used 
in (6). They correspond to our ordinary notion of radial distance 
and time as well as any variables in a non-Euclidean world 
can correspond to words which, as ordinarily used, presuppose 
a Euclidean world. We shall thus call r and t, distance and time. 
But to give names to coordinates does not give more information 
and in this case gives considerably less information than is 
already contained in the formula for ds 2 . If any question arises 
as to the exact significance of r and t it must always be settled 
by reference to equation (6). 

The want of flatness in the gravitational field is indicated by 
the deviation of the coefficient y from unity. If the mass m = 0, 
y = 1, and space- time is perfectly flat. Even in the most intense 
gravitational fields known, the deviation is extremely small. 
For the sun, the quantity m, called the gravitational mass, is 
only 1-47 kilometres*, for the earth it is 5 millimetres. In any 
practical problem the ratio 2m/r must be exceedingly small. 

* Appendix, Note 8. 



vi] AND THE OLD LAW 99 

Yet it is on the small corresponding difference in y that the 
whole of the phenomena of gravitation depend. 

The coefficient y appears twice in the formula, and so modifies 
the flatness of space-time in two ways. But as a rule these two 
ways are by no means equally important. Its appearance as a 
coefficient of dt 2 produces much the most striking effects. 
Suppose that it is wished to measure the interval between two 
events in the history of a planet. If the events are, say 1 second 
apart in time, dt = 1 second = 300,000 kilometres. Thus 
dt 2 = 90,000,000,000 sq. km. Now no planet moves more than 
50 kilometres in a second, so that the change dr associated with 
the lapse of 1 second in the history of the planet will not be 
more than 50 km. Thus dr 2 is not more than 2500 sq. km. 
Evidently the small term 2m/r has a much greater chance of 
making an impression where it is multiplied by dt 2 than where 
it is multiplied by dr 2 . 

Accordingly as a first approximation, we ignore the coefficient 
of dr 2 , and consider only the meaning of 

ds 2 = -dr 2 - r 2 d0 2 + (1 - 2m/r) dt 2 (7). 

We shall now show that particles situated in this kind of space- 
time will appear to be under the influence of an attractive force 
directed towards the origin. 

Let us consider the problem of mapping a small portion of this 
kind of world on a plane. 

It is first necessary to define carefully the distinction which is 
here drawn between a "picture" and a "map." If we are given 
the latitudes and longitudes of a number of places on the earth, 
we can make a picture by taking latitude and longitude as 
vertical and horizontal distances, so that the lines of latitude 
and longitude form a mesh-system of squares ; but that does not 
give a true map. In an ordinary map of Europe the lines of 
longitude run obliquely and the lines of latitude are curved. 
Why is this? Because the map aims at showing as accurately 
as possible all distances in their true proportions *. Distance is 
the important thing which it is desired to represent correctly. 
In four dimensions interval is the analogue of distance, and a 
map of the four-dimensional world will aim at showing all the 

* This is usually the object, though maps are sometimes made for a different 
purpose, e.g. Mercator's Chart. 

72 



100 



THE NEW LAW OF GRAVITATION 



[CH. 



intervals in their correct proportions. Our natural picture of 
space-time takes r and t as horizontal and vertical distances, 
e.g. when we plot the graph of the motion of a particle ; but in 
a true map, representing the intervals in their proper proportions, 
the r and t lines run obliquely or in curves across the map. 

The instructions for drawing latitude and longitude lines (/3, A) 
on a map, are summed up in the formula for ds, p. 79, 




FIG. 14. 

and similarly the instructions for drawing the r and t lines are 
given by the formula (7). 

The map is shown in Fig. 14. It is not difficult to see why the 
Z-lines converge to the left of the diagram. The factor 1 2m/r 
decreases towards the left where r is small; and consequently 
any change of t corresponds to a shorter interval, and must be 
represented in the map by a shorter distance on the left. It is 
less easy to see why the r-lines take the courses shown; by 
analogy with latitude and longitude we, might expect them to 
be curved the other way. But we discussed in Chapter in how 



AND THE OLD LAW 



101 



the slope of the time-direction is connected with the slope of 
the space-direction; and it will be seen that the map gives 
approximately diamond-shaped partitions of the kind repre- 
sented in Fig. 6*. 

Like all maps of curved surfaces, the diagram is only accurate 
in the limit when the area covered is very small. 

It is important to understand clearly the meaning of this map. 
When we speak in the ordinary way of distance from the sun 
and the time at a point in the solar system, we mean the two 
variables r and t. These are not the result of any precise measures 
with scales and clocks made at a point, but are mathematical 
variables most appropriate for describing the whole solar system. 



FIG. 15. 

They represent a compromise, because it is necessary to deal 
with a region too large for accurate representation on a plane 
map. We should naturally picture them as rectangular co- 
ordinates partitioning space-time into square meshes, as in 
Fig. 15; but such a picture is not a true map, because it does 
not represent in their true proportions the intervals between the 
various points in the picture. It is not possible to draw any 
map of the whole curved region without distortion ; but a small 
enough portion can be represented without distortion if the 
partitions of equal r and t are drawn as in Fig. 14. To get back 

* The substitution x=r + $t z m/r 2 , y = t(l-m/r), gives fo*= -dx z + dy*, if 
squares of m are negligible. The map is drawn with x and y as rectangular 
coordinates. 



102 THE NEW LAW OF GRAVITATION [CH. 

from the true map to the customary picture of r and t as per- 
pendicular space and time, we must strain Fig. 14 until all the 
meshes become squares as in Fig. 15. 

Now in the map the geometry is Euclidean and the tracks of 
all material particles will be straight lines. Take such a straight 
track PQ, which will necessarily be nearly vertical, unless the 
velocity is very large. Strain the figure so as to obtain the 
customary representation of r and t (in Fig. 15), and the track 
PQ will become curved curved towards the left, where the sun 
lies. In each successive vertical interval (time), a successively 
greater progress is made to the left horizontally (space). Thus 
the velocity towards the sun increases. We say that the particle 
is attracted to the sun. 

The mathematical reader should find no difficulty in proving 
from the diagram that for a particle with small velocity the 
acceleration towards the sun is approximately m/r 2 , agreeing 
with the Newtonian law. 

Tracks for very high speeds may be affected rather differently. 
The track corresponding to a wave of light is represented by 
a straight line at 45 to the horizontal in Fig. 14. It would 
require very careful drawing to trace what happens to it when 
the strain is made transforming to Fig. 15; but actually, whilst 
becoming more nearly vertical, it receives a curvature in the 
opposite direction. The effect of the gravitation of the sun on 
a light-wave, or very fast particle, proceeding radially is actually 
a repulsion ! 

The track of a transverse light-wave, coming out from the 
plane of the paper, will be affected like that of a particle of 
zero velocity in distorting from Fig. 14 to Fig. 15. Hence the 
sun's influence on a transverse light- wave is always an attraction. 
The acceleration is simply m/r 2 as for a particle at rest. 

The result that the expression found for the geometry of the 
gravitational field of a particle leads to Newton's law of attrac- 
tion is of great importance. It shows that the law, G^ v = 0, 
proposed on theoretical grounds, agrees with observation at 
least approximately. It is no drawback that the Newtonian 
law applies only when the speed is small; all planetary speeds 
are small compared with the velocity of light, and the considera- 
tions mentioned at the beginning of this chapter suggest that 



vi] AND THE OLD LAW 103 

some modification may be needed for speeds comparable with 
that of light. 

Another important point to notice is that the attraction of 
gravitation is simply a geometrical deformation of the straight 
tracks. It makes no difference what body or influence is pursuing 
the track, the deformation is a general discrepancy between the 
"mental picture" and the "true map" of the portion of space- 
time considered. Hence light is subject to the same disturbance 
of path as matter. This is involved in the Principle of Equi- 
valence; otherwise we could distinguish between the acceleration 
of a lift and a true increase of gravitation by optical experi- 
ments ; in that case the observer for whom light-rays appear to 
take straight tracks might be described as absolutely unacceler- 
ated and there could be no relativity theory. Physicists in 
general have been prepared to admit the likelihood of an 
influence of gravitation on light similar to that exerted on 
matter; and the problem whether or not light has "weight" 
has often been considered. 

The appearance of y as the coefficient of dt 2 is responsible for 
the main features of Newtonian gravitation ; the appearance of 
]/y as the coefficient of dr* is responsible for the principal 
deviations of the new law from the old. This classification seems 
to be correct; but the Newtonian law is ambiguous and it is 
difficult to say exactly what are to be regarded as discrepancies 
from it. Leaving aside now the time-term as sufficiently dis- 
cussed, we consider the space- terms alone* 



y 

The expression shows that space considered alone is non- 
Euclidean in the neighbourhood of an attracting particle. This 
is something entirely outside the scope of the old law of gravita- 
tion. Time can only be explored by something moving, whether 
a free particle or the parts of a clock, so that the non-Euclidean 
character of space-time can be covered up by introducing a field 
of force, suitably modifying the motion, as a convenient fiction. 
But space can be explored by static methods ; and theoretically 

* We change the sign of ds 2 , so that ds, when real, means measured space 
instead of measured time. 



104 THE NEW LAW OF GRAVITATION [CH. 

its non-Euclidean character could be ascertained by sufficiently 
precise measures with rigid scales. 

If we lay our measuring scale transversely and proceed to 
measure the circumference of a circle of nominal radius r, we 
see from the formula that the measured length ds is equal to 
rd9, so that, when we have gone right round the circle, 6 has 
increased by 2n and the measured circumference is 27rr. But 
when we lay the scale radially the measured length ds is equal 
to dr/Vy, which is always greater than dr. Thus, in measuring 
a diameter, we obtain a result greater than 2r, each portion being 
greater than the corresponding change of r. 

Thus if we draw a circle, placing a massive particle near the 
centre so as to produce a gravitational field, and measure with 
a rigid scale the circumference and the diameter, the ratio of 
the measured circumference to the measured diameter will not be 
the famous number TT = 3-141592653589793238462643383279... 
but a little smaller. Or if we inscribe a regular hexagon in this 
circle its sides will not be exactly equal to the radius of the 
circle. Placing the particle near, instead of at, the centre, 
avoids measuring the diameter through the particle, and so 
makes the experiment a practical one. But though practical, 
it is not practicable to determine the non-Euclidean character 
of space in this way. Sufficient refinement of measures is not 
attainable. If the mass of a ton were placed inside a circle of 
5 yards radius, the defect in the value of TT would only appear 
in the twenty-fourth or twenty-fifth place of decimals. 

It is of value to put the result in this way, because it shows 
that the relativist is not talking metaphysics when he says that 
space in the gravitational field is non-Euclidean. His statement 
has a plain physical meaning, which we may some day learn how 
to test experimentally. Meanwhile we can test it by indirect 
methods. 

Suppose that a plane field is uniformly studded with hurdles. 
The distance between any two points will be proportional to 
the number of hurdles that must be passed over in getting from 
one point to the other by the straight route in fact the minimum 
number of hurdles. We can use counts of hurdles as the equi- 
valent of distance, and map the field by these counts. The map 
can be drawn on a plane sheet of paper without any inconsis- 



vi] AND THE OLD LAW 105 

tency, since the field is plane. Let us now dismiss from our 
minds all idea of distances in the field or straight lines in the 
field, and assume that distances on the map merely represent 
the minimum number of hurdles between two points; straight 
lines on the map will represent the corresponding routes. This 
has the advantage that if an earthquake occurs, deforming the 
field, the map will still be correct. The path of fewest hurdles 
will still cross the same hurdles as before the earthquake; it 
will be twisted out of the straight line in the field ; but we should 
gain nothing by taking a straighter course, since that would 
lead through a region where the hurdles are more crowded. 
We do not alter the number of hurdles in any path by deforming 
it. 

This can be illustrated by Figs. 14 and 15. Fig. 14 represents 
the original undistorted field with the hurdles uniformly placed. 
The straight line PQ represents the path of fewest hurdles from 
P to Q, and its length is proportional to the number of hurdles. 
Fig. 15 represents the distorted field, with PQ distorted into 
a curve ; but PQ is still the path of fewest hurdles from P to Q, 
and the number of hurdles in the path is the same as before. 
If therefore we map according to hurdle-counts we arrive at 
Fig. 14 again, just as though no deformation had taken place. 

To make any difference in the hurdle-counts, the hurdles 
must be taken up and replanted. Starting from a given point 
as centre, let us arrange them so that they gradually thin out 
towards the boundaries of the field. Now choose a circle with 
this point as centre; but first, what is a circle? It has to be 
defined in terms of hurdle-counts; and clearly it must be a 
curve such that the minimum number of hurdles between any 
point on it and the centre is a constant (the radius). With this 
definition we can defy earthquakes. The number of hurdles in 
the circumference of such a circle will not bear the same pro- 
portion to the number in the radius as in the field of uniform 
hurdles; owing to the crowding near the centre, the ratio will 
be less. Thus we have a suitable analogy for a circle whose 
circumference is less than TT times its diameter. 

This analogy enables us to picture the condition of space 
round a heavy particle, where the ratio of the circumference of 
a circle to the diameter is less than 77. Hurdle-counts will no 



106 THE NEW LAW OF GRAVITATION [CH. 

longer be accurately mappable on a plane sheet of paper, 
because they do not conform to Euclidean geometry. 

Now suppose a heavy particle wishes to cross this field, 
passing near but not through the centre. In Euclidean space, 
with the hurdles uniformly distributed, it travels in a straight 
line, i.e. it goes between any two points by a path giving the 
fewest hurdle jumps. We may assume that in the non-Euclidean 
field with rearranged hurdles, the particle still goes by the path 
of least effort. In fact, in any small portion we cannot distinguish 
between the rearrangement and a distortion ; so we may imagine 
that the particle takes each portion as it comes according to the 
rule, and is not troubled by the rearrangement which is only 
visible to a general survey of the whole field *. 

Now clearly it will pay not to go straight through the dense 
portion, but to keep a little to the outside where the hurdles 
are sparser not too much, or the path will be unduly lengthened. 
The particle's track will thus be a little concave to the centre, 
and an onlooker will say that it has been attracted to the centre. 
It is rather curious that we should call it attraction, when the 
track has rather been avoiding the central region; but it is clear 
that the direction of motion has been bent round in the way 
attributable to an attractive force. 

This bending of the path is additional to that due to the 
Newtonian force of gravitation which depends on the second 
appearance of y in the formula. As already explained it is in 
general a far smaller effect and will appear only as a minute 
correction to Newton's law. The only case where the two rise 
to equal importance is when the track is that of a light- wave, 
or of a particle moving with a speed approaching that of light ; 
for then dr 2 rises to the same order of magnitude as dt 2 . 

To sum up, a ray of light passing near a heavy particle will 
be bent, firstly, owing to the non-Euclidean character of the 
combination of time with space. This bending is equivalent to 
that due to Newtonian gravitation, and may be calculated in 
the ordinary way on the assumption that light has weight like 
a material body. Secondly, it will be bent owing to the non- 

* There must be some absolute track, and if absolute significance can only 
be associated with hurdle-counts and not with distances in the field, the path 
of fewest hurdles is the only track capable of absolute definition. , 



vi] AND THE OLD LAW 107 

Euclidean character of space alone, and this curvature is 
additional to that predicted by Newton's law. If then we can 
observe the amount of curvature of a ray of light, we can make 
a crucial test of whether Einstein's or Newton's theory is 
obeyed. 

This separation of the attraction into two parts is useful in 
a comparison of the new theory with the old; but from the 
point of view of relativity it is artificial. Our view is that light 
is bent just in the same way as the track of a material particle 
moving with the same velocity would be bent. Both causes of 
bending may be ascribed either to weight or to non-Euclidean 
space-time, according to the nomenclature preferred. The only 
difference between the predictions of the old and new theories 
is that in one case the weight is calculated according to Newton's 
law of gravitation, in the other case according to Einstein's. 

There is an alternative way of viewing this effect on light 
according to Einstein's theory, which, for many reasons is to 
be preferred. This depends on the fact that the velocity of 
light in the gravitational field is not a constant (unity) but 
becomes smaller as we approach the sun. This does not mean 
that an observer determining the velocity of light experimentally 
at a spot near the sun would detect the decrease; if he performed 
Fizeau's experiment, his result in kilometres per second would 
be exactly the same as that of a terrestrial observer. It is the 
coordinate velocity that is here referred to, described in terms 
of the quantities r, 6, t, introduced by the observer who is 
contemplating the whole solar system at the same time. 

It will be remembered that in discussing the approximate 
geometry of space-time in Fig. 3, we found that certain events 
like P were in the absolute past or future of O, and others like 
P' were neither before nor after 0, but elsewhere. Analytically 
the distinction is that for the interval OP, ds 2 is positive; for 
OP', ds 2 is negative. In the first case the interval is real or 
"time-like"; in the second it is imaginary or "space-like." The 
two regions are separated by lines (or strictly, cones) in crossing 
which ds 2 changes from positive to negative; and along the lines 
themselves ds is zero. It is clear that these lines must have 
important absolute significance in the geometry of the world. 
Physically their most important property is that pulses of light 



108 THE NEW LAW OF GRAVITATION [CH. 

travel along these tracks, and the motion of a light-pulse is 
always given by the equation ds = 0. 

Using the expression for ds 2 in a gravitational field, we 
accordingly have for light 

= - - dr* - r 2 d6* + ydt 2 . 

y 

For radial motion, dd = 0, and therefore 

dr * 



For transverse motion, dr = 0, and therefore 



Thus the coordinate velocity of light travelling radially is y, 
and of light travelling transversely is \/y, in the coordinates 
chosen. 

The coordinate velocity must depend on the coordinates 
chosen; and it is more convenient to use a slightly different 
system in which the velocity of light is the same in all directions *, 
viz. y or 1 2m/r. This diminishes as we approach the sun 
an illustration of our previous remark that a pulse of light 
proceeding radially is repelled by the sun. 

The wave-motion in a ray of light can be compared to a 
succession of long straight waves rolling onward in the sea. If 
the motion of the waves is slower at one end than the other, the 
whole wave-front must gradually slew round, and the direction 
in which it is rolling must change. In the sea this happens when 
one end of the wave reaches shallow water before the other, 
because the speed in shallow water is slower. It is well known 
that this causes waves proceeding diagonally across a bay to 
slew round and come in parallel to the shore; the advanced end 

* This is obtained by writing r + m instead of r, or diminishing the nominal 
distance of the sun by If kilometres. This change of coordinates simplifies 
the problem, but can, of course, make no difference to anything observable. 
After we have traced the course of the light ray in the coordinates chosen, we 
have to connect the results with experimental measures, using the corresponding 
formula for ds 2 . This final connection of mathematical and experimental results 
is, however, comparatively simple, because it relates to measuring operations 
performed in a terrestrial observatory where the difference of y from unity is 
negligible. 



vi] AND THE OLD LAW 109 

is delayed in the shallow water and waits for the other. In the 
same way when the light waves pass near the sun, the end nearest 
the sun has the smaller velocity and the wave-front slews round ; 
thus the course of the waves is bent. 

Light moves more slowly in a material medium than in 
vacuum, the velocity being inversely proportional to the re- 
fractive index of the medium. The phenomenon of refraction 
is in fact caused by a slewing of the wave-front in passing into 
a region of smaller velocity. We can thus imitate the gravita- 
tional effect on light precisely, if we imagine the space round 
the sun filled with a refracting medium which gives the 
appropriate velocity of light. To give the velocity 1 2m/r, the 
refractive index must be 1/(1 2m/r), or, very approximately, 
1 4- 2m/r. At the surface of the sun, r = 697,000 km., m = 1-47 
km., hence the necessary refractive index is 1-00000424. At a 
height above the sun equal to the radius it is 1-00000212. 

Any problem on the paths of rays near the sun can now be 
solved by the methods of geometrical optics applied to the 
equivalent refracting medium. It is not difficult to show that 
the total deflection of a ray of light passing at a distance r from 
the centre of the sun is (in circular measure) 

4m 



whereas the deflection of the same ray calculated on the 
Newtonian theory would be 

2m 
r 

For a ray grazing the surface of the sun the numerical value 
of this deflection is 

l"-75 (Einstein's theory), 
0"-87 (Newtonian theory). 



CHAPTER VII 
WEIGHING LIGHT 

Query 1. Do not Bodies act upon Light at a distance, and by their action 
bend its Rays, and is not this action (caeteris paribus) strongest at the least 
distance? NEWTON, Opticks. 

WE come now to the experimental test of the influence of 
gravitation on light discussed theoretically in the last chapter. 
It is not the general purpose of this book to enter into details 
of experiments; and if we followed this plan consistently, we 
should, as hitherto, summarise the results of the observations 
in a few lines. But it is this particular test which has turned 
public attention towards the relativity theory, and there appears 
to be widespread desire for information. We shall therefore tell 
the story of the eclipse expeditions in some detail. It will make 
a break in the long theoretical arguments, and will illustrate 
the important applications of this theory to practical obser- 
vations. 

It must be understood that there were two questions to 
answer: firstly, whether light has weight (as suggested by 
Newton), or is indifferent to gravitation; secondly, if it has 
weight, is the amount of the deflection in accordance with 
Einstein's or Newton's laws? 

It was already known that light possesses mass or inertia like 
other forms of electromagnetic energy. This is manifested in 
the phenomena of radiation-pressure. Some force is required to 
stop a beam of light by holding an obstacle in its path; a search- 
light experiences a minute force of recoil just as if it were a 
machine-gun firing material projectiles. The force, which is 
predicted by orthodox electromagnetic theory, is exceedingly 
minute; but delicate experiments have detected it. Probably 
this inertia of radiation is of great cosmical importance, playing 
a great part in the equilibrium of the more diffuse stars. Indeed 
it is probably the agent which has carved the material of the 
universe into stars of roughly uniform mass. Possibly the tails 
of comets are a witness to the power of the momentum of sun- 



CH. vn] WEIGHING LIGHT 111 

light, which drives outwards the smaller or the more absorptive 
particles. 

It is legitimate to speak of a pound of light as we speak of 
a pound of any other substance. The mass of ordinary quantities 
of light is however extremely small, and I have calculated that 
at the low charge of 3d. a unit, an Electric Light Company 
would have to sell light at the rate of 140,000,000 a pound. 
All the sunlight falling on the earth amounts to 160 tons daily. 

It is perhaps not easy to realise how a wave-motion can have 
inertia, and it is still more difficult to understand what is meant 
by its having weight. Perhaps this will be better understood if 
we put the problem in a concrete form. Imagine a hollow body, 
with radiant heat or light- waves traversing the hollow; the 
mass of the body will be the sum of the masses of the material 
and of the radiant energy in the hollow; a greater force will be 
required to shift it because of the light-waves contained in it. 
Now let us weigh it with scales or a spring-balance. Will it also 
weigh heavier on account of the radiation contained, or will the 
weight be that of the solid material alone? If the former, then 
clearly from this aspect light has weight; and it is not difficult 
to deduce the effect of this weight on a freely moving light-beam 
not enclosed within a hollow. 

The effect of weight is that the radiation in the hollow body 
acquires each second a downward momentum proportional to 
its mass. This in the long run is transmitted to the material 
enclosing it. For a free light-wave in space, the added momen- 
tum combines with the original momentum, and the total 
momentum determines the direction of the ray, which is 
accordingly bent. Newton's theory suggests no means for 
bringing about the bending, but contents itself with predicting 
it on general principles. Einstein's theory provides a means, 
viz. the variation of velocity of the waves. 

Hitherto mass and weight have always been found associated 
in strict proportionality. One very important test had already 
shown that this proportionality is not confined to material 
energy. The substance uranium contains a great deal of radio- 
active energy, presumably of an electromagnetic nature, which 
it slowly liberates. The mass of this energy must be an appreciable 
fraction of the whole mass of the substance. But it was shown 



112 WEIGHING LIGHT [CH. 

by experiments with the Eotvos torsion-balance that the ratio 
of weight to mass for uranium is the same as for all other 
substances; so the energy of radio-activity has weight. Still 
even this experiment deals only with bound electromagnetic 
energy, and we are not justified in deducing the properties of 
the free energy of light. 

It is easy to see that a terrestrial experiment has at present 
no chance of success. If the mass and weight of light are in the 
same proportion as for matter, the ray of light will be bent 
just like the trajectory of a material particle. On the earth a 
rifle bullet, like everything else, drops 16 feet in the first second, 
64 feet in two seconds, and so on, below its original line of flight ; 
the rifle must thus be aimed above the target. Light would also 
drop 16 feet in the first second* ; but, since it has travelled 1 86,000 
miles along its course in that time, the bend is inappreciable. 




FIG. 16. 

In fact any terrestrial course is described so quickly that 
gravitation has scarcely had time to accomplish anything. 

The experiment is therefore transferred to the neighbourhood 
of the sun. There we get a pull of gravitation 27 times more 
intense than on the earth; and what is more important the 
greater size of the sun permits a much longer trajectory through- 
out which the gravitation is reasonably powerful. The deflection 
in this case may amount to something of the order of a second 
of arc, which for the astronomer is a fairly large quantity. 

In Fig. 16 the line EFQP shows the track of a ray of light 
from a distant star P which reaches the earth E. The main 
part of the bending of the ray occurs as it passes the sun S; 
and the initial course PQ and the final course FE are practically 
straight. Since the light rays enter the observer's eye or telescope 
in the direction FE, this will be the direction in which the star 
appears. But its true direction from the earth is QP, the initial 

* Or 32 feet according to Einstein's law. The fall increases with the speed of 
the motion. 



vn] WEIGHING LIGHT 113 

course. So the star appears displaced outwards from its true 
position by an angle equal to the total deflection of the light. 

It must be noticed that this is only true because a star is so 
remote that its true direction with respect to the earth E is 
indistinguishable from its direction with respect to the point 
Q. For a source of light within the solar system, the apparent 
displacement of the source is by no means equal to the deflection 
of the light-ray. It is perhaps curious that the attraction of 
light by the sun should produce an apparent displacement of 
the star away from the sun; but the necessity for this is 
clear. 

The bending affects stars seen near the sun, and accordingly 
the only chance of making the observation is during a total 
eclipse when the moon cuts off the dazzling light. Even then 
there is a great deal of light from the sun's corona which stretches 
far above the disc. It is thus necessary to have rather bright 
stars near the sun, which will not be lost in the glare of the 
corona. Further the displacements of these stars can only be 
measured relatively to other stars, preferably more distant from 
the sun and less displaced; we need therefore a reasonable 
number of outer bright stars to serve as reference points. 

