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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
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