1
Eddington, (Sir) Arthur
Stanley
The theory of relativity
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6
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ROMANES
1922
LECTURE
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•^ry 0/* Relativity
and its
T nfluence on Scientific Thought
BY
JR1HUR STANLEY EDDINGTON
M.A., F.R.S.
Plumian Professor 0/ Astronomy ', Cambridge
^resident of the Royal Astronomical Society
DELIVERED
THE SHELDONIAN THEATRE
24 MAY, 1922
OXFORD
T THE CLARENDON PRESS
1922
THE ROMANES LECTURE
i=!W-- 1922
Tk
Theory of Relativity
an
d its
Influence on Scientific Thought
BY
ARTHUR STANLEY EDDINGTON
M.A., F.R.S.
Plumian Professor of Astronomy, Cambridge
President of the Royal Astronomical Society
DELIVERED
IN THE SHELDONIAN THEATRE
24 MAY, 1922
OXFORD CiTVo.AA .
AT THE CLARENDON PRESS
1922
ac
Has not a deeper meditation taught certain of every climate and
age, that the where and the when so mysteriously inseparable
from all our thoughts, are but superficial terrestrial adhesions to
thought ?
Carlyle, Sartor Resartus.
Oxford University Press
London Edinburgh Glasgow Copenhagen
New York Toronto Melbourne Cape Town
Bombay Calcutta Madras Shanghai
Humphrey Milford Publisher to the University
THE THEORY OF RELATIVITY
In the days before Copernicus the earth was, so it
seemed, an immovable foundation on which the whole
structure of the heavens was reared. Man, favourably
situated at the hub of the universe, might well expect
that to him the scheme of nature would unfold itself in
its simplest aspect. But the behaviour of the heavenly
bodies was not at all simple ; and the planets literally
looped the loop in fantastic curves called epicycles. The
cosmogonist had to fill the skies with spheres revolving
upon spheres to bear the planets in their appointed
orbits ; and wheels were added to wheels until the music
of the spheres seemed wellnigh drowned in a discord
of whirling machinery. Then came one of the great
revolutions of scientific thought, which swept aside the
Ptolemaic system of spheres and epicycles, and revealed
the simple plan of the solar system which has endured
to this day.
The revolution consisted in changing the view-point
from which the phenomena were regarded. As pre-
sented to the earth the track of a planet is an elaborate
epicycle ; but Copernicus bade us transfer ourselves to
the sun and look again. Instead of a path with loops
and nodes, the orbit is now seen to be one of the most
elementary curves— an ellipse. We have to realize that
the little planet on which we stand is of no great account
in the general scheme of nature ; to unravel that scheme
we must first disembarrass nature of the distortions
arising from the local point of view from which we
observe it. The sun, not the earth, is the real centre of
the scheme of things — at least of those things in which
A 2
4 THETHEORYOF
astronomers at that time had interested themselves — and
by transferring our view-point to the sun the simplicity
of the planetary system becomes apparent. The need
for a cumbrous machinery of spheres and wheels has
disappeared.
Every one now admits that the Ptolemaic system,
which regarded the earth as the centre of all things,
belongs to the dark ages. But to our dismay we have
discovered that the same geocentric outlook still permeates
modern physics through and through, unsuspected until
recently. It has been left to Einstein to carry forward
the revolution begun by Copernicus — to free our con-
ception of nature from the terrestrial bias imported into
it by the limitations of our earthbound experience. To
achieve a more neutral point of view we have to imagine
a visit to some other heavenly body. That is a theme
which has attracted the popular novelist, and we often
smile at his mistakes when sooner or later he forgets
where he is supposed to be and endows his voyagers
with some purely terrestrial appanage impossible on the
star they are visiting. But scientific men, who have not
the novelist's licence, have made the same blunder.
When, following Copernicus, they station themselves on
the sun, they do not realize that they must leave behind
a certain purely terrestrial appanage, namely, the frame
of space and time in which men on this earth are accus-
tomed to locate the events that happen. It is true that
the observer on the sun will still locate his experiences
in a frame of space and time, if he uses the same faculties
of perception and the same methods of scientific measure-
ment as on the earth ; but the solar frame of space and
time is not precisely the same as the terrestrial frame,
as we shall presently see.
I think you will readily understand what is meant by
a frame of space and time. It is the system of location
RELATIVITY 5
to which we appeal when we state, for example, that one
event is 100 miles distant from and 10 hours later than
another. The terms space and time have not only a
vague descriptive reference to a boundless void and an
ever-rolling stream, but denote an exact quantitative
system of reckoning distances and time-intervals. Ein-
stein's first great discovery was that there are many such
systems of reckoning— many possible frames of space
and time — exactly on all fours with one another. No
one of these can be distinguished as more fundamental
than the rest ; no one frame rather than another can be
identified as the scaffolding used in the construction of
the world. And yet one of them does present itself to
us as being the actual space and time of our experience ;
and we recoil from the other equivalent frames which
seem to us artificial systems in which distance and
duration are mixed up in an extraordinary way. What
is the cause of this invidious selection? It is not
determined by anything distinctive in the frame ; it is
determined by something distinctive in us— by the fact
that our existence is bound to a particular planet and
our motion is the motion of that planet. Nature offers
an infinite choice of frames ; we select the one in which
we and our petty terrestrial concerns take the most
distinguished position. Our mischievous geocentric
outlook has cropped out again unsuspected, persuading
us to insist on this terrestrial space-time frame which in
the general scheme of nature is in no way superior to
other frames.