In a superstitious age a natural philosopher wishing to perform 
an important experiment would consult an astrologer to ascertain 
an auspicious moment for the trial. With better reason, an 
astronomer to-day consulting the stars would announce that the 
most favourable day of the year for weighing light is May 29. 
The reason is that the sun in its annual journey round the 
ecliptic goes through fields of stars of varying richness, but on 
May 29 it is in the midst of a quite exceptional patch of bright 
stars part of the Hyades by far the best star-field encountered. 
Now if this problem had been put forward at some other period 
of history, it might have been necessary to wait some thousands 
of years for a total eclipse of the sun to happen on the lucky 
date. But by strange good fortune an eclipse did happen on 
May 29, 1919. Owing to the curious sequence of eclipses a 
similar opportunity will recur in 1938; we are in the midst of 
the most favourable cycle. It is not suggested that it is im- 
possible to make the test at other eclipses; but the work will 
necessarily be more difficult. 

B.S. 8 



114 WEIGHING LIGHT [CH. 

Attention was called to this remarkable opportunity by the 
Astronomer Royal in March, 1917; and preparations were begun 
by a Committee of the Royal Society and Royal Astronomical 
Society for making the observations. Two expeditions were sent 
to different places on the line of totality to minimise the risk 
of failure by bad weather. Dr A. C. D. Crommelin and Mr C. 
Davidson went to Sobral in North Brazil; Mr E. T. Cottingham 
and the writer went to the Isle of Principe in the Gulf of Guinea, 
West Africa. The instrumental equipment for both expeditions 
was prepared at Greenwich Observatory under the care of the 
Astronomer Royal; and here Mr Davidson made the arrange- 
ments which were the main factor in the success of both 
parties. 

The circumstances of the two expeditions were somewhat 
different and it is scarcely possible to treat them together. We 
shall at first follow the fortunes of the Principe observers. They 
had a telescope of focal length 11 feet 4 inches. On their 
photographs 1 second of arc (which was about the largest dis- 
placement to be measured) corresponds to about y^. inch 
by no means an inappreciable quantity. The aperture of the 
object-glass was 13 inches, but as used it was stopped down to 
8 inches to give sharper images. It is necessary, even when the 
exposure is only a few seconds, to allow for the diurnal motion 
of the stars across the sky, making the telescope move so as to 
follow them. But since it is difficult to mount a long and heavy 
telescope in the necessary manner in a temporary installation 
in a remote part of the globe, the usual practice at eclipses is 
to keep the telescope rigid and reflect the stars into it by a 
coelostat a plane mirror kept revolving at the right rate by 
clock-work. This arrangement was adopted by both expeditions. 

The observers had rather more than a month on the island 
to make their preparations. On the day of the eclipse the 
weather was unfavourable. When totality began the dark disc 
of the moon surrounded by the corona was visible through cloud, 
much as the moon often appears through cloud on a night when 
no stars can be seen. There was nothing for it but to carry out 
the arranged programme and hope for the best. One observer 
was kept occupied changing the plates in rapid succession, whilst 
the other gave the exposures of the required length with a screen 



vii] WEIGHING LIGHT 115 

held in front of the object-glass to avoid shaking the telescope in 
any way. 

For in and out, above, about, below 
'Tis nothing but a Magic Sfadow-show 
Played in a Box whose candle is the Sun 
Round which we Phantom Figures come and go. 

Our shadow-box takes up all our attention. There is a marvellous 
spectacle above, and, as the photographs afterwards revealed, 
a wonderful prominence-flame is poised a hundred thousand 
miles above the surface of the sun. We have no time to snatch 
a glance at it. We are conscious only of the weird half-light of 
the landscape and the hush of nature, broken by the calls of the 
observers, and beat of the metronome ticking out the 302 
seconds of totality. 

Sixteen photographs were obtained, with exposures ranging 
from 2 to 20 seconds. The earlier photographs showed no stars, 
though they portrayed the remarkable prominence; but appar- 
ently the cloud lightened somewhat towards the end of totality, 
and a few images appeared on the later plates. In many cases 
one or other of the most essential stars was missing through 
cloud, and no use could be made of them; but one plate was 
found showing fairly good images of five stars, which were 
suitable for a determination. This was measured on the spot 
a few days after the eclipse in a micrometric measuring-machine. 
The problem was to determine how the apparent positions of 
the stars, affected by the sun's gravitational field, compared 
with the normal positions on a photograph taken when the sun 
was out of the way. Normal photographs for comparison had 
been taken with the same telescope in England in January. 
The eclipse photograph and a comparison photograph were 
placed film to film in the measuring-machine so that corre- 
sponding images fell close together*, and the small distances 
were measured in two rectangular directions. From these the 
relative displacements of the stars could be ascertained. In 
comparing two plates, various allowances have to be made for 
refraction, aberration, plate- orientation, etc.; but since these 
occur equally in determinations of stellar parallax, for which 

* This was possible because at Principe the field of stars was reflected in 
the coelostat mirror, whereas in England it was photographed direct. 

82 



116 WEIGHING LIGHT [CH. 

much greater accuracy is required, the necessary procedure is 
well-known to astronomers. 

The results from this plate gave a definite displacement, in 
good accordance with Einstein's theory and disagreeing with 
the Newtonian prediction. Although the material was very 
meagre compared with what had been hoped for, the writer 
(who it must be admitted was not altogether unbiassed) believed 
it convincing. 

It was not until after the return to England that any further 
confirmation was forthcoming. Four plates were brought home 
undeveloped, as they were of a brand which would not stand 
development in the hot climate. One of these was found to 
show sufficient stars; and on measurement it also showed the 
deflection predicted by Einstein, confirming the other plate. 

The bugbear of possible systematic error affects all investiga- 
tions of this kind. How do you know that there is not something 
in your apparatus responsible for this apparent deflection? 
Your object-glass has been shaken up by travelling; you have 
introduced a mirror into your optical system; perhaps the 50 
rise of temperature between the climate at the equator and 
England in winter has done some kind of mischief. To meet 
this criticism, a different field of stars was photographed at 
night in Principe and also in England at the same altitude as 
the eclipse field. If the deflection were really instrumental, stars 
on these plates should show relative displacements of a similar 
kind to those on the eclipse plates. But on measuring these 
check-plates no appreciable displacements were found. That 
seems to be satisfactory evidence that the displacement observed 
during the eclipse is really due to the sun being in the region, 
and is not due to differences in instrumental conditions between 
England and Principe. Indeed the only possible loophole is a 
difference between the night conditions at Principe when the 
check-plates were taken, and the day, or rather eclipse, con- 
ditions when the eclipse photographs were taken. That seems 
impossible since the temperature at Principe did not vary more 
than 1 between day and night. 

The problem appeared to be settled almost beyond doubt; 
and it was with some confidence that we awaited the return of 
the other expedition from Brazil. The Brazil party had had 



vii] WEIGHING LIGHT 117 

fine weather and had gained far more extensive material on 
their plates. They had remained two months after the eclipse 
to photograph the same region before dawn, when clear of the 
sun, in order that they might have comparison photographs 
taken under exactly the same circumstances. One set of 
photographs was secured with a telescope similar to that used 
at Principe. In addition they used a longer telescope of 4 inches 
aperture and 19 feet focal length*. The photographs obtained 
with the former were disappointing. Although the full number 
of stars expected (about 12) were shown, and numerous plates 
had been obtained, the definition of the images had been spoiled 
by some cause, probably distortion of the coelostat-mirror by 
the heat of the sunshine falling on it. The observers were 
pessimistic as to the value of these photographs ; but they were 
the first to be measured on return to England, and the results 
came as a great surprise after the indications of the Principe 
plates. The measures pointed with all too good agreement to 
the "half-deflection," that is to say, the Newtonian value which 
is one-half the amount required by Einstein's theory. It seemed 
difficult to pit the meagre material of Principe against the wealth 
of data secured from the clear sky of Sobral. It is true the 
Sobral images were condemned, but whether so far as to 
invalidate their testimony on this point was not at first clear; 
besides the Principe images were not particularly well-defined, 
and were much enfeebled by cloud. Certain compensating 
advantages of the latter were better appreciated later. Their 
strong point was the satisfactory check against systematic error 
afforded by the photographs of the check-field; there were 
no check-plates taken at Sobral, and, since it was obvious 
that the discordance of the two results depended on syste- 
matic error and not on the wealth of material, this distinctly 
favoured the Principe results. Further, at Principe there could 
be no evil effects from the sun's rays on the mirror, for the 
sun had withdrawn all too shyly behind the veil of cloud. 
A further advantage was provided by the check-plates at 
Principe, which gave an independent determination of the 

* See Frontispiece. The two telescopes are shown and the backs of the two 
coelostat-mirrors which reflect the sky into them. The clock driving the larger 
mirror is seen on the pedestal on the left. 



118 WEIGHING LIGHT [CH. 

difference of scale of the telescope as used in England and at 
the eclipse; for the Sobral plates this scale-difference was 
eliminated by the method of reduction, with the consequence 
that the results depended on the measurement of a much smaller 
relative displacement. 

There remained a set of seven plates taken at Sobral with the 
4-inch lens ; their measurement had been delayed by the necessity 
of modifying a micrometer to hold them, since they were of 
unusual size. From the first no one entertained any doubt that 
the final decision must rest with them, since the images were 
almost ideal, and they were on a larger scale than the other 
photographs. The use of this instrument must have presented 
considerable difficulties the unwieldy length of the telescope, 
the slower speed of the lens necessitating longer exposures and 
more accurate driving of the clock-work, the larger scale rendering 
the focus more sensitive to disturbances but the observers 
achieved success, and the perfection of the negatives surpassed 
anything that could have been hoped for. 

These plates were now measured and they gave a final verdict 
definitely confirming Einstein'* value of the deflection, in agree- 
ment with the results obtained at Principe. 

It will be remembered that Einstein's theory predicts a 
deflection of l"-74 at the edge of the sun*, the amount falling 
off inversely as the distance from the sun's centre. The simple 
Newtonian deflection is half this, 0"-87. The final results 
(reduced to the edge of the sun) obtained at Sobral and Principe 
with their "probable accidental errors" were 
Sobral l"-98 0"-12, 
Principe 1"-61 0"-30. 

It is usual to allow a margin of safety of about twice the probable 
error on either side of the mean. The evidence of the Principe 
plates is thus just about sufficient to rule out the possibility of 
the "half-deflection," and the Sobral plates exclude it with 
practical certainty. The value of the material found at Principe 
cannot be put higher than about one-sixth of that at Sobral; 
but it certainly makes it less easy to bring criticism against this 
confirmation of Einstein's theory seeing that it was obtained 

* The predicted deflection of light from infinity to infinity is just over I" '745, 
from infinity to the earth it is just under. 



VIl] 



WEIGHING LIGHT 



119 



independently with two different instruments at different places 
and with different kinds of checks. 

The best check on the results obtained with the 4-inch lens 
at Sobral is the striking internal accordance of the measures for 
different stars. The theoretical deflection should vary inversely 
as the distance from the sun's centre; hence, if we plot the mean 
radial displacement found for each star separately against the 
inverse distance, the points should lie on a straight line. This 

Displacement 



i"-oo 

09- 
08- 
07 
06 
05 
04 
03 
02- 
01- 



/ 



/ 



Distance 



I I I I I 
90' 60' 50' 

FIG. 17. 



40' 



I 
30' 



I 
25 X 



is shown in Fig. 17 where the broken line shows the theoretical 
prediction of Einstein, the deviations being within the accidental 
errors of the determinations. A line of half the slope representing 
the half-deflection would clearly be inadmissible. 

Moreover, values of the deflection were deduced from the 
measures in right ascension and declination independently. 
These were in close agreement. 



120 



WEIGHING LIGHT 



[CH. 



A diagram showing the relative positions of the stars is given 
in Fig. 18. 

The square shows the limits of the plates used at Principe, 
and the oblique rectangle the limits with the 4-inch lens at 
Sobral. The centre of the sun moved from S to P in the 2J 

N 






12 



w 



Y 



FIG. 18. 

hours interval between totality at the two stations; the sun is 
here represented for a time about midway between. The stars 
measured on the Principe plates were Nos. 3, 4, 5, 6, 10, 11 ; those 
at Sobral were 11, 10, 6, 5, 4, 2, 3 (in the order of the dots 
from left to right in Fig. 17). None of these were fainter than 
6 m -0, the brightest K 1 Tauri (No. 4) being 4 m -5. 
It has been objected that although the observations establish 



vii] WEIGHING LIGHT 121 

a deflection of light in passing the sun equal to that predicted 
by Einstein, it is not immediately obvious that this deflection 
must necessarily be attributed to the sun's gravitational field. 
It is suggested that it may not be an essential effect of the sun 
as a massive body, but an accidental effect owing to the circum- 
stance that the sun is surrounded by a corona which acts as 
a refracting atmosphere. It would be a strange coincidence if 
this atmosphere imitated the theoretical law in the exact 
quantitative way shown in Fig. 17; and the suggestion appears 
to us far-fetched. However the objection can be met in a more 
direct way. We have already shown that the gravitational 
effect on light is equivalent to that produced by a refracting 
medium round the sun and have calculated the necessary 
refractive index. At a height of 400,000 miles above the surface 
the refractive index required is 1-0000021. This corresponds to 
air at -^ atmosphere, hydrogen at ^ atmosphere, helium at 
fa atmospheric pressure. It seems obvious that there can be no 
material of this order of density at such a distance from the sun. 
The pressure on the sun's surface of the columns of material 
involved would be of the order 10,000 atmospheres; and we 
know from spectroscopic evidence that there is no pressure of 
this order. If it is urged that the mass could perhaps be sup- 
ported by electrical forces, the argument from absorption is 
even more cogent. The light from the stars photographed during 
the eclipse has passed through a depth of at least a million miles 
of material of this order of density or say the equivalent of 
10,000 miles of air at atmospheric density. We know to our 
cost what absorption the earth's 5 miles of homogeneous 
atmosphere can effect. And yet at the eclipse the stars appeared 
on the photographs with their normal brightness. If the irre- 
pressible critic insists that the material round the sun may be 
composed of some new element with properties unlike any 
material known to us, we may reply that the mechanism of 
refraction and of absorption is the same, and there is a limit to 
the possibility of refraction without appreciable absorption. 
Finally it would be necessary to arrange that the density of the 
material falls off inversely as the distance from the sun's centre 
in order to give the required variation of refractive index. 
Several comets have been known to approach the sun within 



122 WEIGHING LIGHT [CH.VII 

the limits of distance here considered. If they had to pass 
through an atmosphere of the density required to account for 
the displacement, they would have suffered enormous resistance. 
Dr Crommelin has shown that a study of these comets sets an 
upper limit to the density of the corona, which makes the 
refractive effect quite negligible. 

Those who regard Einstein's law of gravitation as a natural 
deduction from a theory based on the minimum of hypotheses 
will be satisfied to find that his remarkable prediction is quanti- 
tatively confirmed by observation, and that no unforeseen cause 
has appeared to invalidate the test. 



CHAPTER VIII 
OTHER TESTS OF THE THEORY 

The words of Mercury are harsh after the songs of Apollo. 

Love's Labour's Lost. 

WE have seen that the swift-moving light-waves possess great 
advantages as a means of exploring the non-Euclidean property 
of space. But there is an old fable about the hare and the 
tortoise. The slow-moving planets have qualities which must 
not be overlooked. The light-wave traverses the region in a few 
minutes and makes its report; the planet plods on and on for 
centuries going over the same ground again and again. Eack 
time it goes round it reveals a little about the space, and the 
knowledge slowly accumulates. 

According to Newton's law a planet moves round the sun in 
an ellipse, and if there are no other planets disturbing it, the 
ellipse remains the same for ever. According to Einstein's law 
the path is very nearly an ellipse, but it does not quite close up ; 
and in the next revolution the path has advanced slightly in the 
same direction as that in which the planet was moving. The 
orbit is thus an ellipse which very slowly revolves *. 

The exact prediction of Einstein's law is that in one revolution 
of the planet the orbit will advance through a fraction of a 
revolution equal to 3u 2 /C 2 , where v is the speed of the planet 
and C the speed of light. The earth has 1/10,000 of the speed of 
light; thus in one revolution (one year) the point where the 
earth is at greatest distance from the sun will move on 
3/100,000,000 of a revolution, or 0"-038. We could not detect 
this difference in a year, but we can let it add up for a century 
at least. It would then be observable but for one thing the 
earth's orbit is very blunt, very nearly circular, and so we 
cannot tell accurately enough which way it is pointing and how 
its sharpest apses move. We can choose a planet with higher 
speed so that the effect is increased, not only because v 2 is 
increased, but because the revolutions take less time; but, what 
* Appendix, Note 9. 



124 OTHER TESTS OF THE THEORY [CH. 

is perhaps more important, we need a planet with a sharp 
elliptical orbit, so that it is easy to observe how its apses move 
round. Both these conditions are fulfilled in the case of Mercury. 
It is the fastest of the planets, and the predicted advance of the 
orbit amounts to 43" per century; further the eccentricity of 
its orbit is far greater than that of any of the other seven 
planets. 

Now an unexplained advance of the orbit of Mercury had 
long been known. It had occupied the attention of Le Verrier, 
who, having successfully predicted the planet Neptune from the 
disturbances of Uranus, thought that the anomalous motion of 
Mercury might be due to an interior planet, which was called 
Vulcan in anticipation. But, though thoroughly sought for, 
Vulcan has never turned up. Shortly before Einstein arrived 
at his law of gravitation, the accepted figures were as follows. 
The actual observed advance of the orbit was 574" per century; 
the calculated perturbations produced by all the known planets 
amounted to 532" per century. The excess of 42" per century 
remained to be explained. Although the amount could scarcely 
be relied on to a second of arc, it was at least thirty times as 
great as the probable accidental error. 

The big discrepancy from the Newtonian gravitational theory 
is thus in agreement with Einstein's prediction of an advance 
of 43" per century. 

The derivation of this prediction from Einstein's law can only 
be followed by mathematical analysis; but it may be remarked 
that any slight deviation from the inverse square law is likely 
to cause an advance or recession of the apse of the orbit. That 
a particle, if it does not move in a circle, should oscillate between 
two extreme distances is natural enough; it could scarcely do 
anything else unless it had sufficient speed to break away 
altogether. But the interval between the two extremes will not 
in general be half a revolution. It is only under the exact 
adjustment of the inverse square law that this happens, so that 
the orbit closes up and the next revolution starts at the same 
point. I do not think that any "simple explanation" of this 
property of the inverse-square law has been given ; and it seems 
fair to remind those, who complain of the difficulty of under- 
standing Einstein's prediction of the advance of the perihelion, 



vm] OTHER TESTS OF THE THEORY 125 

that the real trouble is that they have not yet succeeded in 
making clear to the uninitiated this recondite result of the 
Newtonian theory. The slight modifications introduced by 
Einstein's law of gravitation upset this fine adjustment, so that 
the oscillation between the extremes occupies slightly more than 
a revolution. A simple example of this effect of a small deviation 
from the inverse-square law was actually given by Newton. 

It had already been recognised that the change of mass with 
velocity may cause an advance of perihelion; but owing to the 
ambiguity of Newton's law of gravitation the discussion was 
unsatisfactory. It was, however, clear that the effect was too 
small to account for the motion of perihelion of Mercury, the 
prediction being Ju 2 /C 2 , or at most v z /C 2 . Einstein's theory is 
the only one which gives the full amount 3v 2 /C 2 . 

It was suggested by Lodge that, this variation of mass with 
velocity might account for the whole motion of the orbit of 
Mercury, if account were taken of the sun's unknown absolute 
motion through the aether, combining sometimes additively and 
sometimes negatively with the orbital motion. In a discussion 
between him and the writer, it appeared that, if the absolute 
motion were sufficient to produce this effect on Mercury, it 
must give observable effects for Venus and the Earth; and these 
do not exist. Indeed from the close accordance of Venus and 
the Earth with observation, it is possible to conclude that, either 
the sun's motion through the aether is improbably small, or 
gravitation must conform to relativity, in the sense of the 
restricted principle (p. 20), and conceal the effects of the 
increase of mass with speed so far as an additive uniform motion 
is concerned. 

Unfortunately it is not possible to obtain any further test of 
Einstein's law of gravitation from the remaining planets. We 
have to pass over Venus and the Earth, whose orbits are too 
nearly circular to show the advance of the apses observationally. 
Coming next to Mars with a moderately eccentric orbit, the 
speed is very much smaller, and the predicted advance is only 
l"-3 per century. Now the accepted figures show an observed 
advance (additional to that produced by known causes) of 5" 
per century, so that Einstein's correction improves the accord- 
ance of observation with theory; but, since the result for Mars 



126 OTHER TESTS OF THE THEORY [CH. 

is in any case scarcely trustworthy to 5" owing to the inevitable 
errors of observation, the improvement is not very important. 
The main conclusion is that Einstein's theory brings Mercury 
into line, without upsetting the existing good accordance of all 
the other planets. 

We have tested Einstein's law of gravitation for fast move- 
ment (light) and for moderately slow movement (Mercury). 
For very slow movement it agrees with Newton's law, and the 
general accordance of the latter with observation can be trans- 
ferred to Einstein's law. These tests appear to be sufficient to 
establish the law firmly. We can express it in this way. 

Every particle or light-pulse moves so that the quantity s 
measured along its track between two points has the maximum 
possible value, where 

ds 2 = - (1 - 2m/r)~ 1 dr 2 - r 2 dd 2 + (I - 2m/r) dt 2 . 
And the accuracy of the experimental test is sufficient to verify 
the coefficients as far as terms of order m/r in the coefficient of 
dr 2 , and as far as terms of order m z /r 2 in the coefficient of dt 2 *. 

In this form the law appears to be firmly based on experiment, 
and the revision or even the complete abandonment of the 
general ideas of Einstein's theory would scarcely affect it. 

These experimental proofs, that space in the gravitational 
field of the sun is non-Euclidean or curved, have appeared 
puzzling to those unfamiliar with the theory. It is pointed out 
that the experiments show that physical objects or loci are 
"warped" in the sun's field; but it is suggested that there is 
nothing to show that the space in which they exist is warped. 
The answer is that it does not seem possible to draw any dis- 
tinction between the warping of physical space and the warping 
of physical objects which define space. If our purpose were 
merely to call attention to these phenomena of the gravitational 
field as curiosities, it would, no doubt, be preferable to avoid 
using words which are liable to be misconstrued. But if we wish 
to arrive at an understanding of the conditions of the gravita- 
tional field, we cannot throw over the vocabulary appropriate 
for that purpose, merely because there may be some who insist 
on investing the words with a metaphysical meaning which is 
clearly inappropriate to the discussion. 
* Appendix, Note 10. 



vm] OTHER TESTS OF THE THEORY 127 

We come now to another kind of test. In the statement of 
the law of gravitation just given, a quantity s is mentioned; 
and, so far as that statement goes, s is merely an intermediary 
quantity defined mathematically. But in our theory we have 
been identifying s with interval-length, measured with an 
apparatus of scales and clocks ; and it is very desirable to test 
whether this identification can be confirmed whether the 
geometry of scales and clocks is the same as the geometry of 
moving particles and light-pulses. 

The question has been mooted whether we may not divide 
the present theory into two parts. Can we not accept the law 
of gravitation in the form suggested above as a self-contained 
result proved by observation, leaving the further possibility 
that s is to be identified with interval-length open to debate? 
The motive is partly a desire to consolidate our gains, freeing 
them from the least taint of speculation ; but perhaps also it is 
inspired by the wish to leave an opening by which clock-scale 
geometry, i.e. the space and time of ordinary perception, may 
remain Euclidean. Disregarding the connection of s with 
interval-length, there is no object in attributing any significance 
of length to it; it can be regarded as a dynamical quantity like 
Action, and the new law of gravitation can be expressed after 
the traditional manner without dragging in strange theories of 
space and time. Thus interpreted, the law perhaps loses its 
theoretical inevitability; but it remains strongly grounded on 
observation. Unfortunately for this proposal, it is impossible 
to make a clean division of the theory at the point suggested. 
Without some geometrical interpretation of s our conclusions as 
to the courses of planets and light- waves cannot be connected 
with the astronomical measurements which verify them. The 
track of a light- wave in terms of the coordinates r, 6, t cannot 
be tested directly; the coordinates afford only a temporary 
resting-place ; and the measurement of the displacement of the 
star-image on the photographic plate involves a reconversion 
from the coordinates to s, which here appears in its significance 
as the interval in clock-scale geometry. 

Thus even from the experimental standpoint, a rough corre- 
spondence of the quantity s occurring in the law of gravitation 
with the clock-scale interval is an essential feature. We have 



128 OTHER TESTS OF THE THEORY [CH. 

now to examine whether experimental evidence can be found 
as to the exactness of this correspondence. 

It seems reasonable to suppose that a vibrating atom is an 
ideal type of clock. The beginning and end of a single vibration 
constitute two events, and the interval ds between two events 
is an absolute quantity independent of any mesh-system. This 
interval must be determined by the nature of the atom; and 
hence atoms which are absolutely similar will measure by their 
vibrations equal values of the absolute interval ds. Let us 
adopt the usual mesh-system (r, 6, t) for the solar system, so 
that 

ds 2 = - y- 1 dr 2 - r*dd 2 + ycfe 2 . 

Consider an atom momentarily at rest at some point in the solar 
system; we say momentarily, because it must undergo the 
acceleration of the gravitational field where it is. If ds corre- 
sponds to one vibration, then, since the atom has not moved, 
the corresponding dr and d6 will be zero, and we have 

ds 2 = ydt 2 . 

The time of vibration dt is thus l/\/y times the interval of 
vibration ds. 

Accordingly, if we have two similar atoms at rest at different 
points in the system, the interval of vibration will be the same 
for both ; but the time of vibration will be proportional to the 
inverse square-root of y, which differs for the two atoms. Since 

2m 



77? 

1 H -- , very approximately. 

Take an atom on the surface of the sun, and a similar atom 
in a terrestrial laboratory. For the first, 1 + m/r = 1-00000212, 
and for the second 1 + m/r is practically 1. The time of vibration 
of the solar atom is thus longer in the ratio 1*00000212, and it 
might be possible to test this by spectroscopic examination. 

There is one important point to consider. The spectroscopic 
examination must take place in the terrestrial laboratory; and 
we have to test the period of the solar atom by the period of 
the waves emanating from it when they reach the earth. Will 
they carry the period to us unchanged? Clearly they must. 



vm] OTHER TESTS OF THE THEORY 129 

The first and second pulse have to travel the same distance (r), 
and they travel with the same velocity (dr/dt); for the velocity 
of light in the mesh-system used is 1 - 2m/r, and though this 
velocity depends on r, it does not depend on t. Hence the differ- 
ence dt at one end of the waves is the same as that at the other 
end. 

Thus in the laboratory the light from a solar source should 
be of greater period and greater wave-length (i.e. redder) than 
that from a corresponding terrestrial source. Taking blue light 
of wave-length 4000 A, the solar lines should be displaced 
4000 x -00000212, or 0-008 A towards the red end of the 
spectrum. 

