The foundations of Einstein's theory of gravitation

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B M 252 SE3

i&fjiH&BEiKn

THE LIBRARY

OF

THE UNIVERSITY
OF CALIFORNIA

IN MEMORY OF

Professor Richard Fayram
1920-1956

THE FOUNDATIONS OF
EINSTEIN'S THEORY OF GRAVITATION

THE FOUNDATIONS OF

EINSTEIN'S THEORY OF

GRAVITATION

BY

ERWIN FREUNDLIGH

DIRECTOR OF THE EINSTEIN TOWER
WITH A PREFACE BY

ALBERT EINSTEIN

TRANSLATED FROM THE FOURTH GERMAN EDITION, WITH TWO ESSAYS, BY

HENRY L. BROSE

CHRIST CHURCH, OXFORD

WITH AN INTRODUCTION BY

H. H. TURNER, D.Sc., F.R.S.

SAVILIAN PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF OXFORD
WITH FIVE DIAGRAMS

NEW YORK

E. P. DUTTON AND COMPANY
PUBLISHERS

PRINTED IN GREAT BRITAIN

Add to Lib.

PREFACE

DR. FREUNDLICH has undertaken in
the following essay to illumine the
ideas and observations which gave
rise to the general theory of relativity so as
to make them available to a wider circle of
readers.

I have gained the impression in perusing
these pages that the author has succeeded in
rendering the fundamental ideas of the theory
accessible to all who are to some extent con-
versant with the methods of reasoning of the
exact sciences. The relations of the problem
to mathematics, to the theory of knowledge,
physics and astronomy are expounded in a
fascinating style, and the depth of thought of
Riemann, a mathematician so far in advance
of his time, has in particular received warm
appreciation.

Dr. Freundlich is not only highly qualified
as a specialist in the various branches of know-
ledge involved to demonstrate the subject ; he
is also the first amongst fellow-scientists who
has taken pains to put the theory to the test.

May his booklet prove a source of pleasure
to many !

A. EINSTEIN

INTRODUCTION

THE Universe is limited by the properties of
light. Until half a century ago it was
strictly true that we depended upon our
eyes for all our knowledge of the universe, which
extended no further than we could see. Even the
invention of the telescope did not disturb this
proposition, but it is otherwise with the invention
of the photographic plate. It is now conceivable
that a blind man, by taking photographs and ren-
dering their records in some way decipherable by
his fingers, could investigate the universe ; but
still it would remain true, that all his knowledge of
anything outside the earth would be derived from the
use of light and would therefore be limited by its
properties. On this little earth there is, indeed,
a tiny corner of the universe accessible to other
senses : but feeling and taste act only at those
minute distances which separate particles of matter
when " in contact : " smell ranges over, at the
utmost, a mile or two ; and the greatest distance
which sound is ever known to have travelled (when
Krakatoa exploded in 1883) is but a few thousand
miles — a mere fraction of the earth's girdle. The
scale of phenomena manifested through agencies
other than light is so small that we are unlikely
to reach any noteworthy precision by their study.
Few people who are not astronomers have spent
much thought on the limitations introduced by the

vii

viii THEORY OF GRAVITATION

news agency to which we are so profoundly indebted.
Light comes speedily but has far to travel, and some
of the news is thousands of years old before we get
it. Hence our universe is not co-existent : the part
close around us belongs to the peaceful present,
but the nearest star is still in the midst of the late
War, for our news of him is three years old ; other
stars are Elizabethan, others belong to the time of
the Pharaohs ; and we have alongside our modern
civilization yet others of prehistoric date. The
electric telegraph has accustomed us to a world
in which the news is approximately of even date :
but our forefathers must have been better able,
from their daily experience of getting news many
months old, to realize the unequal age of the universe
we know. Nowadays the inequality is almost
entirely the concern of the astronomer, and even he
often neglects or forgets it. But when fundamental
issues are at stake, the time taken by the messenger
is an essential part of the discussion, and we must
be careful to take account of it, with the utmost
precision.

Our knowledge that light had a finite velocity
followed on the invention of the telescope and the
discovery of Jupiter's satellites : the news of their
eclipses came late at times and these times were
identified as those when Jupiter was unusually far
away from us. But the full consequences of the
discovery were not realized at first. One such
consequence is that the stars are not seen in their
true places, that is in the places which they truly
held when the light left them (for what may have
happened to them since we do not know at all — they
may have gone out or exploded). Our earth is
only moving slowly compared with the great haste
of light : but still she is moving, and consequently
there is " aberration " — a displacement due to the

INTRODUCTION ix

ratio of the two velocities, easy enough to recognize
now, but so difficult to apprehend for the first time
that Bradley spent two years in worrying over the
conundrum presented by his observations before he
thought of the solution. It came to him unexpect-
edly, as often happens in such cases. In his own
words — " at last when he despaired of being able
to account for the phenomena which he had ob-
served, a satisfactory explanation of them occurred
to him all at once when he was not in search of it."
He accompanied a pleasure party in a sail upon the
river Thames. The boat in which they were was
provided with a mast which had a vane at the top
of it. It blew a moderate wind, and the party
sailed up and down the river for a considerable
time. Dr. Bradley remarked that every time the
boat put about, the vane at the top of the boat's
mast shifted a little, as if there had been a slight
change in the direction of the wind. The sailors
told him that this was due to the change in the
boat, not the wind : and at once the solution of
his problem was suggested. The earth running
hither and thither round the sun resembles the boat
sailing up and down the river : and the apparent
changes of wind correspond to the apparent changes
in direction of the light of a star. But now comes
a point of detail — does the vane itself affect the wind
just round it ? And, similarly, does the earth
itself by its movement affect the ether just round
it, or the apparent direction of the light waves ?
This question suggested the famous Michelson and
Morley experiment (Phil. Mag., Dec. 1887). It is
curious to think that in the little corner of the
universe represented by the space available in a
laboratory an experiment should be possible which
alters our whole conceptions of what happens in
the profoundest depths of space known to us^but

x THEORY OF GRAVITATION

so it is. The laboratory experiment of Michelson
and Morley was the first step in the great advance
recently made. It discredited the existence of the
virtual stream of ether which is the natural an-
tithesis to the earth's actual motion. It was,
indeed, open to question whether restrictions of a
laboratory might not be responsible for the result :
for the ether stream might exist, but the laboratory
in which it was hoped to detect it might be in a
sheltered eddy. When bodies move through the
air, they encounter an apparent stream of opposing
air, as all motorists know : but by using a glass
screen shelter from the stream can be found. And
even without such special screening, there may be
shelter. When a pendulum is set swinging in
ordinary air, it is found from experiments on clocks
that it carries a certain amount of air along with
it in its movement, although the portion carried
probably clings closely to the surface of the pen-
dulum. A very small insect placed in the region
might be unable to detect the streaming of the air
further out. In a similar way it seemed possible
that as the earth moved through the ether such
tiny insects as the physicists in their laboratories
might be in a part of the ether carried along with
the earth, in which they could not detect the stream-
ing outside. But another laboratory experiment,
this time by Sir Oliver Lodge, discredited this ex-
planation, and it was then suggested as an alternative
that distances were automatically altered by move-
ment.

It may be well to explain briefly the significance
of this alternative. The Michelson-Morley experi-
ment depended on the difference between travelling
up and down stream, and across it. To use a few
figures may be the quickest way of making the point
clear. Suppose a very wide, perfectly smooth stream

INTRODUCTION xi

running at 3 miles an hour, and that oarsmen are to
start from a fixed point O in midstream, row out
in any direction to a distance of 4 miles from O,
and back again to the starting-point O. Which is
the best direction to choose ? We shall probably
all agree that it will be either directly up and down
stream, or directly across it, and we may confine
attention to these two directions. First suppose an
oarsman A starts straight across stream. To keep
straight he must set his boat at an angle to the
stream. If he reaches his 4 mile limit in an hour,
the stream has been virtually carrying him down
3 miles in a direction at right angles to his course :
and the well-known relation between the sides of
a right-angled triangle tells us that he has effectively
pulled 5 miles in the hour. It will take him simi-
larly an hour to come back, and the total journey
will involve an effective pull of 10 miles.

Now suppose another oarsman, B, of equal skill
elects to row up stream. In two hours he could
pull 10 miles if there were no stream ; but since
meantime the stream has pulled him back 6 miles
by " direct action " he will have only just reached
the 4 mile limit from the start, and has still his
return journey to go. No doubt he will accomplish
this pretty quickly with the stream to help him,
but his antagonist has already got home before he
begins the return. We might have let him do his
quick journey down stream first, but it is easy to
see that this would gain him no ultimate advantage.

Michelson and Morley sent two rays of light on
two journeys similar to those of the oarsmen A and
B. The stream was the supposed stream of ether
from east to west which should result from the
earth's movement of rotation from west to east.
They confidently expected the return of A before
that of B, and were quite taken aback to find the

xii THEORY OF GRAVITATION

two reaching the goal together. In the aquatic
analogy of which we have made use, it would no
doubt be suspected that B was really the faster
oar, which might be tested by interchanging the
courses ; but there are no known differences in the
velocity of light which would allow of a parallel
explanation. There was, however, the possibility
that the distances had been marked wrongly, and
this was tested by interchanging them, without
altering the " dead-heat."

Now there are several alternative explanations of
this result. One is that the ether does not itself
exist, and therefore there is no stream of it, actual
or apparent ; and it is to this sweeping conclusion
that modern reasoning, following recent experiments
and observations, is tending. The possibility of sav-
ing the ether by endowing it with four dimensions
instead of three is scarcely calculated to satisfy
those who believed (until recently) that we knew
more about the ether than about matter itself. They
saved the ether for a time by an automatic shortening
of all bodies in the direction of their movement,
which explained the dead-heat puzzle. With the
velocities used above, the goal attained by B must
be automatically moved f of a mile nearer the
starting-point, so that B only rows 3^ miles out and
back instead of 4 miles. So gross a piece of
cheating would enable B to make his dead-heat,
but could scarcely escape detection. The shortening
of the course required in the case of light is very
minute indeed, because the velocities of the heavenly
bodies are so small compared with that of light.
If they could be multiplied a thousand times we
might see some curious things, but we have no
actual experience to guide a forecast.

It is a great triumph for Pure Mathematics that
it should have devised a forecast for us in it^own

INTRODUCTION xiii

peculiar way. Starting from axioms or postu-
lates, Einstein, by sheer mathematical skill, making
full use of the beautiful theoretical apparatus
inherited from his predecessors, pointed ultimately
to three observational tests, three things which
must happen if the axioms and postulates were
well founded. One of the tests — the movement of
the perihelion of Mercury's orbit — had already been
made arid was awaiting explanation as a standing
puzzle. Another — a displacement of lines in the
spectrum of the sun — is still being made, the issue
being not yet clear.

The third suggestion was that the rays of light
from a star would be bent on passing near the sun
by a particular amount, and this test has just
provided a sensational triumph for Einstein. The
application was particularly interesting because it
was not known which of at least three results might
be attained. If light were composed of material
particles as Newton suggested, then in passing the
sun they would suffer a natural deflection (the use
of the adjective is an almost automatic consequence
of modes of thought which we must now abandon)
which we may call N. On Einstein's theory the
deflection would be just twice this amount, E = 2N.
But it was thought quite possible that the result
might be neither N nor E but zero, and Professor
Eddington remarked before setting out on the recent
expedition that a zero result, however disappointing
immediately, might ultimately turn out the most
fruitful of all. That was less than a year ago.
Perhaps a few dates are worth remembering. Eins-
tein's theory was fully developed and stated in
November, 1915, but news of it did not reach Eng-
land (owing to the War) for some months. In
1917 the Astronomer Royal pointed out the special
suitability of the Total Solar Eclipse of May, 1919,

xiv THEORY OF GRAVITATION

as an occasion for testing Einstein's Theory. Pre-
parations for two Expeditions were commenced —
Mr. Hinks described the geographical conditions on
the central line in November, 1917 — but could
not be fully in earnest until the Armistice of Novem-
ber, 1918. In November, 1919, the entirely satis-
factory outcome was announced to the Royal
Society and characterized by the President as
necessitating a veritable revolution in scientific
thought.

But when Mr. Brose brought me his translation
of the pamphlet in the spring of 1919, the issue
was still in doubt. He had become deeply in-
terested in the new theory while interned in Germany
as a civilian prisoner and had there made this trans-
lation. I encouraged him to publish it and opened
negotiations to that end, but it was not until we
enlisted the sympathy of Professor Eddington
(on his return from the Expedition) and approached
the Cambridge Press that a feasible plan of publica-
tion was found. Professor Eddington would have
been a far more appropriate introducer ; and it is
only in deference to his own express wish that I
have ventured to take up the pen that he would
have used to much better purpose. One advantage
I reap from the decision : I can express the thanks
of Mr. Brose and myself to him for his practical
help, and perhaps I may add those of a far wider
circle for his own able expositions of an intricate
theory, which have done so much to make it known
in England.

H. H. TURNER

UNIVERSITY OBSERVATORY,

OXFORD.
November 30, 1919

CONTENTS

PAGE

INTRODUCTION. By Professor H. H. Turner, F.R.S. . vii
BIOGRAPHICAL NOTE ....... xvi

SECT.

1. THE SPECIAL THEORY OF RELATIVITY AS A STEPPING-

STONE TO THE GENERAL THEORY OF RELATIVITY 3

2. TWO FUNDAMENTAL POSTULATES IN THE MATHE-

MATICAL FORMULATION OF PHYSICAL LAWS . IQ

3. CONCERNING THE FULFILMENT OF THE TWO POSTU-

LATES 22

(a) The line-element in the three-dimensional

manifold of points in space, expressed in
a form compatible with the two postu-
lates 23

(b) The line-element in the four-dimensional

manifold of space-time points, expressed
in a form compatible with the two postu-
lates ....... 31

4. THE DIFFICULTIES IN THE PRINCIPLES OF CLASSI-

CAL MECHANICS 37

5. EINSTEIN'S THEORY OF GRAVITATION

(a) THE FUNDAMENTAL LAW OF MOTION AND

THE PRINCIPLE OF EQUIVALENCE OF THE
NEW THEORY 45

(b) RETROSPECT 55

6. THE VERIFICATION OF THE NEW THEORY BY

ACTUAL EXPERIENCE 62

APPENDIX :

Explanatory notes and bibliographical refer-
ences 69

ON THE THEORY OF RELATIVITY. By Henry L. Brose 105

SOME ASPECTS OF RELATIVITY. THE THIRD TEST.

By Henry L. Brose 131

xv

BIOGRAPHICAL NOTE

ALBERT EINSTEIN was born in March, 1879, in the town
Ulm, situated on the banks of the Danube in Wiirtem-
berg, Germany. He attended school at Munich, where
he remained till his sixteenth year.

His university studies extended over the period 1896-
1900 at Zurich, Switzerland. He became a citizen of
Zurich in 1901. During the following seven years he
filled the post of engineer in the Patent Office, Bern.
He accepted a call to Zurich as Professor Extraordinarius
in 1910, which he, however, soon resigned in favour of a
permanent chair in Prague University. In 1911 he
decided to accept a similar post in Zurich. Since 1914
he has continued his researches in Berlin as a member of
the Berlin Academy of Sciences.

His most important achievements are :
1905. The Special Theory of Relativity.

The discovery that all forms of energy possess

inertia.

The law underlying the Brownian movement.
The Quantum-Law of the emission and absorp-
tion of light.
1907. The fundamental notions of the general theory of

relativity.

1912. The recognition of the non-Euclidean nature of
space-determination and its connection with
gravitation.
1915. Gravitational field equations.

Explanation of the motion of Mercury's peri-
helion.

XVI

THE FOUNDATIONS OF EINSTEIN'S
THEORY OF GRAVITATION

INTRODUCTION

' I COWARDS the end of 1915 Albert Einstein
brought to its conclusion a theory of gravi-
tation on the basis of a general principle of
relativity of all motions. His object was to create
not a visual picture of the action of an attractive
force between bodies, but rather a mechanics of the
motions of the bodies relative to one another under
the influence of inertia and gravity. To attain this
difficult goal, it is true, many time-honoured views
had to be sacrificed, but as a reward a standpoint
was reached which had long seemed the highest
aim of all who had occupied their minds with theo-
retical physics. The fact that these sacrifices are
demanded by the new theory must, indeed, inspire
confidence in it. For the unsuccessful attempts
that have been made during the last centuries to
fit the doctrine of gravitation satisfactorily into the
scheme of natural science necessarily lead to the
conclusion that this would not be possible without
giving up many deeply-rooted ideas. As a matter
of fact, Einstein reverted to the foundation pillars
of mechanics as starting-points on which to build
his theory, and he did not satisfy himself by merely
reforming the Newtonian law in order to establish
a link with the more recent views.

2 THE FOUNDATIONS OF EINSTEIN'S

To get at an understanding of Einstein's ideas, we
must compare the fundamental point of view adopted
by Einstein with that of classical mechanics. We
then recognize that a logical development leads from
" the special " principle of relativity to the general
theory, and simultaneously to a theory of gravitation.

THEORY OF GRAVITATION

§1

THE " SPECIAL " THEORY OF RELATIVITY AS
A STEPPING STONE TO THE "GENERAL"
THEORY OF RELATIVITY

THE complete upheaval which we are witness-
ing in the world of physics at the present
time received its impulse from obstacles which
were encountered in the progress of electrodynamics.
Yet the important point in the later development was
that an escape from these difficulties was possible *
only by founding mechanics on a new basis.

The development of electrodynamics took place
essentially without being influenced by the results
of mechanics, and without itself exerting any
influence upon the latter, so long as its range of
investigation remained confined to the electro-
dynamic phenomena of bodies at rest. Only after
Maxwell's equations had furnished a foundation for
these did it become possible to take up the study
of the electrodynamic phenomena of moving media.
All optical occurrences — and according to Maxwell's
theory all these also belong to the sphere of electro-

* Note. — Most of the objections to the new development have, it
is admitted, been raised because a branch of science which was not
considered to have a just claim to deal with questions of mechanics,
asserted the right of exercising a far-reaching influence upon the latter,
extending even to its foundation. If, however, we trace these objec-
tions to their source, we discover that they are due to a wish to give
mechanics the form of a purely mathematical science, similar to geo-
metry, in spite of the fact that it is founded upon hypotheses which
are essentially physical : up to the present, certainly, these hypotheses
have not been recognized to be such.