The more closely we examine the processes by which
events are assigned to their positions in space and time,
the more clearly do we see that our local circumstances
play a considerable part in it. We have no more right
to expect that the space-time frame on the sun will be
identical with our frame on the earth than to expect that
6 THETHEORYOF
the force of gravity will be the same there as here. If
there were no experimental evidence in support of
Einstein's theory, it would nevertheless have made a
notable advance by exposing a fallacy underlying the
older mode of thought — the fallacy of attributing un-
questioningly a more than local significance to our
terrestrial reckoning of space and time. But there is
abundant experimental evidence for detecting and de-
termining the difference between the frames of differently
circumstanced observers. Much of the evidence is too
technical to be discussed here, and I can only refer to
the Michelson-Morley experiment. I fear that some of
you must be getting rather tired of the Michelson-Morley
experiment ; but those who go to a performance of
Hamlet have to put up with the Prince of Denmark.
This famous experiment is a simple test whether light
travels at the same speed in two different directions.
For this purpose an apparatus is constructed with two
equal arms at right angles, providing two equal tracks
for the light. A beam of light is divided into two parts
so that one part travels along one arm and back, and the
other along the other arm and back. The two rays then
re-unite, and by delicate interference tests it is possible
to tell if one has been delayed more than the other ; a
delay of less than a thousand-billionth of a second could
be detected. The experiment is simply a race between
two light-rays with equal tracks, but pointing in different
directions ; the result turns out to be a dead-heat. At
first sight this is just what would be expected ; and one
almost wonders why it should have been thought worth
while to try the experiment. But Michelson, like a
good Copernican, had stationed himself on the sun to
watch the race ; accordingly he realized that the appara-
tus was being borne along by the earth's orbital motion
with a speed of 20 miles a second. Consequently the
RELATIVITY 7
light does not travel exactly the double length of the
arm ; starting at one end it has to go to the turning-mark
at the other end which has moved on a little in the
meantime ; then it returns to the place which the start-
ing-mark has travelled to whilst the race is in progress.
That does not add up to exactly the double-length of
the arm. Making the calculations we easily find that,
although the two arms are equal, the two light -journeys
are unequal ; the competitor whose track lies in the
line of the earth's motion has the longer journey, and is
at a disadvantage. And yet according to the experiment
he does not suffer the expected delay. From our stand-
point on the sun, the experiment seems to have gone
wrong ; Copernicus has met with a rebuff, and Ptolemy
is triumphant.
But that is because we have not admitted the full
consequences of transferring our standpoint to the sun.
We have all the while been keeping one foot on earth.
Of course, the whole experiment turns on the two arms
having been first adjusted to perfect equality. This
could only be ascertained by experiment ; and the test
applied was to rotate the apparatus through a right
angle, so that if, for example, the journey in the line of
the earth's motion had had the advantage of the shorter
arm on one occasion, the transverse journey would have
had it on the repetition. That is a perfectly satisfactory
test for a terrestrial observer ; to turn a rod from one
direction to another is the simple and direct way of
marking out equal lengths. But the test is not satisfac-
tory to an observer on the sun ; he would not think of
attempting to partition equal lengths of space by means
of rods travelling at 20 miles a second. His frame of space
— the space not only of refined measurement, but also of
the cruder measurements made with the sense-organs
of his body which determine his perception of space—
8 THETHEORYOF
is partitioned by appliances at rest relatively to him,
e. g. his own eyes and limbs. Lengths of objects carried
on the earth must be judged by him according to the
room they occupy in his own frame. In the space of the
terrestrial observer the two arms of the apparatus were
adjusted to equal length ; but in the re-partitioned space
of the solar observer they may quite well occupy un-
equal lengths, and when we take the view-point of an
observer on the sun we must not overlook this inequality.
This inequality is not so much an hypothesis proposed
to account for Michelson's result as a direct deduction
from it. The two light-journeys were found to occupy
equal times ; this clearly shows that the arm in the less
favoured direction is shorter than the other so as to
counterbalance the handicap to which I have referred. 1
When the apparatus is turned through a right angle,
the experiment still gives the same result. It does not
matter which of the two arms we place in the line of the
earth's motion ; that arm must be shorter than the other.
In other words each arm must automatically contract
when it is turned from the transverse to the longitudinal
position with respect to its line of motion. This is the
famous FitzGerald contraction of a moving rod. It is
of the same amount whatever the material of the rod,
and depends only on the speed of its motion. For the
earth's orbital motion the contraction amounts to one
part in 200 million ; in fact the earth's diameter in the
direction of its motion is always shortened by 2.\ inches,
the transverse diameter being unaffected.
1 The only alternative is that (relatively to a solar observer) the
velocity of light differs in different directions, at least in the
region where the experiment is conducted. This would pre-
sumably be due to some influence of the moving earth on the
propagation of light (convection of the ether). This explanation
was at one time favoured, but it could not be reconciled with the
observed phenomena of the aberration of light.
RELATIVITY 9
This contraction of a moving material object was first
revealed to us by the Michelson-Morley experiment ;
but it is not at all disagreeable to theoretical anticipations.
We have to remember that a rod consists of a large
number of molecules kept in position by their mutual
forces. The chief force is the force of cohesion, and
there is little doubt that this is of electrical nature. But
when the rod is set in motion, the electrical forces inside
it must change. For example, each electric charge when
put in motion becomes an electric current) and the
currents will exert magnetic attractions on each other
which did not occur in the system at rest. Under the
new system of forces the molecules will have to find
new positions of equilibrium ; they become differently
spaced ; and it is therefore not surprising that the form
of the rod changes. Without going beyond the classical
laws of Maxwell we can anticipate theoretically what
will be the new equilibrium state of the rod, and it
turns out to be contracted to the exact amount required
by the Michelson-Morley result.
The contraction of the moving rod ought not to sur-
prise us ; it would be much more surprising if the rod
were to maintain the same form in spite of the alteration
of the electrical forces which determine the spacing of
the molecules. But the remarkable thing is that the
contraction is only apparent according to the outlook of
the solar observer ; and we on the earth, who travel
with the rod, cannot appreciate it. The fact that the
contraction happens to be very small is irrelevant. For
convenience suppose that the earth's velocity is 8,000
times faster, so that the contraction amounts to some-
thing like a half the original length. We should still
fail to notice it in everyday life. Let us say that the
direction of the earth's motion is vertically upwards.
