(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Relativity For Physics Students"

DAMAGE BOOK 



W > 

u< OU_1 58877 5m 



OUP 391 29-4-72 10,000. 

OSMANIA UNIVERSITY LIBRARY 

Call No. 63'/ Accession No. % S ') 

<r/:r& 

Author 
Title 

This hf)ok should be returned on or before the date last marked below. 



RELATIVITY 

FOR PHYSICS STUDENTS 



KY 

G. B. JRFFERY, M.A., D.Sc. 

PLIXOW OF UNIVERSITY COLLEGE, LONDON, AND PROFESSOR OP 
MATHEMATICS AT KING'S COLLEGE, I ON DON 



\\IIH VOUR DIAGRAMS 



METHUKN & CO. LTD. 

36 ESSEX STREET W.C 

LONDON 



Pint Published in 



PR1NTKD IN GREAT BRITAIN 



CONTENTS 

P*GL 

INTRODUCTION : EINSTKIN'S THEORY OK RELA- 
TIVITY i 

I. THE QRIC.TNS OF HIE THEORY . . 33 
II. THE MICHELSON - MORLEV EXPERIMENT 

AND THK LORHNTZ TRANSFORMATION 56 

IIT THE LAWS OF MOTION AND ELECTRO- 
MAGNETISM 73 

IV. THE RESTRICTED PRINCIPLE OF RELA- 
TIVITY AND SOME CONSEQUENCES . 85 

V. THE EQUIVALENCE HYPOTHESIS . . 98 

VI. THE FOUR-DIMENSIONAL CONTINUUM . 115 

VII. THE GENERAL THEORY .... 133 

INDEX 151 



RELATIVITY FOR PHYSICS 
STUDENTS 

EINSTEIN'S THEORY OF 
RELATIVITY* 

AS I conceive the office of a professor. 
it is that he_shoulj|^^ 
his students_as the 
of those great , men_who in 



l^dge whichj^jms^^ that by 

means of a reverent yet unflinching criti- 
cism he should strive to reveal the workings 
of these master minds, to the end that he 
mayjmgart, not merely knowledge, but that 
more precious gift the art of acquiring 
knowledge, the art of discovery. If we 

An Inaugural Lecture, delivered at King's College, 
London, on gth October, 1922. 
i 



% RELATIVITY FOR PHYSICS STUDENTS 

approach our task in this spirit, we shall 
find the key to the solution of much that is 
difficult and perplexing in our present know- 
ledge, and the inspiration which will lead us 
on to further discoveries. 

It seems natural, therefore, that I should 
seek to illustrate this theme by means of 
the subject which throughout my mathe- 
matical career has inspired me more than 
any other branch of mathematics or physics 
into which my work has led me, and the 
subject which, as far as one may venture to 
prophesy as to the future course of scientific 
thought, seems .marked out fon great ad- 
vances in the immediate future. 

Einstein's theory of relativity has proved 
fuUj)f^ 

It seems to invite us to cut ourselves loose 
from all that has gone before, to scrap all 
our old ideas and to start afresh with new. 
It is true that tL theory does profoundly 
modify our fundamental ideas of space, 
time, and motion, but a deeper study reveals 
the fact that it is nevertheless the natural 



EINSTEIN'S THEORY OF RELATIVITY b 

and almost inevitable sequel to the work of 
the great masters of the past, and more 
particularly to the work of Isaac Newto- 
himself ; how natural and inevitable it wi 
be the main purpose of this lecture to show. 
The story of modern mechanics begins in 
the sixteenth century. Tycho Brahe, with 
no telescope, and the most primitive instru- 
ments in place of the equipment of the 
modern astronomical observatory, sustained 
through years of labour by a most extra- 
ordinary patience, observed night__after 
night; the positions of the planets among 
the surrounding stars. Tycho was one who 
sowed but tfitf j\nt....jc*ap. Those who hav 
any experience of observational astrono^ 
find it difficult to imagine a duller task 
Tycho's the accumulation of volu r 
figures whose meaning it was not 
him to read the construction of - 
Almanac without its beautiful 
order. Nevertheless his work 
the necessary foundation fr 
followed. 



4 RELATIVITY FOR PHYSICS STUDENTS 

The task was taken up by Tycho's pupil 
and assistant, John Kepler. He succeeded 
in clothing his master's data with the form 
of three simple descriptive laws. His was 
a great achievement. All Tycho's volumes 
of figures, all those strange motions of the 
bodies which men have most appropriately 
called wanderers, were summed up in three 
simple statements. Kepler had no theory ; 
he made no attempt to explain the motions 
he studied. The Archangels who kept the 
celestial spheres in motion were dismissed, 
but no subtle scientific hypothesis was im- 
ported to perform their office. In effect he 
: Viewed from the earth, the motions 
the planets are very complicated. Now 
wander forwards ; now they retrace 
teps ; now they move in loops ; and 
ong sweeping curves. But viewed 
nn, these motions are very simple, 
rh planet describes a perfectly 

alileo had brought the same 
on the problems of terres- 



KINST KIN'S THEORY OF i, 

trial dynamics. He investigated u. 
which govern the motions of falling bodies. 
iOSi^LbsSQ .laught thajLeyer^body^iiad its 
" proper place/' The proper place of heavy 
bodies was low down, and the proper place 
of light bodies was highjip. AJbody4^aded 
to_moye_ta ks-~peper-~plae ; the heavier 
a body, the more quickly it fell, since it was 
presumably at a greater distance from its 
proper place. It is a striking commentary 
on medieval thought, that it seems to have 
occurred to nobody before the time of 
Galileo to test this conclusion by means of 
a simple experiment. Galileo made such an 
experiment at the leaning tower of Pisa, and 
found that all bodies, heavy or light, fr 
towards the ground in precisely the,?' 
way. By careful laboratory experir 
he ascertained the law of this fa" 
body falls towards the ground wi' 
which increases in proportion 4 
so that its speed is increas' 
32 feet per second in every 
fall 



. x FOR PHYSICS STUDENTS 

o Newton found ready to hand two 
of descriptive laws : Kepler's laws, 
which embraced the motions of the planets ; 
and Galileo's law, which covered a very 
important case of the motion of terrestrial 
bodies. His first step was to throw the 
laws of Kepler into a different form. No 
doubt he took the hint from Galileo's law 

of falling bodies, and he investigated the 

* 

motion of a planet, moving in accordance 
with Kepler's laws, from the point of view 
of the change of its velocity, or, as we should 
say, its acceleration. He found that Kep- 
ler's laws are equivalent to the statement 
that the acceleration of a planet is always 
r ectly towards the sun, and that this 
Deration depends in no way on the 
but only on its distance from the 
inishing with increasing distance in 
* with the law of the inverse 

served the similarity between 

f the planets round the sun 

of bodies falling towards 



EINSTEIN'S THEORY OF RELATIVITY 

the earth. They too fall with an accelera- 
tion which in no way depends on the falling 
body. Is this gravitation subject to the 
samejaws--^as^ttie ^gravitation ~oj^ JJhe sun 
v/hich keeps the planets in their orbits? 
Do^sTT^Iso"3imimsh as the inverse square 
of the distance ? It is difficult to answer 
these questions in the narrow range of 
height we can employ at the earth's surface, 
but. Newton took the heavens for his 
laboratory. The moon, though somewhat 
disturbed by the sun, moves round the 
earth approximately in accordance with 
Kepler's laws, and has an acceleration 
towards the earth. Is this acceleration 
just what 32 feet per second per second 
would become if it diminished in accordance 
with the inverse square law up to the 
moon's distance ? Newton worked the sum 
and found that it was so. 

Thus the inward nature of gravitation 
was laid bare. There is a gravitation of 
the sun, in virtue of which any planet, 
comet, or meteorite which may happen to 



RELATIVITY FOR PHYSICS STUDENTS 

find itself in a given position experiences an 
acceleration which depends only upon that 
position. There is a gravitation of the 
earth, in virtue of which the moon, or any 
unsupported body near the earth, experi- 
ences an acceleration which again depends 
only upori^lfae "position of "the accelerated 
body. "~~ ~~"~^ 

Thus far Newton was on very safe ground, 
for he was merely expressing the results of 
observation in a concise and compact form. 
He then proceeded to frame a theory which 
should account for the observed facts. 
Here we can trace the influence of Galileo 
very clearly. From his experiments on the 
motion of a body down an inclined plane, 
Galileo inferred that a body moving on a 
horizontal plane would continue to move 
with a constant velocity in a straight line. 
Earlier thinkers had felt the necessity of 
ascribing some cause to the motion of 
bodies ; if a body moves some agency must 
be at work to maintain its motion. The 
experiments of Galileo, and Newton's inter- 



EINSTEIN'S THEORY OF RELATIVITY 

pretation of Kepler's laws, conspired to 
promote the view that it was the change of 
motion, the acceleration of a body, for 
which a cause must be found, rather than 
the motion itself. Newton adopted the 
view that when the motion of a body 
changes it does so because the body is 
acted upon by a force, and that this force 
is measured by the product of the mass of 
the body and its acceleration. Gravitation 
is explained by the action of forces aris- 
ing from, and directed towards, attracting 
bodies. This in the barest outline is the 
Newtonian system of mechanics as com- 
monly understood. Before we proceed to 
criticise it, it may be well for a moment to 
pause to consider the achievement which 
stands to its credit. The motions of the 
planets are not, in fact, quite so simple as 
the laws of Kepler would indicate. Some- 
one has said that if Kepler had possessed a 
modern telescope he would never have dis- 
covered his laws. Nevertheless, with a few 
small outstanding differences, the deviations 



1C RELATIVITY FOR PHYSICS STUDENTS 

from Kepler's laws are all explained when 
we take into account the gravitation of the 
planets upon each other. The history of 
dynamical astronomy has been very largely 
the verification, to an ever-increasing degree 
of refinement, of Newton's law of universal 
gravitation. Cavendish observed the work- 
ings of this same law in the attraction be- 
tween quite small bodies in the laboratory. 
The laws of motion, originally deduced 
from the motions of the planets, are verified 
day by day in every engineering workshop. 
It seems to me that the supposed conflict 
between Newton and Einstein rests very 
largely upon a failure to apprehend a dis- 
tinction upon which Newton was always 
insisting, the distinction between what he 
called mathematical principles and philo- 
sophical principles. Mathematical prin- 
ciples were to Newton, not ultimate causes, 
but merely concise descriptions of the phe- 
nomena of Nature, which could be verified 
by observation and experiment. He distin- 
guishes them very clearly from philosophical 



EINSTEIN'S THEORY OF RELATIVITY 11 

principles, whose function it is to explain 
and interpret phenomena. This distinc- 
tion, maintained in actual scientific work, 
is one of the great debts which we owe to 
Newton. It defines at once the purpose 
and the limitation of Science. When Science 
shall have accomplished its purpose and 
described the whole material universe in 
the simplest way, it must leave us face to 
face .with the philosophical problem of the 
mystery and meaning of the things which 
it has described. But Newton, like many 
of us, had within him something of the 
philosopher. He might jeer at the meta- 
physicians, but at times he could not help 
speculating, and rightly speculating, as to 
the meaning of those great descriptive laws 
which he found running throughout the 
whole fabric of Nature. He was, however, 
always careful to distinguish these specula- 
tions from the formulation of the mathe- 
matical principles which he regarded as 
the main part of his work, and we find 
them for the most part in the scholia in the 



12 RELATIVITY FOR PHYSICS STUDENTS 

Principia, and in the queries in the Optics. 
These speculations have been the subject of 
controversy ever since, and it is towards 
them that the criticism of Relativity is, for 
the most part, directed. 

In a scholium which follows the definitions 
in the Principia, Newton sets forth his views 
on time, sace, and motion. He distin- 
guishes between absolute time and relative 
time which is measured by some motion. 
He says : 

" The natural days, which, commonly, for 
the purpose of the measurement of 
time, are held as equal, are in reality 
unequal. Astronomers correct this 
inequality, in order that they may 
measure by a truer time the celestial 
motions. It may be that there is no 
equable motion, by which time can 
accurately be measured. All motions 
can be accelerated or retarded. But 
the flow of absolute time cannot be 
changed. Duration, or the persis- 
tent existence of things, is always the 



EINSTEIN'S THEORY OF RELATIVITY 13 

same, whether motions be swift or 
slow or null." 

In the same way he distinguishes between 
absolute space and relative space, and be- 
tween absolute motion and relative motion. 
He says : 

" We use in common affairs, instead of 
absolute places and motions, relative 
ones ; and this without any incon- 
venience. But in physical disquisi- 
tions, we should abstract from the 
senses. For it may be that there is 
no body really at rest, to which the 
places and motions of others can be 
referred/* 

Thus we need go no further than Newton 
himself, to find a clear statement of the 
problem to which the theory of relativity 
has attempted to supply an answer. Our 
experience is entirely of relative motions. 
We are at rest relatively to our immediate 
surroundings ; we are moving at a rate of 
100,000 miles an hour relatively to the sun ; 
we are moving relatively to Sirius at such 



14 RELATIVITY FOR PHYSICS STUDENTS 

and such a speed ; but how we are moving 
in an absolute sense, without reference to 
any other body, is a question which experi- 
mental science has often tried, but always 
failed, to answer. The statement that we 
are moving at a rate of 100,000 miles an 
tiour is devoid of all physical meaning what- 
soever, unless we state what we conceive to 
be at rest. This something, which for a 
particular purpose we assume to be at rest, 
we call our " frame of reference." 

Now, if we consider Newton's work in its 
proper setting, there is no doubt at all as to 
what his frame of reference was. It was 
implicit in Tycho's data, and Tycho ob- 
served the motions of the planets relatively 
to the fixed stars. Newton's frame of 
reference was one in which the distant fixed 
stars are at rest. It seems likely that 
Newton, who boasted that he did not frame 
hypotheses, adopted the hypothesis of abso- 
lute space because in the fixed stars he 
found ready to hand a frame of reference 
which transcended the domestic motions of 



EINSTEIN'S THEORY OF RELATIVITY 15 

the solar system the chief objects of his 
study. Was not Newton's absolute space 
after all just the physical space mapped out 
by the fixed stars, rather than the meta- 
physical concept we have usually taken it 
to be? 

In the light of modern knowledge this, 
frame of reference pi ?sents great difficulties. 
We can now, in many cases, measure the 
velocities of these stars relative to each 
other and to our sun. We find that they 
are not fixed, or at least, they are not all 
fixed, for they move relatively to each other 
with widely different velocities. The reason 
why, night after night, they seem to occupy 
the same positions in their constellations is 
the same as that which makes an express 
train seem to move so slowly when viewed 
from a long distance across country. It is 
not that their motions are slow in many 
cases they are almost inconceivably great 
but that the stars themselves are at such 
immense distances from us. Still more 
modern knowledge forbids us to attempt to 



16 RELATIVITY FOR PHYSICS STUDENTS 

surmount this difficulty by supposing that 
the motions of the stars are random, like 
the motions of the atoms of a gas, so that 
we could average them out, in order to 
arrive at our fixed frame of reference. If 
our stellar system has indeed grown out of 
a giant nebula, there may be an ordered 
system in the motions of the stars. 