The properties of a gravitational field of force are similar to 
those of a centrifugal field of force; and it may be of interest 
to see how a corresponding shift of the spectral lines occurs for 
an atom in a field of centrifugal force. Suppose that, as we rotate 
with the earth, we observe a very remote atom momentarily at 
rest relative to our rotating axes. The case is just similar to 
that of the solar atom; both are at rest relative to the respective 
mesh-systems ; the solar atom is in a field of gravitational force, 
and the other is in a field of centrifugal force. The direction of 
the force is in both cases the same from the earth towards the 
atom observed. Hence the atom in the centrifugal field ought 
also to vibrate more slowly, and show a displacement to the red 
in its spectral lines. It does, if the theory hitherto given is 
right. We can abolish the centrifugal force by choosing non- 
rotating axes. But the distant atom was at rest relative to the 
rotating axes, that is to say, it was whizzing round with them. 
Thus from the ordinary standpoint the atom has a large velocity 
relative to the observer, and, in accordance with Chapter i, its 
vibrations slow down just as the aviator's watch did. The shift 
of spectral lines due to a field of centrifugal force is only another 
aspect of a phenomenon already discussed. 

The expected shift of the spectral lines on the sun, compared 
with the corresponding terrestrial lines, has been looked for; 
but it has not been found. 

In estimating the importance of this observational result in 
regard to the relativity theory, we must distinguish between 
a failure of the test and a definite conclusion that the lines are 
E.S. o 



130 OTHER TESTS OF THE THEORY [CH. 

undisplaced. The chief investigators St John, Schwarzschild, 
Evershed, and Grebe and Bachem, seem to be agreed that the 
observed displacement is at any rate less than that predicted 
by the theory. The theory can therefore in no case claim support 
from the present evidence. But something more must be 
established, if the observations are to be regarded as in the 
slightest degree adverse to the theory. If for instance the mean 
deflection is found to be -004 instead of -008 Angstrom units, 
the only possible conclusion is that there are certain causes of 
displacement of the lines, acting in the solar atmosphere and not 
yet identified. No one could be much surprised if this were the 
case; and it would, of course, render the test nugatory. The 
case is not much altered if the observed displacement is -002 
units, provided the latter quantity is above the accidental error 
of measurement; if we have to postulate some unexplained dis- 
turbance, it may just as well produce a displacement -006 as 
+ -002. For this reason Evershed's evidence is by no means 
adverse to the theory, since he finds unexplained displacements 
in any case. One set of lines measured by St John gave a mean 
displacement of -0036 units; and this also shows that the test 
has failed. The only evidence adverse to the theory, and not 
merely neutral, is a series of measures by St John on 17 cyanogen 
lines, which he regarded as most dependable. These gave a mean 
shift of exactly -000. If this stood alone we should certainly be 
disposed to infer that the test had gone against Einstein's 
theory, and that nothing had intervened to cast doubt on the 
validity of the test. The writer is unqualified to criticise these 
mutually contradictory spectroscopic conclusions; but he has 
formed the impression that the last-mentioned result obtained 
by St John has the greatest weight of any investigations up to 
the present*. 

It seems that judgment must be reserved; but it may be well 
to examine how the present theory would stand if the verdict 
of this third crucial experiment finally went against it. 

It has become apparent that there is something illogical in 

* A further paper by Grebe and Bachem (Zeitechrift fur Pkysik, 1920, p, 51), 
received whilst this is passing through the press, makes out a case strongly 
favourable for the Einstein displacement, and reconciles the discordant results 
found by most of the investigators. But it may still be the best counsel to 
"wait and see," and I have made no alteration in. the discussion here given. 



vm] OTHER TESTS OF THE THEORY 131 

the sequence we have followed in developing the theory, owing 
to the necessity of proceeding from the common ideas of space 
and time to the more fundamental properties of the absolute 
world. We started with a definition of the interval by measure- 
ments made with clocks and scales, and afterwards connected 
it with the tracks of moving particles. Clearly this is an inversion 
of the logical order. The simplest kind of clock is an elaborate 
mechanism, and a material scale is a very complex piece of 
apparatus. The best course then is to discover ds by exploration 
of space and time with a moving particle or light-pulse, rather 
than by measures with scales and clocks. On this basis by 
astronomical observation alone the formula for ds in the gravita- 
tional field of the sun has already been established. To proceed 
from this to determine exactly what is measured by a scale and 
a clock, it would at first seem necessary to have a detailed theory 
of the mechanisms involved in a scale and clock. But there is 
a short-cut which seems legitimate. This short-cut is in fact 
the Principle of Equivalence. Whatever the mechanism of the 
clock, whether it is a good clock or a bad clock, the intervals it 
is beating must be something absolute; the clock cannot know 
what mesh-system the observer is using, and therefore its 
absolute rate cannot be altered by position er motion which is 
relative merely to a mesh-system. Thus wherever it is placed, 
and however it moves, provided it is not constrained by impacts 
or electrical forces, it must always beat equal intervals as we 
have previously assumed. Thus a clock may fairly be used to 
measure intervals, even when the interval is defined in the new 
manner; any other result seems to postulate that it pays heed 
to some particular mesh-system*. 

Three modes of escape from this conclusion seem to be left 
open. A clock cannot pay any heed to the mesh-system used; 
but it may be affected by the kind of space-time around itf. 
The terrestrial atom is in a field of gravitation so weak that the 
space-time may be considered practically flat; but the space- 

* Of course, there is always the possibility that this might be the case, 
though it seems unlikely. The essential point of the relativity theory is that 
(contrary to the common opinion) no experiments yet made have revealed any 
mesh-system of an absolute character, not that experiments never will reveal 
such a system. . 

f Appendix, Note 11. 

92 



132 OTHER TESTS OF THE THEORY [CH. 

time round the solar atom is not flat. It may happen that the 
two atoms actually detect this absolute difference in the world 
around them and do not vibrate with the same interval ds 
contrary to our assumption above. Then the prediction of the 
shift of the lines in the solar spectrum is invalidated. Now it is 
very doubtful if an atom can detect the curving of the region it 
occupies, because curvature is only apparent when an extended 
region is considered; still an atom has some extension, and it is 
not impossible that its equations of motion involve the quantities 
B^ va . which distinguish gravitational from flat space-time. An 
apparently insuperable objection to this explanation is that the 
effect of curvature on the period would almost certainly be 
represented by terms of the form m 2 /r 2 , whereas to account for 
a negative result for the shift of the spectral lines terms of much 
greater order of magnitude m\r are needed. 

The second possibility depends on the question whether it is 
possible for an atom at rest on the sun to be precisely similar to 
one on the earth. If an atom fell from the earth to the sun it 
would acquire a velocity of 610 km. per sec., and could only be 
brought to rest by a systematic hammering by other atoms. 
May not this have made a permanent alteration in its time- 
keeping properties? It is true that every atom is continually 
undergoing collisions, but it is just possible that the average 
solar atom has a different period from the average terrestrial 
atom owing to this systematic difference in its history. 

What are the two events which mark the beginning and end 
of an atomic vibration? This question suggests a third possi- 
bility. If they are two absolute events, like the explosions of 
two detonators, then the interval between them will be a definite 
quantity, and our argument applies. But if, for example, an 
atomic vibration is determined by the revolution of an electron 
around a nucleus, it is not marked by any definite events. A 
revolution means a return to the same position as before; but 
we cannot define what is the same position as before without 
reference to some mesh-system. Hence it is not clear that there 
is any absolute interval corresponding to the vibration of an 
atom; an absolute interval only exists between two events 
absolutely defined. 

It is unlikely that any of these three possibilities can negative 



vm] OTHER TESTS OF THE THEORY 133 

the expected shift of the spectral lines. The uncertainties intro- 
duced by them are, so far as we can judge, of a much smaller 
order of magnitude. But it will be realised that this third test 
of Einstein's theory involves rather more complicated considera- 
tions than the two simple tests with light- waves and the moving 
planet. I think that a shift of the Fraunhofer lines is a highly 
probable prediction from the theory and I anticipate that 
experiment will ultimately confirm the prediction ; but it is not 
entirely free from guess-work. These theoretical uncertainties 
are apart altogether from the great practical difficulties of the 
test, including the exact allowance for the unfamiliar circum- 
stances of an absorbing atom in the sun's atmosphere. 

Outside the three leading tests, there appears to be little 
chance of checking the theory unless our present methods of 
measurement are greatly improved. It is not practicable to 
measure the deflection of light by any body other than the sun. 
The apparent displacement of a star just grazing the limb of 
Jupiter should be 0"-017. A hundredth of a second of arc is 
just about within reach of the most refined measurements with 
the largest telescopes. If the observation could be conducted 
under the same conditions as the best parallax measurements, 
the displacement could be detected but not measured with any 
accuracy. The glare from the light of the planet ruins any chance 
of success. 

Most astronomers, who look into the subject, are entrapped 
sooner or later by a fallacy in connection with double stars. 
It is thought that when one component passes behind the other 
it will appear displaced from its true position, like a star passing 
behind the sun; if the size of the occulting star is comparable 
with that of the sun, the displacement should be of the same 
order, l"-7. This would cause a very conspicuous irregularity in 
the apparent orbit of a double star. But reference to p. 113 
shows that an essential point in the argument was the enormous 
ratio of the distance QP of the star from the sun to the distance 
EF of the sun from the earth. It is only in these conditions that 
the apparent displacement of the object is equal to the deflection 
undergone by its light. It is easy to see that where this ratio is 
reversed, as in the case of the double star, the apparent displace- 
ment is an extremely small fraction of the deflection of the light. 
It would be quite imperceptible to observation. 



134 OTHER TESTS OF THE THEORY [CH. 

If two independent stars are seen in the same line of vision 
within about I", one being a great distance behind the other, 
the conditions seem at first more favourable. I do not know if 
any such pairs exist. It would seem that we ought to see the 
more distant star not only by the direct ray, which would be 
practically undisturbed, but also by a ray passing round the 
other side of the nearer star and bent by it to the necessary 
extent. The second image would, of course, be indistinguishable 
from that of the nearer star; but it would give it additional 
brightness, which would disappear in time when the two stars 
receded. But consider a pencil of light coming past the nearer 
star; the inner edge will be bent more than the outer edge, so 
that the divergence is increased. The increase is very small; 
but then the whole divergence of a pencil from a source some 
hundred billion miles away is very minute. It is easily calculated 
that the increased divergence would so weaken the light as to 
make it impossible to detect it when it reached us *. 

If two unconnected stars approached the line of sight still 
more closely, so that one almost occulted the other, observable 
effects might be perceived. When the proximity was such that 
the direct ray from the more distant star passed within about 
100 million kilometres of the nearer star, it would begin to fade 
appreciably. The course of the ray would not yet be appreciably 
deflected, but the divergence of the pencil would be rapidly 
increased, and less light from the star would enter our telescopes. 
The test is scarcely likely to be an important one, since a 
sufficiently close approach is not likely to occur; and in any 
case it would be difficult to feel confident that the fading was 
not due to a nebulous atmosphere around the nearer star. 

The theory gives small corrections to the motion of the moon 
which have been investigated by de Sitter. Both the axis of 
the orbit and its line of intersection with the ecliptic should 
advance about 2" per century more than the Newtonian theory 
indicates. Neither observation nor Newtonian theory are as yet 
pushed to sufficient accuracy to test this; but a comparatively 
small increase in accuracy would make a comparison possible. 

Since certain stars are perhaps ten times more massive than 
the sun, without the radius being unduly increased, they should 
show a greater shift of the spectral lines and might be more 

* Appendix, Note 12. 



viii] OTHER TESTS OF THE THEORY 135 

favourable for the third crucial test. Unfortunately the pre- 
dicted shift is indistinguishable from that caused by a velocity 
of the star in the line-of-sight on Doppler's principle. Thus the 
expected shift on the sun is equivalent to that caused by a re- 
ceding velocity of 0-634 kilometres per second. In the case of the 
sun we know by other evidence exactly what the line-of-sight 
velocity should be; but we have not this knowledge for other 
stars. The only indication that could be obtained would be the 
detection of an average motion of recession of the more massive 
stars. It seems rather unlikely that there should be a real 
preponderance of receding motions among stars taken indis- 
criminately from all parts of the sky; and the apparent effect 
might then be attributed to the Einstein shift. Actually the 
most massive stars (those of spectral type B) have been found 
to show an average velocity of recession of about 4-5 km. per 
sec., which would be explained if the values of m/r for these 
stars are about seven times greater than the value for the sun 
a quite reasonable hypothesis. This phenomenon was well- 
known to astrophysicists some years before Einstein's theory 
was published. But there are so many possible interpretations 
that no stress should be placed on this evidence. Moreover the 
very diffuse "giant" stars of type M have also a considerable 
systematic velocity of recession, and for these mjr must be much 
less than for the sun. 



CHAPTER IX 
MOMENTUM AND ENERGY 

For spirits and men by different standards mete 

The less and greater in the flow of time. 

By sun and moon, primeval ordinances 

By stars which rise and set harmoniously 

By the recurring seasons, and the swing 

This way and that of the suspended rod 

Precise and punctual, men divide the hours, 

Equal, continuous, for their common use. 

Not so with us in the immaterial world; 

But intervals in their succession 

Are measured by the living thought alone 

And grow or wane with its intensity. 

And time is not a common property; 

But what is long is short, and swift is slow 

And near is distant, as received and grasped 

By this mind and by that. NEWMAN, Dream of Oerontius. 

ONE of the most important consequences of the relativity theory 
is the unification of inertia and gravitation. 

The beginner in mechanics does not accept Newton's first law 
of motion without a feeling of hesitation. He readily agrees that 
a body at rest will remain at rest unless something causes it to 
move ; but he is not satisfied that a body in motion will remain 
in uniform motion so long as it is not interfered with. It is 
quite natural to think that motion is an impulse which will 
exhaust itself, and that the body will finally come to a stop. 
The teacher easily disposes of the arguments urged in support 
of this view, pointing out the friction which has to be overcome 
when a train or a bicycle is kept moving uniformly. He shows 
that if the friction is diminished, as when a stone is projected 
across ice, the motion lasts for a longer time, so that if all inter- 
ference by friction were removed uniform motion might con- 
tinue indefinitely. But he glosses over the point that if there 
were no interference with the motion if the ice were abolished 
altogether the motion would be by no means uniform, but like 
that of a falling body. The teacher probably insists that the 
continuance of uniform motion does not require anything that 



CH.IX] MOMENTUM AND ENERGY 137 

can properly be called a cause. The property is given a name 
inertia; but it is thought of as an innate tendency in contrast 
to force which is an active cause. So long as forces are confined 
to the thrusts and tensions of elementary mechanics, where there 
is supposed to be direct contact of material, there is good ground 
for this distinction; we can visualise the active hammering of 
the molecules on the body, causing it to change its motion. But 
when force is extended to include the gravitational field the 
distinction is not so clear. 

For our part we deny the distinction in this last case. Gravita- 
tional force is not an active agent working against the passive 
tendency of inertia. Gravitation and inertia are one. The 
uniform straight track is only relative to some mesh-system, 
which is assigned by arbitrary convention. We cannot imagine 
that a body looks round to see who is observing it and then feels 
an innate tendency to move in that observer's straight line 
probably at the same time feeling an active force compelling 
it to move some other way. If there is anything that can be 
called an innate tendency it is the tendency to follow what we 
have called the natural track the longest track between two 
points. We might restate the first law of motion in the form 
"Every body tends to move in the track in which it actually 
does move, except in so far as it is compelled by material impacts 
to follow some other track than that in which it would otherwise 
move." Probably no one will dispute this profound statement ! 

Whether the natural track is straight or curved, whether the 
motion is uniform or changing, a cause is in any case required. 
This cause is in all cases the combined inertia-gravitation. To 
have given it a name does not excuse us from attempting an 
explanation of it in due time. Meanwhile this identification of 
inertia and gravitation as arbitrary components of one property 
explains why weight is always proportional to inertia. This 
experimental fact verified to a very high degree of accuracy 
would otherwise have to be regarded as a remarkable law of 
nature. 

We have learnt that the natural track is the longest track 
between two points; and since this is the only definable track 
having an absolute significance in nature, we seem to have a 
sufficient explanation of why an undisturbed particle must 



138 MOMENTUM AND ENERGY [CH. 

follow it. That is satisfactory, so far as it goes, but still we should 
naturally wish for a clearer picture of the cause inertia- 
gravitation which propels it in this track. 

It has been seen that the gravitational field round a body 
involves a kind of curvature of space-time, and accordingly 
round each particle there is a minute pucker. Now at each 
successive instant a particle is displaced continuously in time if 
not in space; and so in our four-dimensional representation 
which gives a bird's-eye-view of all time, the pucker has the 
form of a long groove along the track of the particle. Now such 
a groove or pleat in a continuum cannot take an arbitrary 
course as every dress-maker knows. Einstein's law of gravita- 
tion gives the rule according to which the curvatures at any 
point of space-time link on to those at surrounding points ; so 
that when a groove is started in any direction the rest of its 
course can be forecasted. We have hitherto thought of the law 
of gravitation as showing how the pucker spreads out in space, 
cf. Newton's statement that the corresponding force weakens as 
the inverse square of the distance. But the law of Einstein 
equally shows how the gravitational field spreads out in time, 
since there is no absolute distinction of time and space. It can 
be deduced mathematically from Einstein's law that a pucker 
of the form corresponding to a particle necessarily runs along 
the track of greatest interval-length between two points. 

The track of a particle of matter is thus determined by the 
interaction of the minute gravitational field, which surrounds 
and, so far as we know, constitutes it, with the general space- 
time of the region. The various forms which it can take, find 
their explanation in the new law of gravitation. The straight 
tracks of the stars and the curved tracks of the planets are 
placed on the same level, and receive the same kind of explana- 
tion. The one universal law, that the space-time continuum 
can be curved only in the first degree, is sufficient to prescribe 
the forms of all possible grooves crossing it. 

The application of Einstein's law to trace the gravitational 
field not only through space but through time leads to a great 
unification of mechanics. If we have given for a start a narrow 
slice of space-time representing the state of the universe for a few 
seconds, with all the little puckers belonging to particles of matter 



ix] MOMENTUM AND ENERGY 139 

properly described, then step by step all space-time can be linked 
on and the positions of the puckers shown at all subsequent 
times (electrical forces being excluded). Nothing is needed for 
this except the law of gravitation that the curvature is only 
of the first degree and there can thus be nothing in the pre- 
dictions of mechanics which is not comprised in the law of 
gravitation. The conservation of mass, of energy, and of 
momentum must all be contained implicitly in Einstein's law. 

It may seem strange that Einstein's law of gravitation should 
take over responsibility for the whole of mechanics; because in 
many mechanical problems gravitation in the ordinary sense 
can be neglected. But inertia and gravitation are unified; the 
law is also the law of inertia, and inertia or mass appears in all 
mechanical problems. When, as in many problems, we say that 
gravitation is negligible, we mean only that the interaction of 
the minute puckers with one another can be neglected; we do 
not mean that the interaction of the pucker of a particle with 
the general character of the space-time in which it lies can be 
neglected, because this constitutes the inertia of the particle. 

The conservation of energy and the conservation of momentum 
in three independent directions, constitute together four laws 
or equations which are fundamental in all branches of mechanics. 
Although they apply when gravitation in the ordinary sense is 
not acting, they must be deducible like everything else in 
mechanics from the law of gravitation. It is a great triumph for 
Einstein's theory that his law gives correctly these experimental 
principles, which have generally been regarded as unconnected 
with gravitation. We cannot enter into the mathematical 
deduction of these equations; but we shall examine generally 
how they are arrived at. 

It has already been explained that although the values of 
G^ are strictly zero everywhere in space-time, yet if we take 
average values through a small region containing a large number 
of particles of matter their average or "macroscopic" values 
will not be zero*. Expressions for these macroscopic values can 
be found in terms of the number, masses and motions of the 
particles. Since we have averaged the G^, we should also 

* It is the gr's which are first averaged, then the O^v are calculated by the 
formulae in Note 5. 



140 MOMENTUM AND ENERGY [CH. 

average the particles; that is to say, we replace them by a 
distribution of continuous matter having equivalent properties. 
We thus obtain macroscopic equations of the form 

GUV = ^M" 

where on the one side we have the somewhat abstruse quantities 
describing the kind of space-time, and on the other side we have 
well-known physical quantities describing the density, momen- 
tum, energy and internal stresses of the matter present. These 
macroscopic equations are obtained solely from the law of 
gravitation by the process of averaging. 

By an exactly similar process we pass from Laplace's equation 
V 2 (j> = to Poisson's equation for continuous matter V 2 (/> = 47775, 
in the Newtonian theory of gravitation. 

When continuous matter is admitted, any kind of space-time 
becomes possible. The law of gravitation instead of denying the 
possibility of certain kinds, states what values of K^, i.e. what 
distribution and motion of continuous matter in the region, are 
a necessary accompaniment. This is no contradiction with the 
original statement of the law, since that referred to the case in 
which continuous matter is denied or excluded. Any set of 
values of the potentials is now possible; we have only to calculate 
by the formulae the corresponding values of G^, and we at 
once obtain ten equations giving the K^ v which define the 
conditions of the matter necessary to produce these potentials. 
But suppose the necessary distribution of matter through space 
and time is an impossible one, violating the laws of mechanics ! 
No, there is only one law of mechanics, the law of gravitation; 
we have specified the distribution of matter so as to satisfy 
G^ v = K^ vt and there can be no other condition for it to fulfil. 
The distribution must be mechanically possible ; it might, how- 
ever, be unrealisable in practice, involving inordinately high or 
even negative density of matter. 

In connection with the law for empty space, G^ v = 0, it was 
noticed that whereas this apparently forms a set often equations, 
only six of them can be independent. This was because ten 
equations would suffice to determine the ten potentials precisely, 
and so fix not only the kind of space-time but the mesh-system. 
It is clear that we must preserve the right to draw the mesh- 
system as we please; it is fixed by arbitrary choice not by a law 



ix] MOMENTUM AND ENERGY 141 

of nature. To allow for the four-fold arbitrariness of choice, 
there must be four relations always satisfied by the G^, so that 
when six of the equations are given the remaining four become 
tautological. 

These relations must be identities implied in the mathematical 
definition of G^; that is to say, when the G^ have been written 
out in full according to their definition, and the operations 
indicated by the identities carried out, all the terms will cancel, 
leaving only = 0. The essential point is that the four relations 
follow from the mode of formation of the G^ from their simpler 
constituents (g^ and their differential coefficients) and apply 
universally. These four identical relations have actually been 
discovered *. 

When in continuous matter G^ v = K^ v clearly the same four 
relations must exist between the K^, not now as identities, 
but as consequences of the law of gravitation, viz. the equality 
of G^ and K^. 

Thus the four dimensions of the world bring about a four-fold 
arbitrariness of choice of mesh-system ; this in turn necessitates 
four identical relations between the G^; and finally, in conse- 
quence of the law of gravitation, these identities reveal four new 
facts or laws relating to the density, energy, momentum or stress 
of matter, summarised in the expressions K^ v . 

These four laws turn out to be the laws of conservation of 
momentum and energy. 

The argument is so general that we can even assert that 
corresponding to any absolute property of a volume of a world 
of four dimensions (in this case, curvature), there must be four 
relative properties which are conserved. This might be made the 
starting-point of a general inquiry into the necessary qualities 
of a permanent perceptual world, i.e. a world whose substance 
is conserved. 

There is another law of physics which was formerly regarded 
as fundamental the conservation of mass. Modern progress 
has somewhat altered our position with regard to it; not that 
its validity is denied, but it has been reinterpreted, and has 
finally become merged in the conservation of energy. It will be 
desirable to consider this in detail. 

* Appendix, Note 13. 



142 MOMENTUM AND ENERGY [CH. 

It was formerly supposed that the mass of a particle was a 
number attached to the particle, expressing an intrinsic property, 
which remained unaltered in all its vicissitudes. If m is this 
number, and u the velocity of the particle, the momentum is mu; 
and it is through this relation, coupled with the law of conserva- 
tion of momentum that the mass m was denned. Let us take 
for example two particles of masses m 1 = 2 and m 2 = 3, moving 
in the same straight line. In the space-time diagram for an 
observer S the velocity of the first particle will be represented 
by a direction OA (Fig. 19). The first particle moves through 




FIG. 19. 

a space MA in unit time, so that MA is equal to its velocity 
referred to the observer S. Prolonging the line OA to meet the 
second time-partition, NB is equal to the velocity multiplied 
by the mass 2; thus the horizontal distance NB represents the 
momentum. Similarly, starting from B and drawing BC in the 
direction of the velocity of m 2 , prolonged through three 
time-partitions, the horizontal progress from B represents the 
momentum of the second particle. The length PC then repre- 
sents the total momentum of the system of two particles. 

Suppose that some change of their velocities occurs, not 
involving any transference of momentum from outside, e.g. a 
collision. Since the total momentum PC is unaltered, a similar 



ix] MOMENTUM AND ENERGY 143 

construction made with the new velocities must again bring us 
to C; that is to say, the new velocities are represented by the 
directions OB', B'C, where B' is some other point on the line NB. 
Now examine how this will appear to some other observer S 1 
in uniform motion relative to S. His transformation of space 
and time has been described in Chapter in and is represented in 
Fig. 20, which shows how his time-partitions run as compared 
with those of S. The same actual motion is, of course, repre- 
sented by parallel directions in the two diagrams; but the 




FIG. 20. 

interpretation as a velocity MA is different in the two cases. 
Carrying the velocity of m 1 through two time-partitions, and of 
ra 2 through three time-partitions, as before, we find that the total 
momentum for the observer S 1 is represented by PC (Fig. 20); 
but making a similar construction with the velocities after 
collision, we arrive at a different point C'. Thus whilst momen- 
tum is conserved for the observer S, it has altered from PC to 
PC' for the observer S t . 

The discrepancy arises because in the construction the lines 
are prolonged to meet partitions which are different for the two 



144 MOMENTUM AND ENERGY [CH. 

observers. The rule for determining momentum ought to be 
such that both observers make the same construction, inde- 
pendent of their partitions, so that both arrive by the two routes 
at the same point C. Then it will not matter if, through their 
different measures of time, one observer measures momentum 
by horizontal progress and the other by oblique progress; both 
will agree that the momentum has not been altered by the 
collision. To describe such a construction, we must use the 
interval which is alike for both observers; make the interval- 
length of OB equal to 2 units, and that of BC equal to 3 units, 
disregarding the mesh-system altogether. Then both observers 
will make the same diagram and arrive at the same point C 
(different from C or C' in the previous diagrams). Then if 
momentum is conserved for one observer, it will be conserved 
for the other. 

This involves a modified definition of momentum. Momentum 
must now be the mass multiplied by the change of position x 
per lapse of interval Ss, instead of per lapse of time Sz. Thus 

$x 

momentum = m^- 
Ss 

instead of momentum = m ~- , 

ot 

and the mass m still preserves its character as an invariant 
number associated with the particle. 