I turn my arm from horizontal to vertical and it con-
a 3
io THE THEORY OF
tracts to half its length. No, you cannot convince me
I am wrong ; I am not afraid of a yard-measure. Bring
one and measure my arm ; first horizontally, the result
is 30 inches ; now vertically, the result is 30 — half-inches !
Because you must remember that you have turned the
scale into the line of the earth's motion so that each
inch-division contracts to half an inch. ' But we can see
that your arm does not contract. Are we not to trust
our eyes ? ' Certainly not, unless you first correct your
visual impressions for the contraction of the retina in
the vertical direction, and for the effect of our rapid
motion on the apparent direction of propagation of the
waves of light. You will find, when you calculate these
corrections, that they just conceal the contraction.
' But if the contraction takes place, ought one not to feel
it happening to the arm?' Not necessarily; I am an
observer on the earth, and my feelings like other
sense-impressions belong to the geocentric outlook
on nature, which Copernicus has persuaded us to
abandon.
Take a pair of compasses and twiddle them on a sheet
of paper. Is the resulting curve a circle or an ellipse ?
Copernicus from his standpoint on the sun declares that
owing to the FitzGerald contraction the two points
drew nearer together when turned in the direction of
the earth's orbital motion ; hence the curve is flattened
into an ellipse. But here I think Ptolemy has a right
to be heard ; he points out that from the beginning of
geometry circles have always been drawn with compasses
in this way, and that when the word ' circle ' is men-
tioned every intelligent person understands that this is
the curve meant. The same pencil line is in fact a circle
in the space of the terrestrial observer and an ellipse in
the space of a solar observer. It is at the same time
a moving ellipse and a stationary circle. I think that
RELATIVITY n
illustrates as well as possible what we mean by the
relativity of space.
It is sometimes complained that Einstein's conclusion
that the frame of space and time is different ^observers
with different motions tends to make a mystery of
a phenomenon which is not after all intrinsically strange.
We have seen that it depends on a contraction of moving
objects which turns out to be quite in accordance with
Maxwell's classical theory. But even if we have suc-
ceeded in explaining it to ourselves intelligibly, that
does not make the statement any the less true ! A new
result may often be expressed in various ways; one
mode of statement may sound less mysterious; but
another mode may show more clearly what will be the
consequences in amending and extending our know-
ledge. It is for the latter reason that we emphasize the
relativity of space — that lengths and distances differ
according to the observer implied. Distance and dura-
tion are the most fundamental terms in physics ; velocity,
acceleration, force, energy, and so on, all depend on
them ; and we can scarcely make any statement in
physics without direct or indirect reference to them.
Surely then we can best indicate the revolutionary con-
sequences of what we have learnt by the statement that
distance and duration, and all the physical quantities
derived from them, do not as hitherto supposed refer to
anything absolute in the external world, but are relative
quantities which alter when we pass from one observer
to another with different motion. The consequence in
physics of the discovery that a yard is not an absolute
chunk of space, and that what is a yard for one observer
may be eighteen inches for another observer, may be
compared with the consequences in economics of the
discovery that a pound sterling is not an absolute
quantity of wealth, and in certain circumstances may
A 4
12 THETHEORYOF
' really ' be seven and sixpence. The theorist may com-
plain that this last statement tends to make a mystery of
phenomena of currency which have really an intelligible
explanation; but it is a statement which commends
itself to the man who has an eye to the practical applica-
tions of currency.
Ptolemy on the earth and Copernicus on the sun are
both contemplating the same external universe. But
their experiences are different, and it is in the process of
experiencing events that they become fitted into the
frame of space and time— the frame being different
according to the local circumstances of the observer who
is experiencing them. That, I take it, is Kant's doctrine,
1 Space and time are forms of experience.' The frame
then is not in the world ; it is supplied by the observer
and depends on him. And those relations of simplicity,
which we seek when we try to obtain a comprehension
of how the universe functions, must lie in the events
themselves before they have been arbitrarily fitted into
the frame. The most we can hope for from any frame
is that it will not have distorted the simplicity which
was originally present ; whilst an ill-chosen frame may
play havoc with the natural simplicity of things. We
have seen that the simplicity of planetary motions was
obscured in Ptolemy's frame, and became apparent in
Copernicus's frame. But for ordinary terrestrial pheno-
mena the position is reversed and Ptolemy's frame
allows their natural simplicity to become apparent. In
Copernicus's frame the most simple phenomena are
brought about by highly complicated processes which
mutually cancel one another. Ordinary objects contract
and expand as they are moved about, and the changes
are concealed by an elaborate conspiracy in which all
the quantities of nature — electrical, optical, mechanical,
gravitational — have joined. In Copernicus's frame we
RELATIVITY 13
have a great complication of description which has no
counterpart in anything occurring in the external world ;
because the terms of our description refer to the irrele-
vant process of fitting into the selected frame of space
and time. This elaborate Copernican scheme rather
reminds one of the schemes of the White Knight —
But I was thinking of a plan
To dye one's whiskers green,
And always use so large a fan
That they could not be seen.
We do not deny the subtlety and the remarkable
efficiency of the plan ; but we may be allowed to question
whether it is the simplest interpretation of the drab
monotony of the face of nature presented to us. The
simple fact is that a terrestrial or Ptolemaic frame fits
naturally the terrestrial phenomena, and a solar or
Copernican frame fits the phenomena of the solar system ;
but we cannot make one frame serve for both without
introducing irrelevant complications.
We go beyond Copernicus nowadays, and are not
content with a visit to the sun. Why choose the sun
rather than some other star in order to obtain an undis-
torted view of things ? The astronomer now places
himself so as to travel with the centre of gravity of the
stellar universe, and is not even then quite satisfied.