By the time that the discordant motions 
of the stars had been well established, a 
new hope had arisen. The undulatory 
theory of light seemed to call for some 
medium to transmit the light vibrations, 
and the idea of an ether pervading all space 
was developed. Clerk Maxwell showed the 
intimate relation between light and electro- 
magnetism. Later on, the electron theory 
promised to explain the whole of physics in 
terms of electricity. Matter was simply an 
aggregation of electric charge, and elec- 
tricity was a state or singularity of the ether. 
The ether had become fundamental in 
physics. Here it seemed that the solution 
of all our difficulties might lie. A body 



EINSTEIN'S THEORY OF RELATIVITY 

moves when it moves relatively to the ether ; 
our frame of reference is to be fixed, not 
with respect to the so-called fixed stars, but 
with respect to the ether. 

The result did not work out happily. If 
mechanics adopted the ether in order to 
simplify the problem of motion, never was 
foster-parent blessed with a more unruly 
child. If we observe a star, the ether is 
undisturbed by the earth's motion through 
it ; if we fill our telescope with water, the 
water communicates part of its motion to 
the ether ; if we make an interference ex- 
periment in the laboratory, we can only 
conclude that the earth carries the ether 
with it in its motion . Quite apart from the 
logical difficulty as to how the ether, the 
standard of absolute rest, can itself move 
at all, it moves or it does not move in a 
delicate accommodation to the particular 
experiment which we may happen to have 
in hand at the moment. In spite of the 
labours of some of the greatest English 
mathematicians of the latter half of the 



RELATIVITY FOR PHYSICS STUDENTS 

nineteenth century, the situation grew 
steadily worse. 

In the meantime experimental physicists 
had concentrated on the problem of the 
determination of our motion relative to the 
ether. Many different experiments were 
proposed and carried out with all the skill 
and ingenuity of a great generation of 
experimenters. The result was always the 
same. No experiment succeeded in reveal- 
ing our motion through the ether. The 
story is not unlike that of an earlier chapter 
in the history of science, which tells how 
for centuries men tried to construct a per- 
petual motion machine. They failed, and 
out of their failure modern physics has 
erected a great principle. They searched in 
vain, until they were led to deny the very 
possibility of the thing they sought. That 
denial has become the Second Law of 
Thermodynamics, one of the most powerful 
principles of modern physics. Relativity is 
the outcome of the application of the same 
method to our present difficulty. As the 



EINSTEIN'S THEORY OF RELATIVITY 19 

result of repeated failure, it asserts that no 
physical experiment can ever reveal our 
motion through the ether. 

This was the culmination of a long sus- 
tained effort to bring the absolute space of 
Newton within reach of physical experi- 
ment, or perhaps we should say, rather, to 
restore to absolute space the physical 
reality which it lost on the discovery of the 
motions of the fixed stars. It is the starting 
point of the theory of relativity, that no 
method has yet been discovered by which 
this can be accomplished. If this position 
is accepted it constitutes a fatal criticism of 
Newton's laws of motion, at any rate in the 
form in which he stated them, for the very 
terms of those laws motion, change of 
motion have no meaning apart from some 
pre-determined standard of rest or frame of 
reference. It is obvious that the time had 
arrived when some fundamental reconstruc- 
tion of the theory could no longer be delayed. 
Einstein did not bring forth his theory 
merely as an elaboration and refinement of 



20 RELATIVITY FOR PHYSICS STUDENTS 

physical law in order to bring theory into 
accord with a few isolated and newly- 
discovered facts ; he brought it forth to 
meet the situation created by a complete 
theoretical breakdown of the older system. 

If we seek a way out of the difficulty, the 
first suggestion which presents itself is that, 
since our experience is confined to relative 
motions, it ought to be possible to express 
the laws of motion in terms of relative 
motions alone, without any reference to 
absolute motions. 

This in effect is what Einstein has done, 
though he approached the problem from a 
rather different point of view. If we take 
any frame of reference, we can obtain laws 
which will describe the course of natural 
phenomena. Since we have to recognize 
that the choice of a frame of reference is 
arbitrary, we shall expect these descriptive 
laws to be different if we choose another 
frame of reference. In other words, we 
shall expect to find that our descriptive 
laws are relative to the particular frame of 



EINSTEIN'S THEORY OF RELATIVITY 21 

reference which we have chosen. For ex- 
ample, if we take a frame of reference fixed 
with respect to the earth, we shall obtain 
the Ptolemaic system of astronomy with its 
epicycles, etc., whereas if we take a frame 
of reference fixed with respect to the sun, 
we shall obtain the very different descrip- 
tive laws of Kepler. 

The question to which Einstein addresses 
himsfelf is, whether the descriptive laws of 
physics can be framed in such a way that 
if they are true for one frame of reference 
they will also be true for any frame of 
reference whatever. This is essentially a 
mathematical question. If it is answered 
in the affirmative, the experimental ques- 
tion will arise as to whether these general 
laws are in fact true for one frame of refer- 
ence. By the aid of the calculus of tensors, 
Einstein was able to give an answer to the 
mathematical question, and it appears that 
it is possible to frame laws which are absolute 
in the sense that, if they are true at all, 
they are true independently of the particular 



22 RELATIVITY FOR PHYSICS STUDENTS 

frame of reference which we may happen 
to choose. If these laws are verified by 
experiment, we shall have succeeded in dis- 
pensing with absolute space and with all 
the difficulties to which the introduction of 
this concept into our scientific work has 
given rise. 

As so often happens in scientific research, 
Einstein's efforts to clarify our fundamental 
ideas of mechanics led to an important 
extension of knowledge. He was able for 
the first time to bring gravitation into rela- 
tion with other physical phenomena. Let 
us return for a moment to the view of 
gravitation which we have already con- 
sidered. In the Solar system, and in the 
fall of heavy bodies towards the earth, we 
observe the same essential feature, namely, 
that any body placed in a particular position 
experiences an acceleration which depends 
in no way upon itself, but only upon the 
position in which it is placed. In his deter- 
mination to confine himself to the descrip- 
tion of phenomena, Einstein accordingly 



EINSTEIN'S THEORY OF RELATIVITY 28 

regards gravitation as a property of space 
varying from place to place, leaving open 
for the time being the question as to whether 
this property can be expressed in terms of 
the influence of attracting bodies. In this 
sense Einstein's space, unlike that of New- 
ton, is not homogeneous, but differs in its 
properties from place to place. 

We can best explain Einstein's discovery 
by means of a simple, if somewhat fanciful, 
illustration. Imagine a lift working in a 
deep well, and let it be one of the kind 
which is operated, not from within the cage, 
but by a man at the bottom. Suppose that 
within the lift is the ghost of Galileo. He 
will be unconscious of the mechanism of 
modern lifts, but he might well return to 
his old task of the investigation of the laws 
of falling bodies. This he might do by 
allowing a marble to fall through a measured 
height to the floor of the lift and timing its 
fall. To avoid complications, we will allow 
him a stop-watch in place of his water-clock. 
The ghost sits there all day long, condemned 



24 RELATIVITY FOR PHYSICS STUDENTS 

to time the fall of this marble over and over 
again. So long as the lift remains stationary 
he will get the same answer every time. 
But suddenly he finds that the marble is 
falling more quickly, and he will say that 
gravity has increased. The man at the 
bottom knows better. He is sending the 
lift upwards with an accelerating speed. 
The floor of the lift is rising to meet the 
marble, and thus the latter accomplishes 
its measured journey more quickly. The 
man has only to make the lift go upwards 
or downwards with the right acceleration in 
order to make the ghost's gravity anything 
he pleases, downwards or upwards, or nil. 
If the man chooses to play tricks by sending 
the lift now up and now down, the poor 
ghost will find that gravity is fluctuating 
wildly, and will think that some kind of 
gravity storm is in progress. But the man 
on solid earth at the bottom knows that it 
is all due to the motion of the lift. If only 
the ghost would realize that he is being 
fooled, and that his frame of reference is 



EINSTEIN'S THEORY OF RELATIVITY 25 

being accelerated upwards and downwards, 
he would see that gravity has remained the 
same all the time. Einstein is inclined to 
make allowances for the ghost. He claims 
the liberty to take any frame of reference 
he pleases, and he is prepared to allow that 
the ghost was perfectly reasonable in taking 
himself and his immediate surroundings as 
a frame of reference. This fact, however, 
emerges, that if the same phenomenon is 
described from the point of view of different 
frames of reference, the gravitation inferred 
will, in general, be different. This is the 
essence of Einstein's equivalence hypothe- 
sis. He describes a physical phenomenon 
in the absence of gravitation by means of 
an accelerated frame of reference, and thus 
obtains a description of the same phenom- 
enon in the presence of gravitation. It 
was by this method that he was able to 
establish the influence of gravitation on the 
propagation of light. 

It is interesting to note how near Newton 
got to this idea. He lived much closer to 



26 RELATIVITY FOR PHYSICS STUDENTS 

the Copernican controversy than we do. 
Men had only just given up the belief of 
centuries that the stars revolved in their 
courses once a day. In his " System of the 
World " we find him facing the problem 
that the choice between the Ptolemaic and 
Copernican systems could not be settled by 
observation alone, but he points out that, 
whereas on the Copernican system gravita- 
tion can be expressed in terms of forces 
directed towards definite bodies which may 
be regarded as the sources of the gravita- 
tion, on the Ptolemaic theory the forces 
would be directed, not to the earth, but to 
points on the axis of the earth. He says : 
" That forces should be directed to no 
body on which they physically de- 
pend, but to innumerable imaginary 
points on the axe of the earth, is 
an hypothesis too incongruous. Tis 
more incongruous still that those 
forces should increase exactly in 
proportion of the distance from this 
axe. For this is an indication of 



EINSTEIN'S THEORY OF RELATIVITY 27 

an increase to immensity, or rather 
infinity ; whereas the forces of 
natural things commonly decrease in 
receding from the fountain from 
which they flow." 

Newton adopted the Copernican frame 
of reference, not on observational grounds, 
but because that frame of reference possessed 
the peculiar convenience that it enabled 
him* to express gravitation in a simple way 
as arising from the influence of attracting 
bodies. 

It should be pointed out that Einstein 
does not say that it is a matter of indiffer- 
ence as to which frame of reference we adopt. 
A man who attempted to conduct experi- 
ments in an unsprung cart, and who took 
the body of his cart for his frame of refer- 
ence, would be asking for trouble, for he 
would have to deal with a hopelessly 
complicated gravitational field. The im- 
portance of the principle lies in this that 
while one frame of reference may be more 
convenient than another for the discussion 



28 RELATIVITY FOR PHYSICS STUDENTS 

of some particular problem, all frames of 
reference are theoretically admissible. 

Another consequence of the denial of 
absolute motion has been to destroy the 
independence of space and time. There 
has been so much misunderstanding on this 
point that it may be well to state exactly 
what relativity has to say on the matter. 
It may be stated very simply thus : At two 
distant points there is no definite unique 
instant of time at the second which may 
be regarded as simultaneous with a given 
instant at the first. For example, a new 
star bursts suddenly into view, and the as- 
tronomers tell us that, owing to its great 
distance and the time that it takes light 
to travel from it to us, the cataclysm which 
has made it visible must have occurred in 
the time of Newton. Such a statement 
would necessarily be approximate, for we 
have only the roughest notion of the dis- 
tances of stars so remote as this one would 
have to be. But let us in imagination 
concede the astronomer all the accuracy of 



EINSTEIN'S THEORY OF RELATIVITY .., 

his wildest dreams. Could he even then 
assure us, for example, that the cataclysm 
occurred at the precise instant at which 
the famous apple struck the ground ? No, 
for perchance the solar system is moving in 
the direction of this star with a speed 
which we may very appropriately call x. 
If so, we are rushing to meet the light on 
its journey towards us, and we shall receive 
it sooner. How much sooner will depend 
upon x, and % has no meaning apart from 
a frame of reference. Adopt a frame of 
reference in which we are moving in the 
direction of the star, and the apple fell 
before the star burst into flame. Simul- 
taneous and the words before and after, as 
applied to two instants of time at different 
points of space, have no precise scientific 
meaning apart from a specified frame of 
reference. Thus the time of one frame of 
reference depends upon the time and the 
space of another frame of reference. In 
the words of Minkowski : " Time of itself, 
and space of itself, fade into shadows, and 



RELATIVITY FOR PHYSICS STUDENTS 

only a kind of union of the two shall main- 
tain an independent reality/ 1 

Thus the new theory has worked a funda- 
mental change in the concepts of space and 
time. With Newton they were indepen- 
dent, homogeneous, absolute, and infinite ; 
with Einstein they are but different aspects 
of the same continuum, space-time hetero- 
geneous, relative, and possibly finite. 

It is often objected that relativity, pur- 
ports to disprove the existence of the ether, 
and that without the ether phenomena such 
as the propagation of light are inconceivable. 
It is not certain that relativity does do this. 
What has been shown is that the ether 
cannot be made to provide a standard of 
rest, and that the idea of motion of the 
ether is self-contradictory. This may mean 
no more than that the ether is a reality to 
which the idea of motion cannot be applied. 
It may be helpful to remember that pre- 
cisely the same criticism was directed 
against Newton by the Cartesians. Because 
he refused to be drawn into discussions as 



EINSTEIN'S THEORY OF RELATIVITY 31 

to the plenum and its vortices, he was made 
to appear to say that the forces of gravitation 
were transmitted through emptiness from 
one heavenly body to another. Now it is 
quite clear that Newton's space was more 
than mere nothingness, in that it acted as 
the medium for the transmission of gravita- 
tional influences. Yet Newton was right in 
regarding the nature of this space, except in 
so far as it was susceptible to physical 
measurement, as a problem for philosophy 
rather than for science. The present posi- 
tion of the problem of space- time and the 
ether is, I think, very similar. 

Time prevents us from referring to the 
practical achievements of the new theory 
or from exploring its possibilities in the 
regions in which the Newtonian mechanics 
have never yet shed light the regions of 
atomic and sub-atomic structure. The 
formal beauty of the theory can only be 
exhibited by means of mathematical analy- 
sis. 

I said at the beginning of the lecture that 



32 RELATIVITY FOR PHYSICS STUDENTS 

the record of the past would provide the 
key to the solution of much that is difficult 
and perplexing in our present knowledge, 
and I hope that, by attempting to put 
Einstein's work into its proper historical 
setting, I have perhaps made some aspects 
of the theory of relativity a little clearer. 
But I also suggested that the record of the 
past would point us on the way to further 
advance. The work of Newton was carried 
on by the great French school of the 
Revolution period. He laid down the prin- 
ciples, but it was Lagrange, Laplace, Pois- 
son, and others who reduced them to a form 
in which they could readily be applied to 
the solution of physical problems. Again 
Einstein has given us the principles, but it 
is not always easy to see how to apply 
them to all those problems of modern 
physics which are so urgently with us to-day. 
That is the task which now lies before 
mathematics. Einstein has given us the 
" Principia," but " La M6canique Analy- 
tique " has yet to be written. 



I 

THE ORIGINS OF THE THEORY 

THE young scientist can suffer from 
no greater fault than a misunder- 
standing of scientific genius. We are too 
apt to think that advance in scientific 
knowledge is reserved for those who, by 
reason of some special gift, are able for the 
first time in human history to see some- 
thing which others have been too blind to 
see. Now there is a measure of truth in 
this view, but it is important that we 
should see just what that measure is, more 
particularly when we come to the conclusion, 
as most of us must quite young in life, that 
we are very ordinary people with no very 
special gifts. When that moment comes, it 
will depend upon our conception of the 
way in which science advances whether we 
go forward and play our part, such as it 
may be, in the progress of knowledge, or 

3 33 



34 RELATIVITY FOR PHYSICS STUDENTS 

whether in despair we leave the matter to 
those more fortunate ones who have been 
predestinated for the work. 