4 THE FOUNDATIONS OF EINSTEIN'S

dynamics — take place either between stellar bodies
which are in motion relatively to one another, or upon
the earth, which revolves about the sun with a velocity
of about 30 kilometres per second, and performs,
together with the sun, a translational motion of
about the same order of magnitude through the
region of the stellar system. Hence questions of
great fundamental importance at once asserted
themselves. Does the motion of a light-source
leave its trace on the velocity of the light emitted
by it ? And what is the influence of the earth's
motion on the optical phenomena which occur on
its surface, for example, in optical experiments in
a laboratory ? An endeavour was therefore to be
made to find a theory of these phenomena in
which electrodynamic and mechanical effects occurred
simultaneously (vide Note i). Mechanics, which had
long stood as a structure complete in every detail,
had to stand the test as to whether it was capable
of supplying the fitting arguments for a description
of such phenomena. Thus the problem of electro-
dynamic events in the case of moving matter became
at the same time a decisive problem of mechanics.
The first outstanding attempt to describe these
phenomena for moving bodies was made by H.
Hertz. He extended Maxwell's equations by ad-
ditional terms so as also to express the influence of the
motion of matter on electrodynamic phenomena, and
in his extensions he adopted the view, characteristic
for his theory, that the carrier of the electromagnetic
field, the ether, everywhere participates in the
motion of matter. Consequently, in his equations
the state of motion of the ether, as denoting the
state of the ether, occurs as well as the electro-
magnetic field. As is well known, Hertz's extensions
cannot be brought into harmony with the results of
observation, for example, that of Fizeau's experiment

THEORY OF GRAVITATION 5

(Note 2), so that they excite merely an historic
interest as a land-mark on the road to an electro-
dynamics of moving matter. Lorentz was the
first to derive from Maxwell's theory fundamental
electrodynamic equations for moving matter which
were in essential agreement with observation. He,
indeed, succeeded in this only by renouncing a
principle of fundamental importance, namely, by
disallowing that Newton's and Galilei's principle
of relativity of classical mechanics also holds for
electrodynamics. The practical success of Lorentz's
theory at first almost made us fail to see this sacrifice,
but then the disintegration set in at this point
which finally made the position of classical mechanics
untenable. To understand this development we
therefore require a detailed treatment of the principle
of relativity in the fundamental equations of physics.
The principle of relativity of classical mechanics
is understood to signify the consequence, which
arises out of Newton's equations of motion, that
two systems of co-ordinates, moving with uniform
motion in a straight line with respect to one another,
are to be regarded as fully equivalent for the descrip-
tion of events in the domain of mechanics. For
our observations on the earth this means that any
mechanical event on the surface of the earth — for
example, the motion of a projected body — does not
become modified by the circumstance that the
earth is not at rest, but, as is approximately the
case, is moving rectilinearly and uniformly. Yet this
postulate of relativity does not fully characterize
the Newtonian principle of relativity, even if it
expresses that experimental fact which constitutes
the essence of the principle of relativity. The
postulate of relativity has yet to be supplemented
by those formulae of transformation by means
of which the observer is able to transform the

6 THE FOUNDATIONS OF EINSTEIN'S

co-ordinates x, y, z, i that occur in Newton's equations
of motion into those of a system of reference which
is moving uniformly and rectilinearly with respect
to his own and which has the co-ordinates x' , yr , z', t' .
Here the co-ordinates, xt y, z, that occur in the New-
tonian equations denote throughout the results of
measurement (obtained by means of rigid measuring
rods according to the rules of Euclidean geometry),
of the spatial positions of the bodies during the
event in question, and the fourth co-ordinate t
denotes the point of time assigned to the same
event given by the position of the hands of a
clock placed at the point at which the event occurs.
Classical mechanics now supplemented the postulate
of relativity above formulated by equations of
transformation of the form :

Xr = X — vt y' = y z' = Z t' — t

for the cases in which we are dealing with the
co-ordinate relations of two systems of reference
moving with the uniform velocity v in the direction
of the #-axis with respect to each other. This
group of so-called Galilei-transformations is dis-
tinguished, even in the case in which the direction
of motion makes any angle with the co-ordinate axes,
by the circumstance that the time-co-ordinate t
always becomes transformed by the identity t = t'
into the time- values of the second system of reference ;
it is in this that the absolute character of the time-
measures manifests itself in the classical theory.
Newton's equations of mechanics do not alter their
form if we substitute the co-ordinates x' t y', z', t'
in them for x, y, z, t by means of these equations
of transformation. So long as we restrict ourselves
to those systems of reference among all others that
emerge out of each other as a result of transforma-
tions of the above type, there is no sense in talking

THEORY OF GRAVITATION 7

of absolute rest or absolute motion. For we may
freely decide to regard either of two systems moving
in such a way as the one that is at rest or in motion.
According to classical mechanics it was, indeed, be-
lieved that only the Galilei- transformations could
come into question when we were concerned with re-
ferring equivalent systems of reference to each other
according to the principle of relativity. This, how-
ever, is not the case. The recognition of the fact
that other equations of transformation may come
into question for this purpose, and, indeed, may be
chosen to suit the facts of observation which are
to be accounted for, the recognition of this fact
is the characteristic feature of the " special " theory
of relativity of Lorentz-Einstein which replaced
that of Galilei-Newton. Lorentz's fundamental equa-
tions of the electrodynamics of moving matter led
to it. This system of electrodynamics, which is
in satisfactory agreement with observation, is founded,
in contradistinction to Hertz's theory, on the view
of an absolutely rigid ether at rest. Its funda-
mental equations assume as its favoured system the
co-ordinate system that is at rest in the ether.

These fundamental electrodynamical equations
of Lorentz, however, change their form if, in them,
we replace the co-ordinates %, yy z, t of a system
of reference, initially chosen, by the co-ordinates
xr, yr , z', t' of a system moving uniformly and
rectilinear with respect to the former by means
of the transformation relationships. Must we infer
from this that systems of reference which are
moving uniformly and rectilinearly with respect to
each other are not equivalent as regards electro-
dynamic events, and that there is no relativity
principle of electrodynamics ? No, this inference
is not necessary, because, as remarked, the principle
of relativity of classical mechanics with its group

8 THE FOUNDATIONS OF EINSTEIN'S

of equations of transformation does not represent
the only possible way of expressing the equivalence
of systems of reference that are moving uniformly
and rectilinearly with respect to each other. As we
shall show in the sequel, the same postulate of
relativity may be associated with another group of
transformations. Nor did experiment seem to offer
a reason for answering the above question in the
affirmative. For all attempts to prove by optical
experiments in our laboratories on the earth the
progressive motion of the latter gave a negative
result (Note 2). According to our observations
of electrodynamic events in the laboratory the earth
may be regarded equally well as at rest or in motion ;
these two assumptions are equivalent.

This led to the definite conviction that in fact a
principle of relativity holds for all phenomena, be
their character mechanical or electrodynamic. But
there can be only one such principle, and not one
for mechanics and another for electrodynamics.
For two such principles would annul each other's
effects because we should be able to derive a favoured
system from them in the case of events in which
mechanical and electrodynamical events occur in
conjunction, and this favoured system would allow
us to talk with sense of absolute rest or motion with
regard to it.

The one escape from this difficulty is that opened
up by Einstein. In place of the relativity principle
of Galilei and Newton we have to set another which
comprehends the events of mechanics and electro-
dynamics. This may be done, without altering the
postulate of relativity formulated above, by setting
up a new group of transformations, which refer the
co-ordinates of equivalent systems of reference to
one another. The fundamental equations of me-
chanics must, certainly, then be remodelled so that

THEORY OF GRAVITATION 9

they preserve their form when subjected to such a
transformation. Starting-points for this remodel-
ling were already given. For it had been found
empirically that Lorentz's fundamental equations
of electrodynamics allowed new kinds of trans-
formations of co-ordinates, namely, those of the
form

x — vt

nf*

X =

where c = velocity of light in vacuo.

The new principle of relativity set up by Einstein
is as follows : Systems that are moving uniformly and
rectilinearly with respect to each other are completely
equivalent for the description of physical events. The
equations of transformation that allow us to pass from
the co-ordinates of one such system to those of another
possible system, however, are not then (for the case
when both systems are moving parallel to their
#-axes with the constant velocity v) : —

x' = x — vt, y' = y, z' = z, t' = t
but

Thus the Galilei-Newton principle of relativity
of classical mechanics and the Lorentz-Einstein
" special " principle of relativity differ only in the
form of the equations of transformation that effect
the transition to equivalent systems of reference
(Note 3).
Moreover, the relation of these two different

10 THE FOUNDATIONS OF EINSTEIN'S

transformation formulae to each other comes out
clearly in the circumstance that the equations of
transformation of Galilei and Newton may be
derived by a simple passage to the limit from the
new equations of Lorentz and Einstein. For if we
assume the velocity v of each system with respect
to the other to be very small compared with the

velocity of light c, so that the quotient — 2 or ^respec-
tively, may be neglected in comparison with the
remaining terms — an admissible assumption in all
cases with which classical mechanics had so far dealt
— the Lorentz-Einstein transformations pass over
into those of Newton and Galilei.

It immediately suggests itself to us to ask what
it is that compels us to give up the principle of
relativity of classical mechanics, that is, what are
the physical assumptions in its equations of trans-
formation that stand, in contradiction with ex-
perience ? The answer is that the principle of
relativity of Newton and Galilei does not account
for the facts of experience that emerge from Fiz-
eau's and the Michelson-Morley experiment, and
from which it may be inferred that the velocity
of light has the particular character of a universal
constant in the transformation relationships of
the principle of relativity. In how far this peculiar
property of the velocity of light receives expression
in the new equations of transformation requires
the following detailed explanation.

The equations of transformation of the principle
of relativity of Galilei and Newton contain a hypoth-
esis (which had hitherto not been recognized as
such). For it had been tacitly assumed that the
following assumption was fulfilled quite naturally :
if an observer in a co-ordinate system S measure the

THEORY OF GRAVITATION 11

velocity v of the propagation of some effect or other,
for example, a sound wave, then an observer in
another co-ordinate system S' which is moving
relatively to S, necessarily obtains a different
measure for the velocity of propagation of the
same action. This was to hold for every finite
velocity v. Only infinite velocity was to be dis-
tinguished by the singular property that it was to
come out in every system independently of its state
of motion as having exactly the same value in all
the measurements, namely, the value infinity.

This hypothesis — for we are here, of course,
dealing only with a purely physical hypothesis — im-
mediately suggested itself. Without further test
there was no support for supposing that also a
finite velocity, namely, the velocity of light, which
the nai've point of view is inclined to endow with
infinitely great velocity, would manifest the same
singular property.

The fact, however, which the Michelson-Morley
experiment helped us to become aware of was that
the law of propagation for light is, for the observer,
independent of any progressive motion of his
system of reference, and has the property of isotropy
(that is, equivalence of all systems) (cf. Note 2),
so that it immediately suggests itself to us that
the velocity of light is to be considered as having
the same value for all systems of reference. The
recognition of the fact thus arrived at was, without
doubt, a surprise, but it will appear less strange
to those who bear in mind the particular role of
the velocity of light in the equations of Maxwell,
the foundation of our theory of matter.

In consequence of this peculiarity, the velocity
of light occurs in the equations of kinematics as a
universal constant. To understand this better we
pursue the following argument. Long before the

12 THE FOUNDATIONS OF EINSTEIN'S

advent of the questions of electrodynamic phenomena
in moving bodies we might, on grounds of principle,
have suggested quite generally the question : how
are the co-ordinates in two systems of reference
that are moving uniformly and rectilinearly with
respect to each other to be referred to each
other ? We should have been able to attack the
purely mathematical problem with a full consciousness
of the assumptions contained in the hypotheses, as
was actually done later by Frank and Rothe (Note 4).
We then arrive at equations of transformation
that are much more general than those written
down on p. 9. By taking into account the
special conditions that nature manifests to us, for
example the isotropy of space, we may derive from
them particular forms, the hypothetical assumptions
contained in which come clearly to view. Now,
in these general equations of transformation a
quantity occurs that deserves special notice. There
are " invariants " of these equations of transforma-
tion, that is, quantities that preserve their value
even when such a transformation is carried out.
Among these invariants there is a velocity. This
signifies the following : if an effect propagates
itself in one system with the velocity v, then in
general the velocity of propagation of the same
effect in another system is other than v, if the
second system is moving relatively to the first.
Only the invariant velocity preserves its value in
all systems, no matter with what velocity they be
moving relatively to one another. The value of
this invariant velocity enters as a characteristic
constant into the equations of transformation.
Hence, if we wish to find those transformation
relations that hold physically, we must find out the
singular velocity that plays this fundamental part.
To determine it is the task of the experimental

THEORY OF GRAVITATION 13

physicist. If he sets up the hypothesis that a
finite velocity can never be such an invariant,
the general equations of transformation degenerate
into the transformation-relationships of the prin-
ciple of relativity of Galilei and Newton. (This
hypothesis was made, albeit unconsciously, in
Newtonian mechanics.) It had to be discarded
after the results of the Michelson-Morley and
Fizeau's experiment had justified the view that
the velocity of light c plays the part of an in-
variant velocity. Then the general equations
of transformation degenerate into those of the
" special " principle of relativity of Lorentz and
Einstein.

This remodelling of the co-ordinate-transforma-
tions of the principle of relativity led to discoveries
of fundamental importance, as, for example, to the
surprising fact that the conception of the " sim-
ultaneity " of events at different points of space,
the conception on which all time-measurements
are based, has only a relative meaning, that is,
that two events that are simultaneous for one
observer will not, in general, be simultaneous
for another.* This deprived time- values of the

* The assertion, "At a particular point of the earth the sun
rises at 5 o'clock 10' 6"," denotes that " the rising of the sun at a par-
ticular point of the earth is simultaneous with the arrival of the
hands of the clock at the position 5 o'clock 10' 6" at that point of the
earth." In short, the determination of the point of time for the
occurrence of an event is the determination of the simultaneity of
happening of two events, of which one is the arrival of the hands of a
clock at a definite position at the point of observation. The com-
parison of the points of time at which one and the same event occurs,
as noted by several observers situated at different points, requires a
convention concerning the times noted at the different points. The
analysis of the necessary conventions led Einstein to the fundamental
discovery that the conception " simultaneous " is only relative in-
asmuch as the relation of time-measurements to one another in
systems that are moving relatively to one another is dependent on
their state of motion. This was the starting-point for the arguments
that led to the enunciation of the " special principle of relativity."

14 THE FOUNDATIONS OF EINSTEIN'S

absolute character which had previously been a great
point of distinction between them and space co-
ordinates So much has been written in recent
years about this question that we need not treat
it in detail here.

The new form of the equations of transformation
by no means exhausts the whole effect of the prin-
ciple of relativity upon classical mechanics. The
change which it brought about in the conception
of mass was almost still more marked.

Newtonian mechanics attributes to every body a
certain inertial mass, as a property that is in no
wise influenced by the physical conditions to which
the body is subject. Consequently, the Principle of
the Conservation of Mass also appears in classical
mechanics as independent from the Principle of
the Conservation of Energy. The special principle
of relativity shed an entirely new light on these
circumstances when it led to the discovery that
energy also manifests inertial mass, and it hereby
fused together the two laws of conservation, that of
mass and that of energy, to a single principle. The
following circumstance moves us to adopt this new
view of the conception of mass.

The equations of motion of Newtonian mechanics
do not preserve their form when new co-ordinates
have been introduced with the help of the Lorentz-
Einstein transformations. Consequently, the funda-
mental equation of mechanics had to be remodelled
accordingly. It was then found that Newton's
Second Law of Motion : force = mass x accel. can-
not be retained, and that the expression for the
kinetic energy of a body may no longer be furnished
by the simple expression %mv2, which involves
the mass and the velocity. Both these results are
consequences of the change which we found neces-
sary to make in our view of the nature of the mass

THEORY OF GRAVITATION 15

of matter. The new principle of relativity and
the equations of electrodynamics led, rather, to the
fundamentally new discovery that inertial mass is
a property of every kind of energy, and that a point-
mass, in emitting or absorbing energy, decreases or
increases, respectively, in inertial mass, as is shown
in Note 5 for a simple case. The new kinematics
thereby disposes of the simple relation between the
kinetic energy of a body and its velocity relatively
to the system of reference. The simplicity of the
expression for the kinetic energy in Newtonian
mechanics rendered possible the revolution of the
energy of a body into that (kinetic) of its motion
and of the internal energy of the body, which is
independent of the former. Let us consider, for
example, a vessel containing material particles, no
matter of what kind, in motion. If we resolve
the velocity of each particle into two components,
namely, into the velocity, common to all, of the
centre of gravity and the accidental velocity of a
particle relative to the centre of gravity of the
system, then, according to the formulae of classical
mechanics, the kinetic energy divides up into two
parts : one that contains exclusively the velocity
of the centre of gravity and that represents the
usual expression for the kinetic energy of the whole
system (mass of the vessel plus the mass of the
particles), and a second component that involves
only the inner velocities of the system. This
category of internal energy is no longer possible so
long as the expression for the kinetic energy con-
tains the velocity not merely as a quadratic factor ;
so we are led to the view that the internal energy
of the body comes into expression in the energy
due to its progressive motion, and, indeed, as an
increase in the inertial mass of the body.

This discovery of the inertia of energy created

16 THE FOUNDATIONS OF EINSTEIN'S

an entirely new starting-point for erecting the struc-
ture of mechanics. Classical mechanics regards
the inertial mass of a body as an absolute, invariable,
characteristic quantity. The special theory of rela-
tivity, it is true, makes no direct mention of the
inertial mass associated with matter, but it tells
us that every kind of energy has also inertia. But,
as every kind of matter has at all times a probably
enormous amount of latent energy, its inertia is
composed of two components ; the inertia of the
matter and the inertia of its contained energy,
which consequently alters with the amount of
the energy-content. This view leads us naturally
to ascribe the phenomenon of inertia in bodies to
their energy-content altogether.

Thus, there arose the important task of absorbing
these new discoveries concerning the nature of
inert mass into the principles of mechanics. A
difficulty hereby arose which, in a certain sense,
pointed out the limits of achievement of the special
theory of relativity. One of the fundamental
facts of mechanics is the equality of the inertial
and gravitational mass of a body. It is on the
supposition that this is true that we determine
the mass of a body by measuring its weight. The
weight of a body is, however, only denned with
reference to a gravitational field (Note 18) : in our
case, with reference to the earth. The idea of
inertial mass of a body is, however, introduced as
an attribute of matter without any reference what-
soever to physical conditions external to the body.
How does the mysterious coincidence in the values
of the inertial and gravitational mass of a body
come about ?