Whether the momentum as now defined is actually conserved 
or not, is a matter for experiment, or for theoretical deduction 
from the law of gravitation. The point is that with the original 
definition general conservation is impossible, because if it held 
good for one observer it could not hold for another. The new 
definition makes general conservation possible. Actually this 
form of the momentum is the one deduced from the law of 
gravitation through the identities already described. With 
regard to experimental confirmation it is sufficient at present 
to state that in all ordinary cases the interval and the time are 
so nearly equal that such experimental foundation as existed 
for the law of conservation of the old momentum is just as 
applicable to the new momentum. 

Thus in the theory of relativity momentum appears as an 



ix] MOMENTUM AND ENERGY 145 

invariant mass multiplied by a modified velocity 8x/8s. The 
physicist, however, prefers for practical purposes to keep to the 
old definition of momentum as mass multiplied by the velocity 
Sx/St. We have 

8x 8t 8a? 

m TT = m s- . ~T , 
bs 8s 8t 

accordingly the momentum is separated into two factors, the 
velocity Sx/8t, and a mass M = mSt/8s, which is no longer an 
invariant for the particle but depends on its motion relative to 
the observer's space and time. In accordance with the usual 
practice of physicists the mass (unless otherwise qualified) is 
taken to mean the quantity M. 

Using unaccelerated rectangular axes, we have by definition 
of s 

8s 2 = W - Sa? 2 - Sz/ 2 - Ss 2 , 
so that 



= I-u 2 , 

where u is the resultant velocity of the particle (the velocity of 
light being unity). Hence 

-M 

Thus the mass increases as the velocity increases, the factor 
being the same as that which determines the FitzGerald con- 
traction. 

The increase of mass with velocity is a property which chal- 
lenges experimental test. For success it is necessary to be able 
to experiment with high velocities and to apply a known force 
large enough to produce appreciable deflection in the fast- 
moving particle. These conditions are conveniently fulfilled by 
the small negatively charged particles emitted by radioactive 
substances, known as /? particles, or the similar particles which 
constitute cathode rays. They attain speeds up to 0-8 of the 
velocity of light, for which the increase of mass is in the ratio 
1-66; and the negative charge enables a large electric or magnetic 
force to be applied. Modern experiments fully confirm the 
theoretical increase of mass, and show that the factor 1/V(1 w 2 ) 
E. s. 10 



146 MOMENTUM AND ENERGY [CH. 

is at least approximately correct. The experiment was originally 
performed by Kaufmann; but much greater accuracy has been 
obtained by recent modified methods. 

Unless the velocity is very great the mass M may be written 

m/V(l - u 2 ) = m + %mu 2 . 

Thus it consists of two parts, the mass when at rest, together 
with the second term which is simply the energy of the motion. 
If we can say that the term m represents a kind of potential 
energy concealed in the matter, mass can be identified with 
energy. The increase of mass with velocity simply means that 
the energy of motion has been added on. We are emboldened 
to do this because in the case of an electrical charge the electrical 
mass is simply the energy of the static field. Similarly the mass 
of light is simply the electromagnetic energy of the light. 

In our ordinary units the velocity of light is not unity, and 
a rather artificial distinction between mass and energy is intro- 
duced. They are measured by different units, and energy E has 
a mass E/C 2 where C is the velocity of light in the units used. 
But it seems very probable that mass and energy are two ways 
of measuring what is essentially the same thing, in the same 
sense that the parallax and distance of a star are two ways of 
expressing the same property of location. If it is objected that 
they ought not to be confused inasmuch as they are distinct 
properties, it must be pointed out that they are not sense- 
properties, but mathematical terms expressing the dividend 
and product of more immediately apprehensible properties, viz. 
momentum and velocity. They are essentially mathematical 
compositions, and are at the disposal of the mathematician. 

This proof of the variation of mass with velocity is much more 
general than that based on the electrical theory of inertia. It 
applies immediately to matter in bulk. The masses m 1 and m 2 
need not be particles; they can be bodies of any size or com- 
position. On the electrical theory alone, there is no means of 
deducing the variation of mass of a planet from that of an 
electron. 

It has to be remarked that, although the inertial mass of a 
particle only comes under physical measurement in connection 
with a change of its motion, it is just when the motion is changing 
that the conception of its mass is least definite; because it is at 



ix] MOMENTUM AND ENERGY 147 

that time that the kinetic energy, which forms part of the mass, 
is being passed on to another particle or radiated into the 
surrounding field; and it is scarcely possible to define the 
moment at which this energy ceases to be associated with the 
particle and must be reckoned as broken loose. The amount of 
energy or mass in a given region is always a definite quantity; 
but the amount attributable to a particle is only definite when 
the motion is uniform. In rigorous work it is generally necessary 
to consider the mass not of a particle but of a region. 

The motion of matter from one place to another causes an 
alteration of the gravitational field in the surrounding space. 
If the motion is uniform, the field is simply convected; but if 
the motion is accelerated, something of the nature of a gravita- 
tional wave is propagated outwards. The velocity of propagation 
is the velocity of light. The exact laws are not very simple 
because we have seen that the gravitational field modifies the 
velocity of light; and so the disturbance itself modifies the 
velocity with which it is propagated. In the same way the 
exact laws of propagation of sound are highly complicated, 
because the disturbance of the air by sound modifies the speed 
with which it is propagated. But the approximate laws of 
propagation of gravitation are quite simple and are the same as 
those of electromagnetic disturbances. 

After mass and energy there is one physical quantity which 
plays a very fundamental part in modern physics, known as 
Action. Action here is a very technical term, and is not to be 
confused with Newton's " Action and Reaction." In the relativity 
theory in particular this seems in many respects to be the most 
fundamental thing of all. The reason is not difficult to see. If 
we wish to speak of the continuous matter present at any par- 
ticular point of space and time, we must use the term density. 
Density multiplied by volume in space gives us mass or, what 
appears to be the same thing, energy. But from our space-time 
point of view, a far more important thing is density multiplied 
by a four-dimensional volume of space and time; this is action. 
The multiplication by three dimensions gives mass or energy; 
and the fourth multiplication gives mass or energy multiplied 
by time. Action is thus mass multiplied by time, or energy 
multiplied by time, and is more fundamental than either. 

10 2 



148 MOMENTUM AND ENERGY [CH. 

Action is the curvature of the world. It is scarcely possible 
to visualise this statement, because our notion of curvature is 
derived from surfaces of two dimensions in a three-dimensional 
space, and this gives too limited an idea of the possibilities of a 
four-dimensional surface in space of five or more dimensions. 
In two dimensions there is just one total curvature, and if that 
vanishes the surface is flat or at least can be unrolled into a 
plane. In four dimensions there are many coefficients of 
curvature; but there is one curvature par excellence, which is, 
of course, an invariant independent of our mesh-system. It is 
the quantity we have denoted by G. It does not follow that if 
the curvature vanishes space-time is flat; we have seen in fact 
that in a natural gravitational field space-time is not flat 
although there may be no mass or energy and therefore no action 
or curvature. 

Wherever there is matter* there is action and therefore 
curvature ; and it is interesting to notice that in ordinary matter 
the curvature of the space-time world is by no means insignificant. 
For example, in water of ordinary density the curvature is the 
same as that of space in the form of a sphere of radius 570,000,000 
kilometres. The result is even more surprising if expressed in 
time units; the radius is about half-an-hour. 

It is difficult to picture quite what this means; but at least 
we can predict that a globe of water of 570,000,000 km. radius 
would have extraordinary properties. Presumably there must 
be an upper limit to the possible size of a globe of water. So 
far as I can make out a homogeneous mass of water of about 
this size (and no larger) could exist. It would have no centre, 
and no boundary, every point of it being in the same position 
with respect to the whole mass as every other point of it like 
points on the surface of a sphere with respect to the surface. 
Any ray of light after travelling for an hour or two would come 
back to the starting point. Nothing could enter or leave the 
mass, because there is no boundary to enter or leave by; in 
fact, it is coextensive with space. There could not be any other 
world anywhere else, because there isn't an "anywhere else." 

The mass of this volume of water is not so great as the most 

* It is rather curious that there is no action in space containing only light. 
Light has mass (M) of the ordinary kind; but the invariant mass (m) vanishes. 



ix] MOMENTUM AND ENERGY 149 

moderate estimates of the mass of the stellar system. Some 
physicists have predicted a distant future when all energy will 
be degraded, and the stellar universe will gradually fall together 
into one mass. Perhaps then these strange conditions will be 
realised ! 

The law of gravitation, the laws of mechanics, and the laws 
of the electromagnetic field have all been summed up in a single 
Principle of Least Action. For the most part this unification 
was accomplished before the advent of the relativity theory, 
and it is only the addition of gravitation to the scheme which is 
novel. We can see now that if action is something absolute, 
a configuration giving minimum action is capable of absolute 
definition; and accordingly we should expect that the laws of 
the world would be expressible in some such form. The argu- 
ment is similar to that by which we first identified the natural 
tracks of particles with the tracks of greatest interval-length. 
The fact that some such form of law is inevitable, rather dis- 
courages us from seeking in it any clue to the structural details 
of our world. 

Action is one of the two terms in pre-relativity physics which 
survive unmodified in a description of the absolute world. The 
only other survival is entropy. The coming theory of relativity 
had cast its shadow before; and physics was already converging 
to two great generalisations, the principle of least action and 
the second law of thermodynamics or principle of maximum 
entropy. 

We are about to pass on to recent and more shadowy develop- 
ments of this subject; and this is an appropriate place to glance 
back on the chief results that have emerged. The following 
summary will recall some of the salient points. 

1. The order of events in the external world is a four- 
dimensional order. 

2. The observer either intuitively or deliberately constructs 
a system of meshes (space and time partitions) and locates the 
events with respect to these. 

3. Although it seems to be theoretically possible to describe 
phenomena without reference to any mesh-system (by a catalogue 
of coincidences), such a description would be cumbersome. In 



150 MOMENTUM AND ENERGY [CH. 

practice, physics describes the relations of the events to our 
mesh-system; and all the terms of elementary physics and of 
daily life refer to this relative aspect of the world. 

4. Quantities like length, duration, mass, force, etc. have no 
absolute significance; their values will depend on the mesh- 
system to which they are referred. When this fact is realised, 
the results of modern experiments relating to changes of length 
of rigid bodies are no longer paradoxical. 

5. There is no fundamental mesh-system. In particular 
problems, and more particularly in restricted regions, it may 
be possible to choose a mesh-system which follows more or less 
closely the lines of absolute structure in the world, and so 
simplify the phenomena which are related to it. But the world- 
structure is not of a kind which can be traced in an exact way 
by mesh-systems, and in any large region the mesh-system 
drawn must be considered arbitrary. In any case the systems 
used in current physics are arbitrary. 

6. The study of the absolute structure of the world is based 
on the "interval" between two events close together, which is 
an absolute attribute of the events independent of any mesh- 
system. A world-geometry is constructed by adopting the 
interval as the analogue of distance in ordinary geometry. 

7. This world-geometry has a property unlike that of 
Euclidean geometry in that the interval between two ' real 
events may be real or imaginary. The necessity for a physical 
distinction, corresponding to the mathematical distinction be- 
tween real and imaginary intervals, introduces us to the separa- 
tion of the four-dimensional order into time and space. But this 
separation is not unique, and the separation commonly adopted 
depends on the observer's track through the four-dimensional 
world. 

8. The geodesic, or track of maximum or minimum interval- 
length between two distant events, has an absolute significance. 
And since no other kind of track can be defined absolutely, it 
is concluded that the tracks of freely moving particles are 
geodesies. 

9. In Euclidean geometry the geodesies are straight lines. It 
is evidently impossible to choose space and time-reckoning so 
that all free particles in the solar system move in straight lines. 



ix] MOMENTUM AND ENERGY 151 

Hence the geometry must be non-Euclidean in a field of gravita- 
tion. 

10. Since the tracks of particles in a gravitational field are 
evidently governed by some law, the possible geometries must 
be limited to certain types. 

11. The limitation concerns the absolute structure of the 
world, and must be independent of the choice of mesh-system. 
This narrows down the possible discriminating characters. 
Practically the only reasonable suggestion is that the world 
must (in empty space) be "curved no higher than the first 
degree " ; and this is taken as the law of gravitation. 

1 2. The simplest type of hummock with this limited curvature 
has been investigated. It has a kind of infinite chimney at the 
summit, which we must suppose cut out and filled up with a 
region where this law is not obeyed, i.e. with a particle of matter. 

13. The tracks of the geodesies on the hummock are such as 
to give a very close accordance with the tracks computed by 
Newton's law of gravitation. The slight differences from the 
Newtonian law have been experimentally verified by the motion 
of Mercury and the deflection of light. 

14. The hummock might more properly be described as a 
ridge extending linearly. Since the interval-length along it is 
real or time-like, the ridge can be taken as a time-direction. 
Matter has thus a continued existence in time. Further, in 
order to conform with the law, a small ridge must always follow 
a geodesic in the general field of space-time, confirming the con- 
clusion arrived at under (8). 

15. The laws of conservation of energy and momentum in 
mechanics can be deduced from this law of world-curvature. 

16. Certain phenomena such as the FitzGerald contraction 
and the variation of mass with velocity, which were formerly 
thought to depend on the behaviour of electrical forces con- 
cerned, are now seen to be general consequences of the relativity 
of knowledge. That is to say, length and mass being the relations 
of some absolute thing to the observer's mesh-system, we can 
foretell how these relations will be altered when referred to 
another mesh-system. 



CHAPTER X 
TOWARDS INFINITY 

The geometer of to-day knows nothing about the nature of actually existing 
space at an infinite distance; he knows nothing about the properties of this 
present space in a past or a future eternity. He knows, indeed, that the laws 
assumed by Euclid are true with an accuracy that no direct experiment can 
approach, not only in this place where we are, but in places at a distance from 
us that no astronomer has conceived; but he knows this as of Here and Now; 
beyond his range is a There and Then of which he knows nothing at present, 
but may ultimately come to know more. W. K. CLIFFOBD (1873). 

THE great stumbling-block for a philosophy which denies 
absolute space is the experimental detection of absolute rotation. 
The belief that the earth rotates on its axis was suggested by 
the diurnal motions of the heavenly bodies ; this observation is 
essentially one of relative rotation, and, if the matter rested 
there, no difficulty would be felt. But we can detect the same 
rotation, or a rotation very closely equal to it, by methods 
which do not seem to bring the heavenly bodies into considera- 
tion; and such a rotation is apparently absolute. The planet 
Jupiter is covered with cloud, so that an inhabitant would 
probably be unaware of the existence of bodies outside ; yet he 
could quite well measure the rotation of Jupiter. By the gyro- 
compass he would fix two points on the planet the north and 
south poles. Then by Foucault's experiment on the change of 
the 'plane of motion of a freely suspended pendulum, he would 
determine an angular velocity about the poles. Thus there is 
certainly a definite physical constant, an angular velocity about 
an axis, which has a fundamental importance for the inhabitants 
of Jupiter; the only question is whether we are right in giving 
it the name absolute rotation. 

Contrast this with absolute translation. Here it is not a 
question of giving the right name to a physical constant; the 
inhabitants of Jupiter would find no constant to name. We see 
at once that a relativity theory of translation is on a different 
footing from a relativity theory of rotation. The duty of the 



CH. x] TOWARDS INFINITY 153 

former is to explain facts; the duty of the latter is to explaii 
away facts. 

Our present theory seems to make a start at tackling this 
problem, but gives it up. It permits the observer, if he wishes, 
to consider the earth as non-rotating, but surrounded by a field 
of centrifugal force; all the other bodies in the universe are then 
revolving round the earth in orbits mainly controlled by this 
field of centrifugal force. Astronomy on this basis is a little 
cumbersome; but all the phenomena are explained perfectly. 
The centrifugal force is part of the gravitational field, and obeys 
Einstein's law of gravitation, so that the laws of nature are 
completely satisfied by this representation. One awkward 
question remains, What causes the centrifugal force? Certainly 
not the earth which is here represented as non-rotating. As we 
go further into space to look for a cause, the centrifugal force 
becomes greater and greater, so that the more we defer the debt 
the heavier the payment demanded in the end. Our present 
theory is like the debtor who does not mind how big an obligation 
accumulates satisfied that he can always put off the payment. 
It chases the cause away to infinity, content that the laws of 
nature the relations between contiguous parts of the world 
are satisfied all the way. 

One suggested loophole must be explored. Our new law of 
gravitation admits that a rapid motion of the attracting body 
will affect the field of force. If the earth is non-rotating, the 
stars must be going round it with terrific speed. May they not 
in virtue of their high velocities produce gravitationally a 
sensible field of force on the earth, which we recognise as the 
centrifugal field? This would be a genuine elimination of 
absolute rotation, attributing all effects indifferently to the 
rotation of the earth the stars being at rest, or to the revolution 
of the stars the earth being at rest; nothing matters except the 
relative rotation. I doubt whether anyone will persuade himself 
that the stars have anything to do with the phenomenon. We 
do not believe that if the heavenly bodies were all annihilated 
it would upset the gyrocompass. In any case, precise calculation 
shows that the centrifugal force could not be produced by the 
motion of the stars, so far as they are known. 

We are therefore forced to give up the idea that the signs of 



154 TOWARDS INFINITY [CH. 

the earth's rotation the protuberance of its equator, the 
phenomena of the gyrocompass, etc. are due to a rotation 
relative to any matter we can recognise. The philosopher who 
persists that a rotation which is not relative to matter is un- 
thinkable, will no doubt reply that the rotation must then be 
relative to some matter which we have not yet recognised. We 
have hitherto been greatly indebted to the suggestions of 
philosophy in evolving this theory, because the suggestions 
related to the things we know about ; and, as it turned out, they 
were confirmed by experiment. But as physicists we cannot 
take the same interest in the new demand ; we do not necessarily 
challenge it, but it is outside our concern. Physics demands of 
its scheme of nature something else besides truth, namely a 
certain quality that we may call convergence. The law of 
conservation of energy is only strictly true when the whole 
universe is taken into account; but its value in physics lies in 
the fact that it is approximately true for a very limited system. 
Physics is an exact science because the chief essentials of a 
problem are limited to a few conditions; and it draws near to 
the truth with ever-increasing approximation as it widens its 
purview. The approximations of physics form a convergent 
series. History, on the other hand, is very often like a divergent 
series; no approximation to its course is reached until the last 
term of the infinite series has been included in the data of 
prediction. Physics, if it wishes to retain its advantage, must 
take its own course, formulating those laws which are approxi- 
mately true for the limited data of sense, and extending them 
into the unknown. The relativity of rotation is not approxi- 
mately true for the data of sense, although it may possibly be 
true when the unknown as well as the known are included. 

The same considerations that apply to rotation apply to 
acceleration, although the difficulty is less striking. We can if 
we like attribute to the sun some arbitrary acceleration, balancing 
it by introducing a uniform gravitational field. Owing to this 
field the rest of the stars will move with the same acceleration 
and no phenomena will be altered. But then it seems necessary 
to find a cause for this field. It is not produced by the gravitation 
of the stars. Our only course is to pursue the cause further and 
further towards infinity; the further we put it away, the greater 



x] TOWARDS INFINITY 155 

the mass of attracting matter needed to produce it. On the 
other hand, the earth's absolute acceleration does not intrude 
on our attention in the way that its absolute rotation does*. 

We are vaguely conscious of a difficulty in these results ; but 
if we examine it closely, the difficulty does not seem to be a 
very serious one. The theory of relativity, as we have understood 
it, asserts that our partitions of space and time are introduced 
by the observer and are irrelevant to the laws of nature; and 
therefore the current quantities of physics, length, duration, 
mass, force, etc., which are relative to these partitions, are not 
things having an absolute significance in nature. But we have 
never denied that there are features of the world having an 
absolute significance ; in fact, we have spent much time in finding 
such features. The geodesies or natural tracks have been shown 
to have an absolute significance; and it is possible in a limited 
region of the world to choose space and time partitions such 
that all geodesies become approximately straight lines. We may 
call this a "natural" frame for that region, although it is not 
as a rule the space and time adopted in practice ; it is for example 
the space and time of the observers in the falling projectile, not 
of Newton's super-observer. It is capable of absolute definition, 
except that it is ambiguous in regard to uniform motion. Now 
the rotation of the earth determined by Foucault's pendulum 
experiment is the rotation referred to this natural frame. But 
we must have misunderstood our own theory of relativity 
altogether, if we think there is anything inadmissible in an 
absolute rotation of such a kind. 

Material particles and geodesies are both features of the 
absolute structure of the world; and a rotation relative to 
geodesic structure does not seem to be on any different footing 
from a velocity relative to matter. There is, however, the 
striking feature that rotation seems to be relative not merely 
to the local geodesic structure but to a generally accepted 
universal frame; whereas it is necessary to specify precisely 

* To determine even roughly the earth's absolute acceleration we should 
need a fairly full knowledge of the disturbing effects of all the matter in the 
universe. A similar knowledge would be required to determine the absolute 
rotation accurately; but all the matter likely to exist would have so small an 
effect, that we can at once assume that the absolute rotation is very nearly 
the same as the experimentally determined rotation. 



156 TOWARDS INFINITY [CH. 

what matter a velocity is measured with respect to. This is 
largely a question of how much accuracy is needed in specifying 
velocities and rotations, respectively. If in stating the speed of 
a f$ particle we do not mind an error of 10,000 kilometres a 
second, we need not specify precisely what star or planet its 
velocity is referred to. The moon's (local) angular velocity is 
sometimes given to fourteen significant figures; I doubt if 
any universal frame is well-defined enough for this accuracy. 
There is no doubt much greater continuity in the geodesic 
structure in different parts of the world than in the material 
structure; but the difference is in degree rather than in 
principle. 

It is probable that here we part company from many of the 
continental relativists, who give prominent place to a principle 
known as the law of causality that only those things are to be 
regarded as being in causal connection which are capable of 
being actually observed. This seems to be interpreted as placing 
matter on a plane above geodesic structure in regard to the 
formulation of physical laws, though it is not easy to see in 
what sense a distribution of matter can be regarded as more 
observable than the field of influence in surrounding space 
which makes us aware of its existence. The principle itself is 
debateable; that which is observable to us is determined by 
the accident of our own structure, and the law of causality 
seems to impose our own limitations on the free interplay of 
entities in the world outside us. In this book the tradition of 
Faraday and Maxwell still rules our outlook; and for us matter 
and electricity are but incidental points of complexity, the 
activity of nature being primarily in the so-called empty spaces 
between. 

The vague universal frame to which rotation is referred is 
called the inertial frame. It is definite in the flat space-time far 
away from all matter. In the undulating country corresponding 
to the stellar universe it is not a precise conception; it is rather 
a rude outline, arbitrary within reasonable limits, but with the 
general course indicated. The reason for the term inertial frame 
is of interest. We can quite freely use a mesh-system deviating 
widely from the inertial frame (e.g. rotating axes); but we have 
seen that there is a postponed debt to pay in the shape of an 



x] TOWARDS INFINITY 157 

apparently uncaused field of force. But is there no debt to pay, 
even when the inertial frame is used? In that case there is no 
gravitational or centrifugal force at infinity; but there is still 
inertia, which is of the same nature. The distinction between 
force as requiring a cause and inertia as requiring no cause 
cannot be sustained. We shall not become any more solvent by 
commuting our debt into pure inertia. The debt is inevitable 
whatever mesh-system is used; we are only allowed to choose 
the form it shall take. 

The debt after all is a very harmless one. At infinity we have 
the absolute geodesies in space-time, and we have our own 
arbitrarily drawn mesh-system. The relation of the geodesies 
to the mesh-system decides whether our axes shall be termed 
rotating or non-rotating; and ideally it is this relation that is 
determined when a so-called absolute rotation is measured. 
No one could reasonably expect that there would be no deter- 
minable relation. On the other hand uniform translation does 
not affect the relation of the geodesies to the mesh-system if 
they were straight lines originally, they remain straight lines 
thus uniform translation cannot be measured except relative to 
matter. 

We have been supposing that the conditions found in the 
remotest parts of space accessible to observation can be extra- 
polated to infinity; and that there are still definite natural 
tracks in space-time far beyond the influence of matter. Feelings 
of objection to this view arise in certain minds. It is urged that 
as matter influences the course of geodesies it may well be 
responsible for them altogether; so that a region outside the 
field of action of matter could have no geodesies, and conse- 
quently no intervals. All the potentials would then necessarily 
be zero. Various modified forms of this objection arise; but the 
main feeling seems to be that it is unsatisfactory to have certain 
conditions prevailing in the world, which can be traced away to 
infinity and so have, as it were, their source at infinity; and 
there is a desire to find some explanation of the inertial frame 
as built up through conditions at a finite distance. 

Now if all intervals vanished space-time would shrink to a 
point. Then there would be no space, no time, no inertia, no 
anything. Thus a cause which creates intervals and geodesies 



158 TOWARDS INFINITY [CH. 

must, so to speak, extend the world. We can imagine the world 
stretched out like a plane sheet; but then the stretching cause 
the cause of the intervals is relegated beyond the bounds of 
space and time, i.e. to infinity. This is the view objected to, 
though the writer does not consider that the objection has 
much force. An alternative way is to inflate the world from 
inside, as a balloon is blown out. In this case the stretching 
force is not relegated to infinity, and ruled outside the scope of 
experiment; it is acting at every point of space and time, curving 
the world to a sphere. We thus get the idea that space-time 
may have an essential curvature on a great scale independent 
of the small hummocks due to recognised matter. 

It is not necessary to speculate whether the curvature is 
produced (as in the balloon) by some pressure applied from a 
fifth dimension. For us it will appear as an innate tendency of 
four-dimensional space-time to curve. It may be asked, what 
have we gained by substituting a natural curvature of space- 
time for a natural stretched condition corresponding to the 
inertial frame? As an explanation, nothing. But there is this 
difference, that the theory of the inertial frame can now be 
included in the differential law of gravitation instead of re- 
maining outside and additional to the law. 

It will be remembered that one clue by which we previously 
reached the law of gravitation was that flat space-time must be 
compatible with it. But if space-time is to have a small natural 
curvature independent of matter this condition is now altered. 
It is not difficult to find the necessary alteration of the law*. 
It will contain an additional, and at present unknown, constant, 
which determines the size of the world. 

Spherical space is not very easy to imagine. We have to 
think of the properties of the surface of a sphere the two- 
dimensional case and try to conceive something similar applied 
to three-dimensional space. Stationing ourselves at a point let 
us draw a series of spheres of successively greater radii. The 
surface of a sphere of radius r should be proportional to r 2 ; but 
in spherical space the areas of the more distant spheres begin 
to fall below the proper proportion. There is not so much room 
out there as we expected to find. Ultimately we reach a sphere 

* Appendix, Note 14. 



x] TOWARDS INFINITY 159 

of biggest possible area, and beyond it the areas begin to de- 
crease *. The last sphere of all shrinks to a point our antipodes. 
Is there nothing beyond this? Is there a kind of boundary 
there? There is nothing beyond and yet there is no boundary. 
On the earth's surface there is nothing beyond our own antipodes 
but there is no boundary there. 