The view of scientific advance which I 
wish to combat may perhaps be explained 
by an analogy. A region of country has 
been explored and mapped. The rivers 
and mountains, since they must have 
names, are called by the names of those 
who first discovered them. We have some 
knowledge of the geography of the land 
and perhaps we have reached the boundary. 
There we stand facing a mountain precipice, 
vainly seeking some way by which we may 
get a little further, and all the time hoping 
that some super-man may invent an aero- 
plane of thought which shall carry us over 
into the beyond. 

The analogy is incomplete, for it leaves 
out of account the most important thing of 
all the way in which what is now known 
was first discovered. The map may be 
sufficient for the engineer, but the discoverer 
must study much more than the map : he 



THE ORIGINS OF THE THEORY 85 

must be learned in the art and lore of 
exploration. ' Thus if we would become 
discoverers in physics, or even if we would 
in any real sense understand physics as it 
is to-day, the names of Galileo, Huygens, 
Newton, Lagrange, Fresnel, Stokes, Max- 
well must be much more to us than con- 
venient labels for certain laws and experi- 
ments. < 

For this reason we will approach our 
study of Relativity by showing how the 
problems which it attempts to answer have 
gradually arisen. There is another reason 
which prompts us to adopt this course. 
Relativity is commonly supposed to be a 
revolutionary theory. The theory has its 
roots in the very beginnings of modern 
science, but it is nevertheless a revolu- 
tionary theory in that its acceptance com- 
mits us to a radical reconstruction of our 
most fundamental physical concepts. In 
attempting this task of reconstruction it is 
essential that we should understand the 
reasons which led to the formation of the 



36 RELATIVITY FOR PHYSICS STUDENTS 

older concepts, in order that, if these are 
eventually discarded, we may ensure that 
nothing of value is lost. 

Perhaps the most important lesson of the 
history of science is the abiding value of 
the result of a physical experiment care- 
fully and accurately carried out under 
definite conditions. ^The chapter of scien- 
tific history with which we shall be most 
concerned is very largely the story of* an 
always-changing theory based upon a grow- 
ing body of unchanging experimental facts. 

The problem of relativity appears for the 
first time in modern science in the work 
of Newton. The achievements of Galileo, 
Kepler, and Huygens were gathered up into 
a comprehensive theory of motion. It is 
clear that before such a theory can be 
formulated we must have a definition of 
motion, so that an observer may at any 
rate decide whether a particular body is 
moving or not. Now it is a familiar fact 
that we can observe only the relative 
motions of bodies, and that we cannot 



THE ORIGINS OF THE THEORY 37 

observe how any particular body is moving 
without reference to other bodies. In order 
to meet this difficulty, Newton adopted the 
hypothesis of an absolute space. The ve- 
locity and acceleration of a body mean its 
velocity and acceleration relative to absolute 
space, and no plan has yet been revealed by 
which these can be measured. This logical 
defect lies at the root of Newton's theory, 
that the terms in which it is expressed 
cannot be defined in such a way that they 
are unambiguously susceptible to physical 
measurement. Newton himself saw this 
difficulty very clearly, and he certainly 
would not have passed it by if it did not 
seem to him that there was a solution. 
The " fixed " stars are outside the solar 
system, and apparently unaffected by its 
motions. They might be used to define 
absolute space. This was the solution 
which Newton adopted, although with truly 
prophetic foresight he admitted " it may 
be that there is no body really at rest to 
which the places and motions of othfers may 



38 RELATIVITY FOR PHYSICS STUDENTS 

be referred/' Subsequent discovery proved 
the wisdom of this reservation. The stars 
are not fixed even relatively one to another, 
but move with discordant and sometimes 
almost inconceivably great velocities. The 
logical difficulty returned with undiminished 
force, and it is clear that sooner or later it 
had to be faced and solved by some elabora- 
tion or reconstruction of the theory. How- 
ever, the problem was left unanswered until 
our own day, partly because it was hoped 
that a solution would come from other 
branches of physics, but perhaps mainly 
because the stars are at such great distances 
that the ambiguity in the " absolute space " 
which they specify seemed unlikely to 
produce any measurable error in the appli- 
cation of the Newtonian laws to motions 
within the solar system. Thus it came 
about that the theory of relativity in its 
first form did not grow out of mechanics, 
but from other branches of physical know- 
ledge optics, electricity, and magnetism. 
We will endeavour to show how the funda- 



THE ORIGINS OF THE THEORY 39 

mental ideas in these subjects gradually 
changed during the nineteenth century 
until they led to the formulation of the 
relativity theory. It will be convenient to 
begin by forming some idea of the state of 
knowledge in these branches of physics in 
the opening years of the nineteenth century. 
Thanks to the work of Newton and the way 
in which it had been pushed forward by 
the great French school of mathematicians, 
mechanics was very much what it is to-day, 
so that one may say roughly that the whole 
of the mechanics now required for our 
degree examinations was known. Lagrange 
and others had developed the Newtonian 
mechanics into a great and complete system 
which was thought to be capable of compre- 
hending the whole of physics. Given the 
position and motion of all the bodies of 
the universe at any one instant, a master 
mathematician could work out the complete 
history of things past and future. 

In optics the theory of what we now call 
geometrical optics was fairly well advanced, 



40 RELATIVITY FOR PHYSICS STUDENTS 

but practically nothing was known of 
physical optics. The laws of reflection 
and refraction and their application to the 
construction of lenses and telescopes ; the 
phenomena of dispersion, but not the 
Fraunhofer lines in the solar spectrum ; 
Newton's rings and the most elementary 
facts of polarisation, would have been a 
fairly exhaustive syllabus in optics in the 
year 1800. Newton's corpuscular theory of 
light still held the field. It is true that in 
one form or another a wave theory had 
often been proposed, notably by Newton's 
contemporary Huygens, but as men thought 
always of a longitudinal wave, the facts of 
polarisation were held to be an insuperable 
barrier to such a theory. Rather curiously, 
the idea of an ether was already familiar in 
optics. In order to account for Newton's 
rings, Newton invented a theory under 
which his corpuscles suffered from fits of 
easy reflection and easy refraction, which 
were transmitted to the corpuscle through 
an ether filling all space. This same ether 



THE ORIGINS OF THE THEORY 

transmitted instantaneously the forces of 
gravitation between bodies. We have 
omitted to mention one isolated effect 
which was known and destined to play an 
important part in the evolution of later 
optical theories. On the corpuscular theory 
it is clear that, if a fixed star is observed by 
a moving telescope which has a component 
velocity v perpendicular to the direction of 
the star, then the telescope must not be 
pointed directly at the star, but at a point 
whose angular distance from the star is v/c, 
where c is the velocity of light. In con- 
sequence of the motion of the earth in its 
orbit round the sun, the stars will accord- 
ingly appear to describe small ellipses about 
their mean positions. This effect of stellar 
aberration had been observed and explained 
by Bradley in 1728. 

When we turn to electricity and magnet- 
ism, we find that even less was known in 
the year 1800. The lodestone and per- 
manent magnets made by its aid were used 
in navigation. In the reign of Elizabeth, 



42 RELATIVITY FOR PHYSICS STUDENTS 

Gilbert of Colchester had studied magnets, 
and had also discovered a large number 
of substances which became electrified on 
rubbing. Ten years before (1790) Galvani 
had constructed the first galvanic cell, 
and Volta's ''pile" was the latest scientific 
novelty. Little was known of electrostatic 
induction or the properties of electric cur- 
rents and nothing of any connection between 
electricity and magnetism. Although b&th 
a one-fluid and a two-fluid theory of elec- 
tricity had been mooted, and Coulomb and 
Cavendish were laying the foundations for 
future advance by their quantitative inves- 
tigation of the law of attraction, we may 
say that electricity and magnetism consisted 
of a few isolated and mysterious " effects/' 

The early advance of mechanics had an 
important effect which we can trace through 
the greater part of the century. As the 
knowledge of other branches of physics 
increased men tended to explain the new 
knowledge in terms of mechanical theories. 
Thus it was natural that when optics 



THE ORIGINS 

demanded an eth< 
regarded as anoth< 
the laws of Newto 
The opening o 
marked a great 8' 
Young revived tl 
and in the follc 
it by the series 

which we now 



ether assumed 
was natural in 
thought of as ; 
same laws as, at 
only in degree i 
In order to 
aberration whi 
corpuscular th 
suppose that tl 
earth did not p 
earth. Thus th 
was originally 
in its nature th 
interstices of i 
through a grov* 



IVS1CS STUDENTS, 

>sed through the 

ained at rest. It 

mt a d&ficulty in 

theory, refraction 

f the velocity of 

substance to its 

-. Accordingly, 

the fixed ether 

pend upon the 

> tested this by 

n of light from 

find any such 

Ity to Fresnel. 

to Arago, met 

cation in his 

material body 

ain fraction of 

nely, i i//* 2 , 

lex. Since the 

y nearly unity, 

ir is very small, 

is very nearly 

that the ether 

other hand, in 



THE ORIG 

glass of refractive 
carried along wit 
the velocity of tt 
was directly verifi 
Fizeau in 1851. 
an experiment * 
by Boscovich rm 
fore. He obsei^ 
by means of a te 
found that it 
telescope was ft 
foretold by Fre^ 
had been expl 
ging coefficient 
results so well c 
coefficient beca 
had to be satis, 
In the meai 
had been made 
In 1819 Oerste< 
a magnet of a - 
rent. Ampere 
to the discover 
magnetic field c 



S STUDENTS 

^nlarged our know- 
<id discovered the 



upon electric 
^mathematical ex- 
oere and Faraday, 
ise on Electricity 
^ced an important 
ice. During the 
ry a very scanty 
, electricity had 
. comprehensive 
[the influence of 
]attached great 
the medium/' 
( vas still formed 
>ut of the two 
Become the more 
jC strains it pro- 
^orces, and by its 
.,ted light. And 
Aether of Fresnel, 
. * the motion of 
t ether satisfied 
o intal results. 



THE ORIGINS OF THE THEORY 47 

The rift in the lute appeared in 1881, 
when Michelson performed an experiment 
which was originally suggested by Maxwell. 
We shall have to examine this experiment 
in detail later, but for the present it is 
sufficient to note that it was an experiment 
designed to measure the relative velocity 
of the earth with respect to the surrounding 
ether. 

According to Fresnel, the ether inside the 
earth is dragged, but the ether immediately 
outside, e.g. in a laboratory, is at rest, at 
least to the approximation to which the 
refractive index of air is unity. The experi- 
ment was performed and the result was in 
direct contradiction to the predictions of 
Fresnel's theory. It appeared that there 
was no relative velocity as between the 
earth and the ether immediately outside the 
earth. It was clear that some modification 
of the theory was necessary and it may be 
well to recall the three experimental results 
which had to be satisfied by any proposed 
theory :-r- 



48 RELATIVITY FOR PHYSICS STUDENTS 

(1) Stellar aberration of an amount which 

is independent of the medium inside 
the telescope. 

(2) The increase of the velocity of light 

when it is propagated in the direction 
of motion of a moving medium 
(Fizeau's experiment). 

(3) The null result of Michelson's experi- 

ment. 

Some time earlier (1845) the mathe- 
matician vStokes had felt doubts as to 
Fresnel's ether on somewhat theoretical 
grounds. The dragging coefficient imposed 
a discontinuity in the motion of the ether 
at the surface of a moving body. In those 
days discontinuities were less in favour 
among physicists than they are now, and 
Stokes tried to remove the difficulty by 
supposing that the ether was a viscous fluid, 
so that the ether inside the earth is dragged 
along in accordance with Fresnel's coef- 
ficient, but that at the surface the velocity 
does not immediately fall to zero, but gradu- 
ally and continuously falls off just as in the 



THE ORIGINS OF THE THEORY . 4f) 

case of a sphere moving through a viscous 
fluid. The velocity of the ether immediately 
outside the earth is then approximately equal 
to the velocity of the earth and the result 
of Michelson's experiment, had it been 
known, would have suggested Stoke's theory. 
At first sight it would seem, however, that 
such an hypothesis would fail to account 
for aberration. Stokes showed that this 
was not so, that his theory would give the 
correct aberration so long as the motion of 
the ether was of the type which is called 
irrotational in hydrodynamics. 

This brought to light a new difficulty, for 
it appears that there is no possible irro- 
tational motion of a fluid surrounding a 
moving sphere such that the fluid is at rest 
at infinity and there is no slip of the fluid 
over the surface of the sphere. At least, 
such a motion is impossible if the fluid is 
incompressible. The analysis of the motion 
of a compressible viscous fluid is a very 
difficult problem of which very little is 
known even to-day. Thus one method of 



SO RELATIVITY FOR PHYSICS STUDENTS 

adjusting Fresnel's ether to the result of 
Michelson's experiment was disposed of in 
advance. This possibility has, however, 
been returned to in recent years, and Planck 
has shown that if the ether is compressible 
we can make the slip at the earth's surface 
as small as we please, provided that there is 
a condensation of ether round the earth. 
But in order to reduce the slip to i per cent., 
when it would be too small to be measured 
by Michelson's experiment, the density of 
ether immediately outside the earth is about 
80,000 times its density at a great distance. 
Yet this enormous change in the density of 
the ether produces no measurable difference 
in the properties of the ether or in the pro- 
pagation of light. There seems little hope 
of progress in this direction. 

The Michelson experiment was first ex- 
plained by an ad hoc hypothesis suggested 
independently by Fitzgerald and Lorentz 
that a material body moving with velocity 
v through the ether was contracted in the 
ratio i : ^(i v*jc*), where c is the velocity 



THE ORIGINS OF THE THEORY 

of light. This hypothesis explained the 
single result which it was designed to explain, 
but no independent evidence of the existence 
of the contraction could be obtained. In 
fact, there were certain difficulties about 
the conception of the contraction what, 
for example, happens when we rotate a 
wheel at high speed : is its circumference 
contracted without change in its diameter? 
However, the Fitzgerald-Lorentz hypothesis 
might have remained were it not for a bril- 
liant theoretical development by Lorentz 
himself. In order to understand this, we 
must return to the consideration of the 
development of the electromagnetic theory. 
Maxwell's theory was really the analytical 
expression of two physical laws : 

(1) The Law of Faraday. The integral of 
the electric force round any circuit is pro- 
portional to the rate of change of the flux 
of magnetic induction through that circuit. 

(2) The Law of Oersted as Amended by 
Maxwell. The line integral of the magnetic 
force round any circuit is proportional to 



RELATIVITY FOR PHYSICS STUDENTS 

the total flow of current through that circuit. 
The total current includes the displacement 
current which is the rate of change of the 
electric induction. 

Four quantities play an important part 
in these laws the electric force E, the elec- 
tric induction D, the magnetic force H, the 
magnetic induction B. Maxwell assumed 
that these are connected by the empirical 
relations 

B = M H, D -- *E 

where /*, are the magnetic permeability 
and the specific inductive capacity. 

In view of Fresnel's work it was natural 
that the question should arise as to how 
Maxwell's equations were to be applied to 
moving media. Strangely enough, Maxwell 
does not appear to have considered this 
problem in any detail. Hertz, however, took 
Maxwell's equations and assumed that when 
applied to a moving medium the circuits 
referred to above are to be interpreted as 
circuits fixed in the medium, while the 
relations B = /*H and D cE are main- 



THE ORIGINS OF THE THEORY 

tained. These assumptions lead to the 
conclusion that Fresnel's dragging coefficient 
would be unity, and thus Hertz's theory of 
moving media was in direct conflict with 
experiment. 