Nor does the special theory of relativity provide
an answer to this question. The special theory of
relativity does not even preserve the equality in

THEORY OF GRAVITATION 17

the values of inertia and gravitational mass ; a
fact which is to be reckoned amongst the most
firmly established facts in the whole of physics.
For, although the special theory of relativity makes
allowance for an inertia of energy, it makes none
for a gravitation of energy. Consequently, a body
which absorbs energy in any way will register a
gain of inertia but not of weight, thereby trans-
gressing the principle of the equality of inertial and
gravitational mass ; for this purpose a theory of
gravitational phenomena, a theory of gravitation,
is required. The special theory of relativity can,
therefore, be regarded only as a stepping-stone to
a more general principle, which orders gravitational
phenomena satisfactorily into the principles of
mechanics.

This is the point where Einstein's researches
towards establishing a general theory of rela-
tivity set in. He has discovered that, by extend-
ing the application of the relativity-principle to
accelerated motions, and by introducing gravi-
tational phenomena into the consideration of the
fundamental principles of mechanics, a new founda-
tion for mechanics is made possible, in which all
the difficulties occurring up to the present are solved.
Although this theory represents a consistent de-
velopment of the knowledge gathered by means of
the special theory of relativity, it is so deeply
rooted in the substructure of our principles of know-
ing, in their application to physical phenomena,
that it is possible thoroughly to grasp the new
theory only by clearly understanding its attitude
toward these guiding lines provided by the theory
of knowledge.

I shall, therefore, commence the account of his
theory by discussing two general postulates, which
should be fulfilled by every physical law, but neither

18 THE FOUNDATIONS OF EINSTEIN'S

of which is satisfied in classical mechanics : whereas
their strict fulfilment is a characteristic feature of
the new theory. Here we have thus a suitable
point of entry into the essential outlines of the
general theory of relativity.

THEORY OF GRAVITATION 19

§ 2

TWO FUNDAMENTAL POSTULATES IN THE
MATHEMATICAL FORMULATION OF PHY-
SICAL LAWS

NEWTON had established the simple and
fruitful law that two bodies, even when
they are not visibly connected with one
another, as in the case of the heavenly bodies,
exert a mutual influence, attracting one another
with a force directly proportional to the product
of their masses, and inversely proportional to the
square of the distance between them. But Huygens
and Leibniz refused to acknowledge the validity of
this law, on the ground that it did not satisfy a
fundamental condition to which every physical
law is subject, viz. that of continuity (continuity
in the transmission of force, action " by contact "
in contradistinction to action "at a distance ").
How were two bodies to exert an influence upon one
another without a medium between them to transmit
the action ? The demand for a satisfactory answer
to this question became, in fact, so imperative
that finally, in order to satisfy it, the existence of
a substance which pervaded the whole of cosmic
space and permeated all matter — the " luminiferous
ether " — was assumed, although this substance
seemed to be condemned to remain intangible and
invisible (i.e. imperceptible to the senses for all time)
and had to be endowed with all sorts of contradic-
tory properties. In the course of time, however,

20 THE FOUNDATIONS OF EINSTEIN'S

there arose in opposition to such assumptions the
more and more definite demand that, in the formu-
lation of physical laws, only those things were to be
regarded as being in causal connection which were
capable of being actually observed : a demand which
doubtless originates from the same instinct in the
search for knowledge as that of continuity, and
which really gives the law of causality the true
character of an empirical law, i.e. one of actual
experience.

The consistent fulfilment of these two postulates
combined together is, I believe, the mainspring of
Einstein's method of investigation ; this imbues
his results with their far-reaching importance in
the construction of a physical picture of the world.
In this respect his endeavours will probably not
encounter any opposition in the matter of principle
on the part of scientists. For both postulates — (i)
that of continuity and (2) that of causal relationship
between only such things as lie within the realm of
observation — are of an inherent nature, i.e. contained
in the very nature of the problem. The only
question that might be raised is whether it is ex-
pedient to abandon such useful working hypotheses
as " forces at a distance."

The principle of continuity requires that all
physical laws allow of formulation as differential
laws, i.e. physical laws must be expressible in a
form such that the physical state at any point is
completely determined by that of the point in its
immediate neighbourhood. Consequently, the dis-
tances between points, which are at finite distances
from one another, must not occur in these laws,
but only those between points infinitely near to
one another. The law of attraction of Newton
given above, inasmuch as it involves " action at a
distance," disobeys the first postulate.

THEORY OF GRAVITATION 21

The second postulate, that of a stricter form of
expression for causality in its occurrence in physical
laws, is intimately connected with a general theory
of relativity of motions. Such a general principle
of relativity requires that all possible systems of
reference in nature be equivalent for the description
of physical phenomena, and hence it avoids the
introduction of the very questionable conception
of absolute space which, for reasons we know (see
§ 4), could not be circumvented by Newtonian
mechanics. A general theory of relativity would,
in excluding the fictitious quantity " absolute space/'
reduce the laws of mechanics to motions of bodies
relative to one another, which are actually and
exclusively what we observe. Thus, its laws would
be founded on observed facts more completely
than are those of classical mechanics.

The rigorous application of the principles of
continuity and relativity in their general form
penetrates deeply into the problem of the mathe-
matical formulation of physical laws. It will,
therefore, be essential at the outset to enter into a
consideration of the principles involved in the
latter process.

THE FOUNDATIONS OF EINSTEIN'S

§3

CONCERNING THE FULFILMENT OF THE TWO
POSTULATES

A PHYSICAL law is clothed in mathematical
language by setting up a formula. This
comprises, and represents in the form of
an equation, all measurements which numerically
describe the event in question. We make use of
such formulae, not only in cases in which we have
the means of checking the results of our calculations
at any moment actually at our disposal, but also
when the corresponding measurements cannot really
be carried out in practice, but have to be imagined,
i.e. only take place in our minds : e.g. when we
speak of the distance of the moon from the earth,
and express it in metres, as if it were really possible
to measure it by applying a metre-rule end to
end.

By means of this expedient of analysis we have
extended the range of exact scientific research
far beyond the limits of measurement actually
accessible in practice, both in the matter of im-
measurably large, as well as in that of immeasurably
small, quantities. Now, when such a formula is
used to describe an event, symbols occur in it
that stand for those quantities which are, in a certain
sense, the ground elements of the measurements,
with the help of which we endeavour to grip the
event ; thus, for example, in the case of all spatial
measurements, symbols for the " length " of a rod,

THEORY OF GRAVITATION 23

the " volume " of a cube, and so forth. In creating
these ground elements of spatial elements we had
hitherto been led by the idea of a rigid body which
was to be freely movable in space without altering
any of its dimensional relationships. By the re-
peated application of a rigid unit measure along
the body to be measured we obtained information
about its dimensional relationships. This idea of
the ideal rigid measuring rod, which is only partially
realizable in practice, on account of all sorts of
disturbing influences such as the expansion due to
heat, represents the fundamental conception of the
geometry of measure.

The discovery of suitable mathematical terms,
which can be inserted in a formula as symbols
for definite physical magnitudes of measurements,
such as e.g. length of a rod, volume of a cube, etc.,
in order to shift the responsibility, as it were, for
all further deductions upon analysis, is one of the
fundamental problems of theoretical physics and
is intimately connected with the two postulates enun-
ciated in § 2.

To realize this fully, we must revert to the founda-
tions of geometry, and analyse them from the
point of view adopted by Helmholtz in various
essays, and by Riemann in his inaugural disserta-
tion of 1854 : "On the hypotheses which lie at the
bases of geometry." Riemann points almost pro-
phetically to the path now taken by Einstein.

(a) THE LINE-ELEMENT IN THE THREE-DIMENSIONAL
MANIFOLD OF POINTS IN SPACE, EXPRESSED
IN A FORM COMPATIBLE WITH THE Two POS-
TULATES

Every point in space can be singly and unam-
biguously defined by the three numbers xlt x2, x3,

24 THE FOUNDATIONS OF EINSTEIN'S

which may be regarded as the co-ordinates of a
rectangular system of co-ordinates, and which
distinguish it from all other points ; a continuous
variation of these three numbers enables us to
specify every single point of space in turn. The
assemblage of points in space represents, in Rie-
mann's notation, " a multiply extended magnitude "
(an n-iold manifoldness or manifold) between the
single elements (points) of which a continuous
transition is possible. We are familiar with diverse
continuous manifolds, e.g. the system of colours, of
tones and various others. A feature which is
common to all of them is that, in order to specify
a single element out of the entire manifold (to define
a particular point, a particular colour, or a particular
tone), a characteristic number of magnitude-deter-
minations, i.e. co-ordinates, is required : this char-
acteristic number is called the dimensions of the
respective manifold. Its value is three for space,
two for a plane, one for a line. The system of colours
is a continuous manifold of the dimension three,
corresponding to the three " primary " colours,
red, green, and violet, by mixing which in due
proportions every colour can be produced.

But the assumption of continuity for the transition
from one element to another in the same manifold,
and the determination of the dimensions of the
latter, does not give us any information about
the possibility of comparing limited parts of the
same manifold with one another, e.g. about the
possibility of comparing two tones with one another
or two single colours ; i.e. nothing has yet been
stated about the metric relations (measure-condi-
tions) of the manifold, about the nature of the scale,
according to which measurements can be undertaken
within the manifold. In order to be able to do
this, we must allow experience to give us the facts

THEORY OF GRAVITATION 25

from which to establish the metric (measure-) laws
which hold for each particular manifold (space-
points, colours, tones) under various physical con-
ditions ; these metric laws will be different accord-
ing to the set of empirical facts chosen for this
purpose.*

In the case of the manifold of space-points, ex-
perience has taught us that finite rigid point-systems
can be freely moved in space without altering
their form or dimensions ; the conception of " con-
gruence " which has been derived from this fact,
has become a vital factor for a measure-determina-
tion, f It sets us the problem of building up a
mathematical expression from the numbers xlt xz,
xs, and ylt y2, y3> which are assigned to two definite
points in space, and which we may imagine as the
end-points of a rigid measuring rod, such that
this expression may be regarded as a measure of
the distance between them, that is, as an expression
for the length of the rod, and may be introduced as
such into the formulae expressing physical laws.

The equations of physical laws, which — in order
to fulfil the conditions of continuity — must be
differential laws, contain only the distances ds, of
infinitely near points, so-called line-elements. We
must, therefore, inquire whether our two postulates
of § 2 have any influence upon the analytical ex-
pression for the line-element ds, and, if so, which
expression for the latter is compatible with both.
Riemann demands of a line-element in the first
place that it can be compared in respect to its length
with every other line-element independently of its
position and direction. This is a distinguishing
characteristic of the metric conditions (" measure
relations ") prevalent in space ; in practice it

* Vide Note 2. t Vide Note 3.

26 THE FOUNDATIONS OF EINSTEIN'S

denotes that the rods must be freely movable.
This peculiarity does not exist, for instance, in the
manifold of tones or in that of colours (vide Note 7).
Riemann formulates this condition in the words,
" that lines must have a length independent of
their position and that every line is to be measurable
by means of any other." He then discovers that :
if xlf x2, x3 and x± -f- dxlt x2 -f dx2, x3 -f dx3 re-
spectively denote two infinitely near points in space
and if the continuously variable numbers xlf x2, x3
are any co-ordinates whatsoever (not e.g. necessarily
rectilinear), then the square root of an always
positive, integral, homogeneous function of the
second degree in the differentials dxlf dx2, dx3 has
all the properties * which the line-element, being
the expression for the length of an infinitely small
rigid measuring rod, must exhibit. We thus find
that

in which the coefficients g^ are continuous func-
tions of the three variables xlt x2, x3, gives us
an expression for the line-element at the point

lj 2> 3*

In this expression no assumptions are made
concerning the nature of the co-ordinates that
are represented by the three variables, xlf x2, x3,
that is, concerning particular metrical properties
of the manifold that go beyond the postulate of
the freedom of movement of the measuring rods.
But, if we demand, in particular, that each point
of the manifold may be fixed by means of rectangular
Cartesian co-ordinates, whereby particular assump-
tions are made concerning the possible ways of
placing the measuring rods, then the line-element,

* Vide Note 8.

THEORY OF GRAVITATION 27

expressed in these special variables, assumes the
form _

ds = ^dx2 + dy2 + dz*.

Hitherto this expression has always been intro-
duced for the length of the line-element in all
physical laws. It is contained in the more general
expression of Riemann's line-element ds as the
special case

(=itfi = v

SM* \= o, fJi =(= v.

By restricting ourselves to this special form of the
line-element we are enabled to use the measure
laws of Euclidean geometry in all our space-
measurements.

But this particular assumption concerning the
metrical constitution of space contains the hypoth-
esis, as Helmholtz has shown in a detailed discussion,
that finite rigid point-systems, i.e. finite fixed dis-
tances, are capable of unrestrained motion in space,
and can be made (by superposition) to coincide
with other (congruent) point-systems. With respect
to the postulate of continuity, this hypothesis seems
inconsistent, in so far as it introduces implicit
statements about finite distances into purely dif-
ferential laws, in which only line-elements occur ;
but it does not contradict the postulate.

The postulate of the relativity of all motion
adopts a different attitude towards the possibility ef
giving the line-element the Euclidean form in
particular.*

* Strictly speaking, I should at this juncture state in anticipation
that the above investigations can manifestly also be so generalized
as to be valid for the four-dimensional space-time manifold, in which
all events actually take place, and that the transformation-formulae
apply to the four variables of this manifold. In these general remarks
the neglect of the fourth dimension is of no importance. This state-
ment will be justified later in § 3 (£).

28 THE FOUNDATIONS OF EINSTEIN'S

According to the principle of the relativity of all
motions, all systems, which come about owing to
relative motions of bodies towards one another, may
be regarded as fully equivalent. The laws of physics
must, therefore, preserve their form in passing
from one such system to another ; i.e. the trans-
formation-formulae of the variables xlf x2, #3 which
perform this transition to another system, must not
alter the analytical expression for the physical
law under consideration.

This leads us to set up a principle of relativity
which will be called the general principle of relativity
in the sequel. It demands the invariance of
physical laws with respect to arbitrary continuous
substitutions of the four variables. Moreover, the
line-element that occurs in it must preserve its
form when subjected to any arbitrary transformations
whatsoever. This condition is fully satisfied by
the line-element

in which no restrictive reservations of any description
are made as to what the co-ordinates xlt x2, x3 are
to signify. The Euclidean line-element

ds = +/dx* -f dy* + dz2,

on the other hand, preserves its form only for trans-
formations of the special theory of relativity, which
confine themselves to systems moving uniformly
and rectilinearly. Consequently, the element of
arc must be adapted to the further requirements
of a general theory of relativity so that it preserves
its form after any substitutions whatsoever.

3

The choice of the expression ds2 = Zg^x^dx, to

i

THEORY OF GRAVITATION 29

represent the line-element in physical laws is, in
spite of its very general character, still to be re-
garded as a hypothesis, as Riemann has already
pointed out. For there are other functions of the
differentials dxlf dxz, dx3 — such as e.g. the fourth
root of a homogeneous differential expression of
the fourth degree in these variables — which could
provide a measure for the length of the line-element
(vide Note 9). But at present there is no ground
for abandoning the simplest general expression for
the line-element (viz. that of the second degree),
and adopting more complicated functions. Within
the range (of fulfilment) of the two postulates,
which we have imposed upon every description of
physical events, the former expression for ds satisfies
all requirements. Nevertheless, it must never be
forgotten that the choice of an analytical expression
for the line-element always contains a hypothetical
factor ; and it is the duty of the physicist to remain
fully conscious of this fact at all times, without
being in any way prejudiced. It is for this reason
that Riemann closes his essay with the following
remarks, which impress one particularly with their
great importance for the present time : *

" The question of the validity of the hypotheses of
geometry in the infinitely small is bound up with
the question of the ground of the metric relations
of space. In this question, which we may still
regard as belonging to the doctrine of space, is
found the application of the remark made above ;
that in a discrete f manifold, the principle or
character of its metric relations is already given
in the notion of the manifold, whereas in a

* B. Riemann, Uber die Hypothesen, welche der Geometric
zugrunde liegen. New edn., annotated by H. Weyl, Berlin : Springer
& Co., 1919.

t Vide Note 6.

30 THE FOUNDATIONS OF EINSTEIN'S

continuous manifold this ground has to be found
elsewhere, i.e. has to come from outside. Either,
therefore, the reality which underlies space must
form a discrete * manifold, or we must seek the
ground of its metric relations (measure-conditions)
outside it, in binding fo/ces which act upon it.

A decisive answer to these questions can be
obtained only by starting from the conception of
phenomena which has hitherto been justified by
experience, and of which Newton laid the foundation,
and then making in this conception the successive
changes required by facts which admit of no ex-
planation on the old theory ; researches of this
kind, which commence with general notions, cannot
be other than useful in preventing the work from
being hampered by too narrow views, and in keeping
progress in the knowledge of the inter-connections
of things from being checked by traditional pre-
judices.

. This carries us over into the sphere of another
science, that of physics, into which the character
and purpose of the present discussion will not
allow us to enter."

That is to say : according to Riemann's view
these questions are to be solved by starting from
Newton's view of physical phenomena, and com-
pelled by facts which do not allow of any explana-
tion by it, gradually remoulding it. This is what
Einstein has done. The " binding forces," to
which Riemann points, will be found again in
Einstein's theory. As we shall see in the fifth
chapter, Einstein's theory of gravitation is based
upon the view that the gravitational forces are
the " binding forces," i.e. they represent the " inner
ground " of the metric conditions (measure-relations)
in space.

* Vide Note 10.