The difficulty is that we try to realise this spherical world by 
imagining how it would appear to us and to our measurements. 
There has been nothing in our experience to compare it with, 
and it seems fantastic. But if we could get rid of the personal 
point of view, and regard the sphericity of the world as a state- 
ment of the type of order of events outside us, we should think 
that it was a simple and natural order which is as likely as any 
other to occur in the world. 

In such a world there is no difficulty about accumulated debt 
at the boundary. There is no boundary. The centrifugal force 
increases until we reach the sphere of greatest area, and then, 
still obeying the law of gravitation, diminishes to zero at the 
antipodes. The debt has paid itself automatically. 

We must not exaggerate what has been accomplished by this 
modification of the theory. A new constant has been introduced 
into the law of gravitation which gives the world a definite 
extension. Previously there was nothing to fix the scale of the 
world ; it was simply given a priori that it was infinite. Granted 
extension, so that the intervals are not invariably zero, we can 
determine geodesies everywhere, and hence mark out the inertial 
frame. 

Spherical space-time, that is to say a four-dimensional con- 
tinuum of space and imaginary time forming the surface of a 
sphere in five dimensions, has been investigated by Prof, de 
Sitter. If real time is used the world is spherical in its space 
dimensions, but open towards plus and minus infinity in its 
time dimension, like an hyperboloid. This happily relieves us 
of the necessity of supposing that as we progress in time we 
shall ultimately come back to the instant we started from! 
History never repeats itself. But in the space dimensions we 
should, if we went on, ultimately come back to the starting 
point. This would have interesting physical results, and we 

* The area is, of course, to be determined by measurement of some kind. 



160 TOWARDS INFINITY [CH. 

shall see presently that Einstein has a theory of the world in 
which the return can actually happen ; but in de Sitter's theory 
it is rather an abstraction, because, as he says, "all the para- 
doxical phenomena can only happen after the end or before the 
beginning of eternity." 

The reason is this. Owing to curvature in the time dimension, 
as we examine the condition of things further and further from 
our starting point, our time begins to run faster and faster, or 
to put it another way natural phenomena and natural clocks 
slow down. The condition becomes like that described in 
Mr H. G. Wells's story "The new accelerator." 

When we reach half-way to the antipodal point, time stands 
still. Like the Mad Hatter's tea party, it is always 6 o'clock; 
and nothing whatever can happen however long we wait. There 
is no possibility of getting any further, because everything 
including light has come to rest here. All that lies beyond is 
for ever cut off from us by this barrier of time; and light can 
never complete its voyage round the world. 

That is what happens when the world is viewed from one 
station ; but if attracted by such a delightful prospect, we pro- 
ceeded to visit this scene of repose, we should be disappointed. 
We should find nature there as active as ever. We thought time 
was standing still, but it was really proceeding there at the 
usual rate, as if in a fifth dimension of which we had no 
cognisance. Casting an eye back on our old home we should see 
that time apparently had stopped still there. Time in the two 
places is proceeding in directions at right angles, so that the 
progress of time at one point has no relation to the perception 
of time at the other point. The reader will easily see that a being 
confined to the surface of a sphere and not cognisant of a third 
dimension, will, so to speak, lose one of his dimensions altogether 
when he watches things occurring at a point 90 away. He 
regains it if he visits the spot and so adapts himself to the two 
dimensions which prevail there. 

It might seem that this kind of fantastic world-building can 
have little to do with practical problems. But that is not quite 
certain. May we not be able actually to observe the slowing 
down of natural phenomena at great distances from us? The 
most remote objects known are the spiral nebulae, whose 



x] TOWARDS INFINITY 161 

distances may perhaps be of the order a million light years. 
If natural phenomena are slowed down there, the vibrations of 
an atom are slower, and its characteristic spectral lines will 
appear displaced to the red. We should generally interpret this 
as a Doppler effect, implying that the nebula is receding. The 
motions in the line-of-sight of a number of nebulae have been 
determined, chiefly by Prof. Slipher. The data are not so ample 
as we should like; but there is no doubt that large receding 
motions greatly preponderate. This may be a genuine pheno- 
menon in the evolution of the material universe ; but it is also 
possible that the interpretation of spectral displacement as a 
receding velocity is erroneous ; and the effect is really the slowing 
of atomic vibrations predicted by de Sitter's theory. 

Prof. Einstein himself prefers a different theory of curved 
space-time. His world is cylindrical curved in the three space 
dimensions and straight in the time dimension. Since time is no 
longer curved, the slowing of phenomena at great distances 
from the observer disappears, and with it the slight experimental 
support given to the theory by the observations of spiral nebulae. 
There is no longer a barrier of eternal rest, and a ray of light is 
able to go round the world. 

In various ways crude estimates of the size of the world both 
on de Sitter's and Einstein's hypotheses have been made; and 
in both cases the radius is thought to be of the order 10 13 times 
the distance of the earth from the sun. A ray of light from the 
sun would thus take about 1000 million years to go round the 
world; and after the journey the rays would converge again at 
the starting point, and then diverge for the next circuit. The 
convergent would have all the characteristics of a real sun so 
far as light and heat are concerned, only there would be no 
substantial body present. Thus corresponding to the sun we 
might see a series of ghosts occupying the positions where the 
sun was 1000, 2000, 3000, etc., million years ago, if (as seems 
probable) the sun has been luminous for so long. 

It is rather a pleasing speculation that records of the previous 
states of the sidereal universe may be automatically reforming 
themselves on the original sites. Perhaps one or more of the 
many spiral nebulae are really phantoms of our own stellar 
system. Or it may be that only a proportion of the stars are 
E.S. ix 



162 TOWARDS INFINITY [CH. 

substantial bodies; the remainder are optical ghosts revisiting 
their old haunts. It is, however, unlikely that the light rays 
after their long journey would converge with the accuracy which 
this theory would require. The minute deflections by the various 
gravitational fields encountered on the way would turn them 
aside, and the focus would be blurred. Moreover there is a 
likelihood that the light would gradually be absorbed or 
scattered by matter diffused in space, which is encountered on 
the long journey. 

It is sometimes suggested that the return of the light-wave 
to its starting point can most easily be regarded as due to the 
force of gravitation, there being sufficient mass distributed 
through the universe to control its path in a closed orbit. We 
should have no objection in principle to this way of looking at 
it ; but we doubt whether it is correct in fact. It is quite possible 
for light to return to its starting point in a world without 
gravitation. We can roll flat space-time into a cylinder and join 
the edges ; its geometry will still be Euclidean and there will be 
no gravitation ; but a ray of light can go right round the cylinder 
and return to the starting point in space. Similarly in Einstein's 
more complex type of cylinder (three dimensions curved and 
one dimension linear), it seems likely that the return of the 
light is due as much to the connectivity of his space, as to 
the non-Euclidean properties which express the gravitational 
field. 

For Einstein's cylindrical world it is necessary to postulate 
the existence of vast quantities of matter (not needed on de 
Sitter's theory) far in excess of what has been revealed by our 
telescopes. This additional material may either be in the form 
of distant stars and galaxies beyond our limits of vision, or it 
may be uniformly spread through space and escape notice by 
its low density. There is a definite relation between the average 
density of matter and the radius of the world; the greater the 
radius the smaller must be the average density. 

Two objections to this theory may be urged. In the first 
place, absolute space and time are restored for phenomena on 
a cosmical scale. The ghost of a star appears at the spot where 
the star was a certain number of million years ago; and from 
the ghost to the present position of the star is a definite distance 



x] TOWARDS INFINITY 163 

the absolute motion of the star in the meantime*. The world 
taken as a whole has one direction in which it is not curved; 
that direction gives a kind of absolute time distinct from space. 
Relativity is reduced to a local phenomenon ; and although this 
is quite sufficient for the theory hitherto described, we are 
inclined to look on the limitation rather grudgingly. But we 
have already urged that the relativity theory is not concerned 
to deny the possibility of an absolute time, but to deny that it 
is concerned in any experimental knowledge yet found; and it 
need not perturb us if the conception of absolute time turns up 
in a new form in a theory of phenomena on a cosmical scale, 
as to which no experimental knowledge is yet available. Just 
as each limited observer has his own particular separation of 
space and time, so a being coextensive with the world might 
well have a special separation of space and time natural to him. 
It is the time for this being that is here dignified by the title 
"absolute." 

Secondly, the revised law of gravitation involves a new 
constant which depends on the total amount of matter in the 
world; or conversely the total amount of matter in the world 
is determined by the law of gravitation. This seems very hard 
to accept at any rate without some plausible explanation of 
how the adjustment is brought about. We can see that, the 
constant in the law of gravitation being fixed, there may be 
some upper limit to the amount of matter possible; as more 
and more matter is added in the distant parts, space curves 
round and ultimately closes ; the process of adding more matter 
must stop, because there is no more space, and we can only 
return to the region already dealt with. But there seems nothing 
to prevent a defect of matter, leaving space unclosed. Some 
mechanism seems to be needed, whereby either gravitation 
creates matter, or all the matter in the universe conspires to 
define a law of gravitation. 

Although this appears to the writer rather bewildering, it is 
welcomed by those philosophers who follow the lead of Mach. 
For it leads to the result that the extension of space and time 

* The ghost is not formed where the star is now. If two stars were near 
together when the light left them their ghosts must be near together, although 
the stars may now be widely separated. 

II 2 



164 TOWARDS INFINITY [CH. 

depends on the amount of matter in the world partly by its 
direct effect on the curvature and partly by its influence on the 
constant of the law of gravitation. The more matter there is, 
the more space is created to contain it, and if there were no 
matter the world would shrink to a point. 

In the philosophy of Mach a world without matter is unthink- 
able. Matter in Mach's philosophy is not merely required as 
a test body to display properties of something already there, 
which have no physical meaning except in relation to matter; 
it is an essential factor in causing those properties which it is 
able to display. Inertia, for example, would not appear by the 
insertion of one test body in the world ; in some way the presence 
of other matter is a necessary condition. It will be seen how 
welcome to such a philosophy is the theory that space and the 
inertial frame come into being with matter, and grow as it grows. 
Since the laws of inertia are part of the law of gravitation, 
Mach's philosophy was summed up perhaps unconsciously 
in the profound saying " If there were no matter in the universe, 
the law of gravitation would fall to the ground." 

No doubt a world without matter, in which nothing could 
ever happen, would be very uninteresting; and some might deny 
its claim to be regarded as a world at all. But a world uniformly 
filled with matter would be equally dull and unprofitable; so 
there seems to be little object in denying the possibility of the 
former and leaving the latter possible. 

The position can be summed up as follows: in a space 
without absolute features, an absolute rotation would be as 
meaningless as an absolute translation; accordingly, the exist- 
ence of an experimentally determined quantity generally 
identified with absolute rotation requires explanation. It was 
remarked on p. 41 that it would be difficult to devise a plan 
of the world according to which uniform motion has no significance 
but non-uniform motion is significant; but such a world has 
been arrived at a plenum, of which the absolute features are 
intervals and geodesies. In a limited region this plenum gives 
a natural frame with respect to which an acceleration or rotation 
(but not a velocity) capable of absolute definition can be 
measured. In the case of rotation the local distortions of the 
frame are of comparatively little account; and this explains 



x] TOWARDS INFINITY 165 

why in practice rotation appears to have reference to some world- 
wide inertial frame. 

Thus absolute rotation does not indicate any logical flaw in 
the theory hitherto developed; and there is no need to accept 
any modification of our views. Possibly there may be a still 
wider relativity theory, in which our supposed plenum is to be 
regarded as itself an abstraction of the relations of the matter 
distributed throughout the world, and not existent apart from 
such matter. This seems to exalt matter rather unnecessarily. 
It may be true ; but we feel no necessity for it, unless experiment 
points that way. It is with some such underlying idea that 
Einstein's cylindrical space-time was suggested, since this 
cannot exist without matter to keep it stretched. Now we freely 
admit that our assumption of perfect flatness in the remote 
parts of space was arbitrary, and there is no justification for 
insisting on it. A small curvature is possible both conceptually 
and experimentally. The arguments on both sides have hitherto 
been little more than prejudices, which would be dissipated by 
any experimental or theoretical lead in one direction. Weyl's 
theory of the electromagnetic field, discussed in the next 
chapter, assigns a definite function to the curvature of space; 
and this considerably alters the aspect of the question. We are 
scarcely sufficiently advanced to offer a final opinion; but the 
conception of cylindrical space-time seems to be favoured by 
this new development of the theory. 

Some may be inclined to challenge the right of the Einstein 
theory, at least as interpreted in this book, to be called a 
relativity theory. Perhaps it has not all the characteristics 
which have at one time or another been associated with that 
name; but the reader, who has followed us so far, will see how 
our search for an absolute world has been guided by a recognition 
of the relativity of the measurements of physics. It may be 
urged that our geodesies ought not to be regarded as fundamental ; 
a geodesic has no meaning in itself; what we are really concerned 
with is the relation of a particle following a geodesic to all the 
other matter of the world and the geodesic cannot be thought of 
apart from such other matter. We would reply, " Your particle 
of matter is not fundamental ; it has no meaning in itself; what 
you are really concerned with is its ' field ' the relation of the 



166 TOWARDS INFINITY [CH. x 

geodesies about it to the other geodesies in the world and 
matter cannot be thought of apart from its field." It is all 
a tangle of relations; physical theory starts with the simplest 
constituents, philosophical theory with the most familiar con- 
stituents. They may reach the same goal; but their methods 
are often incompatible. 



CHAPTER XI 
ELECTRICITY AND GRAVITATION 

Thou shalt not have in thy bag divers weights, a great and a small. 
Thou shalt not have in thine house divers measures, a great and a small. 
But thou shalt have a perfect and just weight, a perfect and just measure shalt 
thou have. Book of Deuteronomy. 

THE relativity theory deduces from geometrical principles the 
existence of gravitation and the laws of mechanics of matter. 
Mechanics is derived from geometry, not by adding arbitrary 
hypotheses, but by removing unnecessary assumptions, so that 
a geometer like Riemann might almost have foreseen the more 
important features of the actual world. But nature has in 
reserve one great surprise electricity. 

Electrical phenomena are not in any way a misfit in the 
relativity theory, and historically it is through them that it has 
been developed. Yet we cannot rest satisfied until a deeper 
unity between the gravitational and electrical properties of the 
world is apparent. The electron, which seems to be the smallest 
particle of matter, is a singularity in the gravitational field and 
also a singularity in the electrical field. How can these two facts 
be connected? The gravitational field is the expression of some 
state of the world, which also manifests itself in the natural 
geometry determined with measuring appliances; the electric 
field must also express some state of the world, but we have not 
as yet connected it with natural geometry. May there not still 
be unnecessary assumptions to be removed, so that a yet more 
comprehensive geometry can be found, in which gravitational 
and electrical fields both have their place? 

There is an arbitrary assumption in our geometry up to this 
point, which it is desirable now to point out. We have based 
everything on the "interval," which, it has been said, is some- 
thing which all observers, whatever their motion or whatever 
their mesh-system, can measure absolutely, agreeing on the 
result. This assumes that they are provided with identical 
standards of measurement scales and clocks. But if A is in 



168 ELECTRICITY AND GRAVITATION [CH. 

motion relative to B and wishes to hand his standards to B to 
check his measures, he must stop their motion; this means in 
practice that he must bombard his standards with material 
molecules until they come to rest. Is it fair to assume that no 
alteration of the standard is caused by this process? Or if A 
measures time by the vibrations of a hydrogen atom, and space 
by the wave-length of the vibration, still it is necessary to stop 
the atom by a collision in which electrical forces are involved. 

The standard of length in physics is the length in the year 
1799 of a bar deposited at Paris. Obviously no interval is ever 
compared directly with that length ; there must be a continuous 
chain of intermediate steps extending like a geodetic triangula- 
tion through space and time, first along the past history of the 
scale actually used, then through intermediate standards, and 
finally along the history of the Paris metre itself. It may be 
that these intermediate steps are of no importance that the 
same result is reached by whatever route we approach the 
standard; but clearly we ought not to make that assumption 
without due consideration. We ought to construct our geometry 
in such a way as to show that there are intermediate steps, and 
that the comparison of the interval with the ultimate standard 
is not a kind of action at a distance. 

To compare intervals in different directions at a point in 
space and time does not require this comparison with a distant 
standard. The physicist's method of describing phenomena 
near a point P is to lay down for comparison (1) a mesh-system, 
(2) a unit of length (some kind of material standard), which can 
also be used for measuring time, the velocity of light being unity. 
With this system of reference he can measure in terms of his 
unit small intervals PP' running in any direction from P, 
summarising the results in the fundamental formula 

+ g 22 dcc 2 * + ... + 2g 12 dx l dx 2 + .... 



If now he wishes to measure intervals near a distant point Q, he 
must lay down a mesh-system and a unit of measure there. He 
naturally tries to simplify matters by using what he would call 
the same unit of measure at P and Q, either by transporting a 
material rod or some equivalent device. If it is immaterial by 
what route the unit is carried from P to Q, and replicas of the 



xi] ELECTRICITY AND GRAVITATION 169 

unit carried by different routes all agree on arrival at Q, this 
method is at any rate explicit. The question whether the unit 
at Q defined in this way is really the same as that at P is mere 
metaphysics. But if the units carried by different routes dis- 
agree, there is no unambiguous means of identifying a unit at 
Q with the unit at P. Suppose P is an event at Cambridge on 
March 1, and Q at London on May 1 ; we are contemplating the 
possibility that there will be a difference in the results of measures 
made with our standard in London on May 1, according as the 
standard is taken up to London on March 1 and remains there, 
or is left at Cambridge and taken up on May 1. This seems at 
first very improbable; but our reasons for allowing for this 
possibility will appear presently. If there is this ambiguity the 
only possible course is to lay down (1) a mesh-system filling all 
the space and time considered, (2) a definite unit of interval, or 
gauge, at every point of space and time. The geometry of the 
world referred to such a system will be more complicated than 
that of Riemann hitherto used; and we shall see that it is 
necessary to specify not only the 10 g's, but four other functions 
of position, which will be found to have an important physical 
meaning. 

The observer will naturally simplify things by making the 
units of gauge at different points as nearly as possible equal, 
judged by ordinary comparisons. But the fact remains that, 
when the comparison depends on the route taken, exact equality 
is not definable; and we have therefore to admit that the exact 
standards are laid down at every point independently. 

It is the same problem over again as occurs in regard to 
mesh-systems. We lay down particular rectangular axes near 
a point P; presently we make some observations near a distant 
point Q. To what coordinates shall the latter be referred? The 
natural answer is that we must use the same coordinates as we 
were using at P. But, except in the particular case of flat space, 
there is no means of defining exactly what coordinates at Q are 
the same as those at P. In many cases the ambiguity may be 
too trifling to trouble us; but in exact work the only course is 
to lay down a definite mesh-system extending throughout space, 
the precise route of the partitions being necessarily arbitrary. 
We now find that we have to add to this by placing in each 



170 ELECTRICITY AND GRAVITATION [CH. 

mesh a gauge whose precise length must be arbitrary. Having 
done this the next step is to make measurements of intervals 
(using our gauges). This connects the absolute properties of the 
world with our arbitrarily drawn mesh-system and gauge- 
system. And so by measurement we determine the g's and the 
new additional quantities, which determine the geometry of our 
chosen system of reference, and at the same time contain within 
themselves the absolute geometry of the world the kind of 
space-time which exists in the field of our experiments. 

Having laid down a unit-gauge at every point, we can speak 
quite definitely of the change in interval-length of a measuring- 
rod moved from point to point, meaning, of course, the change 
compared with the unit-gauges. Let us take a rod of interval- 
length I at P, and move it successively through the displacements 
dx lf dx 2 , dx 3 , dx^ and let the result be to increase its length 
in terms of the gauges by the amount \l. The change depends 
as much on the difference of the gauges at the two points as 
on the behaviour of the rod; but there is no possibility of 
separating the two factors. It is clear that A will not depend 
on I, because the change of length must be proportional to 
the original length unless indeed our whole idea of measure- 
ment by comparison with a gauge is wrong*. Further it will 
not depend on the direction of the rod either in its initial or 
final positions because the interval-length is independent of 
direction. (Of course, the space-length would change, but that 
is already taken care of by the g's.) A can thus only depend on 
the displacements dx lt dx%, dx 9 , dx, and we may write it 

A = K^dxi + Ac a efo a + K 3 dx s + /c 4 d# 4 , 

so long as the displacements are small. The coefficients K lt K Z , 
K Z , /c 4 apply to the neighbourhood of P, and will in general be 
different in different parts of space. 

This indeed assumes that the result is independent of the 
order of the displacements dx lf dx 2 , dx 3 , dx t that is to say 
that the ambiguity of the comparison by different routes dis- 
appears in the limit when the whole route is sufficiently small. 
It is parallel with our previous implicit assumption that although 
the length of the track from a point P to a distant point Q 

* We refuse to contemplate the idea that when the metre rod changes its 
length to two metres, each centimetre of it changes to three centimetres. 



xi] ELECTRICITY AND GRAVITATION 171 

depends on the route, and no definite meaning can be attached 
to the interval between them without specifying a route, yet in 
the limit there is a definite small interval between P and Q when 
they are sufficiently close together. 

To understand the meaning of these new coefficients K let us 
briefly recapitulate what we understand by the g's. Primarily 
they are quantities derived from experimental measurements of 
intervals, and describe the geometry of the space and time 
partitions which the observer has chosen. As consequential 
properties they describe the field of force, gravitational, centri- 
fugal, etc., with which he perceives himself surrounded. They 
relate to the particular mesh-system of the observer; and by 
altering his mesh-system, he can alter their values, though not 
entirely at will. From their values can be deduced intrinsic 
properties of the world the kind of space-time in which the 
phenomena occur. Further they satisfy a definite condition 
the law of gravitation so that not all mathematically possible 
space-times and not all arbitrary values of the g's are such as 
can occur in nature. 

All this applies equally to the /c's, if we substitute gauge- 
system for mesh-system, and some at present unknown force 
for gravitation. They can theoretically be determined by 
interval-measurement; but they will be more conspicuously 
manifested to the observer through their consequential property 
of describing some kind of field of force surrounding him. The 
K'S refer to the arbitrary gauge-system of the observer; but he 
cannot by altering his gauge-system alter their values entirely 
at will. Intrinsic properties of the world are contained in their 
values, unaffected by any change of gauge-system. Further we 
may expect that they will have to satisfy some law corresponding 
to the law of gravitation, so that not all arbitrary values of the 
K'S are such as can occur in nature. 

It is evident that the /c's must refer to some type of pheno- 
menon which has not hitherto appeared in our discussion; and 
the obvious suggestion is that they refer to the electromagnetic 
field. This hypothesis is strengthened when we recall that the 
electromagnetic field is, in fact, specified at every point by the 
values of four quantities, viz. the three components of electro- 
magnetic vector potential, and the scalar potential of electro- 



172 ELECTRICITY AND GRAVITATION [CH. 

statics. Surely it is more than a coincidence that the physicist 
needs just four more quantities to specify the state of the world 
at a point in space, and four more quantities are provided by 
removing a rather illogical restriction on our system of geometry 
of natural measures. 

[The general reader will perhaps pardon a few words addressed 
especially to the mathematical physicist. Taking the ordinary 
unaccelerated rectangular coordinates a?, y, z, t, let us write 
F 9 G, H, O for fc lf * 2 , /c 3 , * 4 , then 

* = A = Fdx + Gdy + Hdz - <S>dt. 

From which, by integration, 

log I + const. = l(Fdx + Gdy + Hdz - <5>dt). 
The length / will be independent of the route taken if 

Fdx + Gdy + Hdz - dt 
is a perfect differential. The condition for this is 

as_aG_ o <^_^_ a ^_^_ 

dy dz dz dx dx dy 

3<D dF 3O dG aO dH 

~a^~^ =0> T %^S T ~^~^ = 

If F, G, H, O are the potentials of electromagnetic theory, these 
are precisely the expressions for the three components of 
magnetic force and the three components of electric force, given 
in the text-books. Thus the condition that distant intervals can 
be compared directly without specifying a particular route of 
comparison is that the electric and magnetic forces are zero in 
the intervening space and time. 

It may be noted that, even when the coordinate system has 
been denned, the electromagnetic potentials are not unique in 
value; but arbitrary additions can be made provided these 
additions form a perfect differential. It is just this flexibility 
which in our geometrical theory appears in the form of the 
arbitrary choice of gauge-system. The electromagnetic forces 
on the other hand are independent of the gauge-system, which 
is eliminated by "curling."] 

It thus appears that the four new quantities appearing in our 
extended geometry may actually be the four potentials of 



xi] ELECTRICITY AND GRAVITATION 173 

electromagnetic theory; and further, when there is no electro- 
magnetic field our previous geometry is valid. But in the more 
general case we have to adopt the more general geometry in 
which there appear fourteen coefficients, ten describing the 
gravitational and four the electrical conditions of the world. 

We ought now to seek the law of the electromagnetic field 
on the same lines as we sought for the law of gravitation, laying 
down the condition that it must be independent of mesh-system 
and gauge-system since it seeks to limit the possible kinds of 
world which can exist in nature. Happily this presents no 
difficulty, because the law expressed by Maxwell's equations, 
and universally adopted, fulfils the conditions. There is no 
need to modify it fundamentally as we modified the law of 
gravitation. We do, however, generalise it so that it applies 
when a gravitational field is present at the same time not 
merely, as given by Maxwell, for flat space-time. The deflection 
of electromagnetic waves (light) by a gravitational field is duly 
contained in this generalised law. 

Strictly speaking the laws of gravitation and of the electro- 
magnetic field are not two laws but one law, as the geometry 
of the g's and the ic's is one geometry. Although it is often 
convenient to separate them, they are really parts of the general 
condition limiting the possible kinds of metric that can occur in 
empty space. 

It will be remembered that the four-fold arbitrariness of our 
mesh-system involved four identities, which were found to 
express the conservation of energy and momentum. In the new 
geometry there is a fifth arbitrariness, namely that of the selected 
gauge-system, This must also give rise to an identity; and it is 
found that the new identity expresses the law of conservation of 
electric charge. 