Lorentz attacked the problem from an 
entirely new point of view by examining the 
basis of the relations B = ^H and D = eE- 
He assumed that the differences between B 
and H and between D and E were given by 
D-E + P, B = H + M, where P is the 
electric polarisation and M the magnetisation 
of the medium. P and M he regarded as 
due to the influence of E and H upon the 
motion of the electrons contained in the 
atoms of the material. This development 
of the electron theory of matter proved most 
fruitful. A number of hitherto unexplained 
optical effects were accounted for and the 
way was prepared for a theory of moving 
media. The electric induction which plays 
the part of the electric force inside matter 
consists of two parts (i) E residing in the 
ether and unaffected bv the motion of the 



RELATIVITY FOR PHYSICS STUDENTS 

medium ; (2) P arising from the electrons 
of the medium and intimately bound up 
with the motion of the medium. 

There is no need for us now to follow all 
the intricacies of Lorentz's theory. It is 
sufficient to note that he developed a com- 
plete theory of moving media based upon 

(a) Maxwell's equations for free space ; 

(b) his own hypothesis as to the relation 
between E and D and between B and H ; 

(c) the Fitzgerald-Lorentz contraction. The 
equations which expressed this theory were 
naturally more complicated than Maxwell's 
equations, but Lorentz showed that, by 
introducing a new variable r in place of the 
time t, the equations for a moving medium 
took the same form as the equations for 
free space. Lorentz called r the " proper 
time/' but he regarded it as no more than a 
mathematical variable which facilitated the 
solution of the problem of moving media. 
Einstein carried the process further by a 
bold step. Since only relative motions can 
be observed how can we sav whether our 



THE ORIGINS OF THE THEORY 55 

medium is moving or not, and how can we 
distinguish between the fictitious mathe- 
matical " proper time " r and the absolute 
time t ? Einstein assumed that the proper 
time r was the time measured by physical 
observation, and that, therefore, the equa- 
tions for a " moving " medium were in 
relation to the time observed in that 
medium the same as if the medium were 
at vest. On this he based his principle of 
relativity that the laws of nature are such 
that no experiment can reveal an absolute 
velocity, or, what comes to the same thing, 
a velocity relative to the ether. 

We shall have to examine the work of 
Lorentz and Einstein in greater detail, but 
this brief sketch may serve to show how 
their work falls into place in a continuous 
attempt to build up a theory of the ether 
which shall conform to the results of 
physical experiment. 



II 

THE MICHELSON AND MORLEY EX- 
PERIMENT AND THE LORENTZ 
TRANSFORMATION 

IN its essence this experiment was a 
comparison of the velocity of propa- 
gation of light in two mutually perpen- 
dicular directions. A ray of light OA is 
incident at an angle of 45 on a half- 
silvered mirror, so that the reflected and 
transmitted rays are perpendicular. These 
travel along paths AB, AC respectively, 
which in the ideal case may be supposed 
to be exactly equal in length. They are 
incident normally upon plane mirrors at B 
and C, and are reflected back along their 
respective paths, so that both rays arrive 
again at A. The transmitted part of the 
ray originally reflected, and the reflected 

$6 



THE LOHENTZ TRANSFORMATION 57 

part of the ray originally transmitted, will 
then be superposed along AP in a direction 
which is perpendicular to the direction 
of the original ray OA. If the velocity of 
light is the same in the directions AB, AC, 

B 



O 



FIG. i. 



the two rays superposed along AP will be 
in phase ; but if there is a difference of 
velocity in the two directions there will be 
a consequent difference of phase between 
the two rays in AP and this will be made 
manifest by interference. This is, of course, 



58 RELATIVITY FOR PHYSICS STUDENTS 

a very much idealized account of a highly 
technical experiment, but it contains the 
essential principles. 

In order to keep our ideas as definite as 
possible, we will interpret this experiment 
on the basis of Fresnel's fixed ether hy- 
pothesis. If the whole apparatus is at rest 
in the ether, we should expect the velocities 
along AC and AB to be the same. If, 
however, the whole apparatus is moving 
through the ether, say with velocity v in 
the direction AC, then it will appear that 
the time of passage along ACA is greater 
than along ABA by an amount which will 
depend upon v, and which may be measured 
by the interference of the two rays super- 
posed along AP. Thus, on the assumption 
that light is propagated in the ether with 
the same velocity in all directions, the ex- 
periment provides a means of measuring 
the velocity of the apparatus through the 
ether. 

Assuming that the sun is at rest in the 
ether, the earth, owing to its annual motion 



THE LOHENTZ TRANSFORMA 

round the sun, has a velocity of abou 
1 8 miles per second, while owing to the 
diurnal rotation a point on the earth's 
equator has a velocity of about one-third 
of a mile a second. If the experiment is 
performed in a laboratory the apparatus 
is, according to Fresnel's theory, moving 
through the ether with a speed of 18 miles 
per second, and the delicacy of Michelson's 
experiment was such that a velocity of this 
order could be detected. The experiment 
failed to produce any evidence of this or any 
other velocit}^ through the ether. The pro- 
blem is not materially changed if we admit 
the possibility of a motion of the sun 
through the ether. By applying a process 
of averaging to the observed motions of the 
" fixed " stars, astronomers have arrived at 
the conclusion that the whole solar system is 
moving through space with a velocity of 
about 10 miles per second. It is true that 
this might at a particular time reduce the 
velocity of the earth through the ether to 
8 miles per second, but, on the other hand, 



.*fIVITY FOB PHYSICS STUDENTS 

xX months later it would increase it to 
28 miles per second. The only way in 
which we can suppose that our laboratory 
is permanently at rest in a fixed ether is to 
undo the work of Copernicus and Kepler, 
and to return to a Ptolemaic theory of the 
universe, if, on the other hand, we admit 
a motion of the earth through the ether, 
we must suppose that we have left out of 
account some compensating influence which 
prevents Michelson's experiment from de- 
tecting that motion. Such a compensating 
influence was proposed by Fitzgerald and 
Lorentz in their famous contraction hypo- 
thesis : a body moving through the ether 
undergoes a contraction of length in the 
direction of its motion. Thus in Michel- 
son's experiment the path ACA, which, 
owing to the motion through the ether, 
would correspond to the longer time of 
passage, is contracted in length by just such 
an amount that the time of passage is the 
same for the two paths. Such a contraction 
would not be revealed by our ordinary 



THE LORENTZ TRANSFORMATION 



measurements, since presumably our measur- 
ing scales are also contracted in the same 
ratio. The Fitzgerald-Lorentz hypothesis 
introduced a distinction between a measured 
length and a real length. 

We will follow out the implications of the 
result of Michelson's experiment, and it will 








X 



o 

FIG. 2. 



help to keep our ideas clear if we adhere to 
Fresnel's hypothesis of a fixed ether, while 
admitting the distinction between real and 
measured quantities. We will return later to 
a discussion of the meaning of this distinction. 
Suppose that a set of axes o(x,y f z) are 
drawn fixed in our laboratory. The co- 
ordinates % t y, z are measured lengths, and 



<& RELATIVITY FOR PHYSICS STUDENTS 

we also have a means of determining a 
measured time t. We admit that we are 
moving through the ether with an unknown 
velocity, and suppose that this is constant 
and equal to v in the direction ox. We 
take a set of axes O(XYZ) fixed in the 
ether. Since the difference between real and 
measured lengths is due to motion through 
the ether, we may suppose that X, Y, Z are 
real lengths, and further that, corresponding 
to the measured time t, there is a real time T. 
The two sets of axes may be taken to coin- 
cide at time t = o. By Michelson's experi- 
ment, we find that the measured velocity of 
light relatively to our apparatus is the same 
in all directions, and our units may be 
adjusted so that this measured velocity is 
the same as the real velocity c. Let mirrors 
M!, M 2 , M 3 be placed on the axes at equal 
measured distances I l9 1 2 , / 3 ( = /) from o, 
and at time t = o let a pulse of light be 
emitted from o and return to that point 
after reflection at the three mirrors. 
Consider first the ray which passes alon 



THE LOBENTZ TRANSFORMS 

ox ; let it reach M x at measured time t l 
return to o at time tf/'. Then 



If the corresponding real quantities are 
denoted by capital letters, and we note that 
the velocity with which the light approaches 
MX is c v, while that with which it ap- 
proaches o on its return journey is c + v, 
we have 

c v 

T " ^i i k* = _ 2c ^i 

1 c v c + v c 2 v ** 

Next consider the ray which passes along 
oy. With a similar notation for measured 
time, we have 

/ / v ,rt 2/ 

/ 2 __, t, - c . 

The real path of this ray is the hypot- 
enuse of a right-angled triangle of sides 
L 2 and vT t '. Hence c a T 2 /2 - L 2 2 + vT/ 2 , 
or 

LoT 
2 nr " ^-L'2 

i 2 = . 1 2 =- 



-ATIVITY FOR PHYSICS STUDENTS 

ae experimental result is that the two 
ays arrive back at o at the same time. 
Hence T 2 " - T/', or 

/JLi -= L 2 , 

where j9 = 1/^(1 t> 2 /c 2 ), and is therefore a 
fraction greater than unity. 

The original assumption of the Fitzgerald- 
Lorentz hypothesis was that the dimensions 
of a body in a direction perpendicular to the 
direction of motion are unchanged, or, in 
other words, measured and reai_ lengths .are 
the samejn_aiiy_ direction perpendicular. Jto 
that_ of the motion through the ether. It 
has been shown that no material increase of 
generality is obtained by abandoning this 
assumption. Hence we may take L 2 = 
then 



L t - klft - . . . (i) 

Further, the real time T x of the time of 
the double passage along either path is re- 
lated to the corresponding measured time by 

T/'^fo". . . (2) 



THE LORENTZ TRANSFORMATION 65 

These results may be expressed as re- 
lations between the measured co-ordinates 
x> y> z with respect to o and the real co-ordi- 
nates X, Y, Z with respect to O. For the 
real co-ordinates of x, y, z with respect to o 
are #/j8, y, z, which are respectively equal to 
X - vT, Y, Z. Hence 

x - (X - z/T), y = Y, z - Z. 

The relation tfetween the measured time 
t and the real time T may be obtained in a 
similar way, but the argument is clearer if 
we note that the wave surface of a pulse of 
light emitted from the origin at time t = o 
is a sphere with centre o in measured lengths 
and times, whereas it is a sphere with 
centre O in real lengths and times. That 
is to say, that the following two equations 
are equivalent : 

* 2 + y* + z * ^ c * t * 9 x 2 + Y 2 + Z 2 = c 2 T ? 
The first gives 

j3 2 (X - vT) 2 + Y 2 + Z 2 = c 2 * 2 
and using the second we may solv 



(ifi RELATIVITY FOR PHYSICS STUDENTS 

terms of T and X. We thus obtain the 
famous Lorentz transformation 

vT), j>= Y, z = Z t 



From our present point of view these re- 
present the relations between our measured 
lengths and times and the corresponding 
real lengths and times measured with re- 
spect to the fixed ether. 

Perhaps the greatest difficulty which has 

been felt by many in approaching the new 

theory is that %, t each depends upon both 

X and T, so that space and time appear to 

be " mixed up/' This difficulty will dis- 

appear if we are careful to see exactly what 

is implied by these relations. If two events 

take place at the same point in the ether 

they have the same X. The first of the 

orentz relations then asserts that they will 

occur at the same place in our laboratory 

they occur at the same time T. This 

^usly true if in fact we are moving 



THE LORENTZ TRANSFORMATION 67 

through the ether. The fourth of the 
Lorentz relations is not quite so easy to 
dispose of. It asserts that if two events 
occur at the same real time T, i.e. if they 
are really simultaneous, they will not be 
simultaneous in our measured time unless 
they occur at the same place in the ether 
(or at least have the same X). This con- 
tradicts our usual assumption that we can 
determine the simultaneity of events with 
certainty ; that, for example, we can syn- 
chronize two distant clocks. A little reflec- 
tion, however, will show the great difficulty 
of suggesting any means by which this may 
be done without knowing our velocity 
through the ether. The synchronization of 
clocks is a practical problem, and two 
methods have been largely used by astrono- 
mers. Portable clocks are compared in 
turn with the two clocks to be synchronized, 
but in order to do this the portable clocks 
must move through the ether. Their parts 
will be subject to the Fitzgerald-Lorentz 



68 RELATIVITY FOR PHYSICS STUDENTS 

contraction and to the order of accuracy 
with which we are now dealing it would be 
bold to predict what would happen during 
the course of their journey. The more 
modern method is by means of wireless 
signals, and to be exact we must correct for 
the time taken to propagate the signals. 
These, like light waves, are propagated with 
constant velocity through the ether. If 
both our clocks are moving through -the 
ether the correction will depend upon their 
common velocity. For example, if our 
clocks are at o and M t in Fig. 2, and the 
signal is sent from the first to the second, 
the correction for the time of propagation 
would be 

.LL. =..A_ 

c - v ~P(c v) 

and this correction cannot be made unless 
v is known. We can, of course, make the 
clocks synchronous in measured time by 
using the experimental result that the 
velocity of light in measured lengths and 



THE LORENTZ TRANSFORMATION 69 

times is the same in all directions, but 
clocks so synchronized will not be syn- 
chronous in the real time T. 

Corresponding to measured lengths and 
distances there will be measured velocities 
which will in general.be different from the 
true velocities. From equations (3) we 
have 

8* - )8(SX - i;8T), fy = SY, 8* = SZ, 



If the measured velocities of a point are 
given by u x = 8xfit . . . and the corre- 
sponding true velocities by U* = SX/ST 
we have 



70 RELATIVITY FOR PHYSICS STUDENTS 

In a similar way we can obtain the 
relations between the measured and true 
accelerations of a moving point. If 

/, = 8,/ . . . , and F, r = SU*/ST . . . , 



f - 
" 



, 
Jy ' 



F,, 
" 



. (5) 



The formulae (4) and (5) may be used as 
the basis of a complete theory of the kine- 
matics of measured motion, but we will 
note only some of the simpler consequences 
of (4). If the true velocity of a point is 
equal to the velocity of light, say U* = c, 
UK = U* = o, we have u x = c, u y = u z = o. 
More generally, if the true velocity of a 

Mnt is in any direction, but is less than 

r elocity of light, then the measured 

also less than that of light. 



THE LORENTZ TRANSFORMATION 71 

Again, if u x = u y = u z = o, we have U* = v, 
Uy = U* = o, or v is the true velocity 
through the ether of any point fixed with 
respect to the axes o(x,y,z). From this 
point of view it might more properly have 
been denoted by V. The inconsistency is, 
however, removed if we note that when 

U X --= Uy = U s = O, U X = V, lly = ^ = O, 

so that v is also numerically equal to the 
measured velocity of the ether with respect 
to the axes' o(x,y,z), assuming that any 
means could be found by which it could 
be measured. 

Finally, we note a very important pro- 
perty of the Lorentz transformation. Equa- 
tions (3) express the measured co-ordinates 
in terms of the real co-ordinates. If they 
are solved for the latter, we obtain 



T . ( 



/ I *"* \ i \%J / 



Allowing for the fact established above, 
that the measured velocity of the ether 



72 RELATIVITY FOR PHYSICS STUDENTS 

with respect to o is equal and opposite to 
the real velocity of o with respect to the 
ether, we see that the relations between the 
real co-ordinates and the measured co- 
ordinates are completely reciprocal. This 
is the point at which we begin to suspect 
the reality of the real co-ordinates. 