THEORY OF GRAVITATION 31

(b) THE LINE-ELEMENT IN THE FOUR-DIMENSIONAL
MANIFOLD OF SPACE-TIME POINTS, EXPRESSED
IN A FORM COMPATIBLE WITH THE TWO POST-
ULATES

The measure-conditions, which we were to take
as a basis for the formulation of physical laws,
could have been treated immediately in connection
with the four-dimensional manifold of space-time
points. For the special theory of relativity has
led us to make the important discovery that the
space-time-manifold has uniform measure-relations
in its four dimensions. Nevertheless, I wish to
treat time-measurements separately ; for one reason
that it is just this result of the relativity- theory
which has experienced the greatest opposition at
the hands of supporters of classical mechanics ;
and for another that classical mechanics is also
obliged to establish certain conditions about time-
measurement, but that it never succeeded in estab-
lishing agreement on this point. The difficulties
with which classical mechanics had to contend are
contained in its fundamental conceptions. The
law of inertia, particularly, was a permanent factor
of discord that caused the foundations of mechanics
to be incessantly criticised. And since the founda-
tions of time-measurement had been brought into
close relationship with the law of inertia, these
critical attacks applied to them likewise.

In Galilei's law of inertia, a body which is not
subject to external influences continues to move
with uniform motion in a straight line. Two
determining elements are lacking, viz. the reference
of the motion to a definite system of co-ordinates,
and a definite time-measure. Without a time-
measure one cannot speak of a uniform velocity.

32 THE FOUNDATIONS OF EINSTEIN'S

Following a suggestion by C. Neumann,* the law
of inertia has itself been adduced to give a definition
of a time-measure in the form : " Two material
points, both left to themselves, move in such a
way that equal lengths of path of the one correspond
to equal lengths of path of the other." On this
principle, into which time-measure does not enter
explicitly, we can define " equal intervals of time
as such, within which a point, when left to itself,
traverses equal lengths of path."

This is the attitude which was also taken up by
L. Lange, H. Seeliger, and others, in later researches.
Maxwell selected this definition too (in " Matter and
Motion"). On the other hand, H. Streintzf (follow
ing Poisson and d'Alembert) has demanded the
disconnection and independence of the time-measure
from the law of inertia, on the ground that the
roots of the time-concept have a deeper and more
general foundation than the law of inertia. Accord-
ing to his opinion, every physical event, which can
be made to take place again under exactly the same
conditions, can serve for the determination of a
time-measure, inasmuch as every identical event
must claim precisely the same duration of time ;
otherwise, an ordered description of physical events
would be out of the question. In point of fact,
the clock is constructed on this principle. It is
this principle which enables an observer to under-
take a time-measurement at least for his place of
observation. The reduction of time-measurements
to a dependence upon the law of inertia, on the other
hand, leads to an unobjectionable definition of equal
lengths of time ; but the measurement of the equal
paths traversed by uniformly moving bodies, and
the establishment of a unit of time involved therein,
are only then possible for a place of observation,

* Vide Note II. t Vide Note 12.

THEORY OF GRAVITATION 33

when the observer and the moving body are in
constant connection, e.g. by light-signals. It cannot,
however, be straightway assumed that two observers,
who are in rectilinear motion relatively to one
another, and, therefore, according to the law of
inertia, equivalent as reference systems, would
in this manner gain identical results in their time-
measurements. Poisson's idea thus leads to a
satisfactory time-measurement for a given place of
observation itself ; i.e. in a certain sense it allows
the construction of a clock for that place. But it
does not broach the question of the time-relations
of different places with one another at all ; whereas
Neumann's suggestion leads directly to those ques-
tions which have been a centre of discussion since
Einstein's enunciation of the relativity-principle.

In the endeavour to reduce classical mechanics
to as small a number of principles as possible, in
perfect agreement with one another, writers resorted
to ideal-constructions and imaginary experiments.

Yet no one conceived the idea that in fixing a
unit of time on the basis of the law of inertia, that
is, by measuring a length (the distance traversed),
the state of motion of the observer might exert
an influence. It was assumed that the data ob-
tained from the necessary observations had an
absolute meaning quite independent of the conditions
of observation when simultaneous moments were
chosen and a length was evaluated. As Einstein
has shown, however, this is not the case. Rather,
this recognition of the relativity of space- and
time-measurements formed the starting point of
his principle of relativity (Note 13). It is a neces-
sary consequence of the universal significance of
the velocity of light, of which we spoke in the
first section. Its recognition furnished us at once
with the correct formulae of transformation, allowing
3

34 THE FOUNDATIONS OF EINSTEIN'S

us to relate the space-time measurements of systems
moving uniformly and rectilinearly with respect
to each other, and this is what we are concerned
with in Neumann's suggestion of fixing a measure
of time with the aid of the law of inertia. In the
new equations of transformation, t' is not identically
equal to t, but rather

The time-measurements in the second system which
is moving relatively to the first are thus essentially
conditioned by the velocity v of each relative to
the other. Consequently, the fixing of a measure
of time on the basis of the law of inertia, as proposed
by Neumann, does not at all lead to the result
that the time-measurements are entirely indepen-
dent of the state of motion of the systems with
respect to each other, as assumed in classical me-
chanics. Only when the researches of Einstein
concerning the special theory of relativity had
been carried out, did the fundamental assumptions
of our time-measurements become fully cleared up,
and thus a serious shortcoming in classical mechanics
was made good.

That such a fundamental revision of the assump-
tions made regarding time-measurements became
necessary only after so great a lapse of time, is to
be explained by the fact that even the velocities
which occur in astronomy are so small, in comparison
with the velocity of light, that no serious discre-
pancies could arise between theory and observation.
So it occurred that the weaknesses of the theory —
in particular, those due to the motional relations

THEORY OF GRAVITATION 35

of various systems to one another — did not come
to light until the study of electronic motions, in
which velocities of the order of that of light occur,
proved the insufficiency of the existing theory.

The details of the effects, which result from the
relativity of space-time measurements, have so
frequently been discussed in recent years that it is
only possible to repeat what has already often
been said. The essential point in the discussion of
this section is the recognition of the fact that
space and time represent a homogeneous manifold
of " four " dimensions, with homogeneous measure-
relations (vide Note 14). Consequently, to be con-
sistent, we must apply the arguments of the
preceding § 3 (a) about the measure-relations to the
four-dimensional space-time-manifold ; and, in view
of the two fundamental postulates (i) of continuity
and (2) of relativity, and including the time-measure-
ment as the fourth dimension, we must select
for our line-element the expression :

ds* = gudxS + g^xjdxt + . . . +g3idxJXi + gudxf,

in which the g^v (pv = i, 2, 3, 4) are functions of
the variables xlt x2, xs, #4.

Hitherto we have been led to adopt this much
more general attitude towards the questions of the
metric laws involved in physical formulae merely
by the desire not to introduce, from the very outset,
more assumptions into the formulations of physical
laws than are compatible with both postulates,
and to bring about a deeper appreciation of the
points of view, to which the special theory of relativity
has led us.

We can briefly summarize by saying : the adop-
tion of Euclidean metric-conditions (measure-rela-
tions) is compatible with the postulate of continuity ;
though the special assumptions thereby involved

36 THE FOUNDATIONS OF EINSTEIN'S

appear as restrictive or limiting hypotheses, which
need not be made. But the second postulate,
the reduction of all motions to relative motions,
compels us to abandon the Euclidean measure-
determination (cf. p. 43). A description of the
difficulties still remaining in mechanics will make
this step clear.

THEORY OF GRAVITATION 87

§4

THE DIFFICULTIES IN THE PRINCIPLES OF
CLASSICAL MECHANICS

THE foundations of classical mechanics cannot
be exhaustively described in a narrow space.
I can only bring the unfavourable side of
the theory into prominent view for the present
purpose, without being able to do justice to its
great achievements in the past. All doubts about
classical mechanics set in at the very commencement
with the formulation of the law which Newton
places at its head, the formulation of the law of
inertia.

As has already been emphasized on page 31, the
assertion that a point-mass which is left to itself
moves with uniform velocity in a straight line,
omits all reference to a definite co-ordinate system.
An insurmountable difficulty here arises : Nature
gives us actually no co-ordinate system, with
reference to which a uniform rectilinear motion
would be possible. For as soon as we connect a
co-ordinate system with any body such as the earth,
sun, or any other body — and this alone gives it a
physical meaning — the first condition of the law
of inertia (viz. freedom from external influences) is
no longer fulfilled, on account of the mutual gravi-
tational effects of the bodies. One must accordingly
either assign to the motion of the body a meaning
in itself, i.e. grant the existence of motions relative

38 THE FOUNDATIONS OF EINSTEIN'S

to " absolute " space, or have recourse to mental
experiments by following the example of C. Neumann
and introducing a hypothetical body Alpha, relative
to which a system of axes is denned, and with
reference to which the law of inertia is to hold
(Inertial system, vide Note 15). The alternatives
with which one is faced are highly unsatisfactory.
The introduction of absolute space gives rise to
the oft-discussed conceptual difficulties which have
gnawed at the foundations of Newton's mechanics.
The introduction of the system of reference Alpha
certainly takes the relativity of motions so far
into account, that all systems in uniform motion
relative to an Alpha-system are established as
equivalent from the very outset, but we can affirm
with certainty that there is no such thing as a
visible Alpha-system, and that we shall never
succeed in arriving at a final determination of
such a system. (It will, at most, be possible, by
progressively taking account of the influences of
constellations upon the solar system and upon
one another, to approximate to a system of co-
ordinates, which could play the part of such an
inertial system with a sufficient degree of accuracy.)
As a result of this objection, the founder of the view
himself, C. Neumann, admits that it will always be
somewhat unsatisfactory and enigmatical, and that
mechanics, based on this principle, would indeed
be a very peculiar theory.

It therefore seems quite natural that E. Mach
(vide Note 16) should be led to propose that the
law of inertia be so formulated that its relations
to the stellar bodies are directly apparent. " Instead
of saying that the direction and speed of a rrfass
u remains constant in space, we can make use of
the expression that the mean acceleration of the
mass p, relative to the masses m, m', m" ... at

THEORY OF GRAVITATION 39

distances r, rf, r" . . ., respectively, is zero or

d* Smr = Q

The latter expression is equivalent to the former
statement, as soon as a sufficient number and
sufficiently great and extensive masses are taken
into consideration. ..." This formulation cannot
satisfy us. For, in addition to a certain requisite
accuracy, the character of a " contact " law is
lacking, so that its promotion to the rank of a
fundamental law (in place of the law of inertia)
is quite out of the question.

The inner ground of these difficulties is without
doubt to be found in an insufficient connection between
fundamental principles and observation. As a matter
of actual fact, we only observe the motions of bodies
relatively to one another, and these are never ab-
solutely rectilinear nor uniform. Pure inertial mo-
tion is thus a conception deduced by abstraction from
a mental experiment — a mere fiction.

However necessary and fruitful a mental experi-
ment may often be, there is the ever-present danger
that an abstraction which has been carried unduly
far loses sight of the physical contents of its under-
lying notions. And so it is in this case. If there
is no meaning for our understanding in talking of
the " motion of a body " in space, as long as there
is only this one body present, is there any meaning
in granting the body attributes such as inertial
mass, which arise only from our observation of
several bodies, moving relatively to one another ?
If not, then we cannot attach to the conception
" inertial mass of a body/' an absolute significance,
that is, a meaning which is independent of all other
physical conditions, as has hitherto been done.

40 THE FOUNDATIONS OF EINSTEIN'S

Such doubts received fresh strength when the
special theory of relativity endowed every form of
energy with inertia (vide Note 17).

The results of the special theory of relativity
entirely unhinged our view of the inertia of matter,
for they robbed the theorem concerning the equality
of inertial and gravitational mass of its strict validity.
A body was now to have an inertial mass varying
with its contained internal energy, without its
gravitational mass being altered. But the mass of
a body had always been ascertained from its weight,
without any inconsistencies manifesting themselves
(vide Note 18).

A difficulty of such a fundamental character could
come to light only owing to the theorem of the equality
of inertial and gravitational mass not being sufficiently
interwoven with the underlying principles of mechanics,
and because, in the foundations of Newtonian
mechanics, the same importance had not been accorded
to gravitational phenomena as to inertial phenomena,
which, judged from the standpoint of experience,
must be claimed. Gravitation, as a force acting
at a distance, is, on the contrary, introduced only
as a special force for a limited range of phenomena :
and the surprising fact of the equality of inertial
and gravitational mass, valid at all times and in
all places, receives no further attention. One must,
therefore, substitute for the law of inertia a fundamental
law which comprises inertial and gravitational phe-
nomena. This can be brought about by a consistent
application of the principle of the relativity of all
motions, as Einstein has recognized. This, is, there-
fore, the circumstance chosen by Einstein as a nucleus
about which to weave his developments.

The theorem of the equality of inertial and gravita-
tional mass, which reflects the intimate connection
between inertial and gravitational phenomena, may

THEORY OF GRAVITATION 41

be illuminated from another point of view, and
thereby discloses its close relationship (vide page 55)
to the general principle of relativity.

However much the notion of " absolute space "
repelled Newton, he nevertheless believed he had
a strong argument, in support of the existence of
absolute space, in the phenomenon of centrifugal
forces. When a body rotates, centrifugal forces
make their appearance. Their presence in a body
alone, without any other visible body being present,
enables one to demonstrate the fact that it is in
rotation. Even if the earth were perpetually
enveloped in an opaque sheet of cloud, one would
be able to establish its daily rotation about its
axis by means of Foucault's pendulum-experiment.
This peculiarity of rotations led Newton to conclude
that absolute motions exist. From the purely
kinematical point of view, however, the rotation
of the earth is not to be distinguished in any way
from a translation ; in this case, too, we observe
only the relative motions of bodies, and might just
as well imagine that all bodies in the universe
revolve around the earth. E. Mach has, in fact,
affirmed that both events are equivalent, not only
kinematically, but also dynamically : it must,
however, then be assumed that the centrifugal
forces, which are observed at the surface of the earth,
would also arise, equal in quantity and similar in
their manifestations, from the gravitational effect of
all bodies in their entirety, if these revolved around
the supposedly fixed earth (vide Note 19).

The justification for this view, which in the
first place arises out of the kinematical standpoint,
is, in the main, to be sought in the fact, derived
from experience, that inertial and gravitational
mass are equal. According to the conceptions,
which have hitherto prevailed, the centrifugal

42 THE FOUNDATIONS OF EINSTEIN'S

forces are called into play by the inertia of the
rotating body (or rather by the inertia of the separate
points of mass, which continually strive to follow
the bent of their inertia, and, therefore, express the
tendency to fly off at a tangent to the path in which
they are constrained to move). The field of centri-
fugal forces is, therefore, an inertial field (vide
Note 20). The possibility of regarding it equally
well as a gravitational field — and we do that, as soon
as we also assert the relativity of rotations dynami-
cally : for we must then assume that the whole
of the masses describing paths about the (supposed)
fixed body induce the so-called centrifugal forces
by means of their gravitational action — is founded
on the equality of inertial and gravitational mass,
a fact which Eotvos has established with extra-
ordinary precision by making use of the centrifugal
forces of the rotating earth (vide Note 21). From
these considerations one realizes how a general prin-
ciple of the relativity of all motions simultaneously
implies a theory of gravitational fields.

From these remarks one inevitably gains the
impression that a construction of mechanics upon
an entirely new basis is an absolute necessity.
There is no hope of a satisfactory formulation of
the law of inertia without taking into account the
relativity of all motions, and hence just as little
hope of banishing the unwelcome conception of
absolute motion out of mechanics ; moreover, the
discovery of the inertia of energy has taught us facts
which refuse to fit into the existing system, and
necessitate a revision of the foundations of mechanics.
The condition which must be imposed at the very
outset (cf. page 20) is : Elimination of all actions
which are supposed to take place " at a distance "
and of all quantities which are not capable of direct
observation, out of the fundamental laws ; i.e.

THEORY OF GRAVITATION 43

the setting-up of a differential equation which com-
prises the motion of a body under the influence of
both inertia and gravity and symbolically expresses
the relativity of all motions. This condition is
completely satisfied by Einstein's theory of gravita-
tion and the general theory of relativity. The
sacrifice, which we have to make in accepting them,
is to renounce the hypothesis, which is certainly
deeply rooted, that all physical events take place
in space whose measure-relations (geometry) are
given to us a priori, independently of all physical
knowledge. As we shall see in the following section,
the general theory of relativity leads us, rather,
to the view that we may regard the metrical condi-
tions in the neighbourhood of bodies as being con-
ditioned by their gravitation. In this way the
geometry of the measuring physicist becomes in-
timately welded with the other branches of physics.
If we compress into a short statement what we
have so far deduced out of the fundamental postu-
lates formulated at the beginning, we may say :
The postulate of general relativity demands that
the fundamental laws be independent of the par-
ticular choice of the co-ordinates of reference.
But the Euclidean line-element does not preserve
its form after any arbitrary change of the co-
ordinates of reference. We have, therefore, to
substitute in its place the general line-element :

ds* = Sg^dx^dXy.

Whereas, then, the postulate of continuity (cf.
page 20) seemed to render it only advisable not to
introduce the narrowing assumptions of the Euclidean
determination of measure, the principle of general
relativity no longer leaves us any choice.

The reason for so emphasizing the latter principle

44 THE FOUNDATIONS OF EINSTEIN'S

— as, indeed, also the postulate that only observable
quantities are to occur in physical laws — is not to
be sought in any requirement of a merely formal
nature, but rather in an endeavour to invest the
principle of causality with the authority of a law
which holds good in the world of actual physical
experience. The postulate of the relativity of all
motions receives its true value only in the light of
the theory of knowledge (Note 22). One must, above
all, avoid introducing into physical laws, side by
side with observable quantities, hypotheses which
are purely fictitious in character, as e.g. the space
of Newton's mechanics. Otherwise the principle
of causality would not give us any real information
about causes and effects, i.e. the causal relations
of the contents of direct experience ; which is,
presumably, the aim of every physical description of
natural phenomena.

THEORY OF GRAVITATION 45

§5
EINSTEIN'S THEORY OF GRAVITATION

(a) THE FUNDAMENTAL LAW OF MOTION AND THE
PRINCIPLE OF EQUIVALENCE OF THE NEW
THEORY

AFTER the foregoing remarks we shall be able
to proceed to a short account of Einstein's
theory of gravitation. Within the limits of the
mathematics assumed in this book we shall, of course,
only be able to sketch the outlines so far that the as-
sumptions and hypotheses characteristic of the theory
come into clear view and that their relation to the
two fundamental postulates of the second section
becomes manifest. We start out from the funda-
mental law of motion in classical mechanics, the law
of inertia. Since even in the law of inertia all
the weaknesses of the old theory come to light,
a new fundamental law of motion becomes an ab-
solute necessity for the new mechanics. It is
thus natural that we should start building up the
new theory from this point. The new law of motion
must be a differential law, which, in the first place,
describes the motion of a point-mass under the
influence of both inertia and gravity, and which,
secondly, always preserves the same form, irrespec-
tive of the system of co-ordinates to which it be
referred, so that no system of co-ordinates enjoys
a preference to any other. The first condition

46 THE FOUNDATIONS OF EINSTEIN'S

arises from the necessity of ascribing the same
importance to gravitational phenomena as to in-
ertial phenomena in the new process of founding
mechanics — the law must, therefore, also contain
terms which denote the gravitational state of the
field from point to point ; the second condition
is derived from the postulate of the relativity of
all motion.