The physicist dreams of a land of Weissnichtwo, which
shall be truly at rest in the ether. We realize the dis-
tortion imported into the world of nature by the parochial
standpoint from which we observe it, and we try to place
ourselves so as to eliminate this distortion — so as to
observe that which actually is. But it is a vain pursuit.
Wherever we pitch our camera, the photograph is
necessarily a two-dimensional picture distorted accord-
ing to the laws of perspective; it is never a true
semblance of the building itself.
i 4 THETHEORYOF
We must try another plan. I do not think we can ever
eliminate altogether the human element in our conception
of nature; but we can eliminate a particular human
element, namely, this framework of space and time. If our
thought mustbe anthropocentric,it neednot begeocentric.
Nor are we permanently better off if we merely substitute
the space-time frame of some other star or standard of
motion. We must leave the frame entirely indeterminate.
When we do that, we find that the world common to all
observers — in which each observer traces a different
space-time frame according to his own outlook — is a
world of four dimensions. When we look at any object,
say a chair, the impression on our eyes is a two-dimen-
sional picture depending on the position from which we
are looking ; but we have no difficulty in conceiving of
the chair as a solid object, not to be identified with any
one of our two-dimensional pictures of it, but giving rise
to them all as the position of the observer is varied.
We must now realize that this solid chair in three
dimensions is itself only an appearance, which changes
according to the motion of the observer, and that there
is a super-object in four-dimensions, not to be identified
with the three-dimensional chair in Ptolemy's scheme,
or the same chair in Copernicus's scheme, but giving
rise to both these appearances. The synthesis of a
three-dimensional chair from a number of flat pictures
is easy to us because we are accustomed to assume
different positions in rapid succession ; indeed our two
eyes give us slightly different points of view simul-
taneously. By sheer necessity four brains have been
forced to construct the conception of the solid chair to
combine these changing appearances. But we do not
vary our motion to any appreciable extent and our
brains have not hitherto been called upon to combine
the appearances for different motions ; thus the effort
RELATIVITY 15
which we now ask the brain to make is a novel one.
That explains why the result seems to transcend our
ordinary mode of thought.
The discovery, or one should rather say the redis-
covery, of the world of four dimensions is due to
Minkowski. Einstein had worked out fully the relations
between the frames of space and time for observers with
different motions. To the genius of Minkowski we owe
the realization that these frames are merely systems of
partitions arbitrarily drawn across a four-dimensional
world which is common to all observers.
There is a strange delusion that the fourth dimension
must be something wholly beyond the conception of the
ordinary man, and that only the mathematician can be
initiated into its mysteries. It is true that the mathema-
tician has the advantage of understanding the technical
machinery for solving the problems which may arise in
studying the world of four dimensions ; but as regards
the conception of the four dimensions of the world his
point of view is the same as that of anybody else. Is it
supposed that by intense thought he throws himself
into some state of trance in which he perceives some
hitherto unsuspected direction stretching away at right
angles to length, breadth, and thickness ? That would
not be much use. The world of four dimensions, of
which we are now speaking, is perfectly familiar to
everybody. It is obvious to every one— even to the
mathematician — that the world of solid and permanent
objects has three dimensions and no more ; that objects
are arranged in a threefold order, which for any par-
ticular individual may be analysed into right-and-left,
backwards-and-forwards, up-and-down. But it is no less
obvious to every one that the world of events is of four
dimensions ; that events are arranged in a fourfold
order, which in the experience of any particular indi-
16 THE THEORY OF
vidual will be analysed into right-and-left, backwards-
and-forwards, up-and-down, sooner-and-later. The subject
of our study is external nature, which is a world of
events, common to all observers but represented by
them differently in their parochial frames of space and
time ; it is obvious to the most commonplace experience
that this absolute world contains a fourfold order. 1
The news that the events around us form a world of
four dimensions is as stale as the news that Queen Anne
is dead. The reason why the relativist resurrects this
ancient truism is because it is only in this undissected
combination of four dimensions that the experiences of
all observers meet. In our own experience one dimen-
sion is sharply separated from the other three and is
distinguished as time; but our experience is solely
terrestrial, and if we insist on building the scheme of
nature on purely terrestrial experience we are limiting
ourselves to the mediaeval geocentric system of the world.
We have been accustomed to regard the enduring
world as composed of a continuous succession of instan-
taneous states, as though the world of events were
stratified. Each event is supposed to lie in a definite
instant or stratum, and the orderly succession of these
strata makes up the whole of reality. The instant ' now '
represents one such stratum running throughout the
universe. Indeed we are accustomed to extend it beyond
the universe, and we even use the word 'now' with
reference to the existence of those who have passed
away from the material world. The investigations of
the relativity theory show incontrovertibly that this
supposed stratification is an illusion ; there is not the
1 The relativity theory does not suggest that there is such
a thing in nature as a four-dimensional space. The whole object
of the recognition of the four-dimensional world is to eliminate the
harassing frame of space.
RELATIVITY 17
slightest evidence for such a view of world-structure.
The instantaneous state, which we have hitherto taken
to be a natural stratum in the four-dimensional world of
events, is merely an arbitrary partition created by our-
selves to correspond with our geocentric outlook. We
can take a differently inclined partition, 1 that is to say,
a section which includes on the one side of us events
which happened a little while ago and on the other side
of us events which have not yet happened ; such a
farcical combination is in every way equivalent to our
so-called instantaneous state, and indeed it is an instan-
taneous state according to the outlook of some non-
terrestrial observer with suitably assigned motion.