Ill 

THE LAWS OF MOTION AND 
ELECTROMAGNETISM 

WE will retain our distinction be- 
tween the real lengths and times 
measured with respect to the fixed 
ether and the lengths and times actually 
measured in a laboratory moving through 
the ether. We have ascertained the rela- 
tions between the measured lengths and 
times, and the corresponding real lengths 
and times which are dictated by the result 
of the Michelson-Morley experiment, and we 
will now proceed to examine the relations 
between the real and measured values 
of other fundamental physical quantities. 
Prominent among these are mass and force 
in terms of which Newton's laws are ex- 
pressed. We assume that Newton's laws 

73 



74 RELATIVITY FOR PHYSICS STUDENTS 

are true with reference to a set of axes fixed 
in the ether, i.e. our axes of X, Y, Z, making, 
however, this extension, prompted by the 
results j>Jf experiment upon bodies moving 
with high speeds that the mass is not a 
constant but is a function of the speed. 
This at once leads to a difficulty, for we are 
accustomed to express Newton's laws in 
two forms, which are equivalent only so 
long as the mass is constant the law- of 
mass acceleration, 

P - MF, 

and the law of momentum, 



If M is a function of U, and therefore of T, 
the two forms are no longer equivalent. 
We adopt the second. This may^be^ ex- 
pressed as a law of mass acceleration, but if 
this is done the mass of a particle is different 
for forces in the direction of motion from 
what it is for forces at right angles to that 
direction. If the particle is moving in the 



THE LAWS OF MOTION 75 

direction of X with velocity v and accelera- 
tions F A , F y , Fz, we have 



while 

P, = (MV) --= MF>. 



M is spoken of as the " transverse " mass, 
while the " longitudinal " mass is given by 

ii/r TI/T i dM 

Mi == M + v -= . 
a?; 

It should be noted that if we use the 
momentum form for the laws of motion, 
the mass is the same for all directions of 
applied force, and is equal to the transverse 
mass. 

Suppose that a particle of mass m is 
instantaneously at rest in the measured 
co-ordinates, but has accelerations f x> / v , /*. 
The measured forces will be given by 

P* = mf x , Py mfy. 
By formulae (5) of the last lecture, when 

U* = V, Vy = O, lift O, 

/r ~fi*?*Jy - f F y . 



16 RELATIVITY Foil PHYSICS STUDENTS 

Assume that the measured units of force 
and mass are so chosen that the measured 
force is equal to the true force in the direc- 
tion of motion. Then 

P,:= M;F, = */, = M ' A 

Hence 

M/ = mp*, 

or 



+ A 
+ A ' 



This gives 

mv 



- m f 

W J (i - 



Since M = m when v = o, we have for 
the transverse mass 

M = j8w. 
The longitudinal mass is then given by 

M/ - 



av' 

This result has an important consequence. 
The rate at which work is done by the forces 

is 

^ , dv 



THE LAWS OF MOTION 77 

The quantity within brackets is such that 
its rate of increase is equal to the work done 
by the forces. We may clearly add any 
constant without affecting this result, and 
choosing this constant so that the expression 
vanishes with v we have 



T 

T 



I C* al 

= m\ -_ - c z c. 
Vi-^c 2 f 



This is the function which plays the part 
of the kinetic energy when the mass depends 
upon the velocity. Note that if we neglect 
the fourth and higher powers of v/c, T = \mv*. 

Returning to the relations between the 
measured and true forces, we have for a 
particle at rest in the measured system, 



Hence 

p x - P,, p y - j3P^, p, - 

Following the same line of argument, we 
find for a particle at rest in the true co-ordi- 
nates 



78 RELATIVITY FOR PHYSICS STUDENTS 

The equations of the electromagnetic field 
as adopted by Lorentz are 



c aT aY <>Z 

I /^l^r , pjj \ dH^ t)H r 

c VTT */ w ~ az 



aZ 

i 



TT-V 

^4- PTJ ^\ 

IT + v ~ 



c V 3T y J *Z aX 



c aT aX aY 

(S I+PU 0- 

aX" + a V + Tzf = 

^H... ^JcL-i/ ^ii^ n 

<~ -T "iTT " " -v r~w > 



while the force acting on a moving charge 
e is P*, Py, P*, where 

P, = * ( E, + ^ H, ~ "^ H,) 

\ c c * 1 



THE LAWS OF MOTION 79 

In these equations E^, Ey, E* and H*, 
H v , Hz are respectively the components of 
electric and magnetic force ; P is the den- 
sity of charge, and U*, Uy, U A are the com- 
ponents of the velocity *of the charge. 
There are two things to notice about these 
equations. Firstly, the units employed are 
not those belonging to either of the sys- 
tems commonly in use. A unit charge is 
defined to be such that two units at a dis- 
tance apart of I cm. repel each other with 
a force of 1/477 dynes ; it is thus smaller 
than the electrostatic unit in the ratio 
i : v / (4 7r )- The second set of three equa- 
tions expresses the law that the line integral 

of H round any circuit is equal to x total 

current through the circuit. Allowing for 
the change already made in the unit of 
charge, it follows that H is measured in a 
unit which is greater than the electromag- 
netic unit in the ratio ^(4^) : i. These units 
are particularly convenient in theoretical 
work, as they make the equations sym- 



80 RELATIVITY FOR PHYSICS STUDENTS 

metrical and avoid the frequent occurrence 
of the factor 4?r. Secondly, it will be noted 
that the equations ignore the distinction 
between the electric force and induction 
and between the magnetic force and induc- 
tion. On Maxwell's view the equations 
would therefore be those applicable to free 
space unoccupied by matter. Lorentz, how- 
ever, assumed that the above equations are 
strictly true everywhere, even in the interior 
of an electron, and he and Minkowski 
showed that the difference between the 
force and induction in each case could be 
explained as due to interatomic electronic 
motion. It can be shown that the above 
microscopic equations, when averaged over 
a volume sufficiently large to contain many 
electrons, lead to a set of macroscopic equa- 
tions involving the electric and magnetic 
inductions, which differ from the corre- 
sponding forces by terms depending upon 
the polarization of the medium, i.e. the dis- 
tribution and motion of the concealed 
electronic charges. Finally, these equations 



THE LAWS OF MOTION 81 

are supposed to hold for a set of axes fixed 
in the ether, and we have accordingly 
written them in terms of X, Y, Z, T. 

We will find the equations between the 
corresponding measured quantities on the 
assumption that the measured value of an 
electric charge at rest in the measured 
system is equal to its true value. Defining 
the measured electric force as the force on 
a uftit charge at rest in the measured co- 
ordinates, we have 



Similarly, 
ej- 
Thus 



From (3) of the last lecture 
* * V * 



H2 RELATIVITY FOR PHYSICS STUDENTS 

Substituting in Maxwell's equations, as set 
out above, we find that they remain com- 
pletely unchanged save that the true quan- 
tities X, T, E, H, U . . . , are replaced by 
the corresponding measured quantities, pro- 
vided that 



h x = H,, h y = /JH V 4 



The last relation is consistent with the 
assumption that the measured value of a 
charge at rest in the measured co-ordinates 
shall be the same as its true value, for in 
this case U* = v and /> = P/j8, as it should 
be if the measured volume of the element is 
greater than its true volume in the ratio 

The method by which these results are 
established will be sufficiently illustrated 
if we consider the case of the fourth and 



THE LAWS OF MOTION 83 

last of the electromagnetic equations. The 
fourth gives 



The last gives 

B aEv - ^ 
t>x c* 

Hence 



or 

___ / _i_ 

since 



The other equations transform in a similar 
way so that the measured quantities obey 
equations in the measured co-ordinates, 
which are of precisely the same form as the 



84 RELATIVITY FOR PHYSICS STUDENTS 

equations obeyed by the real quantities in 
the real co-ordinates, provided that 

e x = E,, fy - j8(Ev - v c H,), 



A, = H,, A,. - j3(H v + ? E,), 



These relations may be solved so as to 
express the true quantities in terms of the 
measured quantities. Thus 



E z = 



IV 

THE RESTRICTED PRINCIPLE OF 
RELATIVITY AND SOME CONSE- 
^ QUENCES. 

UP to this point we have adhered to 
the hypothesis of a fixed ether 
through which our laboratory is supposed 
to move with a velocity which is definite, 
although so far no way has been discovered 
by which it can be measured. In order to 
account for the result of Michelson's experi- 
ment, we have been led to admit a distinc- 
tion between the actually measured values 
of physical quantities and their true values 
as measured with respect to the fixed ether. 
Assuming that the fundamental laws of 
motion and electromagnetism are true with 

respect to axes fixed in the ether, we have 

85 



86 RELATIVITY FOR PHYSICS STUDENTS 

found the laws governing the corresponding 
quantities as measured with respect to axes 
moving through the ether. On certain 
assumptions, some of which may be avoided 
by a more exhaustive analysis, we have 
found that the physical laws for measured 
quantities are of precisely the same form 
as the laws for the corresponding true quan- 
tities. In other words, the velocity of 
motion through the etlier does not appear 
in the equations for the. measured quantities. 
This result corresponds to the negative fact 
that so far no physical measurement has 
been found which can determine the velocity 
of motion with respect to the fixed ether. 

Further, we have found a complete 
reciprocity in the relations between true 
and measured quantities, so that an equa- 
tion expressing a measured quantity in 
terms of the corresponding true quantity 
can be turned into one expressing the true 
in terms of the measured quantity merely 
by changing the sign of v. In order to see 
the significance of this reciprocity, let us 



RESTRICTED PRINCIPLE OF RELATIVITY 87 

regard our moving axes as fixed in the ether. 
Our measured quantities then become true 
quantities. The axes which were formerly 
regarded as being fixed in the ether are now 
moving with velocity v. If we now 
enquire what will be the values of the 
measured quantities for these axes, we shall 
obtain the values which we have hitherto 
regarded as the true values. The distinc- 
tion between the true value of a physical 
quantity and its measured value a dis- 
tinction which must seem unsatisfactory to 
the physicist now disappears. Both sets 
of quantities are measured, but measured 
with respect to different sets of axes moving 
with different velocities through the ether. 

We may take any set of axes moving with 
a uniform velocity through the ether, and 
regard these as the fixed axes of Newton 
and Maxwell. If we take a second set of 
axes moving uniformly with respect to the 
first, the physical quantities for the two 
sets of axes will be related by the laws given 
above, but the physical laws will be the 



88 RELATIVITY FOR PHYSICS STUDENTS 

same for both sets of axes. If at any time 
It is convenient, we may regard the second 
set of axes as fixed and the first as moving. 
This is the restricted principle of relativity. 
We will now examine some of the conse- 
quences of this point of view, in order to 
show that the principle accounts for the 
governing experimental results. From the 
line of development it is clear that it 
accounts for the nul result of the Michelspn 
experiment. 

STELLAR ABERRATION 

Suppose plane waves of light are received 
from a distant star, fixed with respect to 
the axes of X, Y, Z, in a direction making 
an angle 9 with the axis of X, and in the 
plane of XY. The light disturbance is of 
the form 

/(Xcosfl + Ysinfl + cT). 

With respect to axes %, y, z moving rela- 
tively to X . Y . Z with velocity v in the 
direction of x, 



RESTRICTED PRINCIPLE OF RELATIVITY 89 

X cos e + Y sin + cT 
= p(x + vt) cose + y sin 8 + pc(t + ~) 

= xi p(cos 9 + _ )j + y sin 9 

+ cttp(i 4- ^-cos 0) j 
= j8(i + u cos 0){* cos 0' + y sin 0' + c/} ( 

> C ' 

T. n/ ^ cos + 1> - n/ c sin 

where cos = ------- - - , smfl = ------------- -. 

C + V COS C + V COS 

To an observer moving and measuring 
with the axes x, y, z, the light is received at 
the angle 0' , where 



sin (0 -*') = - 

v J c + v cos e 9 

or, neglecting squares and higher powers of 
v}c, the star is apparently deflected through 
an angle 

v sin e. 
c 

It should be noted that the symbol v now 
has a precise physical significance, namely, 
the velocity of the observer relative to the 



90 RELATIVITY FOR PHYSICS STUDENTS 

star. Consistent with this velocity, we 
may suppose that either the star or the 
observer is at rest in the ether ; the result 
is the same in both cases. 

THE DOPPLER EFFECT 
As a particular case of the above result, 
suppose that the light disturbance is of the 
form 

cos ~ (X cos + Y sin 6 + cT), * 

A 

so that to an observer at rest relatively to 
the star the light is of frequency 



In the co-ordinates %, y, z the disturbance 
becomes 



cos ^ (# cos 6' + y sin 0' + ct) 

A 



where 



A v C 

Apart from terms of the second order, 
this shows that there is an increase in the 



RESTRICTED PRINCIPLE OF RELATIVITY 91 

frequency of the light from a star given by 

(v 9 v)/ v = relative velocity of recession 
o 

of the star in the line of sight. 

FRESNEI/S DRAGGING COEFFICIENT 
The complete investigation of this problem 
demands an examination of the electric 
polarization of the medium, but it is possible 
to get a certain amount of information as 
folfows. Consider a medium moving with 
a constant velocity v in the direction X 
relatively to an observer whose axes are 
X, Y, Z, and let x, y, z be axes fixed in the 
medium. Light is propagated through the 
medium with a velocity c/j* where p, is the 
refractive index. If the propagation is in 
the direction of motion, we have u x cfo 
and 

TJ Ux + v = ^ c "^ ]Lt7; _ 

* ~" I + VUxjC* /*(l + V/fiC)' 

Expanding in powers of vfc and neglecting 
squares and higher powers, this gives 



92 RELATIVITY FOR PHYSICS STUDENTS 
Thus the velocity of propagation is in- 
creased by the amount V(T 
required by Fresnel's formula. 



as s 



THE FIELD OF A MOVING ELECTRON 

Suppose that, as measured in a system 
of co-ordinates relatively to which it is at 
rest, the electron is built up of spherical 
layers of constant charge-density so that the 
electron and its field are symmetrical about 
its centre. Let the electron be at rest at 
the origin of the co-ordinates x, y, z, and 
examine the field as measured in the co- 
ordinates, X, Y, Z as defined above. In 
the latter co-ordinates the electron moves 
with uniform velocity v in the direction 
of X. 

Since the electron is symmetrical and at 
rest in the co-ordinates x, y, z, its field is 
given by 

OC V Z 

e. v = -0, e y = <+, **=-<, 



RESTRICTED PRINCIPLE OF RELATIVITY 95 

where < is the radial electric force and is a 
function of r only, where r a x 2 + y* + z 2 . 
Employing the formulae of transforma- 
tion given in the last lecture, we have for 
the field, as measured in the co-ordinates 
X, Y, Z, 



Or C f 



The electromagnetic energy and momen- 
tum are given by E and G respectively 
where 

E - I f JJ(E, + E/ + E, + H/ + 



G y = 



The integration is in each case through 



94 RELATIVITY FOR PHYSICS STUDENTS 

the whole XYZ space at a constant T. The 
integrands are more simply expressed in 
terms of x, y, z, and we accordingly trans- 
form the integrals so that they are taken 
through the whole x, y y z space. Since 
T = p(t + vxjc*}, the condition T = const. 
implies that t is not constant, so that the 
integrand in x, y, z must be taken for 
different values of t. In the particular case 
under consideration, however, the integrand 
is constant for all values of /, and no com- 
plication arises. Since x = j3(X + vT), 
y = Y, z = Z, we have for T = const., 



- j8 - 1 dxdydz. 
We have 



(/ r 

Transforming to polar co-ordinates defined 

by 

x = r cos0,y = rsinO cos <, 

= r sin sin <, 



RESTRICTED PRINCIPLE OF RELATIVITY 95 

we have 



t 

Wi + v -\ sin 2 0}r*<f>* sin 6dr 



2 

"a" 

It W is the electrostatic energy of the 
electron when at rest, this may be calculated 
in the ordinary way or it may be obtained 
by putting v = o in the above. Then 

W = 27T 

or 



We may note that, neglecting fourth and 
higher powers of vfc, this gives 

E-W + 5W,, 

If we identify the second term with the 
kinetic energy of the electron, and write it 



90 RELATIVITY FOR PHYSICS STUDENTS 



where m is the electromagnetic mass 
of the electron, we have 
W== 5W 

If we conceive the electron to be a sphere 
of radius a with a charge e spread uniformly 

over its surface, W ~ - e - and 

2 a 

m = >--. 