A law of this kind exists in the special theory of
relativity in the equation of motion of a single
point, not subject to any external influence. Ac-
cording to this equation, the path of a point is the
" shortest " or " straightest " line (vide Note 23)
— i.e. the " straight line," if the line-element ds
is Euclidean. Written as an equation of variation
this law is :

= 8 {J V - dx* - dy* - dz* + cHi*} = o.

If the principle of the shortest path, which is to be
followed in actual motions, be elevated in this
form to a general differential law for the motion
in a gravitational field too, with due regard to the
principle of the relativity of all motions, the new
fundamental law must run as follows :

For only this form of the line-element remains
unaltered (invariant) for arbitrary transformations
of the xlf xz, #3, #4. The factors gn . . . g44, which
for the present we leave unexplained, occur in it
as something essentially new. Now, the extra-
ordinarily fruitful idea that occurred to Einstein
was this : Since the new law is to hold for any
arbitrary motions whatsoever, thus also for accelera-
tions, such as we perceive in gravitational fields,
we must make the gravitation field, in which

THEORY OF GRAVITATION 47

observed motion takes place, responsible for the
occurrence of these ten factors g^. These ten
co-efficients g^ which will, in general, be functions
of the variables xlt . . . #4, must, if the new funda-
mental law is to be of use, be able to be brought
into such relationship to the gravitational field, in
which the motion takes place, that they are deter-
mined by the field, and that the motion described
by equation (i) coincides with the observed motion.
This is actually possible. (The g^'s are the gravi-
tational potentials of the new theory, i.e. they
take over the part played by the one gravitational
potential in Newton's theory, without, however,
having the special properties, which according to
our knowledge a potential has, in addition.)
^Corresponding to the measure-relations of a
space-time manifold based upon the line-element :

4

ds* = 2 g^dx^dx,,
i

which is now placed at the foundation of mechanics
by virtue of the relativity of all motions, the remain-
ing physical laws must also be so formulated that
they remain independent of the accidental choice
of the variables. Before we enter into this more
closely, the distinguishing features of the theory
of gravitation characterized by equation (i) will be
considered in greater detail.

The postulate of the new theory, that the laws
of mechanics are only to contain statements about
the relative motions of bodies, and that, in particular,
the motion of a body under the action of the attrac-
tion of the remaining bodies is to be symbolically
described by the formula :

= o,

48 THE FOUNDATIONS OF EINSTEIN'S

is fulfilled in Einstein's theory by a physical hypoth-
esis concerning the nature of gravitational pheno-
mena, which he calls the hypothesis or principle
(respectively) of equivalence (vide Note 24). This
asserts the following :

Any change, which an observer perceives in the
passing of any event to be due to a gravitational field,
would be perceived by him in exactly the same way,
if the gravitational field were not present, provided
that he — the observer — makes his system of reference
move with the acceleration which was characteristic
of the gravitation at his point of observation.

For, if the variables x, y, z, t in the equation of
motion

- dx* - dy* - dz*

= o

of a point-mass moving uniformly and rectilinearly
(i.e. uninfluenced by gravity) be subjected to any
transformation corresponding to the change of the
x, y, z, t into the co-ordinates xlt x2> xs, -^ of a system
of reference which has any accelerated motion
whatsoever with regard to the initial system x, y,
z, t ; then, in general, coefficients g^v will occur
in the transformed expression for ds, and will be
functions of the new variables xlf . . . #4, so that
the transformed equation will be :

Taking into account the extended region of validity
of this equation, one will be able to regard the
g^ which arise from the accelerational transforma-
tion (vide Note 25) just as well, as due to the action
of a gravitational field, which asserts its existence
in effecting just these accelerations. Gravitational
problems thus resolve into the general science of
motion of a relativity -theory of all motions.

By thus accentuating the equivalence of gravita-

THEORY OF GRAVITATION 49

tional and accelerational events, we raise the funda-
mental fact, that all bodies in the gravitational
field of the earth fall with equal acceleration, to a
fundamental assumption for a new theory of gravita-
tional phenomena. This fact, in spite of its being
reckoned amongst the most certain of those gathered
from experience, has hitherto not been allotted any
position whatsoever in the foundations of mechanics.
On the contrary, the Galilean law of inertia makes
an event which had never been actually observed
(the uniform rectilinear motion of a body, which is
not subject to external forces) function as the
main-pillar amongst the fundamental laws of me-
chanics. This brought about the strange view
that inertial and gravitational phenomena, which
are probably not less intimately connected with
one another than electric and magnetic phenomena,
have nothing to do with one another. The phe-
nomenon of inertia is placed at the base of
classical mechanics as the fundamental property
of matter, whereas gravitation is only, as it were,
introduced by Newton's law as one of the many
possible forces of nature. The remarkable fact of the
equality of the inertial and gravitational mass of
bodies only appears as an accidental coincidence.

Einstein's principle of equivalence assigns to
this fact the rank to which it is entitled in the
theory of motional phenomena. The new equation
of motion (i) is intended to describe the relative
motions of bodies with respect to one another
under the influence of both inertia and gravity.
The gravitational and inertial phenomena are amal-
gamated in the one principle that the motion take
place in the geodetic line fflds = o). Since the
element of arc

ds

= J S
* i

50 THE FOUNDATIONS OF EINSTEIN'S

preserves its form after any arbitrary transformation
of the variables, all systems of reference are equally
justified as such, i.e. there is none which is more
important than any other.

The most important part of the problem, with
which Einstein saw himself confronted, was the
setting-up of differential equations for the gravita-
tional potentials g^ of the new theory. With the
help of these differential equations, the gMl/s were
to be unambiguously calculated (i.e. as single-
valued functions) from the distribution of the
quantities exciting the gravitational field ; and the
motion (e.g. of the planets) which was described,
according to equation (i) by inserting these values
for the gnv's, had to agree with the observed motion,
if the theory was to hold true. In setting up these
differential equations for the gravitational potentials
gMV Einstein made use of hints gathered from
Newton's theory, in which the factor which excites
the field in Poisson's equation A</> = — 477/3 for the
Newtonian gravitational potential (viz. the factor
represented by /o, the density of mass in this equa-
tion) is put proportional to a differential expression
of the second order. This circumstance prescribes,
as it were, the method of building up these equations,
taking for granted that they are to assume a form
similar to that of Poisson's equation.

In conformity with the deepened meaning we
have assigned to the mutual relation between
inertia and gravity, as well as to the connection
between the inertia and latent energy of a body,
we find that ten components of the quantity which
determines the " energetic " state at any point of
the field, and which was already introduced by the
special theory of relativity as " stress-energy-tensor,"
duly make their appearance in place of the density
of mass p, in Poisson's equation.

THEORY OF GRAVITATION 51

Concerning the differential expressions of the
second order in the gM,/s which are to correspond
to the A<£ of Poisson's equation, Riemann has shown
the following : the measure-relations of a manifold
based on the line-element

4

ds2 = Zg^dXpdx,,,
i

are in the first place determined by a differential
expression of the fourth degree (the Riemann-
Christoffel Tensor), which is independent of the
arbitrary choice of the variables xlf . . . x± and
from which all other differential expressions which
are likewise independent of the arbitrary choice
of the variables xlt . . . #4 and only contain the
gpv's and their derivatives, can be developed (by
means of algebraical and differential operations).
This differential expression leads unambiguously,
i.e. in only one possible way, to ten differential
expressions in the g^'s. And now, in order to
arrive at the required differential equations, Einstein
puts these ten differential expressions proportional
to the ten components of the stress-energy-tensor,
regarding the latter ten as the quantities exciting
the field. He inserts the gravitational constant
as the constant of gravitation. These differential
equations for the gMV's, together with the principle
of motion given above, represent the fundamental
laws of the new theory. To the first order they,
in point of fact, lead to those forms of motion,
with which Newton's theory has familiarized us
(vide Note 26). More than this, without requiring
the addition of any further hypothesis, they mathe-
matically account for the only phenomenon in the
theory of planetary motion which could not be ex-
plained on the Newtonian theory, viz. the occur-
rence of the remainder-term in the expression

52 THE FOUNDATIONS OF EINSTEIN'S

for the motion of Mercury's perihelion. Yet we
must bear in mind that there is a certain arbitrariness
in these hypotheses just as in that made for the
fundamental law of motion. Only the careful
elaboration of the new theory in all its consequences,
and the experimental testing of it will decide whether
the new laws have received their final forms.

Since the formulae of the new theory are based
upon a space-time-manifold, the line-element of
which has the general form

*=,;

all other physical laws, in order to bring the general
theory of relativity to its logical conclusion, must
receive (see p. 46) a form which, in agreement
with the new measure-conditions, must be indepen-
dent of the arbitrary choice of the four variables

Mathematics has already performed the pre-
liminary work for the solution of this problem
in the calculus of absolute differentials ; Einstein
has elaborated them for his particular purposes
(in his essay " Concerning the formal foundations
of the general theory of relativity * ") ; Gauss
invented the calculus of absolute differentials in
order to study those properties of a surface (in the
theory of surfaces) which are not affected by the
position of the surface in space nor by inelastic
continuous deformations of the surface (deforma-
tions without tearing), so that the value of the
line-element does not alter at any point of the
surface. As such properties depend upon the inner
measure-relations of the surface only, one avoids

* " Uber die formalen Grundlagen der allgemeinen Relativitats-
theorie," Sit*. Ber. d. Kgl. Preuss. Akad. d. Wiss., xli., 1916, S. 1080.

THEORY OF GRAVITATION 53

referring, in the theory of surfaces, to the usual
system of co-ordinates, i.e. one avoids reference to
points which do not themselves lie on the surface.
Instead of this, every point in the surface is fixed,
by covering the surface with a net-work, consisting
of two intersecting arbitrary systems of curves,
in which each curve is characterized by a parameter ;
every point of the surface is then unambiguously,
i.e. singly, defined by the two parameters of the
two curves (one from each system) which pass
through it. According to this view of surfaces, a
cylindrical envelope and a plane, for instance,
are not to be regarded as different configurations :
for each can be unfolded upon the other without
stretching, and accordingly the same planimetry
holds for both — a criterion that the inner measure-
relations of these two manifolds are the same
(vide Note 27). The general theory of relativity
is based upon the same view ; but now not as
applied to the two-dimensional manifold of surfaces,
but with respect to the four-dimensional space-time
manifold. As the four space-time variables are
devoid of all physical meaning, and are only to be
regarded as four parameters, it will be natural
to choose a representation of the physical laws,
which provides us with differential laws which
are independent of the chance choice of the xlt
%2> xs> %\ '> this is what is done by the calculus of
absolute differentials. The results of the preceding
paragraphs, the far-reaching consequences of which
can be fully recognized only by a detailed study
of the mathematical developments involved, may
be summarized as follows :

A mechanics of the relative motions of bodies,
which is in harmony with the two fundamental
postulates of continuity and relativity, can be
built up only on a fundamental law of motion

54 THE FOUNDATIONS OF EINSTEIN'S

that preserves its form independently of the kind
of motion the system is undergoing. An available
law of this kind is given if we raise the law of motion
along a geodetic line, which, in the special theory
of relativity, holds only for a body moving under
no forces, to the rank of a general differential law
of the motion in the gravitational field, too. In
this general law, we must, it is true, give the line-
element of the orbit of the moving body the general
form :

ds =

at which we arrive in the second section, using as
our basis the two fundamental postulates. The
new functions g^ that now occur may be -inter-
preted as the potentials of the gravitational field,
if we take our stand on the hypothesis of equivalence.
To calculate the quantities gMV from the factors
determining the gravitational field, namely, matter
and energy, it immediately suggests itself to us
to assume a system of differential equations of the
second order, that are built up analogously to
Poisson's differential equation for the Newtonian
gravitational potential. These differential equa-
tions, together with the fundamental law of motion,
represent the fundamental equations of the new
mechanics and the theory of gravitation.

Since the new theory uses the generalized curvi-
linear co-ordinates xlt x2) x3, #4, and not the Car-
tesian co-ordinates of Euclidean geometry, all the
other physical laws must also receive a general
form that is independent of the special choice of
co-ordinates. The mathematical instrument for re-
moulding these formulae is given by*the*general
calculus of differentials.

This theory, which is built up from the most

THEORY OF GRAVITATION 55

general assumptions, leads, for a first approxi-
mation, to Newton's laws of motion. Wherever
deviations from the theory hitherto accepted reveal
themselves, we have possibilities of testing the new
theory experimentally. Before we turn to this
question, let us look back, and become clear as to
the attitude which the general theory of relativity
compels us to adopt towards the various questions
of principle we have touched upon in the course
of this essay.

(b) RETROSPECT

i. The conceptions " inertia! " and " gravita-
tional " (heavy) mass no longer have the absolute
meaning which was assigned to them in Newton's
mechanics. The " mass " of a body depends, on
the contrary, exclusively upon the presence and
relative position of the remaining bodies in the
universe. The equality of inertial and gravita-
tional mass is put at the head of the theory as a
rigorously valid principle. The hypothesis of equi-
valence herein supplements the deduction of the
special theory of relativity, that all energy possesses
inertia, by investing all energy with a corresponding
gravitation. It becomes possible — on the basis
(be it said) of certain special assumptions into which
we cannot enter here — to regard rotations un-
restrictedly as relative motions too, so that the
centrifugal field around a rotating body can be
interpreted as a gravitational field, produced by
the revolution of all the masses in the universe
about the non-rotating body in question. In
this manner mechanics becomes a perfectly general
theory of relative motions. As, our statements
are concerned only with observations of relative
motions, the new mechanics fulfils the postulate

56 THE FOUNDATIONS OF EINSTEIN'S

that in physical laws observable things only are
to be brought into causal connection with one
another. It also fulfils the postulate of continuity ;
since the new fundamental laws of mechanics are
differential laws, which contain only the line-
element ds and no finite distances between bodies.

2. The principle of the constancy of the velocity
of light in vacuo, which was of particular importance
in the special theory of relativity, is no longer
valid in the general theory of relativity. It pre-
serves its validity only in regions in which the
gravitational potentials are constant, finite portions
of which we can never meet with in reality. The
gravitational field upon the earth's surface is cer-
tainly so far constant that the velocity of light,
within the limits of accuracy of our measurements,
had to appear to be a universal constant in the
results of Michelson's experiments. In a gravita-
tional field, however, in which the gravitational
potentials vary from place to place, the velocity
of light is not constant ; the geodetic lines, along
which light propagates itself, will thus in general
be curved. The proof of the curvature of a ray of
light, which passes by in close proximity to the sun,
offers us one of the most important possibilities of
confirming the new theory.

3. The greatest change has been brought about
by the general theory of relativity in our concep-
tions of space and time.*

According to Riemann the expression for the
line-element, viz.

4

1

* This aspect of the problem has been treated with particular
clearness and detail in the book " Raum und Zeit in der gegenwartigen
Physik," by Moritz Schlick, published by Jul. Springer, Berlin. The
Clarendon Press has published an English rendering under the title :
" Space and Time in Contemporary Physics."

THEORY OF GRAVITATION 57

determines, in our case, the measure-relations of
the continuous space-time manifold ; and according
to Einstein the coefficients g^ of the line-element
ds have, in the general theory of relativity, the
significance of gravitational potentials. Quantities,
which hitherto had only a purely geometrical
import, for the first time became animated with
physical meaning. It seems quite natural that
gravitation should herein play the fundamental
part, viz. that of predominating over the measure-
laws of space and time. For there is no physical
event in which it does not co-operate, inasmuch
as it rules wherever matter and energy come into
play. Moreover, it is the only force, according to
our present knowledge, which expresses itself quite
independently of the physical and chemical constitu-
tion of bodies. It therefore without doubt occupies
a unique position, in its outstanding importance
for the construction of a physical picture of the
world.

According to Einstein's theory, then, gravitation
is the " inner ground of the metric relations of space
and time " in Riemann's sense (vide the final para-
graph of Riemann's essay " On the hypotheses
which lie at the bases of geometry " quoted on
p. 29). If we uphold the view that the space-
time manifold is continuously connected, its measure-
relations are not then already contained in its
definition as being a continuous manifold of the
dimensions " four." These have, on the contrary,
yet to be gathered from experience. And it is,
according to Riemann, the task of the physicist
finally to seek the inner ground of these measure-
relations in " binding forces which act upon it."
Einstein has discovered in his theory of gravitation
a solution to this problem, which was presumably
first put forward in such clear terms by Riemann.

58 THE FOUNDATIONS OF EINSTEIN'S

At the same time he gives an answer to the question
of the true geometry of physical space, a question
which has exercised physicists for the last century,
— but an answer, it is true, of a sort quite different
from that which had been expected.

The alternative, Euclidean or non-Euclidean
geometry, is not decided in favour of either one or
the other ; but rather space, as a physical thing
with given geometrical properties, is banished out
of physical laws altogether : just as ether was
eliminated out of the laws of electrodynamics by
the Lorentz-Einstein special theory of relativity.
This, too, is a further step in the sense of the
postulate that only observable things are to have
a place in physical laws. The inner ground of
metric relations of the space- time manifold, in
which all physical events take place, lies, according
to Einstein's view, in the gravitational conditions.
Owing to the continual motion of bodies relatively
to one another, these gravitational conditions are
continually altering ; and, therefore, one cannot
speak of an invariable given geometry of measure
or distance — whether Euclidean or non-Euclidean.
As the laws of physics preserve their form in the
general theory of relativity, independent of how
the four variables xlt . . . #4 may chance to be
chosen, the latter have no absolute physical meaning.
Accordingly xlt xz, x3, for instance, will not in
general denote three distances in space which can
be measured with a metre rule, nor will %± denote
a moment of time which can be ascertained by
means of a clock. The four variables have only
the character of numbers, parameters, and do
not immediately allow of an objective interpretation.
Time and space have, therefore, not the meaning
of real physical things in the description of the
events of physical nature.