It is so contrary to our natural prejudices to recognize
that the world-wide instant now is created by ourselves
and has no existence apart from our geocentric outlook,
that I will spend a few moments trying to show its
artificiality. When I say that I am conscious of an
instant now, I am only conscious of it in so far as it is
here — inside me. What then has led me to imagine
that there exists a continuation of i.t outside me ? It is
because I look out on the world and see various events
1 The inclination must not exceed a certain limit. This limiting
angle may be regarded as a fundamental constant of the world-
structure, and owing to its fundamental character it appears in
many kinds of phenomena ; for example, it determines the velocity
of propagation of light. The instant on the sun which is simulta-
neous with a given instant on the earth is indeterminate (varying
according to the space and time frame employed) but only within
a range of 16 minutes. Any event on the sun happening before
this 16 minutes is absolutely in the past, all observers agreeing on
this point ; in fact it would be possible for us to have already
received a wireless message announcing its occurrence. Events
after the 16 minutes are in the absolute future. The neutral zone
which is (absolutely) neither past nor future becomes propor-
tionately wider as the distance increases ; at the nearest fixed
star it extends to 8 years, and at the most distant stars yet known
it reaches 400,000 years.
18 THE THEORY OF
happening ' now ', so that I jump to the conclusion that
this instant of which I am conscious has to be extended
to include them. But that idea is another inheritance
from the dark ages, overthrown by Romer in 1675. It
is not the events themselves but the sense-impressions
to which they give rise which are happening in the
instant now. So my justification for placing the events
outside me in the instants of which I am conscious has
entirely disappeared. Unfortunately, however, the crude
outlook was not abolished, but patched up ; it was found
that the immediate difficulties could be met by locating
the external events not in the instant of our visual
perception of them but in an instant which we had
experienced a little time back — allowing, as we say, for
the time of propagation of light. Thus our instants
were still made to extend through space ; but they were
carried like partitions among the events by an artificial
process of computation, and no longer by immediate
intuition. The relativity theory recognizes these world-
wide instants for what they are — artificial partitions con-
structed for purposes of calculation. I may add that it
in no way tampers with the local instants which form the
stream of our consciousness; it fully recognizes that
the chain of events in such a time-succession is a series
of an entirely distinctive character from the succession
of points along a line in space. Those who suspect that
Einstein's theory is playing unjustifiable tricks with
time should realize that it leaves entirely untouched
that time-succession of which we have intuitive know-
ledge, and confines itself to overhauling the artificial
scheme of time which Romer first introduced into
physics.
The study of the four-dimensional world of events
gives us a new insight into the processes of nature
because it removes the irrelevant stratification in a par-
RELATIVITY 19
ticular direction — the instantaneous states — which we
have so unnecessarily introduced in our customary out-
look. When this stratification is ignored we are enabled
to see the processes in their simplest aspect, though not,
of course, in their most familiar aspect. We must dis-
tinguish between 'simplicity and familiarity ; a pig may
be most familiar to us in the form of rashers, but the
unstratified pig is a simpler object of study to the
biologist who wishes to understand how the animal
functions.
I will conclude this part of the argument with an
experimental application which illustrates the power of
Einstein's method. Much study has of late been given
to electrons moving with very high speeds ; for example,
the /3 particles shot off from radioactive substances are
negative electrons which sometimes attain speeds of
100,000 miles a second. It is found by experiment that
the rapid motion produces an increase of mass of these
particles. I want to show that the theory of relativity
gives a very simple explanation of just how this increase
of mass occurs. But I must first remark that an ex-
planation had been previously given which had generally
been accepted as satisfactory. The phenomenon was
actually predicted by J. J. Thomson before relativity was
thought of; because, assuming that the mass of a /?
particle is of electrical origin, an application of Maxwell's
equations shows that it ought to increase with velocity.
But the precise law of increase cannot be predicted on
this basis, since various plausible assumptions lead to
slightly different results. Moreover, Maxwell's equa-
tions are after all only empirical laws, with a mystery of
their own ; it was a notable advance to connect the
change of mass at high speeds with other phenomena
whose strangeness has disappeared by long familiarity,
but there is still scope for a more far-reaching explana-
20 THETHEORYOF
tion. Einstein takes us straight to the root of the
mystery, and he clears up one point which was mislead-
ing, if not actually wrong, in the older explanation.
The change of mass does not in any way depend on
whether the mass is of electrical origin or not ; it arises
simply from the fact that mass is a relative quantity,
depending by its definition on the relative quantities
length and time. Let us look at the £ particle from its
own point of view ; it is just an ordinary electron in no
way different from any other. ' But it is travelling
unusually rapidly ? ' ' That ', says the electron, ' is a
matter of opinion. So far as I am aware I am at rest, if
the word " rest " has any meaning. In fact I was just
contemplating with amazement your extraordinary speed
of 100,000 miles a second with which you are shooting
past me.' Of course our motion is of no particular con-
cern to the electron, and it will not modify its constitution
on our account; so it keeps its mass, radius, electric
field, &c, equal to the standard constants applying to
electrons in general. These terms are relative, and
refer therefore to some particular frame of space and
time— clearly the frame appropriate to an electron in
self-contemplation, viz. the one with respect to which it
is at rest. But this frame is not the usual geocentric
frame to which we refer quantities such as length, time,
and mass ; there is a difference of 100,000 miles a second
between our station of observation and that of the P
particle in self-contemplation. It is a mere matter of
geometry to discover what the /? particle's lengths and
times become when referred to the partitions which we
have drawn across the world. But when we calculate
the consequential change of mass resulting from the
changes of length and time, we find that it should be
increased in precisely the proportion indicated by the
most refined experiments.
RELATIVITY 21
The point is that every electron, at rest or in motion,
is a perfectly constant structure; but we distort it by
fitting it into the space-time frame appropriate to our
own motion with which the electron has no concern.
The greater our motion with respect to the electron, the
greater will be the distortion. The distortion is not
produced by any physical agency at work in the electron ;
it is a purely subjective distortion depending on our trans-
formation of the reference frame of space and time.
This distortion involves a change in our physical de-
scription of the electron in terms of mass, shape, size ;
and in particular the change of mass agrees precisely
with that found experimentally.