A grasp of the new geometry may perhaps be assisted by a 
further comparison. Suppose an observer has laid down a line 
of a certain length and in a certain direction at a point P, and 
he wishes to lay down an exactly similar line at a distant point 
Q. If he is in flat space there will be no difficulty; he will have 
to proceed by steps, a kind of triangulation, but the route chosen 
is of no importance. We know definitely that there is just one 
direction at Q parallel to the original direction at P; and it is 



174 ELECTRICITY AND GRAVITATION [CH. 

in ordinary geometry supposed that the length is equally 
determinate. But if space is not flat the case is different. 
Imagine a two-dimensional observer confined to the curved 
surface of the earth trying to perform this task. As he does not 
appreciate the third dimension he will not immediately perceive 
the impossibility; but he will find that the direction which he has 
transferred to Q differs according to the route chosen. Or if he 
went round a complete circuit he would find on arriving back 
at P that the direction he had so carefully tried to preserve on 
the journey did not agree with that originally drawn*. We 
describe this by saying that in curved space, direction is not 
integrable; and it is this non-integrability of direction which 
characterises the gravitational field. In the case considered the 
length would be preserved throughout the circuit; but it is 
possible to conceive a more general kind of space in which the 
length which it was attempted to preserve throughout the 
circuit, as well as the direction, disagreed on return to the starting 
point with that originally drawn. In that case length is not 
integrable; and the non-integrability of length characterises the 
electromagnetic field. Length associated with direction is called 
a vector; and the combined gravitational and electric field 
describe that influence of the world on our measurements by 
which a vector carried by physical measurement round a closed 
circuit changes insensibly into a different vector. 

The welding together of electricity and gravitation into one 
geometry is the work of Prof. H. Weyl, first published in 1918 f. 
It appears to the writer to carry conviction, although up to the 
present no experimental test has been proposed. It need scarcely 
be said that the inconsistency of length for an ordinary circuit 
would be extremely minute {, and the ordinary manifestations 
of the electromagnetic field are the consequential results of 

* It might be thought that if the observer preserved mentally the original 
direction in three-dimensional space, and obtained the direction at any point 
in his two-dimensional space by projecting it, there would be no ambiguity. 
But the three-dimensional space in which a curved two-dimensional space is 
conceived to exist is quite arbitrary. A two-dimensional observer cannot 
ascertain by any observation whether he is on a plane or a cylinder, a sphere 
or any other convex surface of the same total curvature. 

t Appendix, Note 15. 

j I do not think that any numerical estimate has been made. 



xi] ELECTRICITY AND GRAVITATION 175 

changes which would be imperceptible to direct measurement. 
It will be remembered that the gravitational field is likewise 
perceived by the consequential effects, and not by direct interval- 
measurement. 

But the theory does appear to require that, for example, the 
time of vibration of an atom is not quite independent of its 
previous history. It may be assumed that the previous histories 
of terrestrial atoms are so much alike that there are no significant 
differences in their periods. The possibility that the systematic 
difference of history oT solar and terrestrial atoms may have an 
effect on the expected shift of the spectral lines on the sun has 
already been alluded to. It seems doubtful, however, whether 
the effect could attain the necessary magnitude. 

It may seem difficult to identify these abstract geometrical 
qualities of the world with the physical forces of electricity and 
magnetism. How, for instance, can the change in the length of 
a rod taken round a circuit in space and time be responsible for 
the sensations of an electric shock? The geometrical potentials 
(K) obey the recognised laws of electromagnetic potentials, and 
each entity in the physical theory charge, electric force, 
magnetic element, light, etc. has its exact analogue in the 
geometrical theory; but is this formal correspondence a sufficient 
ground for identification? The doubt which arises in our minds 
is due to a failure to recognise the formalism of all physical 
knowledge. The suggestion " This is not the thing I am speaking 
of, though it behaves exactly like it in all respects" carries no 
physical meaning. Anything which behaves exactly like 
electricity must manifest itself to us as electricity. Distinction 
of form is the only distinction that physics can recognise; and 
distinction of individuality, if it has any meaning at all, has no 
bearing on physical manifestations. 

We can only explore the world with apparatus, which is itself 
part of the world. Our idealised apparatus is reduced to a few 
simple types a neutral particle, a charged particle, a rigid 
scale, etc. The absolute constituents of the world are related in 
various ways, which we have studied, to the indications of these 
test-bodies. The main features of the absolute world are so 
simple that there is a redundancy of apparatus at our disposal ; 
and probably all that there is to be known could theoretically 



176 ELECTRICITY AND GRAVITATION [CH. 

be found out by exploration with an uncharged particle. Actually 
we prefer to look at the world as revealed by exploration with 
scales and clocks the former for measuring so-called imaginary 
intervals, and the latter for real intervals ; this gives us a unified 
geometrical conception of the world. Presumably, we could obtain 
a unified mechanical conception by taking the moving uncharged 
particle as standard indicator; or a unified electrical conception 
by taking the charged particle. For particular purposes one 
test-body is generally better adapted than others. The gravita- 
tional field is more sensitively explored with a moving particle 
than a scale. Although the electrical field can theoretically be 
explored by the change of length of a scale taken round a circuit, 
a far more sensitive way is to use a little bit of the scale an 
electron. And in general for practical efficiency, we do not use 
any simple type of apparatus, but a complicated construction 
built up with a view to a particular experiment. The reason for 
emphasising the theoretical interchangeability of test-bodies is 
that it brings out the unity and simplicity of the world ; and for 
that reason there is an importance in characterising the electro- 
magnetic condition of the world by reference to the indications 
of a scale and clock, however inappropriate they may be as 
practical test-bodies. 

Weyl's theory opens up interesting avenues for development. 
The details of the further steps involve difficult mathematics; 
but a general outline is possible. As on Einstein's more limited 
theory there is at any point an important property of the world 
called the curvature ; but on the new theory it is not an absolute 
quantity in the strictest sense of the word. It is independent of 
the observer's mesh-system, but it depends on his gauge. It is 
obvious that the number expressing the radius of curvature of 
the world at a point must depend on the unit of length; so we 
cannot say that the curvatures at two points are absolutely 
equal, because they depend on the gauges assigned at the two 
points. Conversely the radius of curvature of the world provides 
a natural and absolute gauge at every point; and it will pre- 
sumably introduce the greatest possible symmetry into our laws 
if the observer chooses this, or some definite fraction of it, as 
his gauge. He, so to speak, forces the world to be spherical by 
adopting at every point a unit of length which will make it so. 



xi] ELECTRICITY AND GRAVITATION 177 

Actual rods as they are moved about change their lengths com- 
pared with this absolute unit according to the route taken, and 
the differences correspond to the electromagnetic field. Einstein's 
curved space appears in a perfectly natural manner in this 
theory; no part of space- time is flat, even in the absence of 
ordinary matter, for that would mean infinite radius of curva- 
ture, and there would be no natural gauge to determine, for 
example, the dimensions of an electron the electron could not 
know how large it ought to be, unless it had something to 
measure itself against. 

The connection between the form of the law of gravitation 
and the total amount of matter in the world now appears less 
mysterious. The curvature of space indirectly provides the 
gauge which we use for measuring the amount of matter in the 
world. 

Since the curvature is not independent of the gauge, Weyl 
does not identify it with the most fundamental quantity in 
nature. There is, however, a slightly more complicated invariant 
which is a pure number, and this is taken to be Action*. We 
can thus mark out a definite volume of space and time, and 
say that the action within it is 5, without troubling to define 
coordinates or the unit of measure ! It might be expected that 
the action represented by the number 1 would have specially 
interesting properties; it might, for instance, be an atom of 
action and indivisible. Experiment has isolated what are be- 
lieved to be units of action, which at least in many phenomena 
behave as indivisible atoms called quanta; but the theory, as 
at present developed, does not permit us to represent the 
quantum of action by the number 1. The quantum is a very 
minute fraction of the absolute unit. 

When we come across a pure number having some absolute 
significance in the world it is natural to speculate on its possible 
interpretation. It might represent a number of discrete entities ; 
but in that case it must necessarily be an integer, and it seems 
clear that action can have fractional values. An angle is com- 
monly represented as a pure number, but it has not really this 
character; an angle can only be measured in terms of a unit of 
angle, just as a length is measured in terms of a unit of length. 

* Appendix, Note 16. 
E. s. 12 



178 ELECTRICITY AND GRAVITATION [CH. 

I can only think of one interpretation of a fractional number 
which can have an absolute significance, though doubtless there 
are others. The number may represent the probability of some- 
thing, or some function of a probability. The precise function 
is easily found. We combine probabilities by multiplying, but 
we combine the actions in two regions by adding; hence the 
logarithm of a probability is indicated. Further, since the 
logarithm of a probability is necessarily negative, we may 
identify action provisionally with minus the logarithm of 
the statistical probability of the state of the world which 
exists. 

The suggestion is particularly attractive because the Principle 
of Least Action now becomes the Principle of Greatest Proba- 
bility. The law of nature is that the actual state of the world is 
that which is statistically most probable. 

Weyl's theory also shows that the mass of a portion of matter 
is necessarily positive ; on the original theory no adequate reason 
is given why negative matter should not exist. It is further 
claimed that the theory shows to some extent why the world 
is four-dimensional. To the mathematician it seems so easy to 
generalise geometry to n dimensions, that we naturally expect 
a world of four dimensions to have an analogue in five dimensions. 
Apparently this is not the case, and there are some essential 
properties, without which it could scarcely be a world, which 
exist only for four dimensions. Perhaps this may be compared 
with the well-known difficulty of generalising the idea of a knot; 
a knot can exist in space of any odd number of dimensions, but 
not in space of an even number. 

Finally the theory suggests a mode of attacking the problem 
of how the electric charge of an electron is held together; at 
least it gives an explanation of why the gravitational force is so 
extremely weak compared with the electric force. It will be 
remembered that associated with the mass of the sun is a certain 
length, called the gravitational mass, which is equal to 1-5 kilo- 
metres. In the same way the gravitational mass or radius of an 
electron is 7-10~ 56 cms. Its electrical properties are similarly 
associated with a length 2-1 0" 33 cms., which is called the electrical 
radius. The latter is generally supposed to correspond to the 
electron's actual dimensions. The theory suggests that the ratio 



xi] ELECTRICITY AND GRAVITATION 179 

of the gravitational to the electrical radius, 3-10 42 , ought to be 
of the same order as the ratio of the latter to the radius of 
curvature of the world. This would require the radius of space 
to be of the order 6-10 29 cms., or 2-10 11 parsecs., which though 
somewhat larger than the provisional estimates made by de 
Sitter, is within the realm of possibility. 



122 



CHAPTER XII 
ON THE NATURE OF THINGS 

Hippolyta. This is the silliest stuff that ever I heard. 
Theseus. The best in this kind are but shadows; and the worst are 
no worse, if imagination amend them. 

A Midsummer -Night's Dream. 

THE constructive results of the theory of relativity are based 
on two principles which have been enunciated the restricted 
principle of relativity, and the principle of equivalence. These 
may be summed up in the statement that uniform motion and 
fields of force are purely relative. In their more formal enuncia- 
tions they are experimental generalisations, which can be 
admitted or denied; if admitted, all the observational results 
obtained by us can be deduced mathematically without any 
reference to the views of space, time, or force, described in this 
book. In many respects this is the most attractive aspect of 
Einstein's work; it deduces a great number of remarkable 
phenomena solely from two general principles, aided by a 
mathematical calculus of great power; and it leaves aside as 
irrelevant all questions of mechanism . But this mode of develop- 
ment of the theory cannot be described in a non-technical book. 

To avoid mathematical analysis we have had to resort to 
geometrical illustrations, which run parallel with the mathe- 
matical development and enable its processes to be understood 
to some extent. The question arises, are these merely illustrations 
of the mathematical argument, or illustrations of the actual 
processes of nature. No doubt the safest course is to avoid the 
thorny questions raised by the latter suggestion, and to say 
that it is quite sufficient that the illustrations should correctly 
replace the mathematical argument. But I think that this 
would give a misleading view of what the theory of relativity 
has accomplished in science. 

The physicist, so long as he thinks as a physicist, has a definite 
belief in a real world outside him. For instance, he believes that 
atoms and molecules really exist; they are not mere inventions 



CH.XII] ON THE NATURE OF THINGS 181 

that enable him to grasp certain laws of chemical combination. 
That suggestion might have sufficed in the early days of the 
atomic theory ; but now the existence of atoms as entities in the 
real world of physics is fully demonstrated. This confident 
assertion is not inconsistent with philosophic doubts as to the 
meaning of ultimate reality. 

When therefore we are asked whether the four-dimensional 
world may not be regarded merely as an illustration of mathe- 
matical processes, we must bear in mind that our questioner has 
probably an ulterior motive. He has already a belief in a real 
world of three Euclidean dimensions, and he hopes to be allowed 
to continue in this belief undisturbed. In that case our answer 
must be definite; the real three-dimensional world is obsolete, 
and must be replaced by the four-dimensional space-time with 
non-Euclidean properties. In this book we have sometimes 
employed illustrations which certainly do not correspond to any 
physical reality imaginary time, and an unperceived fifth 
dimension. But the four-dimensional world is no mere illustra- 
tion; it is the real world of physics, arrived at in the recognised 
way by which physics has always (rightly or wrongly) sought for 
reality. 

I hold a certain object before me, and see an outline of the 
figure of Britannia; another observer on the other side sees a 
picture of a monarch ; a third observer sees only a thin rectangle. 
Am I to say that the figure of Britannia is the real object; and 
that the crude impressions of the other observers must be 
corrected to make allowance for their positions? All the appear- 
ances can be accounted for if we are all looking at a three- 
dimensional object a penny and no reasonable person can 
doubt that the penny is the corresponding physical reality. 
Similarly, an observer on the earth sees and measures an oblong 
block; an observer on another star contemplating the same 
object finds it to be a cube. Shall we say that the oblong block 
is the real thing, and that the other observer must correct his 
measures to make allowance for his motion? All the appearances 
are accounted for if the real object is four-dimensional, and the 
observers are merely measuring different three-dimensional 
appearances or sections; and it seems impossible to doubt that 
this is the true explanation. He who doubts the reality of the 



182 ON THE NATURE OF THINGS [CH. 

four-dimensional world (for logical, as distinct from experi- 
mental, reasons) can only be compared to a man who doubts the 
reality of the penny, and prefers to regard one of its innumerable 
appearances as the real object. 

Physical reality is the synthesis of all possible physical aspects 
of nature. An illustration may be taken from the phenomena of 
radiant-energy, or light. In a very large number of phenomena 
the light coming from an atom appears to be a series of spreading 
waves, extending so as to be capable of filling the largest 
telescope yet made. In many other phenomena the light coming 
from an atom appears to remain a minute bundle of energy, all 
of which can enter and blow up a single atom. There may be 
some illusion in these experimental deductions; but if not, it 
must be admitted that the physical reality corresponding to 
light must be some synthesis comprehending both these appear- 
ances. How to make this synthesis has hitherto baffled con- 
ception. But the lesson is that a vast number of appearances 
may be combined into one consistent whole perhaps all 
appearances that are directly perceived by terrestrial observers 
and yet the result may still be only an appearance. Reality 
is only obtained when all conceivable points of view have been 
combined. 

That is why it has been necessary to give up the reality of 
the everyday world of three dimensions. Until recently it com- 
prised all the possible appearances that had been considered. 
But now it has been discovered that there are new points of 
view with new appearances ; and the reality must contain them 
all. It is by bringing in all these new points of view that we 
have been able to learn the nature of the real world of 
physics. 

Let us briefly recapitulate the steps of our synthesis. We 
found one step already accomplished. The immediate perception 
of the world with one eye is a two-dimensional appearance. But 
we have two eyes, and these combine the appearances of the 
world as seen from two positions ; in some mysterious way the 
brain makes the synthesis by suggesting solid relief, and we 
obtain the familiar appearance of a three-dimensional world. 
This suffices for all possible positions of the observer within the 
parts of space hitherto explored. The next step was to combine 



xn] ON THE NATURE OF THINGS 183 

the appearances for all possible states of uniform motion of the 
observer. The result was to add another dimension to the world, 
making it four-dimensional. Next the synthesis was extended 
to include all possible variable motions of the observer. The 
process of adding dimensions stopped, but the world became 
non-Euclidean; a new geometry called Riemannian geometry 
was adopted. Finally the points of view of observers varying 
in size in any way were added; and the result was to replace 
the Riemannian geometry by a still more general geometry 
described in the last chapter. 

The search for physical reality is not necessarily utilitarian, 
but it has been by no means profitless. As the geometry became 
more complex, the physics became simpler; until finally it 
almost appears that the physics has been absorbed into the 
geometry. We did not consciously set out to construct a 
geometrical theory of the world; we were seeking physical 
reality by approved methods, and this is what has happened. 

Is the point now reached the ultimate goal? Have the points 
of view of all conceivable observers now been absorbed? We do 
not assert that they have. But it seems as though a definite 
task has been rounded off, and a natural halting-place reached. 
So far as we know, the different possible impersonal points of 
view have been exhausted those for which the observer can be 
regarded as a mechanical automaton, and can be replaced by 
scientific measuring-appliances. A variety of more personal 
points of view may indeed be needed for an ultimate reality; 
but they can scarcely be incorporated in a real world of physics. 
There is thus justification for stopping at this point but not for 
stopping earlier. 

It may be asked whether it is necessary to take into account 
all conceivable observers, many of whom, we suspect, have no 
existence. Is not the real world that which comprehends the 
appearances to all real observers? Whether or not it is a tenable 
hypothesis that that which no one observes does not exist, 
science uncompromisingly rejects it. If we deny the rights of 
extra-terrestrial observers, we must take the side of the Inquisi- 
tion against Galileo. And if extra-terrestrial observers are 
admitted, the other observers, whose results are here combined, 
cannot be excluded. 



184 ON THE NATURE OF THINGS [CH. 

Our inquiry into the nature of things is subject to certain 
limitations which it is important to realise. The best comparison 
I can offer is with a future antiquarian investigation, which may 
be dated about the year 5000 A.D. An interesting find has been 
made relating to a vanished civilisation which flourished about 
the twentieth century, namely a volume containing a large 
number of games of chess, written out in the obscure symbolism 
usually adopted for that purpose. The antiquarians, to whom 
the game was hitherto unknown, manage to discover certain 
uniformities; and by long research they at last succeed in 
establishing beyond doubt the nature of the moves and rules of 
the game. But it is obvious that no amount of study of the 
volume will reveal the true nature either of the participants in 
the game the chessmen or the field of the game the chess- 
board. With regard to the former, all that is possible is to give 
arbitrary names distinguishing the chessmen according to their 
properties; but with regard to the chess-board something more 
can be stated. The material of the board is unknown, so too 
are the shapes of the meshes whether squares or diamonds; 
but it is ascertainable that the different points of the field are 
connected with one another by relations of two-dimensional 
order, and a large number of hypothetical types of chess-board 
satisfying these relations of order can be constructed. In 
spite of these gaps in their knowledge, our antiquarians may 
fairly claim that they thoroughly understand the game of 
chess. 

The application of this analogy is as follows. The recorded 
games are our physical experiments. The rules of the game, 
ascertained by study of them, are the laws of physics. The 
hypothetical chess-board of 64 squares is the space and time of 
some particular observer or player; whilst the more general 
relations of two-fold order, are the absolute relations of order 
in space-time which we have been studying. The chessmen are 
the entities of physics electrons, particles, or point-events; and 
the range of movement may perhaps be compared to the fields 
of relation radiating from them electric and gravitational 
fields, or intervals. By no amount of study of the experiments 
can the absolute nature or appearance of these participants be 
deduced; nor is this knowledge relevant, for without it we may 



xii] ON THE NATURE OF THINGS 185 

yet learn "the game" in all its intricacy. Our knowledge of the 
nature of things must be like the antiquarians' knowledge of 
the nature of chessmen, viz. their nature as pawns and pieces 
in the game, not as carved shapes of wood. In the latter aspect 
they may have relations and significance transcending anything 
dreamt of in physics. 

It is believed that the familiar things of experience are very 
complex; and the scientific method is to analyse them into 
simpler elements. Theories and laws of behaviour of these 
simpler constituents are studied; and from these it becomes 
possible to predict and explain phenomena. It seems a natural 
procedure to explain the complex in terms of the simple, but 
it carries with it the necessity of explaining the familiar in terms 
of the unfamiliar. 

There are thus two reasons why the ultimate constituents of 
the real world must be of an unfamiliar nature. Firstly, all 
familiar objects are of a much too complex character. Secondly, 
familiar objects belong not to the real world of physics, but to 
a much earlier stage in the synthesis of appearances. The 
ultimate elements in a theory of the world must be of a nature 
impossible to define in terms recognisable to the mind. 

The fact that he has to deal with entities of unknown nature 
presents no difficulty to the mathematician. As the mathe- 
matician in the Prologue explained, he is never so happy as 
when he does not know what he is talking about. But we our- 
selves cannot take any interest in the chain of reasoning he is 
producing, unless we can give it some meaning a meaning, 
which we find by experiment, it will bear. We have to be in 
a position to make a sort of running comment on his work. 
At first his symbols bring no picture of anything before our 
eyes, and we watch in silence. Presently we can say "Now he 
is talking about a particle of matter "..."Now he is talking 
about another particle "..."Now he is saying where they will 
be at a certain time of day "..."Now he says that they will be 
in the same spot at a certain time." We watch to see. "Yes. 
The two particles have collided. For once he is speaking about 
something familiar, and speaking the truth, although, of course, 
he does not know it." Evidently his chain of symbols can be 
interpreted as describing what occurs in the world; we need not, 



186 ON THE NATURE OF THINGS [CH. 

and do not, form any idea of the meaning of each individual 
symbol ; it is only certain elaborate combinations of them that 
we recognise. 

Thus, although the elementary concepts of the theory are of 
undefined nature, at some later stage we must link the derivative 
concepts to the familiar objects of experience. 

We shall now collect the results arrived at in the previous 
chapters by successive steps, and set the theory out in more 
logical order. The extension in Chapter xi will not be considered 
here, partly because it would increase the difficulty of grasping 
the main ideas, partly because it is less certainly established. 

In the relativity theory of nature the most elementary concept 
is the point-event. In ordinary language a point-event is an 
instant of time at a point of space ; but this is only one aspect 
of the point-event, and it must not be taken as a definition. 
Time and space the familiar terms are derived concepts to 
be introduced much later in our theory. The first simple con- 
cepts are necessarily undefinable, and their nature is beyond 
human understanding. The aggregate of all the point-events is 
called the world. It is postulated that the world is four-dimen- 
sional, which means that a particular point-event has to be 
specified by the values of four variables or coordinates, though 
there is entire freedom as to the way in which these four identi- 
fying numbers are to be assigned. 

The meaning of the statement that the world is four-dimen- 
sional is not so clear as it appears at first. An aggregate of a 
large number of things has in itself no particular number of 
dimensions. Consider, for example, the words on this page. To 
a casual glance they form a two-dimensional distribution; but 
they were written in the hope that the reader would regard 
them as a one-dimensional distribution. In order to define the 
number of dimensions we have to postulate some ordering 
relation; and the result depends entirely on what this ordering 
relation is whether the words are ordered according to sense 
or to position on the page. Thus the statement that the world 
is four-dimensional contains an implicit reference to some ordering 
relation. This relation appears to be the interval, though I am 
not sure whether that alone suffices without some relation 
corresponding to proximity. It must be remembered that if the 



xn] ON THE NATURE OF THINGS 187 

interval s between two events is small, the events are not 
necessarily near together in the ordinary sense. 

Between any two neighbouring point-events there exists a 
certain relation known as the interval between them. The 
relation is a quantitative one which can be measured on a 
definite scale of numerical values*. But the term "interval" 
is not to be taken as a guide to the real nature of the relation, 
which is altogether beyond our conception. Its geometrical 
properties, which we have dwelt on so often in the previous 
chapters, can only represent one aspect of the relation. It may 
have other aspects associated with features of the world outside 
the scope of physics. But in physics we are concerned not with 
the nature of the relation but with the number assigned to 
express its intensity; and this suggests a graphical representa- 
tion, leading to a geometrical theory of the world of physics. 

What we have here called the world might perhaps have been 
legitimately called the aether; at least it is the universal sub- 
stratum of things which the relativity theory gives us in place 
of the aether. 

We have seen that the number expressing the intensity of 
the interval-relation can be measured practically with scales and 
clocks. Now, I think it is improbable that our coarse measures 
can really get hold of the individual intervals of point-events; 
our measures are not sufficiently microscopic for that. The 
interval which has appeared in our analysis must be a macro- 
scopic value ; and the potentials and kinds of space deduced from 
it are averaged properties of regions, perhaps small in comparison 
even with the electron, but containing vast numbers of the 
primitive intervals. We shall therefore pass at once to the 
consideration of the macroscopic interval; but we shall not 
forestall later results by assuming that it is measurable with 
a scale and clock. That property must be introduced in its 
logical order. 

Consider a small portion of the world. It consists of a large 
(possibly infinite) number of point-events between every two of 
which an interval exists. If we are given the intervals between 

* There is also a qualitative distinction into two kinds, ultimately identified 
as time-like and space-like, which for mathematical treatment are distinguished 
by real and imaginary numbers. 



188 ON THE NATURE OF THINGS [CH. 

a point A and a sufficient number of other points, and also 
between B and the same points, can we calculate what will be 
the interval between A and .B? In ordinary geometry this 
would be possible; but, since in the present case we know nothing 
of the relation signified by the word interval, it is impossible to 
predict any law a priori. But we have found in our previous 
work that there is such a rule, expressed by the formula 



This means that, having assigned our identification numbers 
(x lt x 2 > x a> ^4) to the point-events, we have only to measure 
ten different intervals to enable us to determine the ten coeffi- 
cients, g llt etc., which in a small region may be considered to be 
constants ; then all other intervals in this region can be predicted 
from the formula. For any other region we must make fresh 
measures, and determine the coefficients for a new formula. 

I think it is unlikely that the individual interval-relations of 
point-events follow any such definite rule. A microscopic 
examination would probably show them as quite arbitrary, the 
relations of so-called intermediate points being not necessarily 
intermediate. Perhaps even the primitive interval is not 
quantitative, but simply 1 for certain pairs of point-events and 
for others. The formula given is just an average summary 
which suffices for our coarse methods of investigation, and holds 
true only statistically. Just as statistical averages of one com- 
munity may differ from those of another, so may this statistical 
formula for one region of the world differ from that of another. 
This is the starting point of the infinite variety of nature. 

Perhaps an example may make this clearer. Compare the 
point-events to persons, and the intervals to the degree of 
acquaintance between them. There is no means of forecasting 
the degree of acquaintance between A and B from a knowledge 
of the familiarity of both with C, Z>, E, etc. But a statistician 
may compute in any community a kind of average rule. In 
most cases if A and B both know C, it slightly increases the 
probability of their knowing one another. A community in 
which this correlation was very high would be described as 
cliquish. There may be differences among communities in this 
respect, corresponding to their degree of cliquishness ; and so 



xn] ON THE NATURE OF THINGS 189 

the statistical laws may be the means of expressing intrinsic 
differences in communities. 

Now comes the difficulty which is by this time familiar to us. 
The ten g's are concerned, not only with intrinsic properties of 
the world, but with our arbitrary system of identification- 
numbers for the point-events; or, as we have previously ex- 
pressed it, they describe not only the kind of space-time, but 
the nature of the arbitrary mesh-system that is used. Mathe- 
matics shows the way of steering through this difficulty by fixing 
attention on expressions called tensors, of which B vtr and G^ v 
are examples. 