On the other hand, if the charge is spread 
uniformly through the volume of the elec- 
tron W == - e - and 

m = . 
ac 2 

Returning to the momentum, it is easily 
seen that G y = G* = o, while 



i /? r w 

.2.?fr s 
3Jo 



RESTRICTED PRINCIPLE OF RELATIVITY ^q 

Tf we make the improbable assumption 
that the field of an electron moving with 
variable velocity is at every instant the 
same as if the electron were moving with a 
constant velocity equal to its instantaneous 
velocity, then 

d tr \ d ir \ dv 

j, \{*x) = -y- (Cr*) j 

at dv ^ 'at 

so that the longitudinal mass is given by 



as is easily seen to be true without approxi- 
mation. The rest mass is obtained by 
dividing by 3 , and is accordingly 

4 W 

3 <?' 



V 
THE EQUIVALENCE HYPOTHESIS 

WE are now in a position to state the 
problem of relativity in its general 
form, and to indicate the kind of solution 
which Einstein has proposed. 

The laws of physics need for their ^mathe- 
matical statement a set^of axes in space, or 
a "frame of reference/' .H^we^ take, .two 
such 'frames of reference, one moving rela- 
tively to the other, we should expect that 
the corresponding physical laws would be 
different. The classical view was that we 
should obtain the physical laws in their 
simplest form by choosing for our frame of 
reference a set of axes " at rest/' The 
specification of this set of axes at rest has 
proved historically a matter of very great 

difficulty. All our physical observations 

98 



THE EQUIVALENCE HYPOTHESIS 99 

are of relative motions, and cannot of them- 
selves lead to the determination of absolute 
rest. One great historical effort to solve 
the dilemma was the development of the 
theory of the ether. Although not itself 
susceptible to physical observation, it might 
yet serve as a standard of absolute rest. 
Since the ether is the seat of all physical 
phenomena, the laws of physics might well 
asstime a peculiar simplicity when they are 
stated with reference to a set of axes at 
rest in the ether. The proposed solution 
was briefly this : the absolute frame of 
reference is that for which Newton's laws 
of motion and Maxwell's laws of electro- 
magnetism are accurately satisfied ; the 
motion of any other frame of reference will 
be revealed by complication of these laws. 
The proposed solution failed because the 
Michelson and other experiments compelled 
us to assume that, if we take a moving 
frame of reference, the very motion of the 
frame gives rise to certain compensations 
which prevent us from detecting the motion. 



TOO RELATIVITY FOR PHYSICS STUDENTS 

The laws of physics are not more complicated 
for a " moving " frame than they are for 
a " fixed " frame ; they are precisely the 
same. This failure drives us back to the 
original difficulty. We can find no meaning 
in physical experience for absolute motion, 
nor can we determine the frame of reference 
which is at rest. 

It may be that experiment may yet dis- 
cover some answer to the problem and sdme 
means of measuring our motion through 
space. At the same time, a great body of 
evidence suggests that it would be well to 
face the possibility of ultimate failure. 

This is the standpoint of the theory of 
relativity. It assumes that, of the infinity 
of possible frames of reference, each moves 
relatively to the others, but none is "at 
rest " in any absolute or unique sense. We 
may select any frame of reference, but we 
must recognize that it is only one of an 
infinite number of equally eligible frames. 
We may by experiment determine the laws 
of physics for our selected frame, but they 



THE EQUIVALENCE HYPOTHESIS 101 

will be relative to that frame and, if we 
choose a different frame, the corresponding 
laws will be different. But we think of 
physical phenomena as pursuing their course 
independently of our measurement or de- 
scription, and if this be so, there ought to 
be certain physical laws which are indepen- 
dent of the particular frame of reference 
which we may happen to have chosen. We 
thiis arrive at the great problem of relativity : 
is it possible to express the laws of physics 
in a form which is independent of our choice 
of a frame of reference ? Such laws of 
physics, if they exist, may well be called 
the absolute laws of physics. 

The restricted theory of relativity has 
supplied a partial solution to the problem. 
If we confine ourselves to frames of reference 
wElch are moving relatively one to another 
wSfi TorisIaFf Velocity in a straight line, 
we have seen, for example, that the equa- 
tions of the electromagnetic field have 
precisely the same form for all such frames 
of reference. Thus in this restricted sense 



102 RELATIVITY FOR PHYSIC'S STUDENTS 

Maxwell's equations express absolute laws 
of physics. This is clearly only a partial 
solution, which falls short of the full re- 
quirements of relativity. The restriction to 
frames of reference moving relatively one 
to another with uniform velocity was felt 
to be arbitrary, and many attempts were 
made to remove it. This w r as accomplished 
in a very complete manner by Einstein in 
his general theory of relativity. He showed 
that by taking gravitation into account the 
laws of physics may be expressed in the 
same form for all frames of reference. 

In order to see how this was possible, we 
will examine briefly some of the outstand- 
ing features of gravitation. Newton inter- 
preted gravitation as arising from the 
mutual attraction of bodies. Between any 
two bodies there is a force which is propor- 
tional to the product of their masses and 
inversely proportional to the square of their 
distance apart. Thus, according to Newton, 
gravitation was a~ mutual action between 
the attracting body and the body attracted. 



THE EQUIVALENCE HYPOTHESIS 108 

Against this view we may note that the 
force is observable only through the accelera- 
tion which it produces in the attracted body, 
and this, being equal to the force divided 
by the mass, is independent of the body 
attracted. Just as Maxwell transferred the 
emphasis from attracting charges to the 
electromagnetic field, so Einstein directed 
attention to the gravitational field itself 
rather than to attracting bodies. This 
change of view-point brings to light a fun- 
damental simplicity of gravitational fields 
which was somewhat obscured by the New- 
tonian presentation. A gravitational( field 

^ ,^"W""~ *" * " - "" ^> "" " " 

impresses upon a body placed in it an accel- 
eration which is quite independent of the 
bodyjtself . Thus the uniform^gravitational 
field which we experience in a limited region 
at the earth's surface means that any body 
free to move in it has a doyraward jaccelera- 
tion~ of approximately 32 ft. /sec. 2 . The 
gravitational field of the sun means^that a 
planet atji given distance from the sun has 
an acceleration which depends on the sun 



104 RELATIVITY FOR PHYSICS STUDENTS 

and not jon the planet. It was in this 
description of gravitation in terms of ac- 
celerations that Einstein found the way to 
the extension of relativity. 

In the first place, it suggests a means by 
which all the appearance of a gravitational 
field may be produced artificially. Suppose 
that there is no gravitation but that an 
observer works in a room which is moving 
" upwards " with an acceleration g. All 
his observations inside the room will lead 
him to the conclusion that there is a gravita- 
tion al field of the type familiar to us at the 
earth's surface. A body left free to move 
will in reality remain at rest or in uniform 
motion in a straight line. Suppose it is 
at rest. It will appear to the observer 
to fall downwards with an acceleration g 
which is the same for all bodies. Ij_hej3rp- 
jects a particle, L it wilLappear to .describe a 
garabola^ A pendulum would execute oscil- 
lations in conformity with the usual formula. 
In short, by every test that the observer 
could make, he is at rest in a uniform 



TIN; EQUIVALENCE HYPOTHESIS 105 
ield. The classical view draws 



a sharp distinction between an " artificial " 
gravitational field of this kind and a " true " 
gravitational field. Relativity denies the 
distinction because it cannot be tested by 
physical experiment. It denies that the 
observer and his room are moving in any 
absolute sense, but suggests rather that the 
observed facts may be interpreted in, among 
others, two ways (i) the room is at rest 
and is occupied by a uniform gravitational 
field of intensity g ; (2) there is no gravita- 
tional field, but the room is moving with an 
acceleration g. This liberty of interpreta- 
tion is the essence of Einstein's " equivalence 
hypothesis/' It does not imply that gravi- 
tation is merely an appearance arising from 
acceleration of our frame of reference, 
neither does it imply that for any given 
problem the two interpretations are equally 
simple or convenient. It merely insists 
that the two interpretations are equally 
true to the observable facts. It points the 
way to a complete solution of the problem 



106 RELATIVITY FOR PHYSICS STUDENTS 

of the choice of a frame of reference. We 
may select any frame of reference and ob- 
serve the gravitational field ; we may with 
equal justification select any other frame of 
reference, but the gravitational field will 
then, in general, be different. All frames of 
reference are equally valid ; the only dis- 
tinction between them is that for one the 
gravitational field may be simpler than for 
another. Thus, by bringing gravitationnnto 
account, Einstem was able to extend the 
theory of relativity to systems in relative 
acceleration. 

It should be pointed out that the problem 
is not always so simple as in the case of the 
uniform gravitational field considered above. 
The gravitational field of the earth as a 
whole cannot be interpreted on the assump- 
tion that the radius of the earth is increasing 
at an accelerated rate of 32 ft. /sec. 2 . This 
merely means that there are some gravita- 
tional fields such that, of the infinity of 
possible frames of reference, there is no one 
for which the gravitation is everywhere nil. 



THE EQUIVALENCE HYPOTHESIS 107 

We will return later to some discussion of 
the methods used in such cases. 

The inclusion of gravitation in the new 
theory was the source of some of its greatest 
achievements, for gravitation had so far 
occupied a very isolated position in the 
scheme of physics. There appeared to be 
no interconnection between it and other 
physical phenomena. We will conclude 
this* lecture by two examples of the way in 
which the liberty of interpretation permitted 
by the equivalence hypothesis enables us 
to establish such interconnections. The 
methods which we shall use are open to 
criticism at several points, but they serve 
to bring out the true nature of the equiva- 
lence hypothesis, and the results may be 
verified by more rigorous analysis. 

Our first problem will be the effect of a 
gravitational field on the path of a ray of 
light. Suppose there are two sets of parallel 
axes Oxyz and OVyY, and that O has an 
acceleration g relative to O' in the positive 
direction of z, and that, if O' is regarded as 



108 RELATIVITY FOR PHYSICS STUDENTS 

being at rest, there is no gravitation (see 
Fig. 3). Light is emitted from O and is 
received at a point P, on the axis of x, by a 
telescope inclined at an angle a to the axis 
of x. We will interpret this result firstly 
from the point of view of the " fixed " 



FIG. 3. 

frame of reference O'x'y'z. There will then 
be no question of gravitation, but there will 
be an aberration effect due to the relative 
motion as between O at the instant of 
emission and P at the instant of reception. 
If c is the velocity of light and OP = /, 
this relative velocity is gl/c. The small 



THE EQUIVALENCE HYPOTHESIS 109 

angle of aberration will be this divided by 
the velocity of light, i.e., 

a = glfc*. 

We will now interpret the same pheno- 
menon from the point of view of the frame 
of reference Oxyz. There will now be a 
field of gravitation of intensity g downwards. 
For simplicity, we will adopt a corpuscular 
theory of light, and admit the possibility 
that the corpuscles have weight, so that they 
have a downward acceleration G. Follow- 
ing the ordinary theory of projectiles, a 
corpuscle, projected from O and arriving 
at P with an angle of descent a, has its 
^-component of velocity reversed in time Ifc 
approximately. Hence IG/c = 20 sin a, or 
approximately 

a = G//2C 2 . 

Comparing the two interpretations we see 
that G = 2g. That is to say, the light 
corpuscles are subject to the influence of 
gravitation, and experience an acceleration, 



110 RELATIVITY FOR PHYSICS STUDENTS 

which is twice that experienced by a 
material particle. 

A material comet, which at a great dis- 
tance from the sun is moving with a high 
velocity c along a line whose perpendicular 
distance from the sun is p, is deflected 
by the sun's attraction through an angle 
2yM//? 2 , where M is the mass of the sun and 
y is the constant of gravitation. Accord- 
ingly, if we are justified in applying to" the 
sun's gravitational field the result which 
we have proved above for a uniform gravi- 
tational field, a light corpuscle coming from 
a distant star will be deflected through an 
angle 

4yM 
pc* ' 

If the ray of light just grazes the limb of 
the sun, so that p is the sun's radius, this 
formula gives an angle of 1*73", which 
agrees with the results obtained by obser- 
vation. 

Our second problem will be the effect of 
a gravitational field on the observed fre- 



THE EQUIVALENCE HYPOTHESIS 111 

quency of the radiation emitted by a 
vibrating atom. In Fig. 4 let Qxyz and 
Q'x'y'z be two sets of parallel axes, and let 
O have an acceleration g relative to O' in 
the direction of z. Let two precisely similar 
atoms, Si and S 2 , be fixed relatively to O, 




FIG. 4. 

and situated at the points (o, o, o) and 
(o, o, z) respectively. The frequencies of 
their emitted radiation are observed at O'. 
For simplicity we shall assume that the 
relative velocity of O and O' vanishes at 
the instant at which the figure is drawn. 
Firstly, we take the point of view that O' 



RELATIVITY FOR PHYSICS STUDENTS 

is at rest, and that there is no gravitational 
field. Although the atoms are at rest rela- 
tively to O' when the radiation is received, 
they had a relative velocity when the 
radiation was emitted, and this will give 
rise to a Doppler effect. If v is the fre- 
quency of the emitted radiation, i>,', r 2 ' the 
respective observed frequencies at O', and 
if O'O = z , the velocity of S 2 at the moment 
of emission was (z + z^g/c towards O'' and 
hence 

v ' = - c - 

v * ~ c~ (z'+~z }g/c 

Since *>/ is obtained from this by setting 

z = o, we have 

* 

*V = f_nM/?_- 

Vi - (Z + ^ )g/^' 

We will now calculate the same ratio 
from the point of view that O is at rest in a 
uniform gravitational field of intensity g in 
the negative direction of z. We will leave 
open the question as to whether the fre- 
quencies of the atoms are affected by the 



THE EQUIVALENCE HYPOTHESIS 1W 

gravitational field and suppose that these 
are respectively ^ and v 2 . As O' is at rest 
at the instant of reception, we have 



Comparing the values of the ratio cal- 
culated from the two points of view, we see 
that v l 4= ^2, and, in fact, 



_-. 
vj c- (z + z )g/c 

If < is the gravitational potential with 
the arbitrary constant adjusted so that 
< vanishes at O', we have <f> t = z g, 
<f>* = - (z + z Q )g, and 



or 

^(i + ^/c 2 ) = const. 

Accordingly the frequency of the radiation 
emitted by an atom in a place of high 
gravitational potential is less than that 
emitted by a similar atom in a place of low 
gravitational potential. Thus the lines of 



114 RELATIVITY FOll PHYSICS STUDENTS 

the solar spectrum should be displaced 
slightly towards the red as compared with 
the corresponding lines of atoms vibrating 
in the comparatively low gravitational po- 
tential of a terrestrial laboratory. 