You see that it is not altogether idle discussing the
natural space-time frames for observers moving with
huge velocities. We know of no animate observers
with these speeds ; but we do know of inanimate material
objects. Their common resemblance is obscured when
we refer them indiscriminately to our irrelevant geocen-
tric frame ; we think they have altered their properties,
varied in mass, and so on ; but the resemblance is
restored when we refer each individual to the frame
appropriate to it, and so describe them all in comparable
terms.
Our measurements of distance in space are found to
be subject to certain laws — the laws of geometry. But it
has now become impossible to regard the subject of
space-geometry as complete in itself. Consider a triangle
formed by three points (or events) in the four-dimen-
sional world ; if we happen to have drawn our instan-
taneous strata so that the three points lie in one stratum,
then the triangle is a space-triangle and its properties
fall within the scope of our classical geometry. But
another observer will draw his strata in a different
direction, and for him the triangle would be partly in
22 THETHEORYOF
space and partly in time, so that it would not be a fit
subject for space-geometry. The subject of geometry
is in a desperate condition, because Copernicus and
Ptolemy not merely disagree as to the geometry of a
configuration ; they even disagree as to whether a given
configuration is one to which space-geometry is applic-
able. It is clear that to save it we must extend our
geometry so as to include time as well as space. Let
me give an illustration of this extension. The terrestrial
observer can have a space-triangle (formed by three
points or events at the same instant) whose sides he can
measure with scales ; he can also have a ' time-triangle ',
formed by three events on different dates, whose sides
he must measure with clocks. 1 You all know the law of
the space-triangle — that if you measure with a scale from
A to B and from B to C the sum of the readings is
always greater than the measure from A to C. It is not
so well known that there is a precisely analogous law for
the time-triangle — that if you measure with a clock from
A to B and from B to C the sum of the readings is
always less than the reading of a clock measuring directly
from A to C. In the space-triangle any two sides are
together greater than the third side ; in the time-triangle
two sides are together less than the third side. 2 Both
these laws must be combined in our general geometry
1 The three events must not be at the same place since that
would give a t\mt-line not a triangle. The clock must move so
that the two events whose time-distance is to be determined both
happen where it is, just as the scale must be directed so that the
two points fall on it. You are not allowed to ' bend ' the clock,
i. e. apply force so as to make it move with other than uniform
velocity, any more than you are allowed to bend the scale by
applying force.
2 Of course, it is not true that any two sides are less than the
third side. A clock, unlike a scale, can only measure in one
direction, viz. from past to future, so that the sides AB + BC and
AC can be chosen in only one way.
RELATIVITY 23
of four dimensions, so that it will not be quite so simple
a geometry as that to which we are accustomed. 1
But the point to which I would especially direct
attention is this. Evidently the proposition which I
have given you about time-triangles cannot be dissociated
from the corresponding proposition about space-triangles.
When we give up the mediaeval geocentric standpoint,
we must recognize that they belong to one geometry, of
which our ordinary space-geometry is only a part or
projection. But if you examine the proposition about
time-triangles, you will see that it is a statement about
the behaviour of clocks when they move about, a subject
which obviously comes under the heading of mechanics,
When we deal with the four-dimensional world we can
no longer distinguish between geometry and mechanics.
They become the same subject. When we have com-
pletely mastered the geometry of the world of events,
we shall have inevitably learnt the mechanics of it.
That is why Einstein, studying the geometry of the
world and discovering that it was strictly non-Euclidean,
found that he was at the same time studying the
mechanical force of gravitation. And when he had
made up his mind which of the possible varieties of non-
Euclidean geometry was obeyed, and so settled the laws
of the new geometry, the same decision settled the law
of gravitation— a law approximating to, but not identical
with, the law which Newton had given.
Here a wide vista opens before us. We see that two
great divisions of mathematical physics, viz. geometry
and mechanics, have met in the four-dimensional
world. It is not merely that mechanical problems can
be treated by formulae originally belonging to pure
1 This involves only a comparatively trifling generalization of
Euclidean geometry, not to be confused with the ' non-Euclidean '
geometry introduced later in the lecture.
24 THE THEORY OF
geometry ; that device has long been in use. Experi-
mental geometry and mechanics actually relate to the
same subject-matter; and the young student who dis-
covers experimental laws with ruler and compasses and
cardboard figures, and later goes on to pendulums and
spring-balances, is developing a single subject which
cannot be divided any more than the subject of magnet-
ism can be divided from electricity.
It is through this unification of geometry and mechanics
that I should like to approach the problem of gravita-
tion, showing that a field of force is a manifestation of
the geometry of space and time. But I fear that that
would be too technical ; so we will approach it from a
different angle.
We have shown that the contemplation of the world
from the standpoint of a single observer is liable to dis-
tort its simplicity, and we have tried to obtain a juster
idea by taking into account and combining other points
of view. The more standpoints the better. Let us now
consider another point of view, which we have not
previously thought about — the point of view of an ob-
server who has tumbled out of an aeroplane and is
falling headlong. In many respects his is an ideal situa-
tion — temporarily. Unfortunately on terra firma we are
continually subjected to a very disturbing influence ; we
undergo a terrific bombardment by the molecules of the
ground, which are hammering on the soles of our boots
with a total force of some ten stone weight pressing us
upwards. Now our bodies are the scientific appliances
which we use to make our common observations of the
world. I am sure that no physicist would permit any
one to enter his laboratory and hammer on his clocks
and galvanometers whilst he was observing with them ;
at any rate he would think it necessary to apply some
corrections for the effect of the disturbance. Let us then
RELATIVITY 25
allow ourselves to fall freely in vacuo ; then we shall be
free from this disturbing bombardment and able to
take a much more natural view of what is going on
around us.
Whilst falling, we perform the experiment of letting
go an apple held in the hand. The apple is now free,
but it cannot fall any more than it was falling already ;
consequently it remains poised in contact with our hand.