THEORY OF GRAVITATION 59

And yet it seems as if the new theory may even
be able to give a definite answer in favour of one
or other of the above alternatives, if, namely, we
postulate their validity for the world as a whole.
The application of the formulae of the new theory
to the world as a whole at first led to the same
difficulties as those revealed in classical mechanics.
Boundary conditions for what is infinitely distant
could not be set up entirely satisfactorily and at
the same time satisfy the condition of general
relativity. Yet Einstein * succeeded in extending
the differential equations for the gravitational
potentials g^ in such a way that it became possible
to apply his theory of gravitation to the universe.
The difficulties that arose for the boundary conditions
at infinity here vanished, for an extraordinarily
interesting reason. For it was shown that in
these new formulae a space that is filled uniformly
with matter which is at rest would, to a first approxi-
mation, be built up like an, indeed, unbounded, but
finitely closed space, so that boundary conditions
would not appear at all for infinity. Even if the
assumptions that would lead to this result are not
fulfilled in the world, yet it must be remembered
that the velocities of matter as ascertained in the
case of the stars are extraordinarily small com-
pared with the velocity of light which we now
take as our unit. Nor does the distribution of the
matter so far show, in the main, irregularities
sufficient to place Einstein's view of a stationary,
uniformly-filled world quite out of the realm of
possible truth. Thus this deduction of the theory
would answer our above alternative in this sense :
the geometry that we must use as our basis of
spatial happening is, indeed, neither Euclidean nor

* " Kosmologische Betrachtungen zur allgemeinen Relativitats-
theorie " Sitz. Ber. d. Preuss. Akad. der Wiss., 1917, p. 142.

60 THE FOUNDATIONS OF EINSTEIN'S

non-Euclidean, but, as stated above, conditioned
by the gravitational states from place to place.
But a world built up according to the simplest
scheme would in the new theory behave on the
whole like a finite closed manifold, that is, as if it
were non-Euclidean. Even if this result is only
of theoretical importance for the present, since the
stellar system that we see around us does not fulfil
Einstein's assumptions — in particular, the scarcely-
to-be-doubted flattening of the Milky Way is not
compatible with these simple assumptions — and
since we have at present no knowledge of the
stellar systems outside the Milky Way, yet this
aspect of the theory opens up undreamed-of per-
spectives for our view of the world as a whole.

4. The gravitational theory, which emerges out
of the general theory of relativity, is, in contra-
distinction to the Newtonian theory, built up, not
upon an elementary law of the gravitational forces,
but upon an elementary law of the motion of a body
in the gravitational field. Consequently, the ex-
pressions which would be interpreted as gravitational
forces in the new theory play only a minor part in
the building-up of the theory (as indeed the con-
ception of force in mechanics altogether is to be
regarded as only an auxiliary or derived conception,
if we regard it as the object of mechanics to give
a flawless description of the motions occurring in
physical events).

Nor does Einstein's theory endeavour to explain
the nature of gravitation ; it does not seek to give
a mechanical model, which would symbolize the
gravitational effect of two masses upon one another.
This is what the various theories involving ether-
impulses attempted to do, by freely using hypothet-
ical quantities which had never been actually
observed, such as ether-atoms. It is very doubtful
whether such endeavours will ever lead to a satis-

THEORY OF GRAVITATION 61

factory theory of gravitation. For, the difficulties
of Newton's mechanics are not contained only in
the fact that it formulates the law of gravitation
as a law of forces acting at a distance. Two much
more serious points are : first, that the close rela-
tionship existing between inertial and gravitational
phenomena receives no recognition whatsoever,
although Newton was already aware of the fact
that inertial and gravitational mass are equal ;
and second, that Newton's mechanics does not
present us with a theory of the relative motions
of bodies, although we only observe relative motions
of bodies with respect to one another. Re-moulding
Newton's law of gravitational force, in order to
make the attraction of matter more feasible, would
therefore not have helped us finally to a satisfactory
theory of the phenomena of motion (vide Note 28).

What distinguishes the Newtonian theory, above
all, is the extraordinary simplicity of its mathe-
matical form. Classical mechanics, which is built
up on Newton's initial construction, will, for this
reason, never lose its importance as an excellent
mathematical theory for arithmetically following
the observed phenomena of motion.

Einstein's theory, on the other hand, as far as
the uniformity of its conceptual foundations is
concerned, satisfies all the conditions for a physical
theory. The fact that (by abandoning the Euclidean
measure of distance) it cuts its connection with the
familiar representation by means of Cartesian co-
ordinates, will not be felt to be a disturbing factor,
as soon as the analytical appliances, which have
been called into use as a help, have been more
generally adopted. This mathematical elaboration
of the theory at the same time gives to the astro-
nomer the task of testing the theory experimentally
in those phenomena in which measurable deviations
from the results of the classical theory arise.

62 THE FOUNDATIONS OF EINSTEIN'S

§6

THE VERIFICATION OF THE NEW THEORY BY
ACTUAL EXPERIENCE

AS far as can be seen at present, there are
three possible experiments for verifying Eins-
tein's theory of gravitation ; all three can
be performed only by the agency of astronomy.
One of them — arising from the deviation of the
motion of a material point in the gravitational
field according to Einstein's theory, as compared
with that required by Newton's theory — has already
decided in favour of the new theory : not less so
one of the other two that arise through a com-
bination of electromagnetic and gravitational phen-
omena.

Since the sun far exceeds all other bodies of the
solar system in mass, the motion of each particular
planet is primarily conditioned by the gravitational
field of the sun. Under its action the planet de-
scribes, according to Newton's theory, an ellipse
(Kepler's law), the major axis of which — defined
by connecting the point of its path nearest the sun
(perihelion) with the farthest point (aphelion) — is
at rest, relative to the stellar system. Upon this
elliptic motion of a planet there are superimposed
more or less considerable influences (disturbances)
due to the remaining planets, which do not, how-
ever, appreciably alter the elliptic form ; these
influences partly only call forth periodical fluctua-

THEORY OF GRAVITATION 63

tions in the defining elements of the initial ellipse
(i.e. major axis, eccentricity, etc.), partly cause a
continual increase or decrease of the latter. In
this second kind of disturbance are also to be classed
the slow rotation of the major axis, and consequently
also of the corresponding perihelion, relative to
the stellar system ; which has been observed in
the case of all planets. For all the larger planets,
the observed motions of the perihelion agree with
those calculated from the disturbing effects (except
for small deviations which have not been definitely
established, as in the case of Mars) ; on the other
hand, in the case of Mercury the calculations give
a value which is too small by 43" per 100 years.
Hypotheses of the most diverse description have
been evolved to explain this difference ; but all
of them are unsatisfactory. They oblige one to
resort to still unknown masses in the solar system :
and, as all the searches for masses large enough to
explain this anomalous behaviour of Mercury prove
fruitless, one is compelled to make assumptions
about the distribution of these hypothetical masses,
in order to excuse their invisibility. In view of
these circumstances, there is no shade of probability
in these hypotheses.

According to Einstein's theory, a planet, at the
distance of Mercury for instance, moves, under
the action of the sun's attraction, along the " straight-
est path," according to the equation :

The g^'s can be derived from the differential
equations, which were given for them above, and
which result from the assumed sole presence of the
sun and the planet being regarded as a mass concen-
trated at a point. Einstein's developments give the

64 THE FOUNDATIONS OF EINSTEIN'S

ellipse of Kepler too as a first approximation for
the path of the planet : at a higher degree of ap-
proximation, however, it is found that the radius
vector from the sun to the planet, between two
consecutive passages through perihelion and aphe-
lion, sweeps out an angle, which is about 0-05"
greater than 180° ; so that, for each complete
revolution of the planet in its path, the major axis
of the path — i.e. the straight line connecting perihe-
lion with aphelion — turns through about O'i" in
the sense in which the path is described. There-
fore, in 100 years — Mercury completes a revolution
in 88 days — the major axis will have turned through
43". The new theory, therefore, actually explains
the hitherto inexplicable amount, 43 seconds per
100 years, in the motion of Mercury's perihelion,
merely from the effect of the sun's gravitation.
(The deviations due to such disturbances would
only differ very inappreciably from the values
obtained by Newton's theory in the case of the
remaining planets.) The only arbitrary constant
which enters into these calculations is the gravita-
tional constant which figures in the differential
equations for the gravitational potentials g^, as has
already been mentioned on page 50. This achieve-
ment of the new theory can scarcely be estimated
too highly.

The reason that a measurable deviation from the
results according to Newton's theory occurs in the
case of Mercury, the planet nearest to the sun,
but not in the case of the planets more distant
from the sun, is that this deviation decreases rapidly
with increasing distance of the planet from the sun,
so that it already becomes imperceptible at the
distance of the earth. In the case of Venus, the
eccentricity of the path is, unfortunately, so small,
that it scarcely differs from a circle : and the

THEORY OF GRAVITATION 65

position of the perihelion can, therefore, only be
determined with great uncertainty.

Of the other two possibilities of verifying the
theory, one arises from the influence of gravitation
upon the time an event takes to pass. How such
an influence can come about, will be evident from
the following example : According to the new
theory, an observer cannot immediately distinguish
whether a change, which he observes during the
passage of a certain event, is due to a gravitational
field or to a corresponding acceleration of his place
of observation (his system of reference). Let us
assume an invariable gravitational field, denoted
by parallel lines of force in the negative direction
of the 2-axis, and having a constant value y for the
acceleration with which all bodies in the field fall
(i.e. characterized by conditions which approxi-
mately exist on the surface of the earth). According
to Einstein's theory, any event will take place in
this field in just the same way as it appears to occur
when referred to a co-ordinate system which has
an acceleration y in the positive direction of the
<z-axis. Now if a ray of light, the time of oscillation
of which is vlf travels from a point A — which is to be
conveniently supposed at rest relatively to the cor-
responding co-ordinate system at the moment of
departure of the ray — in the direction of the z-axis
for a distance h to a point B, then an observer
at B will, owing to his own acceleration, y, have
attained a velocity y . h/c at the instant the ray
of light reaches him (c denotes the velocity of light).
According to the usual Doppler Principle, he will
assign a time of oscillation v2 = ^ (i -f y . h/c2)
to the ray of light as a first approximation, instead
of i/j. If we transfer the same event to the equivalent
gravitational field, this result assumes the following
form : The time of oscillation v2 of a ray of light
5

66 THE FOUNDATIONS OF EINSTEIN'S

at a place B, the gravitational potential of which
differs from that of a place A by the amount + <£,
is connected with the time of oscillation there
observed by the relation :

according to the principle of equivalence of Einstein's
theory of gravitation.

This special case shows how the duration of an
event is to be understood as being dependent upon
the gravitational condition.

Moreover, one can regard every vibrating system
(which emits a spectral line) as a clock, the motion
of which, according to the investigation made just
above, depends upon the gravitational potentials
of the place where it is stationed. This same
" clock " will have a different time of oscillation
at another place in the field according to the gravita-
tional potential, i.e. it will go at a different rate.
Consequently, a particular line in the spectrum of
the light which comes from the sun, e.g. an Fe-line
(iron), must appear to be shifted in comparison
with the corresponding line as produced by a
source of light (arc-lamp) on the earth ; the gravita-
tional potential at the surface of the sun has, cor-
responding to the latter's great mass, a different
value from that at the surface of the earth, and a
definite time of oscillation (colour) is characterized
in the spectrum by a definite position (Fraunhofer
line). It has not yet been possible to observe
this effect, which amounts to about o-oo8A * for
a wave-length of 400^ with certainty. For the
conditions of emission of the light from the sun's
surface have not yet been sufficiently investigated,
and the systematic errors in the wave-lengths in

* A = Angstrom unit = jo ~ 8 cm.

THEORY OF GRAVITATION 67

the light from the source used for comparison on
the earth, the arc-lamp, are not yet sufficiently
known to allow the negative results of observation
hitherto obtained to be regarded as giving binding
decisions. This is the more true inasmuch as in
the case of the fixed stars there are, doubtless,
signs of the presence of a gravitational shift of the
spectral lines (vide the closing essay of this book).
It is a particularly important task of astronomy to
establish this effect with certainty, for this gravita-
tional displacement of the spectral lines is a direct
consequence of the hypothesis of equivalence, and
does not assume the other hypotheses of the theory
such as, for example, the differential equations of
the gravitational field.

The third and particularly important inference
from Einstein's theory is the dependence of the
velocity of light upon the gravitational potential,
and the resultant curvature (based upon Huygens'
principle) of a ray of light in passing through a
gravitational field. The theory asserts that a ray
of light, coming e.g. from a fixed star, and which
passes in close proximity to the sun, has a curved
path. As a consequence of this curvature, the star
must appear displaced from its true position in
the heavens by an amount which attains the value
17" at the edge of the sun's disc, and decreases
in proportion to the distance from the centre of
the sun. But since a ray of light which comes
from a fixed star and passes by the sun can be
caught only when the light of the sun, which over-
powers all else by its brilliancy, is intercepted
before its entrance into our atmosphere, only the
rare moments of a total eclipse come into account
for this observation and for the solution of the
problem. The solar eclipse of 2Qth May, 1919,
during which photographs were taken at two widely-

68 THE FOUNDATIONS OF EINSTEIN'S

separated stations, for the purpose of this test, has,
as far as the results of measurement allow us to
pass definite judgment, decided in favour of the
general theory of relativity.*

The experimental verification of Einstein's theory
of gravitation has thus not reached completion.
But if, in spite of this, the theory can, even at this
early stage, justly claim general attention, the
reason is to be found in the unusual unity and logical
structure of the ideas underlying it. In truth, it
solves, at one stroke, all the riddles, concerning the
motions of bodies, which have presented them-
s*elves since the time of Newton, as the result of the
conventional view about the meaning of space and
time in the physical description of natural pheno-
mena.

* The results were made public at the meeting of the Royal
Society on the 6th Nov., 1919.— H. L. B.

THEORY OF GRAVITATION 69

APPENDIX

Note i (p. 4). So long as the universal signi-
ficance of the velocity of light remained unknown,
two conjectures were possible in the question as
to whether, under certain circumstances, the motion
of the source of light would make itself observable
in the velocity of propagation of light. It might
be surmised that the velocity of the source simply
added itself to that velocity of light which is char-
acteristic for the propagation of the light from a source
at rest. Or, it might be conceived that the motion
of the source has no influence at all on the velocity
of the light emitted by it. In the second case it
was imagined that the source of light only excites
the periodically changing states of the luminiferous
ether, which is at rest, that is, which does not
share in the motion of the matter (source of light),
and ^ that these states then propagate themselves
with a velocity that is characteristic of the ether, and
with a velocity that makes these states perceptible to
us as light waves. This view had finally apparently
won the day. It was the advent of the special
theory of relativity and the quantum hypothesis
that made this view impossible. For the special
theory of relativity, in robbing the assertion :
" the ether is at rest " of its significance, since we
may arbitrarily define any system as being at rest in
the ether, as far as uniform translations are concerned,
and in depriving the luminiferous ether of its exis-
tence, deprived light-waves of their carrying or

70 THE FOUNDATIONS OF EINSTEIN'S

transmitting medium. The quantum hypothesis, in
raising light-quanta to the rank of self-supporting
individuals, deprived the velocity of light of its
character as a constant that is characteristic of the
ether. Thus, our view of light-quanta again leads
to a kind of emission theory of light. According
to classical mechanics it would have been typical
of a theory of emission for the velocity of the source
in motion to have added itself to the velocity of the
light from the source at rest. We thus revert
to the conjecture which we quoted first above.
Now, such a superposition of velocities would
necessarily cause quite remarkable phenomena in
the case of spectroscopic binary stars (de Sitter,
" Phys. Zeitschrift," 14, 429). For if two stars move
in circular Kepler orbits around each other, and if
our line of sight lies in the common plane of the
orbits, then we should necessarily perceive the
following : if 2T is the time of revolution of the
system, v the orbital velocity of the one (bright)
component, A the distance of the whole system
from the earth, and, finally, c, the velocity in vacuo
of the light from the source which is at rest, then
the velocity of light at the epoch of greatest positive
velocity in the direction of vision is c -\- v, and
c — v in the other direction, respectively. Con-
sequently the time-interval between two such suc-

cessive positions would have the values T -f — —

and T — — alternately, for the observer on the
c2

earth, as a simple calculation shows. Since, on
account of the gigantic distances between the

fixed stars, the member - may become very

c

great, indeed, greater than T, we should be able to
observe definite anomalies in the case of the spectro-

THEORY OF GRAVITATION 71

scopic binaries. For the time intervals between
two such successive epochs in the orbit should be
able to contract to nil, indeed, even become negative,
and we should not be able to interpret the measured
Doppler effects by means of motions in the Kepler
ellipses. In reality, however, these anomalies have
never manifested themselves. Observation of these
very sensitive subjects of test (spectroscopic binaries)
teaches us that the motion of the source of light
does not make itself remarked in the propagation
of the light. This renders our first view likewise
untenable. The special principle of relativity, alone
in postulating the constancy of the velocity of light,
and in putting forward a new addition theorem
of velocities, has led us to an attitude in this
question that is free from inner contradictions
and compatible with experience. „ (Cf. Note 2.)

Note 2 (p. 5). There are essentially two funda-
mental optical experiments on which our view of
the distinctive significance of the velocity of light
in physical nature is founded : Fizeau's experiment
concerning the velocity of light in flowing water,
and the Michelson-Morley experiment. Aberration,
on the other hand, has nothing to do directly with
the question whether it is possible to prove by
means of optical experiments in the laboratory a
motion of the earth relative to the ether. The
aberration in the case of stars states merely that
the motion of the earth relatively to the star under
consideration changes periodically in the course of
a year. If, however, we hold the view that an
all-pervading ether is the carrier for the propagation
of the light, the phenomenon of aberration may
be satisfactorily explained only if we assume that
this ether does not participate in the motion of the
earth.