As we have already remarked, the solu- 
tions which we have given to these two 
problems are very open to criticism, but 
they serve to show the essence of Einstein's 
method, which is to describe one and the 
same physical phenomenon from the point 
of view of two frames of reference. 



VI 

THE FOUR-DIMENSIONAL 
CONTINUUM 

OUR last lecture was devoted to a 
discussion of the powerful method 
by which Einstein brought gravitation into 
relation with other physical phenomena. 
But this was only a stepping-stone to the 
accomplishment of his main purpose, which 
was to supply a complete answer to the diffi- 
culty of the choice of a frame of reference 
by formulating the laws of physics in such 
a way that they are true for all possible 
frames. Before we can follow him further 
in this direction, we must make the ac- 
quaintance of some of the mathematical 
methods which he employed. 

One of the most important of these is 
related to the idea of space-time. It is 

"5 



T16 RELATIVITY FOR PHYSICS STUDENTS 

commonly stated that the theory of rela- 
tivity assumes that space and time as we 
ordinarily understand them are not essen- 
tially distinct, but that they are merely 
special aspects of a more fundamental four- 
dimensional space-time. It has always 
seemed to me that this is a strong suspicion 
which might occur to one after a deep study 
of the theory rather than a dogma which 
must be accepted at the beginning. It is 
sufficient at this stage if we accept the four- 
dimensional continuum as a convenient 
mathematical representation. It is not 
necessary to assume that pressure and 
volume are fundamentally of the same 
nature before we can plot pressure against 
volume, or, as we may say, draw the iso- 
thermals of a gas in a two-dimensional 
pressure- volume space. In the same way 
we can represent an event occurring at a 
given place (%, y, z) at a given time (t) by a 
point (#, y, z, t) in a four-dimensional space. 
It will be easier to grasp the simplicity of 
this representation if, for the moment, we 



FOUR-DIMENSIONAL CONTINUUM 117 

reduce space to two dimensions and repre- 
sent the motion of a point in the plane 
x, y by means of the three-dimensional 
space-time x, y, t. The career of a particle, 
moving with velocity u, v in a straight line 
and passing through the origin at time 
/ = o, is represented by the straight line 

x/u yfv = /. 

A particle describing the circle # 2 +V 2 # 2 
with constant angular velocity o> is repre- 
sented by the helix 

% ~ a cos c*t, y = a sin a>t. 

Whatever the motion of a particle in the 
plane x, y, its complete history, past, pres- 
ent, and future, will be represented by a 
" world-line " in the space x, y, t. The re- 
presentation of the motion of a particle in 
three-dimensional space x, y, z requires a 
four-dimensional space-time x, y, z, t. The 
fact that we cannot represent such a space in 
three dimensions that we cannot, for ex- 
ample, draw four lines each perpendicular to 
the other three is of little more account 



118 RELATIVITY FOR PHYSICS STUDENTS 

than the familiar difficulty of representing 
three-dimensional figures on two-dimen- 
sional paper. 

It being granted that the representation 
is possible, our next step is to show that it 
is convenient in that it supplies a simple 
interpretation of known results. For this 
purpose we will employ r = ict for the fourth 
co-ordinate, where c is a constant ultimately 
identified with the velocity of light. The 
introduction of the imaginary i at this stage 
is by no means necessary, and is not resorted 
to in modern presentations of the theory. 
In our space x, y, z y r, suppose the axes of % 
and r are rotated through an angle 0, the 
other axes remaining as before. The for- 
mulae of transformation for two axes about 
their origin are 

%' = % cos + r sin 0, T' = x sin + r cos 0. 

Now suppose cos ~ ]8, sin = ivp[c, 
where j3 2 = i/(i v 2 lc 2 ) ; is then a pure 
imaginary. The formulae become 

), r' = p(r ivxjc), 



FOUH-DIMENSIONAL CONTINUUM 119 

or, expressed in terms of t, 

%' - fi(x - vt), t' = p(t - vxlc 2 ). 

Thus the Lorentz transformation admits 
of a simple, if somewhat strange, interpre- 
tation. It corresponds in the four-dimen- 
sional space x, y, z, r to a mere transforma- 
tion to new axes, obtained by rotating the 
axes of x, r in their own plane through an 
imaginary angle. 

Let us consider from this point of view 
the rather complicated formulae for the 
transformation of velocities which we ob- 
tained in Lecture II. We have 

dx ___ - A% 

**-^ &'-"-& 

so that if x * s the angle between the axis of 
% and the projection of the tangent to the 
world-line on the plane of XT, u x = ic cot # 
Now let the axes of x and T be turned through 
the angle 9, as described above If x = 
X 0, the new component of velocity is 



cot 6 cot 



120 RELATIVITY * OR PHYSICS STUDENTS 

Substituting for x an( 3 0> this gives the 
result already established, 



/ = U x V 
* I - 



We are now in a position to see a reason 
for the complexity of this result. The com- 
ponents of velocity serve to define the 
direction of the tangent to the world-line. 
For this purpose we have used u x = ic cot x 
and similar expressions for u y and u z . A 
more symmetrical procedure would be to 

i J.I. j j.- dx dy dz 

employ the direction-cosines _ , -/, , , 

as as as 

/, where ds is the length of an element of 
as 

arc of the world-line and 

ds 2 = d%* + dy 2 + dz* + dr\ 
We accordingly take 

dx -dy - dz . dr 

1C ~j~, 1C j^ 9 ic 1C T 

ds ds ds ds 

as the components of a four-dimensional 
velocity vector (the factor ic is inserted so 
that for slow motion these approximate to 



FOUR-DIMENSIONAL CONTINUUM 121 

M X , Uy, u s , i). These suggest a four-dimen- 
sional acceleration vector with components 



_ ,2 c , _ ,2 

<fe 8 ' rfs 8 "' " d&' d^ 

The velocity and acceleration vectors are 
then simple aspects of the geometry of the 
world-line of the particle. Without imply- 
ing too much by the names, we may say, 
on the analogy of the three-dimensional 
geometry of curves, that the velocity defines 
the direction of the tangent to the world-line, 
while the acceleration defines the magnitude 
and direction of its curvature. 

Consider the equations 



--*.. 



ds* 



2 

- me* ~ 
as 2 



where m is the constant mass of the particle 



122 RELATIVITY FOR PHYSICS STUDENTS 

and k is a vector whose first three com- 
ponents are the components of force, while 
& 4 is at present undefined. If we can show 
that these equations express the law of 
motion of the particle, we may say that 
the four-dimensional view has restored to 
Newton's laws the simplicity which they 
seemed to lose under the Lorentz theory. 
To do this we must translate the above 
equations back into the ordinary three- 
dimensional notation. If u x = dxjdt, . . ., 
and if u* 2 + u y 2 + u z 2 = v 2 , so that v is the 
resultant velocity in the ordinarj 7 sense, we 
have ds* = (v* - c*)dt*, or 

^L = iP d 
ds c df 

The first of the above equations then 
becomes 



This, together with the second and third 
equations, expresses the law of momentum 
with the variable mass $m. 

Multiply the equations by dxjds, . . ., 



FOUR-DIMENSIONAL CONTINUUM 128 

and add. The left-hand side becomes the 
differential of a constant, and hence 

7 doc . i dy , i dz , 7 dr 



or 

/t 



Ujc f 

c 



The fourth equation then gives 





This expresses the law of energy since 

me 2 : j8, or mc 2 (jS i), is the form already 
found for the kinetic energy. 

We cannot now follow the expression of 
the whole of the restricted theory of rela- 
tivity in terms of four-dimensional space- 
time, but the above discussion of the theory 
of variable mass may serve to indicate the 
simplification which follows the adoption of 
this point of view. This was Minkowski's 
great contribution to the development of 
the theory of Relativity.* 

* For a translation of Minkowski's classical memoir 
" Space and Time/' see "The Principle of Relativity," 
pp. 75-96 (Methuen, 19-23)- 



RELATIVITY FOR PHYSICS STUDENTS 

The presentation of the Lorentz trans- 
formation as the effect of a change of rect- 
angular axes in four-dimensional space-time 
points the way to the formulation of the 
absolute laws of physics. We are familiar 
with the simplicity which is introduced into 
physics by the introduction of the idea of a 
vector. Let us return again to the law of 
motion. We may express this as follows : 
There is a certain vector, the mass accelera- 
tion, and there is another vector, the force, 
and these two vectors are equal in magni- 
tude and direction. We usually express 
this law in a somewhat round-about way 
by resolving each vector along three direc- 
tions and asserting the equality of the 
respective components. The equations thus 
obtained will depend upon our choice of 
axes, but the law in its vector form is 
independent of this choice. Thus it is in 
the development of the vector point of view 
that we shall seek the emancipation of the 
laws of physics. After what we have seen, 
we shall be prepared to find that the 



FOUR-DIMENSIONAL CONTINUUM 125 

appropriate vectors are four-dimensional 
vectors in space-time rather than the ordi- 
nary three-dimensional vectors. 

Our next step is to obtain an appropriate 
generalization of the idea of a vector to 
four dimensions. In three dimensions we 
define a vector as a quantity having magni- 
tude and direction, and such that two vec- 
tors add by the parallelogram law. The 
fact that it will often be difficult to form 
images of the geometrical relations in four 
dimensions suggests that it will be con- 
venient to express this definition in ana- 
lytical rather than geometrical form. Sup- 
pose that there are two sets of rectangular 
axes (x l9 x 2 , # 3 ) and (#/, # 2 ', # 3 ') having a 
common origin and that the direction 
cosines of one set with respect to the other 
are such that the formulae of transformation 
are 



186 RELATIVITY FOR PHYSICS STUDENTS 

or, what is the same thing, 

^1 ~ *l^l I 'if -^2 I" '3^3 



A vector which has components A x , A 2 , A 3 
along the axes of x lt x 2 , x% will have com- 
ponents along Xi, x 2 ', x 3 ' equal to A/, 
A 2 ', A 3 ; , where 

A/ = ^Ax + w^ + WjAa, etc. 
This may be expressed in either of the 
two forms 






or 



4^ 2 A 4 * 3 A 
--, A 2 + -7 A 3 . 

oi OXi 0X1 

Each of these is typical of three equations 
which may be written in condensed form 



A ' 

A 



A 

A " 






= i, 2, 3). 



FOUR-DIMENSIONAL CONTINUUM 127 

A three-dimensional vector may be defined 
as a quantity having three components A x , 
A 2 , A 3 , which transform from one set of 
co-ordinates to another by either of the 
above formulae of transformation. It can 
be shown that this analytical definition is 
equivalent to the more usual geometrical 
definition. 

The extension to four dimensions is now 
obvious. Before stating it, we will note that 
the equivalence of the above two forms of 
the transformation law is a special property 
of transformations with rectangular axes. 
For other transformations it is necessary to 
distinguish between the vectors which obey 
the one or the other. This is done by the 
position of the index indicating the com- 
ponent. We define as follows : 

A Co-variant Vector is a quantity having 
four components A^ (p = i, 2, 3, 4), which 
transforms from one set of co-ordinates to 
another by the law 

A/ === 



128 RELATIVITY FOR PHYSIOS STUDENTS 

A Contravariant Vector has four com- 
ponents A^(/i =-- i, 2, 3, 4), which transform 
by the law 



The distinction between the two types of 
vector does not arise if we restrict ourselves 
to changes of orthogonal axes. 

Consider two of the vectors which we 
introduced earlier in the lecture. The velo- 
city vector has components which we may 
now write as proportional to dxjds = A*. 
Then, taking the contra variant law, 

4 

Y ft * n Y 

*IL U'A'V ">*,, 



PS' = , ~~ - - 

^ tx v ds ds ' 

V ~\ 

Hence the velocity is a contra variant 
vector. 

On the other hand, the acceleration vector 



ds* 



ds ds ' 



FOUR-DIMENSIONAL CONTINUUM 

is not a vector in the full sense. However, 
for any transformation in which the new 
co-ordinates are linear functions of the old, 
the second differentials in the double sum 
vanish and the acceleration behaves as a 
contravariant vector. If, in addition, the 
transformation is orthogonal (like the 
Lorentz transformation), 8x^1 8x v = d% v \A%J 
and the acceleration behaves as either a 
contravariant or a covariant vector. 

In three dimensions we sometimes meet 
quantities with properties similar to those 
of a vector, but having more than three 
components. For example, if in an elastic 
solid %y denotes the ^-component of force 
across a unit area normal to the #-axis, the 
stress is completely specified by nine (3 2 ) 
components, xx, xy, . . . , some of which 
are equal. A typical equation for their 
transformation for rectangular axes is 



%'y' = IJ^xx + m^m^yy + n^n^zz + 

(m^n* + 



130 RELATIVITY FOR PHYSICS STUDENTS 

If we write xx S u , xy = S 12 , . . . , this 
becomes 






This suggests a definition for a " tensor " 
of the second rank as a quantity with 
4 2 = 16 components, which transforms ac- 
cording to a law corresponding to its 
character. A tensor of the second rank 
may be covariant, contravariant, or mixed. 
The law of transformation for a mixed 
tensor is 



From this it is easy to frame the defini- 
tions and laws of transformation of a co- 
variant (A /xl ,) or a contravariant (A 1 ") 
tensor of the second rank, or of a tensor of 
any higher rank or character (e.g., A^ T ). 
Vectors take their place as tensors of the 
first rank, and we may complete the scheme 
by defining a tensor of zero rank as a quan- 



FOUR-DIMENSIONAL CONTINUUM 181 

tity with a single component which is 
unchanged by any change of co-ordinates 
an invariant. 

It is easy to prove from the definitions 
that the components of two tensors of the 
same rank and character may be added (or 
subtracted) to form the components of a 
new tensor. Rules can be laid down for 
the multiplication of tensors, and a consis- 
teftt algebra of tensors can be built up. 

The application of the theory of tensors 
to the problem of relativity arises in this 
way. Suppose it is possible to express a 
physical law by the equality of two tensors 
of the same rank and character, e.g. 
mAf = P 4 , where A* is the acceleration 
vector and P* the force vector. By trans- 
ferring both terms to the same side of the 
equation, and using the rules for addition 
of tensors, the law may be expressed by 
saying that a certain tensor vanishes, that 
is to say, that each of its components 
vanishes. If we now take a new system of 
co-ordinates, each of the new components 



182 RELATIVITY FOR PHYSICS STUDENTS 

of the tensor is the sum of multiples of the 
old components, and therefore vanishes. If 
the components of a tensor vanish in one 
system of co-ordinates, they will vanish in 
all systems. 

The way to make the laws of physics 
independent of our choice of co-ordinates is 
to express these laws as relations between 
tensors. 



VII 
THE GENERAL THEORY 

IN the last lecture we saw that, from 
the point of view of four-dimensional 
space-time, the Lorentz transformation 
corresponds to a change of rectangular axes 
in which two of the axes are rotated through 
a certain angle in their own plane, while the 
other two are unaltered. From the ordinary 
three-dimensional point of view it corre- 
sponds to a change from one frame of 
reference to another, which moves rela- 
tively to the first with uniform velocity in 
the direction of one of the axes. The prob- 
lem of relativity demands a much more 
general change for the frame of reference, 
namely, from one to another moving rela- 
tively to it in any manner. It might be 

thought that such a general change of the 

133 



134 RELATIVITY FOR PHYSICS STUDENTS 

frame of reference might correspond to a 
more general change of axes in space-time. 
However, so long as we confine ourselves to 
rectangular Cartesian axes in space-time, 
this is not the case. The most general 
change of rectangular axes about the same 
origin corresponds to frames of reference 
which are moving relatively the one to the 
other with a velocity which is uniform but 
not necessarily in the direction of any uf 
the axes ; a change to parallel axes through 
a new origin merely alters the origin in 
space and the zero instant in time. The 
most general change of rectangular axes in 
space-time thus gives no more than the 
Lorentz transformation. 