In our new outlook — in our new frame of space and
time — an apple does not drop. There is no mysterious
force accelerating it. And remember that this new
frame of space and time is the natural frame of a free
observer ; whereas the old frame, in which the mysterious
accelerating force occurred, was the frame of a very
much disturbed observer. It is true that when we look
down at the earth we see trees and houses rushing up
to meet us ; but there is no mystery about that. There is
an obvious cause for it ; plainly they are being propelled
upwards from below by that molecular bombardment
which I have mentioned. You see that the apple's view
of things is simpler than Newton's. Newton had to
invent a mysterious force dragging the apple down ; the
apple observes only a familiar physical agency propelling
Newton up.
It is not my purpose to emphasize unduly the superior-
ity of the apple's view over Newton's, but rather to
regard both on an equal footing. I have perhaps been
a little unfair to Newton. His position on the surface
of the earth was unfortunate, but he would have been
perfectly content to be at the centre of the earth, where
he could have remained without support, i. e. without
disturbance by molecular bombardment. From there
he would still have observed the well-known acceleration
of the apple; and the apple would have observed
a corresponding acceleration of Newton without any
26 THETHEORYOF
molecular bombardment causing it. From either point
of view there is a mysterious agent at work. How
shall we picture to ourselves this agent? Shall we
picture it as a force— a tug of some kind ? But if so, to
which of them is the tug applied ? If we take the stand-
point of Newton the tug is applied to the apple, if the
standpoint of the apple the tug is applied to Newton ; so
that in our synthesis of all standpoints we cannot decide
which is being tugged, and the picture of gravitation as
a tugging agent becomes impossible. Einstein replaces
it by a different picture, which we shall perhaps better
understand if we compare it with a very similar revolu-
tion of scientific thought which occurred long ago.
The ancients believed that the earth was flat. The
small portion of its surface with which they were chiefly
concerned could be represented without serious dis-
tortion on a flat map. As more distant countries were
added, it would be natural to think that they also could
be included in the flat map. You have all seen such
maps of the world, e. g. Mercator's projection, and you
will remember how Greenland appears enormously
exaggerated in size. Now those who adhered to the
flat-earth theory must hold that the flat map gives the
true size of Greenland. How then would they explain
that travellers in that country reported that the distances
were much shorter? They would, I suppose, invent
a theory that a demon resided in -that country who
helped travellers on their way, making the journeys
appear much shorter than they ' really ' were. No
doubt the scientists would preserve their self-respect by
using some Graeco-Latin polysyllable instead of the word
' demon ', but that must not disguise from us the fact
that they were appealing to a deus ex machina. The
name demon is rather suitable, however, because he has
the impish characteristic that we cannot pin him down
RELATIVITY 27
to any particular locality. We might equally well start
our flat map with its centre in Greenland ; then it would
be found that journeys there were quite normal, and
that the activities of the demon were disturbing travellers
in Europe. We now recognize that the true explana-
tion is that the earth's surface is curved ; and the
demoniacal complications appeared because we were
forcing the earth's surface into an inappropriate flat
frame which distorts the simplicity of things.
What has happened in the case of the earth has hap-
pened also in the case of the world, and a similar
revolution of thought is needed. An observer, say at
the centre of the earth, finds that there is a frame of
space and time — a flat or Euclidean frame — in which he
can locate things happening in his neighbourhood with-
out distorting their natural simplicity. There is no
gravitation, no tendency of bodies to fall, so long as the
observer confines his observations to his immediate
neighbourhood. He extends this frame of space and
time to greater distances, and ultimately to the earth's
surface where he encounters the phenomenon of falling
apples. This new phenomenon must be accounted for,
so he invents a deus ex machina which he calls gravita-
tion to whose activities the disturbance is attributed.
But we have seen that we may just as well start with
the falling apple. It has a flat frame of space and time
into which phenomena in its neighbourhood fit without
distortion ; and from its point of view bodies near it do
not undergo any acceleration. But when it extends this
frame farther afield, the simplicity is lost ; and it too has
to postulate the demon force of gravitation existing in
distant parts, and for example causing undisturbed
objects at the centre of the earth to fall towards it. As
we change from one observer to another — from one flat
space-time frame to another — so we have to change the
28 THE THEORY OF
region of activity of this demon. Is not the solution
now apparent ? The demon is simply the complication
which arises when we force the world into a flat Euclidean
space-time frame into which it does not fit without
distortion. It does not fit the frame, because it is not
a Euclidean or flat world. Admit a curvature of the
world and the mysterious disturbance disappears.
Einstein has exorcized the demon.
Einstein, recognizing that in the phenomena of gravi-
tation he was not dealing with a ' tug ' but with a curva-
ture of the world, had to reconsider the law of gravitation.
He could not make any possible law of curvature
correspond exactly with the previously assumed law of
tugging. Thus he was led to propound a new law of
gravitation — a law which in most practical cases differs
very little from that of Newton, although it has an
essentially different foundation. I need not here dwell
on the very remarkable way in which Einstein's emenda-
tion of the law of gravitation has been confirmed both
by the anomalous secular change in the orbit of the
planet Mercury, and by the observed displacement of
the stars near the sun during the total eclipse of 1919.