Fizeau's experiment was designed to decide

n THE FOUNDATIONS OF EINSTEIN'S

finally whether moving matter influences the ether
and to determine the value of the velocity of light
in moving matter with respect to the observer.
Michelson and Morley repeated the experiment in
the following improved form. A beam of light
from a source on the earth is sent through a U-shaped
tube, through which water flows, in the direction
of both limbs. After each part of the beam has
traversed the flowing water, the one in the direction of
the current, the other contrary to it, the two beams,
are made to interfere. The light and the water move
in the same direction in the one limb, and oppo-

FlG. I.

sitely in the other. Now, there immediately ap-
pear to be two possibilities. Either the water that
flows with a velocity v with respect to the walls
of the tube drags along the carrier that effects
the transmission of the light, namely the ether ;

in this case, the velocity of the light is - -+- v in the

n

one limb, and — — v in the other, for, on account
n

of the coefficient of refraction n of the water, - is

n

the velocity of light in resting water. Or, the
motion of the water has no influence at all on the

THEORY OF GRAVITATION 73

ether which transmits light and which permeates

the water. In this case the velocity of light is -

n

in both limbs. According, as the one or the other
of these two assumptions is valid, the interference
fringes would have to become displaced or remain
at rest when the direction of the current is reversed.
The experiment decided in favour of neither of these
possibilities. The interference fringes did, indeed,
become shifted, not to the expected amount, how-
ever, but only to an amount that would result if

the ether assumes the velocity v(i — -ij in water,

and not the full value v. This value of the con-
vection of the ether is called Fresnel's convection
coefficient. Yet this term is capable of being mis-
understood inasmuch as in the electrodynamics
developed by Lorentz, the result of Fizeau's experi-
ment speaks in favour of an ether that is absolutely
at rest, and the so-called convection coefficient is
only a consequence of the structure of matter, in
particular of the interaction between electrons and
matter, a question into which we cannot enter here.
At any rate, at the time preceding the Michelson-
Morley experiment aberration, as well as Fizeau's
experiment, appeared to speak in favour of an ether
that was absolutely at rest.

Now, the Michelson-Morley experiment was to
establish the existence of the current of ether
(ether " wind ") through which the earth continually
moves, since the ether is supposed not to participate
in the motion of the earth. The scheme of the
experiment is as on p. 74.

A ray of light, starting out from L, traverses the
course LP + PSX + SXP + PFl : here Sj and S2 are
two mirrors, on to which the ray falls perpendicu-
larly ; P is a glass plate that reflects one half of

74 THE FOUNDATIONS OF EINSTEIN'S

the light and allows the remainder to pass through ;
F is the telescope of the observer. Another ray of
light traverses the course LP + PS2 + S2 P + PF.
Let PSi = PS2 = /. Further, let FSi be in the
direction of the earth's motion. Our assumption is
that the ether does not share in the earth's motion.
Let the velocity of the earth be q. Then the

FIG. 2.

velocity of the light relative to the instrument
(earth) is as follows in the directions specified :

PSt : c - q, thus the light- time for the journey is

PS2 :
S2P :

Consequently, the course PS± + SXP is traversed in
the time

THEORY OF GRAVITATION 75

and the course PS2 + S2P in the time

t 2l _
^-x/crzr^- c

The difference of these two times is

If we exchange the positions of Sx and S2 by turning
the whole apparatus through 90°, then

If we make these two rays of light interfere at
F then, when the apparatus is turned through 90°,
the interference fringes should become shifted.
The amount of this displacement may easily be
calculated. If we denote by r the vibration fre-
quency of the light-ray used in the experiment,
then CT = A is the corresponding wave-length.
Thus, expressed in fractions of the interval .between
the fringes, the expected displacement becomes
equal to

_
r Ac2'

By causing the light to be reflected many times

2l

2l was magnified to such an extent that -^ became

A

of the order io7. If, for example, 2l = 30
metres = 3O-io2 cms., A = 6-io~5 cms. = the wave-

2l

length of sodium light, then — = 5«io7cms. On the

A

other hand £ is of the order ( 3Q kilometres \«

c2 \300,ooo kilometres/

that is, io~ 8. The expected displacement of the

76 THE FOUNDATIONS OF EINSTEIN'S

fringes would thus have to be about 0-56 of the
breadth of a fringe. Actually, an amount of the
order 0-02 of the breadth of a fringe was observed.
Thus, the ether wind did not make itself remarked
optically in the motion of the earth. By carrying
out the experiment at different times of the year
the possible objection that the motion of translation
of the entire solar system might have counter-
balanced the motion of the earth in her orbit was
removed.

The Michelson-Morley experiment has shown
conclusively that there is no physical sense in
talking of absolute rest or of a translation relative
to absolute space, since all systems that move
rectilinearly and uniformly with respect to one
another are of equal value for describing natural
phenomena. It is thus a matter of convention
which system we are to regard as at rest and which
as being in motion. We may assign the same value
to the velocity of light in all systems. A detailed
theory of these fundamental experiments may be
found in all comprehensive accounts of the special
theory of relativity. We here merely mention the
original paper by A. Einstein (Annalen der Physik,
Bd. 17, 1905, p. 891), and the booklet, " Einfiihrung
in die Relativitatstheorie," by Dr. W. Block, out
of the series " Aus Natur und Geisteswelt," Teubner,
1918.

Note 3 (p. 9). Abolishing the transformations
of Newton's principle of relativity and replacing
them by the so-called Lorentz-Einstein transforma-
tions signified a step of extraordinarily far-reaching
consequence. It was justified in that the new
theory of relativity which followed as a result of
it, confirmed, without difficulty, the results of all
the fundamental experiments of optics and electro-
dynamics. Concerning the Michelson-Morley ex-

THEORY OF GRAVITATION 77

periment, Lorentz, to account for its negative
result within the realm of electrodynamics, had
been compelled to set up the hypothesis that the
dimensions of all bodies contract in the direction of
their motion. But Einstein now showed that if
we define the conception of simultaneity rigorously,
taking into account the postulate of the constancy
of the velocity of light, the Lorentz-transformations,
which had been found empirically, followed neces-
sarily as those equations of transformation that
must hold between the co-ordinates of two systems
moving uniformly and rectilinearly with respect to
each other. And without the help of any further
hypothesis there appears as a direct consequence of
this transformation just that contraction of lengths
which Lorentz had adduced to explain the result
of the Michelson-Morley experiment. This con-
traction of a length / in the direction of motion of

/ v*

an object to the value Mi — _ is, however, in the

* c

new theory the expression of the general fact that
the dimensions of a body have only a relative
meaning, that is, that their values depend on the
state of motion of the observer, which determines
the dimensions of the body in question. This
holds for the extension of bodies in time as well
as in space. From the point of view of the new
principle of relativity the negative result of the
Michelson-Morley experiment was self-explained.
But what was the position with regard to the other
fundamental facts of optics and electrodynamics ?
The result of Fizeau's experiment concerning the
velocity of light in flowing water became a direct
test of the kinematics arising out of the new formulae.
According to the Lorentz transformation the two
velocities, q and v with which, for example, two
locomotives approach each other, do not merely

78 THE FOUNDATIONS OF EINSTEIN'S

become added, so that q -f v would be the relative
velocity of each with respect to the other, but
rather each engine-driver will find as the velocity
with which he passes the other driver, the value

according to the new formulae. This is the addition
theorem of velocities according to the new theory.
It gives us immediately the amount observed in
Fizeau's experiment for the velocity of light in
flowing water. Aberration and the Doppler effect
follow just as readily to the correct amount. A
detailed discussion of these questions is to be found
in every account of the " special " theory of relativity
(cf. the references given in Note 2).

Note 4 (p. 12). Ph. Frank and H. Rothe,
Ann. d. Phys., 4 Folge, Bd. 34, p. 825.

The assumptions for the general equations of
transformation by which two systems S and S'
that move uniformly and rectilinearly with the
velocity q with respect to each other are connected
are as follows : —

1. The equations of transformation form a linear
homogeneous group in the variable parameter q.
This means that the successive application of two
equations of transformation, of which the one refers
the system S to the system S', and the second S' to
S" (S is to have the constant velocity q with respect
to S', and S' the constant velocity q' with respect to
S") again leads to an equation of transformation
of the same form as that of the initial equations.
The parameter q" that occurs in the new equation
depends in a definite way on q' and q.

2. The contractions of the lengths depend only
on the value of the parameter q. We must, of

THEORY OF GRAVITATION 79

course, from the very outset reckon with the possi-
bility that the length of a rod measured in the
system that is at rest comes out differently when
measured in the moving system. Now, condition 2
requires that if contractions occur (that is, changes
of length in these various methods of determination)
values are to depend only on the magnitude of the
velocity of both systems and not on the direction
of their motion in space. Thus this postulate
endows space with the property of isotropy, and is
in fair correspondence with the postulate of section
3#, which states that it must be possible to compare
each line-element with every other in length in-
dependently of its position in space, and its direction.

An essential feature is that the constancy of the
velocity of light is not demanded in either of the
postulates i and 2. Rather, the distinguishing pro-
perty of a definite velocity in virtue of which it pre-
serves its value in all systems that emerge out of one
another through such transformations is a direct
corollary to these two general postulates, and the
result of the Michelson-Morley experiment merely
determines the value of this special velocity which
could, of course, be found only from observation.

Note 5 (p. 15). Einstein has shown in a simple
example how, on the basis of the formulae of the
special theory of relativity, a point-mass loses
inertial mass when it radiates out energy.

We assume that a point-mass emits a light-wave

of energy - in a certain direction, and a light-wave

of the same energy --in the opposite direction.

Then, in view of the symmetry of the process of
emission with respect to the system of reference of
the co-ordinates x, y, z, t originally chosen, the
point-mass remains at rest. Let the total energy

80 THE FOUNDATIONS OF EINSTEIN'S

of the point-mass be E0 referred to this system, but
H0 referred to a second system which we suppose
moving with the uniform velocity v with respect
to the first. We shall apply the principle of energy
to this process. If V and A are the frequency
and amplitude of the light-wave in the initial
system, V', A', x',y', z', t' the frequency, amplitude,
and co-ordinates in the second (the moving) system,
further, (f> the angle between the wave-normals
and the line connecting the point-mass with the
observer, then Doppler's principle gives for the
frequency of the light-wave in the moving system :

v
i — ~ . cos (f>

t C

" ="

The formulae of the special theory of relativity
give us, correspondingly, for the amplitude in the
moving system

I — - cos <f>

A' = A .

According to Maxwell's theory the energy of the
light-wave per unit volume is ^— . A2. We now wish

O7T

to calculate the corresponding energy-density also
with respect to the moving system. We must here
take into account that, in consequence of the con-
traction of the lengths according to the Lorentz-
Einstein transformation formulae, the volume V of a
sphere in the resting system becomes transformed
into that of an ellipsoid as measured from the
moving system V ; indeed, this volume of the ellip-
soid is

THEORY OF GRAVITATION

81

V' = V.

V ,

-COS </>

c

Hence the energy-densities in the accented and un-
accented system are in the ratio :

U
L

A'2.V

Sir

877

.A2.V

If we now designate the energy-content of the
point-mass after the emission by Ex, and the cor-
responding quantity referred to the moving system
by H!, then we have :

whereas

/ D2 + 2 / V*

V1-^ V1-^

From this we get directly that
[H0 - E0] - [H, - EJ= L

What does this equation assert ?

H and E are the energy- values of the same point-
mass, in the first place referred to a system with
respect to which the point-mass moves, and in the
second related to a system in which the point-mass
6

82 THE FOUNDATIONS OF EINSTEIN'S

is at rest. Hence the difference H — E, except
for an additive constant, must be equal to the
kinetic energy of the point-mass referred to the
moving system. Thus, we may write

H0 - E0 = K0 + C Hx - E! = Kx + C

wherein C denotes a constant which does not alter
during the light-emission of the point-mass, since,
owing to the symmetry of the process, the point-
mass remains at rest with respect to the initial
system. So we arrive at the relation :

In words this equation states that owing to the
point-mass emitting the energy L as light, its
kinetic energy referred to a moving system sinks
from the value K0 to the value K, corresponding to

a loss in inertial mass of the amount — . For, accor-

c2

ding to classical mechanics, the expression \ m . vz
in which m is the inertial mass of the observed
body, is a measure of the kinetic energy of this
body referred to a system with respect to which

it moves with the velocity v. Thus — must be

c

taken as standing for the inertial mass of an amount
of energy L.

Note 6 (p. 29). The facts that every pair of points
(point-pair) in space have the same magnitude-
relation (viz. the same expression for the mutual
distance between them) and that with the aid of
this relation, every point-pair can be compared
with every other, constitute the characteristic
feature which distinguishes space from the remaining

THEORY OF GRAVITATION 83

continuous manifolds which are known to us. We
measure the mutual distance between two points on
the floor of a room, and the mutual distance between
two points which lie vertically above one another
on the wall, with the same measuring-scale, which
we thus apply in any direction at pleasure. This
enables us to " compare " the mutual distance of
a point-pair on the floor with the mutual distance
of any other pair of points on the wall.

In the system of tones, on the contrary, quite
different conditions prevail. The system of tones
represents a manifold of two dimensions, if one
distinguishes every tone from the remaining tones
by its pitch and its intensity. It is, however, not
possible to compare the " distance " between two
tones of the same pitch but different intensity (ana-
logous to the two points on the floor) with the
" distance " between two tones of different pitch
but equal intensity (analogous to the two points on
the wall). The measure-conditions are thus quite
different in this manifold.

In the system of colours, too, the measure-relations
have their own peculiarity. The dimensions of the
manifold of colours are the same as those of space,
as each colour can be produced by mixing the
three " primary " colours. But there is no relation
between two arbitrary colours, which would corre-
spond to the distance between two points in
space. Only when a third colour is derived by
mixing these two, does one obtain an equation
between these three colours similar to that which
connects three points in space lying in one straight
line.

These examples, which are borrowed from
Helmholtz's essays, serve to show that the measure-
relations of a continuous manifold are not already
given in its definition as a continuous manifold,

84 THE FOUNDATIONS OF EINSTEIN'S

nor by fixing its dimensions. A continuous mani-
fold generally allows of various measure-relations.
It is only experience which enables us to derive
the measure-laws which are valid for each particular
manifold. The fact, discovered by experience,
that the dimensions of bodies are independent of
their particular position and motion, led to the
laws of Euclidean geometry where congruence is the
deciding factor in comparing various portions of
space. These questions have been exhaustively
treated by Helmholtz in various essays. Refer-
ences : —

Riemann, " Uber die Hypothesen, welche der
Geometric zugrunde liegen " (1854). Newly pub-
lished and annotated by H. Weyl, Berlin, 1919.

Helmholtz. " Ueber die tatsachlichen Grund-
lagen der Geometric," Wiss. Abh. 2, S. 10.

Helmholtz. " Ueber die Tatsachen, welche der
Geometrie zugrunde liegen," Wiss. Abh. 2, S. 618.

Helmholtz. " Ueber den Ursprung und die Be-
deutung der geometrischen Axiome," Vortrage und
Reden, Bd. 2, S. i.

Note 7 (p. 26). The postulate that finite rigid
bodies are to be capable of free motions, can be
most strikingly illustrated in the realm of two-
dimensions. Let us imagine a triangle to be drawn
upon a sphere, and also upon a plane : the former
being bounded by arcs of great circles and the
latter by straight lines ; one can then slide these
triangles over their respective surfaces at will, and
can make them coincide with other triangles,
without thereby altering the lengths of the sides
or the angles. Gauss has shown that this is possible
because the curvature at every point of the sphere
(or the plane, respectively) has exactly the same
value. And yet the geometry of curves traced
upon a sphere is different from that of curves traced

THEORY OF GRAVITATION 85

upon a plane, for the reason that these two con-
figurations cannot be deformed into one another
without tearing (vide Note 27). But upon both of
them planimetrical figures can be freely shifted
about, and, therefore, theorems of congruence hold
upon them. If, however, we were to define a cur-
vilinear triangle upon an egg-shaped surface by
the three shortest lines connecting three given
points upon it, we should find that triangles could
be constructed at different places on this surface,
having the same lengths for the sides ; but these
sides would enclose angles different from those
included by the corresponding sides of the initial
triangle,^ and, consequently, such triangles would
not be congruent, in spite of the fact that corre-
sponding sides are equal. Figures upon an egg-
shaped surface cannot, therefore, be made to slide
over the surface without altering their dimensions :
and in studying the geometrical conditions upon
such a surface, we do not arrive at the usual theorems
of congruence. Quite analogous arguments can
be applied to three- and four-dimensional realms :
but the latter cases offer no corresponding pictures
to the mind. If we demand that bodies are to be
freely movable in space without suffering a change
of dimensions, the " curvature " of the space must
be the same at every point. The conception of
curvature, as applied to any manifold of more than
two dimensions, allows of strict mathematical
formulation ; the term itself only hints at its
analogous meaning, as compared with the conception
of curvature of a surface. In three-dimensional
space, too, various cases can be distinguished,
similarly to plane- and spherical-geometry in two-
dimensional space. Corresponding to the sphere,
we have a non-Euclidean space with constant
positive curvature ; corresponding to the plane we

86 THE FOUNDATIONS OF EINSTEIN'S

have Euclidean space with curvature zero. In both
these spaces bodies can be moved about without
their dimensions altering ; but Euclidean space is
furthermore infinitely extended : whereas " spher-
ical " space, though unbounded, like the surface
of a sphere, is not infinitely extended. These
questions are to be found extensively treated in
a very attractive fashion in Helmholtz's familiar
essay : " Ueber den Ursprung und die Bedeutung
der geometrischen Axiome " (Vortrdge und Reden,
Bd. 2, S. i).

Note 8 (p. 26). The properties, which the ana-
lytical expression for the length of the line-element
must have, may be understood from the following :

Let the numbers xlt x2 denote any point of any
continuous two-dimensional manifold, e.g. a surface.
Then, together with this point, a certain " domain "
around the point is given, which includes points all
of which lie in the plane. — D. Hilbert has strictly
defined the conception of a multiply-extended
magnitude (i.e. a manifold) upon the basis of the
theory of aggregates in his " Grundlagen der Geo-
metric " (p. 177). In this definition the conception
of the " domain " encircling a point is made to give
Riemann's postulate of the continuous connection
existing between the elements of a manifold and a
strict form.