A tensor does not express explicitly the measure of an intrinsic 
quality of the world, for some kind of mesh-system is essential 
to the idea of measurement of a property, except in certain very 
special cases where the property is expressed by a single number 
termed an invariant, e.g. the interval, or the total curvature. 
But to state that a tensor vanishes, or that it is equal to another 
tensor in the same region, is a statement of intrinsic property, 
quite independent of the mesh-system chosen. Thus by keeping 
entirely to tensors, we contrive that there shall be behind our 
formulae an undercurrent of information having reference to the 
intrinsic state of the world. 

In this way we have found two absolute formulae, which 
appear to be fully confirmed by observation, namely 

in empty space, G^ = 0, 

in space containing matter, G^ = K^ v , 

where K^ v contains only physical quantities which are perfectly 
familiar to us, viz. the density and state of motion of the matter 
in the region. 

I think the usual view of these equations would be that the 
first expresses some law existing in the world, so that the point- 
events by natural necessity tend to arrange their relations in 
conformity with this equation. But when matter intrudes it 
causes a disturbance or strain of the natural linkages; and a 
rearrangement takes place to the extent indicated by the second 
equation. 

But let us examine more closely what the equation G^ v = 
tells us. We have been giving the mathematician a free hand 



190 ON THE NATURE OF THINGS [CH. 

with his indefinable intervals and point-events. He has arrived 
at the quantity G^; but as yet this means to us absolutely 
nothing. The pure mathematician left to himself never " deviates 
into sense." His work can never relate to the familiar things 
around us, unless we boldly lay hold of some of his symbols and 
give them an intelligible meaning tentatively at first, and then 
definitely as we find that they satisfy all experimental know- 
ledge. We have decided that in empty space G^ vanishes. Here 
is our opportunity. In default of any other suggestion as to 
what the vanishing of G^ v might mean, let us say that the 
vanishing of G^ v means emptiness; so that G^, if it does not 
vanish, is a condition of the world which distinguishes space 
said to be occupied from space said to be empty. Hitherto G^ 
was merely a formal outline to be filled with some undefined 
contents ; we are as far as ever from being able to explain what 
those contents are; but we have now given a recognisable 
meaning to the completed picture, so that we shall know it 
when we come across it in the familiar world of experience. 

The two equations are accordingly merely definitions 
definitions of the way in which certain states of the world 
(described in terms of the indefinables) impress themselves on 
our perceptions. When we perceive that a certain region of the 
world is empty, that is merely the mode in which our senses 
recognise that it is curved no higher than the first degree. 
When we perceive that a region contains matter we are recognising 
the intrinsic curvature of the world; and when we believe we 
are measuring the mass and momentum of the matter (relative 
to some axes of reference) we are measuring certain components 
of world-curvature (referred to those axes). The statistical 
averages of something unknown, which have been used to 
describe the state of the world, vary from point to point; and 
it is out of these that the mind has constructed the familiar 
notions of matter and emptiness. 

The law of gravitation is not a law in the sense that it restricts 
the possible behaviour of the substratum of the world; it is 
merely the definition of a vacuum. We need not regard matter 
as a foreign entity causing a disturbance in the gravitational 
field; the disturbance is matter. In the same way we do not 
regard light as an intruder in the electromagnetic field, causing 



xn] ON THE NATURE OF THINGS 191 

the electromagnetic force to oscillate along its path; the oscilla- 
tions constitute the light. Nor is heat a fluid causing agitation 
of the molecules of a body ; the agitation is heat. 

This view, that matter is a symptom and not a cause, seems 
so natural that it is surprising that it should be obscured in the 
usual presentation of the theory. The reason is that the con- 
nection of mathematical analysis with the things of experience 
is usually made, not by determining what matter is, but by 
what certain combinations of matter do. Hence the interval is 
at once identified with something familiar to experience, namely 
the thing that a scale and a clock measure. However advan- 
tageous that may be for the sake of bringing the theory into 
touch with experiment at the outset, we can scarcely hope to 
build up a theory of the nature of things if we take a scale and 
clock as the simplest unanalysable concepts. The result of this 
logical inversion is that by the time the equation G^ v = K^ v is 
encountered, both sides of the equation are well-defined 
quantities. Their necessary identity is overlooked, and the 
equation is regarded as a new law of nature. This is the fault 
of introducing the scale and clock prematurely. For our part 
we prefer first to define what matter is in terms of the elementary 
concepts of the theory; then we can introduce any kind of 
scientific apparatus; and finally determine what property of the 
world that apparatus will measure. 

Matter defined in this way obeys all the laws of mechanics, 
including conservation of energy and momentum. Proceeding 
with a similar development of Weyl's more general theory of 
the combined gravitational and electrical fields, we should find 
that it has the familiar electrical and optical properties. It is 
purely gratuitous to suppose that there is anything else present, 
controlling but not to be identified with the relations of the 
fourteen potentials (g's and fc's). 

There is only one further requirement that can be demanded 
from matter. Our brains are constituted of matter, and they 
feel and think or at least feeling and thinking are closely 
associated with motions or changes of the matter of the brain. 
It would be difficult to say that any hypothesis as to the nature 
of matter makes this process less or more easily understood; 
and a brain constituted out of differential coefficients of g's can 



192 ON THE NATURE OF THINGS [CH. 

scarcely be said to be less adapted to the purposes of thought 
than one made, say, out of tiny billiard balls ! But I think we 
may even go a little beyond this negative justification. The 
primary interval relation is of an undefined nature, and the 
g's contain this undefinable element. The expression G^ is 
therefore of defined form, but of undefined content. By its form 
alone it is fitted to account for all the physical properties of 
matter; and physical investigation can never penetrate beneath 
the form. The matter of the brain in its physical aspects is 
merely the form; but the reality of the brain includes the 
content. We cannot expect the form to explain the activities of 
the content, any more than we can expect the number 4 to 
explain the activities of the Big Four at Versailles. 

Some of these views of matter were anticipated with marvellous 
foresight by W. K. Clifford forty years ago. Whilst other English 
physicists were distracted by vortex-atoms and other will-o'- 
the-wisps, Clifford was convinced that matter and the motion 
of matter were aspects of space-curvature and nothing more. 
And he was no less convinced that these geometrical notions 
were only partial aspects of the relations of what he calls 
"elements of feeling." "The reality corresponding to our per- 
ception of the motion of matter is an element of the complex 
thing we call feeling. W 7 hat we might perceive as a plexus of 
nerve-disturbances is really in itself a feeling; and the succession 
of feelings which constitutes a man's consciousness is the reality 
which produces in our minds the perception of the motions of 
his brain. These elements of feeling have relations of nextness 
or contiguity in space, which are exemplified by the sight- 
perceptions of contiguous points ; and relations of succession in 
time which are exemplified by all perceptions. Out of these two 
relations the future theorist has to build up the world as best 
he may. Two things may perhaps help him. There are many 
lines of mathematical thought which indicate that distance or 
quantity may come to be expressed in terms of position in the 
wide sense of the analysis situs. And the theory of space- 
curvature hints at a possibility of describing matter and motion 
in terms of extension only." (Fortnightly Review, 1875.) 

The equation G^ v = K^ v is a kind of dictionary explaining 
what the different components of world-curvature mean in 



xn] ON THE NATURE OF THINGS 193 

terms ordinarily used in mechanics. If we write it in the slightly 
modified, but equivalent, form 



we have the following scheme of interpretation 



I* > 4 = Pll + PU> Pu + P, PlS 

T 23 , T 24 Pn + pv 2 , Pw + pvw, - pv, 

T 33 , T M pw + pw*, -pw, 



Here we are using the partitions of space and time adopted in 
ordinary mechanics; p is the density of the matter, u, v, w its 
component velocities, and p llf p l2 , ... p 33 , the components of 
the internal stresses which are believed to be analysable into 
molecular movements. 

Now the question arises, is it legitimate to make identifications 
on such a wholesale scale? Having identified T 44 as density, 
can we go on to identify another quantity T 34 as density 
multiplied by velocity? It is as though we identified one " thing " 
as air, and a quite different "thing" as wind. Yes, it is legiti- 
mate, because we have not hitherto explained what is to be the 
counterpart of velocity in our scheme of the world ; and this is 
the way we choose to introduce it. All identifications are at 
this stage provisional, being subject to subsequent test by 
observation. 

A definition of the velocity of matter in some such terms as 
'''wind divided by air," does not correspond to the way in 
which motion primarily manifests itself in our experience. 
Motion is generally recognised by the disappearance of a particle 
at one point of space and the appearance of an apparently 
identical particle at a neighbouring point. This manifestation 
of motion can be deduced mathematically from the identifying 
definition here adopted. Remembering that in physical theory 
it is necessary to proceed from the simple to the complex, 
which is often opposed to the instinctive desire to proceed from 
the familiar to the unfamiliar, this inversion of the order in 
which the manifestations of motion appear need occasion no 
surprise. Permanent identity of particles of matter (without 
which the ordinary notion of velocity fails) is a very familiar 
idea, but it appears to be a very complex feature of the world. 
E.S. 13 



194 ON THE NATURE OF THINGS [CH. 

A simple instance may be given where the familiar kinematical 
conception of motion is insufficient. Suppose a perfectly homo- 
geneous continuous ring is rotating like a wheel, what meaning 
can we attach to its motion? The kinematical conception of 
motion implies change disappearance at one point and reap- 
pearance at another point but no change is detectable. The 
state at any one moment is the same as at a previous moment, 
and the matter occupying one position now is indistinguishable 
from the matter in the same position a moment ago. At the 
most it can only differ in a mysterious non-physical quality 
that of identity ; but if, as most physicists are willing to believe, 
matter is some state in the aether, what can we mean by saying 
that two states are exactly alike, but are not identical? Is the 
hotness of the room equal to, but not identical with, its hotness 
yesterday? Considered kinematically, the rotation of the ring 
appears to have no meaning; yet the revolving ring differs 
mechanically from a stationary ring. For example, it has 
gyros tatic properties. The fact that in nature a ring has atomic 
and not continuous structure is scarcely relevant. A conception 
of motion which affords a distinction between a rotating and 
non-rotating continuous ring must be possible; otherwise this 
would amount to an a priori proof that matter is atomic. 
According to the conception now proposed, velocity of matter 
is as much a static quality as density. Generally velocity is 
accompanied by changes in the physical state of the world, 
which afford the usual means of recognising its existence; but 
the foregoing illustration shows that these symptoms do not 
always occur. 

This definition of velocity enables us to understand why 
velocity except in reference to matter is meaningless, whereas 
acceleration and rotation have a meaning. The philosophical 
argument, that velocity through space is meaningless, ceases to 
apply as soon as we admit any kind of structure or aether in 
empty regions; consequently the problem is by no means so 
simple as is often supposed. But our definition of velocity is 
dynamical, not kinematical. Velocity is the ratio of certain 
components of T^ vt and only exists when T 44 is not zero. Thus 
matter (or electromagnetic energy) is the only thing that can 
have a velocity relative to the frame of reference. The velocity 



xii] ON THE NATURE OF THINGS 195 

of the world-structure or aether, where the T^ v vanish, is always 
of the indeterminate form 0^-0. On the other hand acceleration 
and rotation are defined by means of the g^ v and exist wherever 
these exist * ; so that the acceleration and rotation of the world- 
structure or aether relative to the frame of reference are deter- 
minate. Notice that acceleration is not defined as change of 
velocity; it is an independent entity, much simpler and more 
universal than velocity. It is from a comparison of these two 
entities that we ultimately obtain the definition of time. 

This finally resolves the difficulty encountered in Chapter x 
the apparent difference in the Principle of Relativity as 
applied to uniform and non-uniform motion. Fundamentally 
velocity and acceleration are both static qualities of a region 
of the world (referred to some mesh-system). Acceleration is a 
comparatively simple quality present wherever there is geodesic 
structure, that is to say everywhere. Velocity is a highly com- 
plex quality existing only where the structure is itself more 
than ordinarily complicated, viz. in matter. Both these qualities 
commonly give physical manifestations, to which the terms 
acceleration and velocity are more particularly applied; but it 
is by examining their more fundamental meaning that we can 
understand the universality of the one and the localisation of 
the other. 

It has been shown that there are four identical relations 
between the ten qualities of a piece of matter here identified, 
which depend solely on the way the G^ were by definition 
constructed out of simpler elements. These four relations state 
that, provided the mesh-system is drawn in one of a certain number 
of ways, mass (or energy) and momentum will be conserved. 
The conservation of mass is of great importance; matter will 
be permanent, and for every particle disappearing at any point 
a corresponding mass will appear at a neighbouring point; the 
change consists in the displacement of matter, not its creation 
or destruction. This gives matter the right to be regarded, not 
as a mere assemblage of symbols, but as the substance of a 

* Even in Newtonian mechanics we speak of the "field of acceleration," and 
think of it as existing even when there is no test body to display the accelera- 
tion. In the present theory this field of acceleration is described by the g^ v . 
There is no such thing as a "field of velocity" in empty space; but there is in 
a material ocean. 

132 



196 ON THE NATURE OF THINGS [CH. 

permanent world. But the permanent world so found demands 
the partitioning of space-time in one of a certain number of 
ways, viz. those discussed in Chapter in* ; from these a particular 
space and time are selected, because the observer wishes to 
consider himself, or some arbitrary body, at rest. This gives 
the space and time used for ordinary descriptions of experience. 
In this way we are able to introduce perceptual space and time 
into the four-dimensional world, as derived concepts depending 
on our desire that the new-found matter should be permanent. 

I think it is now possible to discern something of the reason 
why the world must of necessity be as we have described it. 
When the eye surveys the tossing waters of the ocean, the 
eddying particles of water leave little impression; it is the waves 
that strike the attention, because they have a certain degree 
of permanence. The motion particularly noticed is the motion of 
the wave-form, which is not a motion of the water at all. So 
the mind surveying the world of point-events looks for the 
permanent things. The simpler relations, the intervals and 
potentials, are transient, and are not the stuff out of which 
mind can build a habitation for itself. But the thing that has 
been identified with matter is permanent, and because of its 
permanence it must be for mind the substance of the world. 
Practically no other choice was possible. 

It must be recognised that the conservation of mass is not 
exactly equivalent to the permanence of matter. If a loaf of 
bread suddenly transforms into a cabbage, our surprise is not 
diminished by the fact that there may have been no change of 
weight. It is not very easy to define this extra element of 
permanence required, because we accept as quite natural 
apparently similar transformations an egg into an omelette, 
or radium into lead. But at least it seems clear that some degree 
of permanence of one quality, mass, would be the primary 
property looked for in matter, and this gives sufficient reason 
for the particular choice. 

We see now that the choice of a permanent substance for the 

* When the kind of space-time is such that a strict partition of this kind is 
impossible, strict conservation does not exist; but we retain the principle as 
formally satisfied by attributing energy and momentum to the gravitational 
field. 



xn] ON THE NATURE OF THINGS 197 

world of perception necessarily carries with it the law of gravita- 
tion, all the laws of mechanics, and the introduction of the 
ordinary space and time of experience. Our whole theory has 
really been a discussion of the most general way in which 
permanent substance can be built up out of relations ; and it is 
the mind which, by insisting on regarding only the things that 
are permanent, has actually imposed these laws on an indifferent 
world. Nature has had very little to do with the matter; she 
had to provide a basis point-events; but practically anything 
would do for that purpose if the relations were of a reasonable 
degree of complexity. The relativity theory of physics reduces 
everything to relations; that is to say, it is structure, not 
material, which counts. The structure cannot be built up with- 
out material ; but the nature of the material is of no importance. 
We may quote a passage from Bertrand Russell's Introduction 
to Mathematical Philosophy. 

"There has been a great deal of speculation in traditional 
philosophy which might have been avoided if the importance 
of structure, and the difficulty of getting behind it, had been 
realised. For example it is often said that space and time are 
subjective, but they have objective counterparts; or that 
phenomena are subjective, but are caused by things in them- 
selves, which must have differences inter se corresponding with 
the differences in the phenomena to which they give rise. Where 
such hypotheses are made, it is generally supposed that we can 
know very little about the objective counterparts. In actual 
fact, however, if the hypotheses as stated were correct, the 
objective counterparts would form a world having the same 
structure as the phenomenal world.... In short, every proposition 
having a communicable significance must be true of both worlds 
or of neither: the only difference must lie in just that essence 
of individuality which always eludes words and baffles descrip- 
tion, but which for that very reason is irrelevant to science." 

This is how our theory now stands. We have a world of 
point-events with their primary interval-relations. Out of these 
an unlimited number of more complicated relations and qualities 
can be built up mathematically, describing various features of 
the state of the world. These exist in nature in the same sense 
as an unlimited number of walks exist on an open moor. But 



198 ON THE NATURE OF THINGS [CH. 

the existence is, as it were, latent unless someone gives a signifi- 
cance to the walk by following it; and in the same way the 
existence of any one of these qualities of the world only acquires 
significance above its fellows, if a mind singles it out for 
recognition. Mind filters out matter from the meaningless 
jumble of qualities, as the prism filters out the colours of the 
rainbow from the chaotic pulsations of white light. Mind exalts 
the permanent and ignores the transitory; and it appears from 
the mathematical study of relations that the only way in which 
mind can achieve her object is by picking out one particular 
quality as the permanent substance of the perceptual world, 
partitioning a perceptual time and space for it to be permanent 
in, and, as a necessary consequence of this Hobson's choice, the 
laws of gravitation and mechanics and geometry have to be 
obeyed. Is it too much to say that mind's search for permanence 
has created the world of physics? So that the world we 
perceive around us could scarcely have been other than it is*? 

The last sentence possibly goes too far, but it illustrates the 
direction in which these views are tending. With Weyl's more 
general theory of interval-relations, the laws of electrodynamics 
appear in like manner to depend merely on the identification 
of another permanent thing electric charge. In this case the 
identification is due, not to the rudimentary instinct of the 
savage or the animal, but the more developed reasoning-power 
of the scientist. But the conclusion is that the whole of those 
laws of nature which have been woven into a unified scheme 
mechanics, gravitation, electrodynamics and optics have their 
origin, not in any special mechanism of nature, but in the 
workings of the mind. 

"Give me matter and motion," said Descartes, "and I will 
construct the universe." The mind reverses this. "Give me a 
world a world in which there are relations and I will construct 
matter and motion." 

Are there then no genuine laws in the external world? Laws 
inherent in the substratum of events, which break through into 

* This summary is intended to indicate the direction in which the views 
suggested by the relativity theory appear to me to be tending, rather than to 
be a precise statement of what has been established. I am aware that there 
are at present many gaps in the argument. Indeed the whole of this part of 
the discussion should be regarded as suggestive rather than dogmatic. 



xn] ON THE NATURE OF THINGS 199 

the phenomena otherwise regulated by the despotism of the 
mind? We cannot foretell what the final answer will be; but, 
at present, we have to admit that there are laws which appear 
to have their seat in external nature. The most important of 
these, if not the only law, is a law of atomicity. Why does that 
quality of the world which distinguishes matter from emptiness 
exist only in certain lumps called atoms or electrons, all of 
comparable mass? Whence arises this discontinuity? At 
present, there seems no ground for believing that discontinuity 
is a law due to the mind; indeed the mind seems rather to take 
pains to smooth the discontinuities of nature into continuous 
perception. We can only suppose that there is something in 
the nature of things that causes this aggregation into atoms. 
Probably our analysis into point-events is not final; and if it 
could be pushed further to reach something still more funda- 
mental, then atomicity and the remaining laws of physics would 
be seen as identities. This indeed is the only kind of explanation 
that a physicist could accept as ultimate. But this more ultimate 
analysis stands on a different plane from that by which the 
point-events were reached. The world may be so constituted 
that the laws of atomicity must necessarily hold; but, so far as 
the mind is concerned, there seems no reason why it should 
have been constituted in that way. We can conceive a world 
constituted otherwise. But our argument hitherto has been 
that, however the world is constituted, the necessary combina- 
tions of things can be found which obey the laws of mechanics, 
gravitation and electrodynamics, and these combinations are 
ready to play the part of the world of perception for any mind 
that is tuned to appreciate them; and further, any world of 
perception of a different character would be rejected by the 
mind as unsubstantial. 

If atomicity depends on laws inherent in nature, it seems at 
first difficult to understand why it should relate to matter 
especially; since matter is not of any great account in the 
analytical scheme, and owes its importance to irrelevant con- 
siderations introduced by the mind. It has appeared, however, 
that atomicity is by no means confined to matter and electricity ; 
the quantum, which plays so great a part in recent physics, is 
apparently an atom of action. So nature cannot be accused of 



200 ON THE NATURE OF THINGS [CH. 

connivance with mind in singling out matter for special distinc- 
tion. Action is generally regarded as the most fundamental 
thing in the real world of physics, although the mind passes it 
over because of its lack of permanence ; and it is vaguely believed 
that the atomicity of action is the general law, and the appear- 
ance of electrons is in some way dependent on this. But the 
precise formulation of the theory of quanta of action has hitherto 
baffled physicists. 

There is a striking contrast between the triumph of the 
scientific mind in formulating the great general scheme of 
natural laws, nowadays summed up in the principle of least 
action, and its present defeat by the newly discovered but equally 
general phenomena depending on the laws of atomicity of 
quanta. It is too early to cry failure in the latter case; but 
possibly the contrast is significant. It is one thing for the human 
mind to extract from the phenomena of nature the laws which 
it has itself put into them; it may be a far harder thing to 
extract laws over which it has had no control. It is even possible 
that laws which have not their origin in the mind may be 
irrational, and we can never succeed in formulating them. This 
is, however, only a remote possibility; probably if they were 
really irrational it would not have been possible to make the 
limited progress that has been achieved. But if the laws of 
quanta do indeed differentiate the actual world from other 
worlds possible to the mind, we may expect the task of formu- 
lating them to be far harder than anything yet accomplished 
by physics. 

The theory of relativity has passed in review the whole subject- 
matter of physics. It has unified the great laws, which by the 
precision of their formulation and the exactness of their applica- 
tion have won the proud place in human knowledge which 
physical science holds to-day. And yet, in regard to the nature 
of things, this knowledge is only an empty shell a form of 
symbols. It is knowledge of structural form, and not knowledge 
of content. All through the physical world runs that unknown 
content, which must surely be the stuff of our consciousness. 
Here is a hint of aspects deep within the world of physics, and 
yet unattainable by the methods of physics. And, moreover, 
we have found that where science has progressed the farthest, 



xn] ON THE NATURE OF THINGS 201 

the mind has but regained from nature that which the mind has 
put into nature. 

We have found a strange foot-print on the shores of the 
unknown. We have devised profound theories, one after 
another, to account for its origin. At last, we have succeeded 
in reconstructing the creature that made the foot-print. And 
Lo ! it is our own. 

5 " 



APPENDIX 
MATHEMATICAL NOTES 

THE references marked "Report" are to the writer's "Report 
on the Relativity Theory of Gravitation" for the Physical 
Society of London (Fleet way Press), where fuller mathematical 
details are given. 

Probably the most complete treatise on the mathematical 
theory of the subject is H. Weyl's Raum, Zeit, Materie (Julius 
Springer, Berlin). 

Note 1 (p. 20). 

It is not possible to predict the contraction rigorously from 
the universally accepted electromagnetic equations, because 
these do not cover the whole ground. There must be other forces 
or conditions which govern the form and size of an electron; 
under electromagnetic forces alone it would expand indefinitely. 
The old electrodynamics is entirely vague as to these forces. 

The theory of Larmor and Lorentz shows that if any system 
at rest in the aether is in equilibrium, a similar system in 
uniform motion through the aether, but with all lengths in the 
direction of motion diminished in FitzGerald's ratio, will also 
be in equilibrium so far as the differential equations of the 
electromagnetic field are concerned. There is thus a general 
theoretical agreement with the observed contraction, provided 
the boundary conditions at the surface of an electron behave in 
the same way. The latter suggestion is confirmed by experiments 
on isolated electrons in rapid motion (Kaufmann's experiment). 
It turns out that this requires an electron to suffer the same 
kind of contraction as a material rod; and thus, although the 
theory throws light on the adjustments involved in material 
contraction, it can scarcely be said to give an explanation of the 
occurrence of contraction generally. 



MATHEMATICAL NOTES 203 

Note 2 (p. 47). 

Suppose a particle moves from (sc lf */i, #1, t^) to (x. 2> y z , z z , t z ), 
its velocity u is given by 



Hence from the formula for 



_ 

(We omit a V 1, as the sign of s 2 is changed later in the 
chapter.) 

If we take t- L and t 2 to be the start and finish of the aviator's 
cigar (Chapter i), then as judged by a terrestrial observer, 

t 2 t = 60 minutes, \/(l u 2 ) = FitzGerald contraction = J. 

As judged by the aviator, 

*2 - *i = 30 minutes, */(! - u 2 ) = 1. 

Thus for both observers 5 = 30 minutes, verifying that it is 
an absolute quantity independent of the observer. 

Note 3 (p. 48). 

The formulae of transformation to axes with a different 
orientation are 

x = x' cos 6 T' sin 6, y = y f , z = z', r x' sin 6 + r' cos 6, 
where 9 is the angle turned through in the plane XT. 

Let u = i tan 6, so that cos 6 = (1 - u 2 ) ~ * = /?, say. The 
formulae become 

x = P(x'-Mr'), y = y', z = z f , r = (r'+ iux'), 
or, reverting to real time by setting ir = t, 



which gives the relation between the estimates of space and 
time by two different observers. 

The factor ft gives in the first equation the FitzGerald con- 
traction, and in the fourth equation the retardation of time. 
The terms ut' and ux' correspond to the changed conventions 
as to rest and simultaneity. 

A point at rest, x = const., for the first observer corresponds 
to a point moving with velocity u 9 x' ut'= const., for the second 
observer. Hence their relative velocity is u. 



204 APPENDIX 

Note 4 (p. 81). 
The condition for flat space in two dimensions is 

3 



git 



, 3 



Note 5 (p. 89). 

Let g be the determinant of four rows and columns formed 
with the elements g^ v . 

Let g be the minor of g^ v , divided by g. 
Let the "3-index symbol" {/>tv, A} denote 



summed for values of a from 1 to 4. There will be 40 different 
3-index symbols. 

Then the Riemann-Christoffel tensor is 

o o 

B ^ Va - = (P** ) {" P) ~ {/*" ) ( C7 ' P) + g^- &>P} ~ ^ {^'P}' 

the terms containing e being summed for values of e from 1 to 4. 
The "contracted" Riemann-Christoffel tensor G^ v can be 
reduced to 

a} 



-I - log? \/ { uuv. a\ ~ loer V' . 

ox ox ox 

where in accordance with a general convention in this subject, 
each term containing a suffix twice over (a and /?) must be 
summed for the values 1, 2, 3, 4 of that suffix. 