I rrught, however, remind you that in the latter observa-
tion the point at issue between Newton's and Einstein's
theory was not the existence of a deflexion of light-rays
passing near the sun but the amount of the deflexion,
Einstein predicting twice the deflexion possible on the
Newtonian theory. The larger deflexion was quantita-
tively confirmed by the eclipse observations. Einstein's
main achievement is a new law, not a new explanation,
of gravitation. He attributes the gravitation of massive
bodies to a curvature of the world in the region sur-
rounding them and so throws a flood of light on the
whole problem ; but he is not primarily concerned to
explain how material bodies produce (or are associated
RELATIVITY 29
with) this curvature of the world around them, nor how
this curvature is made subject to a law. Although it
would be an entire misunderstanding of Einstein's atti-
tude in propounding the general relativity theory to
regard it as a search for an explanation of gravitation,
nevertheless I think that the further following up of his
ideas has led to a genuine explanation as complete as
could be desired. But I am not going to give you the
explanation in this lecture ; sometimes an explanation
requires a great deal of explaining. 1
I think that we can without mathematics form a general
1 The following brief outline will give a hint of the nature
of the explanation. Einstein's law of gravitation is usually ex-
pressed as a set of ten very lengthy differential equations ; these
equations are exactly equivalent to the geometrical statement that
'the radius of spherical curvature of any 3-dimensional section of
the 4-dimensional world is a universal constant length, the same
for all points of the world and for all directions of the section '.
The law therefore implies that the world has a certain type of
homogeneity and isotropy (not, however, the complete homogeneity
and isotropy of a sphere). To explain the law of gravitation and
the phenomena governed by it, we have to explain how this
isotropy and homogeneity is secured. Our explanation is that the
homogeneity and isotropy is not initially in the external world,
but in the measurements which we make of it. It is introduced
in all our operations of measurement, because the appliances
which we use for measurement are themselves part of the world.
In the earlier part of this lecture we saw that the contraction ot
the arm turned from horizontal to vertical is not detected by
measurements made with a yard-measure which shares the con-
traction ; in the same way any anisotropy of the world does not
appear in measurements of it by appliances which, being part of
the world, share the same anisotropy. The law of gravitation
therefore arises from the fact that a certain type of non-homogeneity
and non-isotropy of the world cannot come into observational
experience, because it is necessarily eliminated in all observations
and measurements made with material appliances. The orderly
phenomena of gravitation are due to the absence of certain con-
ceivable effects. We have been trying to find a key to the mystery ;
but the secret of the lock lies not in the key but in the wards.
3 o THETHEORYOF
idea of why Einstein found it necessary to amend
Newton's law of gravitation. Let us return to the
illustration of the pig, and imagine that we wish to dis-
cover the law governing the distribution of fat and lean
in the animal. From the breakfast-table standpoint a
plausible type of law would be that half of each rasher
is fat and the other half lean ; and if this turned out to
to be confirmed very approximately by observation we
might well imagine that we had discovered the exact
law of porcine structure. But the case is altered if
we give up the breakfast-table standpoint and contem-
plate the animal in a more general way, remembering
that he has not been designed with any particular
reference to the series of rashers into which our grocer
has chosen to slice him. We must now look for a
different type of law altogether. Two possibilities may
arise. We may find that our proposed law, although
expressed in breakfast-table parlance, is nevertheless
equivalent to a possible biological law ; it may be imme-
diately capable of translation into a more general state-
ment which makes no reference to a particular stratifica-
tion. But on the other hand, it may happen that the
suggested law cannot be freed from this reference to a
particular system of slicing. In that case we can only
regard it as approximate, perhaps holding fairly well for
the slices of which we have most experience but
becoming less and less accurate in the more tortuous
parts of the animal. Both these cases are illustrated in
Einstein's modifications of classical theory. Newton's
law of gravitation explicitly refers to a space-time frame
and therefore to a world stratified into instantaneous
states. It proves to be impossible to free it from this
reference to a particular stratification without modifying
it. In fact if the crucial astronomical observations had
shown that Newton's law and not Einstein's was the
RELATIVITY 31
exact law df gravitation, this would have been evidence
of a real stratification of the structure of the world — a
stratification revealed by no other phenomena. Einstein's
law is the simpler law because it is consistent with what
we now know of the general plan of world-structure ;
Newton's law could only be made possible by intro-
ducing a novel and specialized feature — a stratified
arrangement of structure — which is not revealed in any
other phenomena.
Maxwell's laws of electromagnetism afford an example
of the other type. These, it is true, are stated as
relating to the particular slices of the world of events,
which are served up to us like rashers instant by instant.
But they can be restated, without alteration of effect, in
a form making no reference to slices. This is a very
remarkable property of Maxwell's equations which was
quite unknown at the time they were first put forward.
It was brought to light much later by the researches of
Larmor and Lorentz. In consequence of this Einstein
is able to take over the whole classical theory of electro-
magnetism unaltered ; he restates it so as to show how
it applies generally and is not bound up with the purely
terrestrial point of view, but he does not amend the
laws. He metes out different treatment to the gravita-
tional laws and electromagnetic laws, because he finds
the latter already adapted to his scheme.
If I have succeeded in my object, you will have
realized that the present revolution of scientific thought
follows in natural sequence on the great revolutions at
earlier epochs in the history of science. Einstein's
special theory of relativity, which explains the indeter-
minateness of the frame of space and time, crowns the
work of Copernicus who first led us to give up our
insistence on a geocentric outlook on nature ; Einstein's
general theory of relativity, which reveals the curvature
32 THEORY OF RELATIVITY
or non-Euclidean geometry of space and time, carries
forward the rudimentary thought of those earlier astro-
nomers who first contemplated the possibility that their
existence lay on something which was not flat. These
earlier revolutions are still a source of perplexity in
childhood, which we soon outgrow ; and a time will
come when Einstein's amazing revelations have likewise
sunk into the commonplaces of educated thought.
To free our thought from the fetters of space and
time is an aspiration of the poet and the mystic, viewed
somewhat coldly by the scientist who has too good
reason to fear the confusion of loose ideas likely to
ensue. If others have had a suspicion of the end to be
desired, it has been left to Einstein to show the way to
rid ourselves of these ' terrestrial adhesions to thought '.
And in removing our fetters he leaves us, not (as might
have been feared) vague generalities for the ecstatic
contemplation of the mystic, but a precise scheme of
world-structure to engage the mathematical physicist.
Printed in England at the Oxford University Press
PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET
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QC Eddington, (Sir) Arthur
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E4-2 The theory of relativity
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