If we look back over the last lecture we 
see how little depends upon the axes being 
rectilinear and rectangular, and Einstein's 
next step was to pass at once to a much 
more general conception of co-ordinates in 
space-time. Just as we frequently em- 
ploy curvilinear co-ordinates for special 
purposes in three-dimensional space, so we 



THE GENERAL THEORY 135 

employ curvilinear co-ordinates in space- 
time. In place of x, y, z, t we may take 
any four independent functions of these, 
XD %*> x*> %*> and these will serve to specify 
the point of space-time corresponding to a 
physical event. These relations may be 
expressed 




(I) 

/ Y /y <Y 

I / 4 %, , .T 

If /i, /2> /3> /4 are arbitrary functions, we 
have the most general system of co-ordinates 
in space-time. The theory of tensors de- 
veloped in the last lecture is applicable 
without modification. Physical laws, ex- 
pressed in tensor form, are equally valid 
whether the co-ordinates are x, y, z, t or 

In the development of the theory great 
importance attaches to the form for ds 2 , 
the square of the length of an element of 
arc in space-time. Expressing it in terms 



136 RELATIVITY FOR PHYSICS STUDENTS 

of t instead of r, and making an unimportant 
change of sign, it is 

ds 2 = - dx 2 - dy 2 - dz 2 + c 2 dt 2 . (2) 

From (i) we obtain equations of the type 
dx - *& dx, + & dx, + -V* dx, + ^ dx,, 

^X l ^X 2 " t>#3 ^#4 

and these may be used to transform (2) to 
the new variables. We obtain 

4 4 

guJxudx, . . (3) 



There is no loss of generality in assuming 
that g^ = g^, and these coefficients are 
then definite functions of the differential 
coefficients of f lf f, / 3 , / 4 , and are therefore 
functions of x l9 x 2 , x 3 , x. 

It has been known since the time of 
Riemann that the form for ds 2 can be used 
as the basis for the geometry of the space 
to which it belongs. This method stands in 
sharp contrast to the more familiar methods 
of Euclidean and projective geometry. In 
these " finite " geometries we start with 



THE GENERAL THEORY 137 

definitions of figures such as lines, planes, 
spheres, etc., and there is usually the 
underlying assumption that the space is 
homogeneous ; that its properties are every- 
where exactly the same. Differential geo- 
metry, on the other hand, starts with the 
relations between infinitely near points as 
expressed by the form for ds 2 , and, provided 
that suitable assumptions are made, arrives 
at conclusions consistent with those of the 
finite geometries. But differential geo- 
metry has this advantage, that it is much 
more readily applicable to the study of 
heterogeneous space whose properties are 
different from point to point. There is 
nothing strange in the idea of heterogeneous 
space. The surface of an ellipsoid is a two- 
dimensional heterogeneous space, and the 
surface of the earth is more so. The form 
(2) corresponds to a space which is homo- 
geneous and, moreover, with respect to its 
three dimensions %, y, z, isotropic. In (3) 
the coefficients are functions of the co- 
ordinates and the space is heterogeneous. 



138 RELATIVITY FOR PHYSICS STUDENTS 

The space-time of Einstein is hetero- 
geneous. We have pointed out that gravi- 
tation is an effect by which any body placed 
in a given position acquires an acceleration 
which depends upon the position and not 
upon the accelerated body. Thus, in a 
sense, we may say that gravitation is a 
property of space-time and that the latter 
is therefore obviously heterogeneous. Some 
would prefer to say that the space-time 
considered by Einstein is not so much space- 
time as space- time cum gravitation. 

It is easily proved, by making a further 
change of co-ordinates in (3), that the g^ 
are the components of a covariant tensor of 
the second rank. This tensor defines the 
gravitational field, and its components are 
sometimes spoken of as the gravitational 
potentials. 

The homogeneous and isotropic form (2) 
can correspond only to a space in which there 
is no gravitation, and, since (3) has been 
obtained from (2) by a change of co-ordi- 
nates, the gravitation expressed by (3) may 



THE GENERAL THEORY 139 

be said to be fictitious, and to be an appear- 
ance arising from our choice of co-ordinates. 
Thus, when we were considering a uniform 
gravitational field, we were able to choose a 
new frame of reference in such a way that 
the gravitation disappeared. This is not 
always possible. A substitution will always 
bring the form (2) to the form (3), but it 
does not follow that, if we start from (3) 
with given functions of the co-ordinates for 
g^, we can find a substitution which will 
bring us back to (2). This may be expressed 
by saying that there are some gravitational 
fields which cannot be " transformed away " 
by a change in the frame of reference. If 
we have a given gravitational field expressed 
by (3), it is clearly a question of fact, inde- 
pendent of the co-ordinates employed, as to 
whether there is a frame of reference for 
which ds 2 is given by (2). Accordingly this 
condition ought to be expressible by the 
vanishing of a tensor. This is the case, 
and the condition is B*,, p = o, where B^ p 
is a mixed tensor of the fourth rank, called 



140 RELATIVITY FOR PHYSICS STUDENTS 

the Riemann-Christoffel tensor. It has 
4 4 == 256 components, but fortunately only 
twenty of these are independent and these 
are known functions of the g^ and their 
differential coefficients. 

The general method of applying the equi- 
valence hypothesis to this theory is as fol- 
lows : Suppose we know some physical law 
for the case when there is no gravitational 
field. We express it in tensor (or invariant) 
form in the variables x, y, z, t applicable to 
the space (2). We now make the change of 
co-ordinates which brings (2) to (3), and 
we shall expect that the g^, expressing the 
resultant " fictitious " gravitational field, 
will appear in our tensor equations. Ein- 
stein assumes that the equations thus ob- 
tained will be valid even when the gravita- 
tional field is not " fictitious " ; that the 
g^ enter into the expression of physical 
laws in precisely the same way whether they 
represent a gravitational field which can be 
" transformed away " or whether they re- 
present one which cannot. 



THE GENERAL THEORY 141 

For example, the law of motion for a 
freely moving particle when there is no 
gravitational field is that it moves with 
uniform velocity in a straight line. We 
have seen that this motion corresponds to 
a straight world-line in space-time. Adopt- 
ing the definition of a straight line as the 
shortest distance between two points, we 

may express the law by saying that ds 
9 J A 

is a minimum where the integral is taken 
along the world-line between any two 
points A and B. Again, this may be written 

8 ("ds = o, . . . (4) 

J A 

where the left-hand side means the change 
in the value of the integral when it is taken 
along a curve joining A and B, but differing 
slightly from the world-line. Since ds is an 
invariant, the condition expressed in this 
form is independent of the co-ordinates 
employed. Accordingly Einstein assumes 
(4) to be the law of motion for all systems 
of co-ordinates, and for any gravitational 



142 RELATIVITY FOR PHYSICS STUDENTS 

field. The integral form (4) may be re- 
placed by equivalent differential equations. 
If this is done it will be found, as we should 
expect, that these equations express the 
vanishing of a tensor. The carrying through 
of this work requires some slight further 
development of the theory of tensors, but 
we will show that (4) provides a practicable 
method of determining the motion of a 
particle by applying it to the form 'of 
equation (3), which Einstein found for the 
gravitational field of the sun. 

This is most conveniently expressed in 
spherical polar co-ordinates f, 0, <, and then 



ds* - - y - l dr 2 - r 2 d0* - f 2 sin 2 

v c*dt* . (5) 
where 

2/cM 
V = l ~ c*r' 

and M is the mass of the sun and K the 
constant of gravitation. 

If the particle (by which, of course, we 
mean a planet) is moving in the equatorial 



THE GENERAL THEORY 148 

plane " , dQ o and (4) may be written 



It will simplify our formulae if we consider 
the special cases of this equation in w r hich 
<j> and t separately have values slightly 
different from those appropriate to the 
actual path. First let <f> be subject to a 



small variation S<, then s - (8<f>), and 

\ds/ as 

(6) gives 



ds _ 

" ' 



The denominator is equal to unit} 7 by (5), 
and hence 



P d + * (8<f>)ds = o. 
JA ds ds v ' 

Integrating by parts, 



RELATIVITY FOR PHYSICS STUDENTS 

The variations of the path being subject 
to the condition that the path passes 
through A and B, we have 8< = o at A and 
at B. Hence the first term vanishes. 
Apart from these terminal values, S< is 
arbitrary, and hence the integral will vanish 
only if 

-f- (> f} - - 

as \ ds/ 
or 



where h is a constant. 

Allowing t to take a small variation, and 
following precisely the same method, we 
find 

At dt\ Q 

ds \ ds/ 

or 

yf = c, . . . (8) 

ds N ' 

where C is a constant. 

A third equation can be obtained by 
allowing v to take a small variation ; this 



THE GENERAL THEORY 145 

equation is more complicated, and we can 
proceed without it. For the case, = |TT, 
(5) is equivalent to 



Using (7) and (8) to eliminate s and t t 
writing i/r = u, and remembering the value 
of y, this equation becomes 



Differentiating with respect to <, and 
dividing by 2dujd^ y we have 



The second term on the right-hand side 
of this equation is, in practical cases, very 
small compared with the first. For a cir- 
cular orbit the ratio is three times the 
square of the ratio of the velocity of the 
planet to that of light. If, for the moment, 
we neglect this small term, the equation 
is identical with the well-known differential 
equation for central orbits under a central 



146 RELATIVITY FOR PHYSICS STUDENTS 

force KM.fr 2 , with h for Kepler's areal 
constant. Thus an approximation to New- 
ton's law of the inverse square appears as 
a consequence of Einstein's method. It is 
important to notice that we did not pos- 
tulate any " force of gravitational attrac- 
tion " in order to obtain this result. The 
particle was supposed to be unacted upon 
by any force, and to move " with uniform 
velocity in a straight line," but the idea bf 
a straight line was modified to allow for 
the heterogeneity of space-time which marks 
the existence of a gravitational field. This 
non-appearance of gravitational force is 
characteristic of Einstein's theory. For the 
purpose of comparison with the older theory, 
however, we may interpret Einstein's re- 
sult in terms of gravitational force. We 
will consider a simple case in which the 
particle moves in a straight line through 
the origin, so that d(f>/ds = o. Equation 
(9) then becomes 

= C *C* - i + 2 * M . 
c*r 



THE GENERAL THEORY 147 

On differentiation this becomes 



d 2 r 
ds* 



If the left-hand side is taken as the 
acceleration, we have an accurate inverse 
square law ; but if we take the more usual 
d 2 r/dt 2 as the acceleration and calculate its 
value by the aid of (8), it will be found to 
contain a term varying as the inverse square 
of the distance, and, in addition, terms 
varying as the inverse third and fourth 
powers. 

The second term on the right-hand side 
of (n), though small, has important conse- 
quences. Its effect is that the orbit is not 
accurately a conic, but may be represented 
by, for example, an ellipse whose major axis 
rotates slowly. 

Let u l9 u* be two of the values of u for 
which du/d<f> vanishes; they are the recip- 
rocals of the apsidal distances. The con- 
stants C and h can be expressed in terms of 
%, MI, and (10) becomes 



148 RELATIVITY FOR PHYSICS STUDENTS 



being small, this gives approxi- 
mately 

, ^ f[Lil^M( w 



If we write u = ^ cos 2 $ + u 2 sin 2 <, 
then as ^ goes from o to TT, starts from 
its apsidal value j, and returns to that 
value. The angle <f> described between 
corresponding apsidal positions is thus 

~r K 2 (^1 + ^2+^1 cos 2 1/> + 2^2 sin 2 



The excess of this value over four right- 
angles represents a rotation of the apse 
line which has occurred during one revolu- 
tion. If a, e are respectively the major 
semi-axis and eccentricity of the orbit, 
u t = i/#(i 0), ^ 2 = i/#(i + 2), and 
Ul + HZ = 2/a(i ^ 2 ) = 2//, where / is the 
semi-latus rectum. The rotation of the 



THE GENERAL THEORY 149 



apse line for one revolution is thus 
This result has been verified by observations 
on the orbit of the planet Mercury, for which 
it amounts to about 43" per century. 

We must now bring our account of the 
theory of relativity to a close. We have 
endeavoured to trace the development of 
the great work of Fresnel, and to show how 
the difficulty of determining an absolute 
frame of reference dogged the footsteps of 
science, and to explain in some measure the 
way in which Einstein has supplied an 
answer to that difficulty. It is clear that 
the subject merits a more thorough and 
systematic treatment than we have been 
able to give in these lectures. There is a 
mistaken idea that this treatment can be 
accomplished only by the aid of very diffi- 
cult mathematics. It is true that one must 
master the theory of tensors and gain a 
certain facility in their manipulation, but 
this is by no means so difficult as is com- 
monly supposed. A very clear, if somewhat 
compressed, account of the mathematical 



150 RELATIVITY FOR PHYSICS STUDENTS 

theory will be found in Eddington's " Report 
to the Physical Society on the Relativity 
Theory of Gravitation " (Fleet way Press) ; 
a fuller and more systematic treatment is 
given by the same author in " The Mathe- 
matical Theory of Relativity " (Cambridge 
University Press) ; translations of the clas- 
sical papers on the subject by Lorentz, 
Einstein, Minkowski, and Weyl are pub- 
lished under the title of " The Principle of 
Relativity " (Methuen). Every young Eng- 
lish physicist should study at least the first 
of these books, and it is hoped that these 
lectures will help to smooth the way by a 
preliminary exploration of the ground. 



INDEX 



ABERRATION, 41, 88 
Absolute space and time, 12 
Airey, 45 

BENDING of a light ray, 107 
Brtihe, 3 

CAVENDISH, 10, 42 
Contraction hypothesis, 50 
Contravarient vectors, 128 
Covarient vectors, 127 

DIFFERENTIAL geometry, 136 
Displacement of lines in solar 

spectrum, no 
Doppler effect, 90 

ELECTROMAGNETIC mass, 97 
Electromagnetism, 41, 45, 51, 78 
Electron, field of a moving, 92 
Equivalence hypothesis, 98 
Ether, 16, 40 

FIXED stars, 14 

Four-dimensional continuum, 
113 

FRESNBL; fixed ether, 43; 
dragging coefficient, 44, 91 

GALILEO, 4 

Gravitation, 4, 7, 22, 103 
Gravitational field of the sun, 
142 



INVARIANT laws, 131 

KEPLER, 4 
Kinetic energy, 77 

LONGITUDINAL mass, 75 
Lorentz, 50, 53, 66 

MASS, 75, 97 
Maxwell, 16, 46 
Michelson and Mnrlcy's experi- 
ment, 47, 56 
Motion of a planet. 142 

Newton, 6, 25, 37 
OPTICS, 39 

PERIHELION of Mercury, 149 
STOKES, 48 

TENSORS, 130 
Transverse mass, 75 

VECTORS in four dimensions, 

I2 5 

Velocity, of the earth, 58; 
transformation of, 69; as a 
four- vector, 120 

WATER telescope, 45 
World-line, 117 



PRINTED IN GREAT BRITAIN AT 
THE UNIVERSITY PRESS, ABERDEEN