The curvature G = g^G^, summed in accordance with the 
foregoing convention. 

Note 6 (p. 94). 
The electric potential due to a charge e is 

t = [r(l-v r IC)Y 



MATHEMATICAL NOTES 205 

where v r is the velocity of the charge in the direction of r, C the 
velocity of light, and the square bracket signifies antedated 
values. To the first order of v r /C, the denominator is equal to 
the present distance r, so the expression reduces to e/r in spite 
of the time of propagation. The foregoing formula for the 
potential was found by Lienard and Wiechert. 

Note 7 (p. 97). 

It is found that the following scheme of potentials rigorously 
satisfies the equations G^ v = 0, according to the values of G^ v 
in Note 5, _ ^ Q Q Q 

- a^ 2 

- a^ 2 sin 2 a? 2 2 

y 

where y = 1 K/x lt and K is any constant (see Report, 28). 
Hence these potentials describe a kind of space-time which can 
occur in nature referred to a possible mesh-system. If K = 0, 
the potentials reduce to those for flat space-time referred to 
polar coordinates; and, since in the applications required K will 
always be extremely small, our coordinates can scarcely be 
distinguished from polar coordinates. We can therefore use the 
familiar symbols r, 6, (f>, t, instead of x lt cc 2 , x 3 , a? 4 . It must, 
however, be remembered that the identification with polar 
coordinates is only approximate; and, for example, an equally 
good approximation is obtained if we write x = r + JAC, a sub- 
stitution often used instead of x r since it has the advantage 
of making the coordinate- velocity of light more symmetrical. 

We next work out analytically all the mechanical and optical 
properties of this kind of space-time, and find that they agree 
observationally with those existing round a particle at rest at 
the origin with gravitational mass \K. The conclusion is that 
the gravitational field here described is produced by a particle 
of mass \K or, if preferred, a particle of matter at rest is 
produced by the kind of space-time here described. 

Note 8 (p. 98). 

Setting the gravitational constant equal to unity, we have for 
a circular orbit m i r z _ V 2i r ^ 

so that m = i?V. 



206 APPENDIX 

The earth's speed, r>, is approximately 30 km. per sec., or 
TTT^TTTF ^ terms of the velocity of light. The radius of its orbit, 
r, is about 1-5 . 10 8 km. Hence, m, the gravitational mass of the 
sun is approximately 1-5 km. 

The radius of the sun is 697,000 kms., so that the quantity 
2m/r occurring in the formulae is, for the sun's surface, -00000424 
or 0"-87. 

Note 9 (p. 123). 

See Report, 29, 30. The general equations of a geodesic are 



From the formula for the line-element 

ds*= - y-'dr 2 - rW 2 + ydP ............ (1), 

we calculate the three-index symbols and it is found that two 
of the equations of the geodesic take the rather simple form 

M % <bM_ 

j n "T" j j - ", 

as* r as as 

d 2 * d(logy) dr<tt_n 
5^ + dr 'dsds~ ' 

which can be integrated giving 



* e 

ds y 

where h and c are constants of integration. 

Eliminating dt and d# from (1), (2) and (3), we have 



or writing u 1/r, 

f du\ 2 c 2 - 1 2mu 



Differentiating with respect to & 
d?u m 



MATHEMATICAL NOTES 207 

which gives the equation of the orbit in the usual form in particle 
dynamics. It differs from the equation of the Newtonian orbit 
by the small term 3mu 2 , which is easily shown to give the motion 
of perihelion. 

The track of a ray of light is also obtained from this formula, 
since by the principle of equivalence it agrees with that of a 
material particle moving with the speed of light. This case is 
given by ds = 0, and therefore h = oo . The differentia] equation 
for the path of a light-ray is thus 

d?u 
JM + u 

An approximate solution is 



neglecting the very small quantity m?/R 2 . Converting to 
Cartesian coordinates, this becomes 



m 

M\i - ~=T 



The asymptotes of the light-track are found by taking y 
very large compared with x, giving 



so that the angle between them is 4<m/R. 

Note 10 (p. 126). 
Writing the line element in the form 

+ b + c r ^+ ...W 

the approximate Newtonian attraction fixes b equal to 2; 
then the observed deflection of light fixes a equal to + 2 ; and 
with these values the observed motion of Mercury fixes c equal 
to 0. 

To insert an arbitrary coefficient of r 2 d0 2 would merely vary 
the coordinate system. We cannot arrive at any intrinsically 
different kind of space-time in that way. Hence, within the 
limits of accuracy mentioned, the expression found by Einstein 
is completely determinable by observation. 



208 APPENDIX 

It may be mentioned that the line-element 

ds* = - dr* - rW + (1 - 2m/r) dt*, 

gives one-half the observed deflection of light, and one-third 
the motion of perihelion of Mercury. As both these can be 
obtained on older theories, taking account of the variation of 
mass with velocity, the coefficient y -1 of dr* is the essentially 
novel point in Einstein's theory. 

Note 1 1 (p. 131). 

It is often supposed that by the Principle of Equivalence any 
invariant property which holds outside a gravitational field also 
holds in a gravitational field; but there is necessarily some 
limitation on this equivalence. Consider for instance the two 
invariant equations 

ds z = 1, 



where k is some constant having the dimensions of a length. 
Since Z?J W vanishes outside a gravitational field, if one of these 
equations is true the other will be. But they cannot both hold 
in a gravitational field, since there B^B^" does not vanish, 
and is in fact equal to 24ra 2 /r 6 . (I believe that the numerical 
factor 24 is correct; but there are 65,536 terms in the expression, 
and the terms which do not vanish have to be picked out. 

This ambiguity of the Principle of Equivalence is referred to 
in Report, 14, 27; and an enunciation is given which makes 
it definite. The enunciation however is merely an explicit state- 
ment, and not a defence, of the assumptions commonly made in 
applying the principle. 

So far as general reasoning goes there seems no ground for 
choosing ds 2 rather than ds 2 (1 + 24& 4 ra 2 /r 6 ), or any similar ex- 
pression, as the constant character in the vibration of an atom. 

Note 12 (p. 134). 

Let two rays diverging from a point at a distance R pass at 
distances r and r + dr from a star of mass m. The deflection 
being 4ra/r, their divergence will be increased by 4>mdr/r 2 . This 
increase will be equal to the original divergence dr/R if 
r = V4<mR. Take for instance 4m = 10 km., R = 10 15 km., then 
r = 10 8 km. So that the divergence of the light will be doubled, 



MATHEMATICAL NOTES 209 

when the actual deflection of the ray is only 10" 7 , or 0"-02. 
In the case of a star seen behind the sun the added divergence 
has no time to take effect; but when the light has to travel a 
stellar distance after the divergence is produced, it becomes 
weakened by it. Generally in stellar phenomena the weakening 
of the light should be more prominent than the actual deflection. 

Note 13 (p. 141). 
The relations are (Report, 39) 

^-iJI (A* = 1.2, 8, 4), V ' 

where G"^ is the (contracted) covariant derivative of G, or 



I doubt whether anyone has performed the laborious task of 
verifying these identities by straightforward algebra. 

Note 14 (p. 158). 
The modified law for spherical space-time is in empty space 

G> = A M ,. 

In cylindrical space- time, matter is essential. The law in space 
occupied by matter is 



the term 2A being the only modification. Spherical space-time 
of radius R is given by A = 3/R 2 ; cylindrical space-time by 
A = l/R 2 provided matter of average density p = l/47rR 2 is 
present. (See Report, 50, 51.) The total mass of matter in 
the cylindrical world is JrrR. This must be enormous, seeing 
that the sun's mass is only Ij kilometres. 

Note 15 (p. 174). 

WeyPs theory is given in Berlin. Sitzungsberichte, 30 May, 1918 ; 
Annalen der Physik, Bd. 59 (1919), p. 101. 

Note 16 (p. 177). 

The argument is rather more complicated than appears in 
the text, where the distinction between action-density and 
action in a region, curvature and total curvature in a region, 
has not been elaborated. Taking a definitely marked out region 
in space and time, its measured volume will be increased 16-fold 
E. s. 14 



210 APPENDIX 

by halving the gauge. Therefore for action-density we must 
take an expression which will be diminished 16-fold by halving 
the gauge. Now G is proportional to l/R 2 , where R is the radius 
of curvature, and so is diminished 4-fold. The invariant B^ V<T B* V " 
has the same gauge-dimensions as G z ; and hence when integrated 
through a volume gives a pure number independent of the gauge. 
In WeyPs theory this is only the gravitational part of the com- 
plete invariant 



which reduces to 



The second term gives actually the well-known expression for 
the action-density of the electromagnetic field, and this evidently 
strengthens the identification of this invariant with action- 
density. 

Einstein's theory, on the other hand, creates a difficulty here, 
because although there may be action in an electromagnetic 
field without electrons, the curvature is zero. 

HISTORICAL NOTE 

Before the Michelson-Morley experiment the question had 
been widely discussed whether the aether in and near the earth 
was carried along by the earth in its motion, or whether it 
slipped through the interstices between the atoms. Astronomical 
aberration pointed decidedly to a stagnant aether; but the 
experiments of Arago and Fizeau on the effect of motion of 
transparent media on the velocity of light in those media, 
suggested a partial convection of the aether in such cases. These 
experiments were first-order experiments, i.e they depended on 
the. ratio of the velocity of the transparent body to the velocity 
of light. The Michelson-Morley experiment is the first example 
of an experiment delicate enough to detect second-order effects, 
depending on the square of the above ratio; the result, that no 
current of aether past terrestrial objects could be detected, 
appeared favourable to the view that the aether must be con- 
vected by the earth. The difficulty of reconciling this with 
astronomical aberration was recognised. 



HISTORICAL NOTE 211 

An attempt was made by Stokes to reconcile mathematically 
a convection of aether by the earth with the accurately verified 
facts of astronomical aberration; but his theory cannot be 
regarded as tenable. Lodge investigated experimentally the 
question whether smaller bodies carried the aether with them 
in their motion, and showed that the aether between two 
whirling steel discs was undisturbed. 

The controversy, stagnant versus convected aether, had now 
reached an intensely interesting stage. In 1895, Lorentz dis- 
cussed the problem from the point of view of the electrical 
theory of light and matter. By his famous transformation of 
the electromagnetic equations, he cleared up the difficulties 
associated with the first-order effects, showing that they could 
all be reconciled with a stagnant aether. In 1900, Larmor carried 
the theory as far as second-order effects, and obtained an exact 
theoretical foundation for FitzGerald's hypothesis of contrac- 
tion, which had been suggested in 1892 as an explanation of the 
Michelson-Morley experiment. The theory of a stagnant aether 
was thus reconciled with all observational results; and hence- 
forward it held the field. 

Further second-order experiments were performed by Rayleigh 
and Brace on double refraction (1902, 1904), Trouton and Noble 
on a torsional effect on a charged condenser (1903), and Trouton 
and Rankine on electric conductivity (1908). All showed that 
the earth's motion has no effect on the phenomena. On the 
theoretical side, Lorentz (1902) showed that the indifference of 
the equations of the electromagnetic field to any velocity of the 
axes of reference, which he had previously established to the 
first order, and Larmor to the second order, was exact to all 
orders. He was not, however, able to establish with the same 
exactness a corresponding transformation for bodies containing 
electrons. 

Both Larmor and Lorentz had introduced a "local time" for 
the moving system. It was clear that for many phenomena this 
local time would replace the "real" time; but it was not 
suggested that the observer in the moving system would be 
deceived into thinking that it was the real time. Einstein, in 
1905 founded the modern principle of relativity by postulating 
that this local time was the time for the moving observer; no 



212 APPENDIX 

real or absolute time existed, but only the local times, different 
for different observers. He showed that absolute simultaneity 
and absolute location in space are inextricably bound together, 
and the denial of the latter carries with it the denial of the 
former. By realising that an observer in the moving system 
would measure all velocities in terms of the local space and time 
of that system, Einstein removed the last discrepancies from 
Lorentz's transformation. 

The relation between the space and time coordinates in two 
systems in relative motion was now obtained immediately from 
the principles of space and time-measurement. It must hold 
for all phenomena provided they do not postulate a medium 
which can serve as a standard for absolute location and simul- 
taneity. The previous deduction of these formulae by lengthy 
transformation of the electromagnetic equations now appears 
as a particular case; it shows that electromagnetic phenomena 
have no reference to a medium with such properties. 

The combination of the local spaces and times of Einstein 
into an absolute space-time of four dimensions is the work of 
Minkowski (1908). Chapter in is largely based on his researches. 
Much progress was made in the four-dimensional vector- 
analysis of the world; but the whole problem was greatly 
simplified when Einstein and Grossmann introduced for this 
purpose the more powerful mathematical calculus of Riemann, 
Ricci, and Levi-Civita. 

In 1911, Einstein put forward the Principle of Equivalence, 
thus turning the subject towards gravitation for the first time. 
By postulating that not only mechanical but optical and 
electrical phenomena in a field of gravitation and in a field 
produced by acceleration of the observer were equivalent, he 
deduced the displacement of the spectral lines on the sun and 
the displacement of a star during a total eclipse. In the latter 
case, however, he predicted only the half-deflection, since he 
was still working with Newton's law of gravitation. Freundlich 
at once examined plates obtained at previous eclipses, but failed 
to find sufficient data; he also prepared to observe the eclipse 
of 1914 in Russia with this object, but was stopped by the out- 
break of war. Another attempt was made by the Lick Observa- 
tory at the not very favourable eclipse of 1918. Only preliminary 



HISTORICAL NOTE 213 

results have been published ; according to the information given, 
the probable accidental error of the mean result (reduced to the 
sun's limb) was about l"-6, so that no conclusion was permissible. 
The principle of equivalence opened up the possibility of a 
general theory of relativity not confined to uniform motion, for 
it pointed a way out of the objections which had been urged 
against such an extension from the time of Newton. At first 
the opening seemed a very narrow one, merely indicating that 
the objections could not be considered final until the possibilities 
of complications by gravitation had been more fully exhausted. 
By 1913, Einstein had surmounted the main difficulties. His 
theory in a complete form was published in 1915; but it was not 
generally accessible in England until a year or two later. As 
this theory forms the main subject-matter of the book, we may 
leave our historical survey at this point. 



143 



INDEX 



Absolute, approached through the 

relative, 82 

Absolute acceleration, 68, 154, 194 
Absolute past and future, 50 
Absolute rotation, 152, 164, 194 
Absolute simultaneity, 12, 51 
Absolute time, in cylindrical world, 

163 

Acceleration, a simpler quality than 
velocity, 195; modifies FitzGerald 
contraction, 75 
Action, 147; atomicity of, 199; on 

Weyl's theory, 177 
Action, Principle of Least, 149, 178 
Addition of velocities, 59 
Aether, a plenum with geodesic struc- 
ture, 164; identified with the 
"world," 187; non-material nature 
of, 39; stagnant, 210 
Artificial fields of force, 64 
Atom, vibrating on sun, 128 
Atomicity, law of, 199; of Action, 177 
Aviator, space and time-reckoning of, 
23 

Bending of light, effect on star's 
position, 112; observational results, 
118; theory of, 107,207 

Beta particles, 59, 145 

Brain, constitution of, 191 

Brazil, eclipse expedition to, 117 

Causality, law of, 156 

Causation and free will, 51 

Centrifugal Force, compared with 
gravitation, 41, 65; debt at infinity, 
157; not caused by stars, 153; 
vibrating atom in field of, 129 

Chess, analogy of, 184 

Christoffel, 89 

Circle in non-Euclidean space, 104 

Clifford, 77, 152, 192 

Cliquishness, 188 



Clock, affected by velocity, 58; on sun, 
74, 128; perfect, 13; recording 
proper- time, 71 

Clock-scale, 58 

Clock-scale geometry, not fundamental, 
73, 131, 191 

Coincidences, 87 

Comets, motion through coronal me- 
dium, 121; radiation-pressure in, 110 

Conservation of electric charge, 173; 
of energy and momentum, 139; of 
mass, 141, 196 

Content contrasted with structural 
form, 192, 200 

Continuous matter, 91, 140 

Contraction, FitzGerald, 19, 54 

Convergence of physical approximations, 
154 

Coordinates, 77 

Coordinate velocity, 107 

Corona, refraction by, 121 

Cottingham, 114 

Crommelin, 114, 122 

Curvature, degrees of, 91; identified 
with action, 148; merely illustra- 
tive, 84; of a globe of water, 148; 
of space and time, 158; on Weyl's 
theory, 176; perception of, 190 

Cylinder and plane, indistinguishable 
in two dimensions, 81 

Cylindrical world, Einstein's, 161, 177 

Davidson, 114 

Deflection of light, effect on star's 

position, 112; observational results, 

118; theory of, 107,207 
Density, effect of motion on, 62 
Displacement of spectral lines, 129; in 

nebulae, 161 ; in stars, 135 
Displacement of star-images, 112, 115 
Double stars and Einstein effect, 133 
Duration, not inherent in external 

world, 34 



216 



INDEX 



Eclipse, observations during, 113 
Ehrenfest's paradox, 75 
Electrical theory of inertia, 61 
Electricity and gravitation, 167 
Electromagnetic potentials and forces, 

172 

Electron, dimensions of, 177 ; geometry 
inside, 91; gravitational mass of, 
178; inertia of, 61; Kaufmann's 
experiment on, 62, 146; singularity 
in field, 167 
"Elsewhere," 50 
Emptiness, perception of, 190 
Energy, conservation of, 139; identi 
fied with mass, 146; inertia of, 61, 
146; weight of radio-active, 112 
Entropy, 149 

Eotvos torsion- balance, 112 
Equivalence, Principle of, 76, 131, 212 
Euclidean geometry, 1, 47, 73 
Euclidean space of five dimensions, 84 
Event, definition of, 45, 186 
Evershed, 130 
Extension in four dimensions, 37, 46 

Feeling, elements of, 192 

Fields of force, artificial, 64; due to 
disturbance of observer, 69; elec- 
tromagnetic, 171 ; relativity of, 67 

Field of velocity, 195 

FitzGerald contraction, 19; conse- 
quences of, 22; modified by 
acceleration, 75; relativity ex- 
planation of. 54 

Flat space in two dimensions, 80 

Flat space-time, 83; at infinity, 84; 
conditions for, 89 

Flatfish, analogy of, 96 

Flatland, 57 

Force, compared with inertia, 137; 
electromagnetic, 172; elementary 
conception of, 63; fields of, 64; 
relativity of, 43, 67, 76 

Form contrasted with content, 192, 200 

Formalism of knowledge, 175 

Foucault's pendulum, 152 

Four-dimensional order, 35, 56, 186 

Four- dimensional space -time, geo- 
metry of, 45, 82; reality of, 181 

Fourth dimension, 13 



Frame, inertial, 156 

Frames of reference, "right" and 

"wrong," 42 
Freewill, 51 
Freundlich, 212 
Future, absolute, 50 

Galilean potentials, 83 

Gauge, effect on observations, 31; 
provided by radius of space, 177 

Gauge-system, 169 

Geodesic, absolute significance of, 70, 
150; definition of, 75; motion of 
particles in, 138, 151; in regions at 
infinity, 157 

Geodesic structure, absolute character 
of, 155, 164; acceleration of, 195 

Geometrical conception of the world, 
176, 183 

Geometry, Euclidean, 1; hyperbolic, 
47; Lobatchewskian, 1, 9; natural, 
2; non-Euclidean, or Riemannian, 
6, 73, 84, 90; non-Riemannian, 169 ; 
semi-Euclidean, 47 

Ghosts of stars, 161 

Globe of water, limit to size of, 148 

Gravitation, Einstein's law of, differ- 
ential formula, 90; integrated for- 
mula for a particle, 97 ; macroscopic 
equations, 140, 193 

Gravitation, Newton's law of, ambig- 
uity of, 93; approximation to 
Einstein's law, 103; deflection of 
light, 109, 111 

Gravitation, propagation with velocity 
of light, 94, 147; relativity for 
uniform motion, 21, 125 

Gravitational field of Sun, 97; de- 
flection of light, 107, 118, 207; dis- 
placement of spectral lines, 129; 
motion of perihelion, 122, 206; 
Newtonian attraction, 102; result 
of observational verification, 126 

Grebe and Bachem, 130 

Greenwich, Royal Observatory. 114 

Gyro-compass, 152 

Hummock in space-time, 97 
Hurdles, analogy of counts of, 104 
Hyperbolic geometry, 47 



INDEX 



217 



Identities connecting G^v, 141 

Identity permanent, 40, 193 

Imaginary intervals, 150, 187 

Imaginary time, 48, 181 

Inertia, compared with force, 137; 
electrical theory of, 61 ; in regions 
at infinity, 157; infinite, 56; 
Mach's views, 164; of light, 110; 
relativity theory of, 139 

Inertia-gravitation, 137 

Inertial frame, 156 

Infinity, conditions at, 157 

Integrability of length and direction, 
174 

Interval, 46, 150, 187; general ex- 
pression for, 82; practical measure- 
ment of, 58, 75 

Interval-length, geometrical signifi- 
cance essential, 127 ; identified with 
proper time, 71; tracks of maximum, 
70, 150 ; zero for velocity of light, 71 

Invariant mass, 145; of light, 148 

Jupiter, deflection of light by, 133 

Kaufmann's experiment, 62, 146 
Kinds of space, 81 

Laplace's equation, 96, 140 

Larmor, 19, 211 

Length, definition of, 2; effect of 
motion on, 19; relativity of, 34 

Le Verrier, 124 

Levi-Civita, 89 

Lift, accelerated, 64 

Light, bending of, 107, 112, 118, 207; 
coordinate velocity of, 107; mass 
of, 62, 110, 148; voyage round the 
world, 161; weight of, 111 

Light, velocity of, an absolute velocity, 
59; importance of, 60; system 
moving with, 26, 56 

Lobatchewsky, 1, 9 

Lodge, 32, 125, 211 

Longest tracks, 70 

Lorentz, 19, 211 

Mach's philosophy, 163 
Macroscopic equations, 92, 139; in- 
terval, 187 



Map of sun's gravitational field, 99 

Mass, conservation of, 141, 195; 
electrical theory of, 61; gravita- 
tional, 98; identified with energy, 
146; invariant, 145; of light, 62, 
110, 148; variation with velocity, 
145 

Mathematics, Russell's description of, 
14 

Matter, continuous, 91 ; definition of a 
particle, 98; extensional relations 
of, 8; gravitational equations in, 
141; perception of, 190; physical 
and psychological aspects, 192 

Mercury, perihelion of, 123, 125 

Mesh-systems, 77; irrelevance to laws 
of nature, 87 

Michelson-Morley experiment, 18 

Minkowski, 30, 212 

Mirror, distortion by moving, 22 

Momentum, conservation of, 141; re- 
definition of, 144; of light, 111 

Moon, motion of, 93, 134 

Motion, insufficiency of kinematical 
conception, 194; Newton's first 
law, 136 

Natural frame, 155 

Natural gauge, 176 

Natural geometry, 2 

Natural tracks, 70 

Nebulae, atomic vibrations in, 161 

Newton, absolute rotation, 41; bend- 
ing of light, 110; law of gravitation, 
93; law of motion, 136; relativity 
for uniform motion, 40; super- 
observer, 68 

Non-Euclidean geometry, 6, 73, 84, 90 

Non-Riemannian geometry, 169 

Observer, an unsymmetrical object, 57 
Observer and observed, 30 
Orbits under Einstein's law, 123 
Order and dimensions, 14, 186 
Ordering of events in external world, 
35, 54, 184 

Past, absolute, 50 

Perceptions, as crude measures, 10 ; 15, 
31 



218 



INDEX 



Perihelia of planets, motions of, 123 
Permanence of matter, 196 
Permanent identity. 40, 193 
Permanent perceptual world, 141, 198 
Poincare, 9 
Point-event, 45, 186 
Potentials, 80; Galilean values, 83 
Potentials, electromagnetic, 172 
Principe, eclipse expedition to, 114 
Principle of Equivalence, 76, 131, 212 
Principle of Least Action, 149, 178 
Principle of Relativity (restricted), 20 
Probability, a pure number, 178 
Projectile, Jules Verne's, 65 
Propagation of Gravitation, 94, 147 
Proper-length, 11 
Proper- time, 71 
Pucker in space-time, 85 

Quanta, 60, 177, 182, 200 

Radiation-pressure, 110 
Real world of physics, 37, 181 
Receding velocities of B- type stars, 

135; of spiral nebulae, 161 
Reflection by moving mirror, 22 
Refracting medium equivalent to 

gravitational field, 109 
Refraction of light in corona, 121 
Relativity of force, 43, 76; of length 
and duration, 34; of motion, 38; of 
rotation, 152, 155; of size, 33 
Relativity, Newtonian, 40; restricted 

Principle of, 20 ; standpoint of, 28 
Repulsion of light proceeding radially, 

102, 108 

Retardation of time, 24, 55; in centri- 
fugal field, 129; in spherical world, 
160 

Ricci, 89 

Riemann, 2, 89, 167 
Riemann-Christoffel tensor, 89 
Riemannian, or non-Euclidean, geo- 
metry, 6, 73, 84, 90 
Rigid scale, definition of, 3 
Rotation, absolute, 152, 164, 194 
Rotation of a continuous ring. 194 
Russell, 14, 197 

St John, 130 



Semi-Euclidean geometry, 47 

Simultaneity, 12, 51 

de Sitter, 134, 159, 179 

Sobral, eclipse expedition to. 117 

Space, conventional, 9; kinds of, 81; 

meaning of, 3, 8, 15; relativity of, 

34 

Space-like intervals, 60, 187 
Space -time, 45 ; due to Minkowski, 212 ; 

partitions of, 54; 
Spherical space-time, 159 
Standard metre, comparison with, 168 
Stresses in continuous matter, 193 
Structure opposed to content, 197, 200 
Structure, geodesic, absolute character 

of, 155, 164; acceleration of, 195; 

behaviour at infinity, 157 
Super-observer, Newton's, 68 
Synthesis of appearances, 31, 182 

Tensors, 89, 189 

Thomson, J. J., 61 

Time, absolute, 163; depends on 
observer's track, 38, 57; for moving 
observer, 24; imaginary, 48; mea- 
surement of, 13; past and future, 
51; "standing still," 26, 160 

Time-like intervals, 60, 187 

Tracks, natural, 70 

Vacuum, defined by law of gravita- 
tion, 190 

Vector, non-integrable on Weyl's 
theory, 174 

Velocity, addition-law, 59; definition 
of, 193; static character, 194 

Velocity of gravitation, 94, 147 

Velocity of light, importance of, 60; 
in gravitational field, 108; system 
moving with, 26, 56 

Warping of space, 8, 126 

Wave-front, slewing of, 108 

Weight, of light, 107, 111; of radio- 
active energy, 111; proportional to 
inertia, 137; vanishes inside free 
projectile, 65 

Weyl, 174 

World, 186, 187 

World-line, 87 



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