Setting out from the point xlt x2 we can contin-
uously pass into its domain, and at any point, e.g.
#1 + <foi» xz + dxz> inquire as to the " distance " of
this point from the starting-point. The function
which measures this distance will depend upon the
values of xlt xz, dxlt dx2, and for every intermediate
point of the path which has conducted us from xlt x2
to the point x± -f- dxlf x2 +*dx2 will successively
assume certain continually changing, and, as we
may suppose, continually increasing, values. At

THEORY OF GRAVITATION 87

the point xlt x2 itself it will assume the value zero,
and for every other point of the domain its value
must be positive. Moreover, we shall expect to
find that, for any intermediate point, denoted by

required function which measures the distance of
this point from xlf x2, will, at this point, have a
value half that of its value for the point xt + dxlt
x2 -f- dx2. Under these assumptions, the function
will be homogeneous and of the first degree in the
dx's ; its value will then appear multiplied by that
factor in proportion to which the dx's were increased.
In addition, it must itself vanish if all the dx's are
zero ; and if they all change their sign it must not
alter its value, which always remains positive. It
will immediately be evident that the function

ds = t/gudxf + gl2dx^x2 -f g22dx22

fulfils all these requirements ; but it is by no means
the only function of this kind.

Note 9 (p. 29). But the expression of the fourth
degree for the line element would not permit of any
geometrical interpretation of the formula, such as
is possible with the expression

which latter may be regarded as a general case of
Pythagoras' theorem.

'Note 10 (p. 30). By a "discrete" manifold
we mean one in which no continuous transition of
the single elements from one to another is possible,
but each element to a certain extent represents an
independent entity. The aggregate of all whole
numbers, for instance, is a manifold of this type,
or the aggregate of all planets in our solar system,
etc., and many other examples may be found ; and

88 THE FOUNDATIONS OF EINSTEIN'S

indeed all finite aggregates in the theory of aggre-
gates are such discrete manifolds. " Measuring,"
in the case of discrete manifolds, is performed
merely by " counting," and does not present any
special difficulties, as all manifolds of this type
are subject to the same principle of measurement.
When Riemann then proceeds to say : " Either,
therefore, the reality which underlies space must
form a discrete manifold, or we must seek the ground
of its metric relations outside it, in binding forces
which act upon it," he only wishes to hint at a
possibility, which is at present still remote, but
which must, in principle, always be left open. In
just the last few years a similar change of view
has actually occurred in the case of another mani-
fold which plays a very important part in physics,
viz. " energy " ; the meaning of the hint Riemann
gives will become clearer if we consider this
example.

Up till a few years ago, the energy which a body
emanates by radiation was regarded as a contin-
uously variable quantity : and attempts were there-
fore made to measure its amount at any particular
moment by means of a continuously varying
sequence of numbers. The researches of Max
Planck have, however, led to the view that this
energy is emitted in " quanta," and that therefore
the " measuring " of its amount is performed by
counting the number of " quanta." The reality
underlying radiant energy, according to this, is a
discrete and not a continuous manifold. If we now
suppose that the view were gradually to take root
that, on the one hand, all measurements in space
only have to do with distances between ether-atoms ;
and that, on the other hand, the distances of single
ether-atoms from one another can only assume
certain definite values, all distances in space would

THEORY OF GRAVITATION 89

be obtained by " counting " these values, and we
should have to regard space as a discrete manifold.

Note ii (p. 32). C. Neumann. " Ueber die
Prinzipien der Galilei-Newtonschen Theorie," Leipzig
1870, S. 18.

Note 12 (p. 32). H. Streintz. " Die physikali-
schen Grundlagen der Mechanik," Leipzig, 1883.

Note 13 (p. 33). A. Einstein. " Annalen der
Physik," 4 Folge, Bd. 17, S. 891.

Note 14 (p. 35). Minkowski was the first to
call particular attention to this deduction of the
special principle of relativity.

Note 15 (p, 38). The term " inertial system "
was originally not associated with the system, which
Neumann attached to the hypothetical body Alpha.
Nowadays it is generally understood to signify a
rectilinear system of co-ordinates, relatively to which
a point-mass, which is only subject to its own
inertia, moves uniformly in a straight line. Whereas
C. Neumann only invented the body Alpha, as an
absolutely hypothetical configuration, in order to
be able to formulate the law of inertia, later re-
searches, especially those of Lange, tended to show
that, on the basis of rigorous kinematical considera-
tions, a co-ordinate system could be derived, which
would possess the properties of such an inertial
system. However, as C. Neumann and J. Petzoldt
have demonstrated, these developments contain
faulty assumptions, and give the law of inertia no
firmer basis than the body Alpha introduced by
Neumann.

Such an inertial system is determined by the
straight lines which connect three point-masses
infinitely distant from one another (and thus unable
to exert a mutual influence upon one another) and
which are not subject to any other forces. This
definition makes it evident why no inertial system

90 THE FOUNDATIONS OF EINSTEIN'S

will be discoverable in nature, and why, consequently,
the law of inertia will never be able to be formulated
so as to satisfy the physicist. References : —

C. Neumann. " Ueber die Prinzipien der Galilei-
Newtonschen Theorie," Leipzig, 1870.

L. Lange. " Berichte der Kgl. Sachs. Ges. d.
Wissenschaften. Math.-phil. Klasse," 1885.

L. Lange. " Die Geschichte der Entwickelung
des Bewegungsbegriffes," Leipzig, 1886.

H. Seeliger. " Ber. der Bayr. Akademie," 1906,
Heft i.

C. Neumann. " Ber der Kgl. Sachs. Ges. d. Wiss.
Math.-phys. Klasse," 1910, Bd. 62, S. 69 and 383.

J. Petzoldt. " Ann. der Naturphilosophie," Bd. 7.

Note 1 6 (p. 38). E. Mach. " Die Mechanik in
in ihrer Entwickelung," 4 Aufl. S. 244.

Note 17 (p. 40). The new points of view as
to the nature of inertia are based upon the study
of the electromagnetic phenomena of radiation.
The special theory of relativity, by stating the
theorem of the inertia of energy, organically grafted
these views on to the existing structure of theo-
retical physics. The dynamics of cavity-radiation,
i.e. the dynamics of a space enclosed by walls
without mass, and filled with electromagnetic
radiation, taught us that a system of this kind
opposes a resistance to every change of its motion,
just like a heavy body in motion. The study of
electrons (free electric charges) in a state of free
motion, e.g. in a cathode-tube, taught us likewise
that these exceedingly small particles behave like
inert bodies ; that their inertia is not, however,
conditioned by the matter to which they might
happen to be attached, but rather by the electro-
magnetic effects of the field to which the moving
electron is subject. This gave rise to the concep-
tion of the apparent (electromagnetic) mass of an

THEORY OF GRAVITATION 91

electron. The special theory of relativity finally
led to the conclusion that to all energy must be
accorded the property of inertia.

Every body contains energy (e.g. a certain definite
amount in the form of heat-radiation internally).
The inertia, which the body reveals, is thus partly
to be debited to the account of this contained
energy. As this share of inertia is, according to
the special theory of relativity, relative (i.e. re-
presents a quantity which depends upon the choice
of the system of reference), the whole amount of
the inertial mass of the body has no absolute value,
but only a relative one. This energy-content of
radiant heat is distributed throughout the whole
volume of each particular body ; one can thus
speak of the energy-content of unit volume. This
enables us to derive the notion of density of energy.
The density of the energy (i.e. amount per unit
volume) is thus a quantity, the value of which is
also dependent upon the system of reference.
References : —

M. Planck. " Ann. der Phys.," 4 Folge, Bd. 26.

M. Abraham. " Elect romagnetische Energie der
Strahlung," 4 Aufl., 1908.

Note 1 8 (p. 40). The determination of the inertial
mass of a body by measuring its^weight is rendered
possible only by the experimental fact that all
bodies fall with equal acceleration in the gravita-
tional field at the earth's surface. If p and p'
denote the pressures of two bodies upon the same
support (i.e. their respective weights), and g denote
the acceleration due to the earth's gravitational field
at the point in question, then p = m . g dynes and
p' = m' . g dynes, respectively, where m and m'
are the factors of proportionality, and are called
the masses of the two bodies, respectively. As g has
the same value in both equations, we have

92 THE FOUNDATIONS OF EINSTEIN'S

p m

and we can accordingly measure the masses of two
bodies at the same place, by determining their
weights.

Although Galilei and Newton had already known
that all bodies at the same place fall with the same
velocity (if the resistance of the air be eliminated),
this very remarkable fact has not received any
recognition in the foundations of mechanics. Ein-
stein's principle of equivalence is the first to assign
to it the position to which it is, beyond doubt,
entitled.

Note 19 (p. 41). Arguing along the same lines
B. and J. Friedlander have suggested an ex-
periment to show the relativity of rotational
motions, and, accordingly, the reversibility of
centrifugal phenomena (*' Absolute and Relative
Motion/' Berlin, Leonhard Simion, 1896). On
account of the smallness of the effect, the experiment
cannot, at present, be performed successfully ;
but it is quite appropriate for making the physical
content of this postulate more evident. The fol-
lowing remarks may be quoted :

" The torsion-balance is the most sensitive of
all instruments. The largest rotating-masses, with
which we can experiment, are probably the large
fly-wheels in rolling-mills and other big factories.
The centrifugal forces assert themselves as a pressure
which tends from the axis of rotation. If, therefore,
we set up a torsion-balance in somewhat close
proximity to one of these large fly-wheels, in such a
position that the point of suspension of the movable
part of the torsion-balance (the needle) lies exactly,
or as nearly as possible, in the continuation of the
axis of the fly-wheel, the needle should endeavour

THEORY OF GRAVITATION 93

to set itself parallel to the plane of the fly-wheel,
if it is not originally so, and should register a corre-
sponding displacement. For centrifugal force acts
upon every portion of mass which does not lie exactly
in the axis of rotation, in such a way as to tend to
increase the distance of the mass from the axis.
It is immediately apparent that the greatest possible
displacement-effect is attained when the needle is
parallel to the plane of the wheel."

This proposed experiment of B. and J. Friedlander
is only a variation of the experiment which per-
suaded Newton to his view of the absolute character
of rotation. Newton suspended a cylindrical vessel
filled with water by a thread, and turned it about
the axis denned by the thread till the thread became
quite stiff. After the vessel and the contained
liquid had completely come to rest, he allowed the
thread to untwist itself again, whereby the vessel
and the liquid started to rotate rapidly. He thereby
made the following observations. Immediately after
its release the vessel alone assumed a motion of rota-
tion, since the friction (viscosity) of the water was
not sufficient to transmit the rotation immediately
to the water. So long as this state of affairs pre-
vailed, the surface of the water remained a hori-
zontal plane. But the more rapidly the water was
carried along by the rotating walls of the vessel, the
more definitely did the centrifugal forces assert
themselves, and drive the water up the walls, so
that finally its free surface assumed the form of a
paraboloid of revolution. From these observations
Newton concluded that the rotation of the walls
of the vessel relative to the water does not call up
forces in the latter. Only when the water itself
shares in the rotation, do the centrifugal forces
make their appearance. From this he came to his
conclusion of the absolute character of rotations.

94 THE FOUNDATIONS OF EINSTEIN'S

This experiment became a subject of frequent
discussion later : and E. Mach long ago objected
to Newton's deduction, and pointed out that it
cannot be straightway affirmed that the rotation
of the walls of the vessel relative to the water is
entirely without effect upon the latter. He regards
it as quite conceivable that, provided the mass of
the vessel were large enough, e.g. if its walls
were many kilometres thick, then the free surface
of the water which is at rest in the rotating vessel
would not remain plane. This objection is quite
in keeping with the view entailed by the general
theory of relativity. According to the latter, the
centrifugal forces can also be regarded as gravita-
tional forces, which the total sum of the masses
rotating around the water exerts upon it. The
gravitational effect of the walls of the vessel upon
the enclosed liquid is, of course, vanishingly small
compared with that of all the masses in the universe.
It is only when the water is in rotation relatively
to all these masses that perceptible centrifugal
forces are to be expected. The experiment of B.
and J. Friedlander was intended to refine the ex-
periment performed by Newton, by using a sensitive
torsion-balance susceptible to exceedingly small
forces in place of the water, and by substituting a
huge fly-wheel for the vessel which contained the
water. But this arrangement, too, can lead to
no positive result, as even the greatest fly-wheel at
present available represents only a vanishingly
small mass compared with the sum-total of masses
in the universe.

Note 20 (p. 42). We use the term " field of
force " to denote a field in which the force in question
varies continuously from place to place, and is given
for each point in the field by the value of some
function of the place. The centrifugal forces in the

THEORY OF GRAVITATION 95

interior and on the outer surface of a rotating body
are so distributed as to compose a field of this kind
throughout the whole volume of the body, and there
is nothing to hinder us from imagining this field
to extend outwards beyond the outer surface of
the body, e.g. beyond the surface of the earth into
its own atmosphere. We can thus briefly speak of
the whole field as the centrifugal field of the earth ;
and, as the centrifugal field, according to the older
views, is conditioned only by the inertia of bodies,
and not by their gravitation, we can further speak
of it as an inertia! field, in centra-distinction to
the gravitational field, under the influence of which
all bodies which are not suspended or supported
fall to earth.

Accordingly the effects of various fields of force
are superposed at the earth's surface : (i) the effect
of the gravitational field, due to the gravitation of
the particles of the earth's mass towards one another,
and which is directed towards the centre of the earth ;
(2) the effect of the centrifugal field, which, accord-
ing to Einstein's view, can be regarded as a gravita-
tional field, and the direction of action of which is
outwards and parallel to the plane of the meridian
of latitude ; finally (3) the effect of the gravitational
field, due to the various heavenly bodies, foremost
amongst them, the sun and the moon.

Note 21 (p. 42). Eotvos has published the re-
sults of his measurements in the " Mathematische
und Naturwissenschaftliche Berichte aus Ungarn,"
Bd. 8, S. 64, 1891. A detailed account is given by
D. Pekar, " Das Gesetz der Proportionality von
Tragheit und Gravitation." "Die Naturwissen-
schaft," 1919, 7, p. 327.

Whereas the earlier investigations of Newton and
Bessel (" Astr. Nachr." 10, S. 97, and " Abhand-
lungen von Bessel," Bd. 3, S. 217), about the

96 THE FOUNDATIONS OF EINSTEIN'S

attractive effect of the earth upon various sub-
stances, are based upon observations with a pendu-
lum, Eotvos worked with sensitive torsion-balances.

The force, in consequence of which bodies fall, is
composed of two components : first the attractive
force of the earth, which (except for deviations
which may, for the present, be neglected) is directed
towards the centre of the earth ; and, second, the
centrifugal force, which is directed outwards parallel
to the meridians of latitude. If the attraction of
the earth upon two bodies of equal mass but of
different substance were different, the resultant of
the attractive and centrifugal forces would point in
a different direction for each body. Eotvos then
states: "By calculation we find that if the attractive
effect of the earth upon two bodies of equal mass,
but composed of different substance, differed by
a thousandth, the directions of the gravitational
forces acting upon the two bodies respectively
would make an angle of 0-356 second, i.e. about
a third of a second with one another ; " and if the
difference in the attractive force were to amount
to a twenty-millionth, this angle would have to be
seconds ; that is, slightly more than
of a second ; and later :

" I attached separate bodies of about 30 grms.
weight to the end of a balance-beam about 25 to
30 cms. long, suspended by a thin platinum wire
in my torsion-balance. After the beam had been
placed in a position perpendicular to the meridian,
I determined its position exactly by means of two
mirrors, one fixed to it and another fastened to the
case of the instrument. I then turned the instru-
ment, together with the case, through 180°, so that
the body which was originally at the east end of
the beam now arrived at the west end : I then de-
termined the position of the beam again, relative

THEORY OF GRAVITATION 97

to the instrument. If the resultant weights of the
bodies attached to both sides pointed in different
directions, a torsion of the suspending wire should
ensue. But this did not occur in the cases in which
a brass sphere was constantly attached to the one
side, and glass, cork, or crystal antimony was
attached to the other ; and yet a deviation of
^.^ooth of a second in the direction of the
gravitational force would have produced a torsion
of one minute, and this would have been observed
accurately."

Eotvos thus attained a degree of accuracy, such
as is approximately reached in weighing ; and this
was his aim : for his method of determining the
mass of bodies by weighing is founded upon the
axiom that the attraction exerted by the earth
upon various bodies depends only upon their mass,
and not upon the substance composing them. This
axiom had, therefore, to be verified with the same
degree of accuracy as is attained in weighing. If a
difference of this kind in the gravitation of various
bodies having the same mass but being composed of
different substance exists at all, it is, according to
Eotvos, less than a twenty-millionth for brass, glass,
antimonite, cork, and less than a hundred-thousandth
for air.

Note 22 (p. 44). Vide also A. Einstein, " Grund-
lagen der allgemeinen Relativitatstheorie," "Ann. d.
Phys.," 4 Folge, Bd. 49, S. 769.

Note 23 (p. 46). The equation 8{$ds} = o asserts
that the variation in the length of path between
two sufficiently near points of the path vanishes
for the path actually traversed ; i.e. the path
actually chosen between two such points is the
shortest of all possible ones. If we retain the view
of classical mechanics for a moment, the following
example will give us the sense of the principle
7

98 THE FOUNDATIONS OF EINSTEIN'S

clearly : In the case of the motion of a point-
mass, free to move about in space, the straight
line is always the shortest connecting line between
two points in space : and the point-mass will move
from the one point to the other along this straight
line, provided no other disturbing influences come
into play (Law of inertia). If the point-mass is
constrained to move over any curved surface, it
will pass from one point to another along a geodetic
line to the surface, since the geodetic lines represent
the shortest connecting lines between points on the
surface. In Einstein's theory there is a fully corre-
sponding principle, but of a much more general form.
Under the influence of inertia and gravitation every
point-mass passes along the geodetic lines of the
space-time-manifold. The fact of these lines not,
in general, being straight lines, is due to the gravi-
tational field, in a certain sense, putting the point-
mass under a sort of constraint, similar to that
imposed upon the freedom of motion of the point-
mass by a curved surface. A principle in every
way corresponding had already been installed in
mechanics as a fundamental principle for all motions
by Heinrich Hertz.

Note 24 (p. 48). Vide A. Einstein, "Ann. d.
Phys.," 4 Folge, Bd. 35, S. 898.

Note 25 (p. 48). The expression " accelera-
tion-transformation " means that the equations
giving the transformation from the variables x, y,
z, t to the system of variables xlt x2, x3, #4, which is
the basis of our discussion, can be regarded as giving
the relations between two systems of reference
which are moving with an accelerated motion rela-
tively to one another. The nature of the state of
motion of two systems of reference relative to one
another finds its expression in the analytical form
of the equations of transformation of their co-
ordinates.

THEORY OF GRAVITATION 99

Note 26 (p. 51). Two things are to be under-
taken in the following : (i) the fundamental equa-
tions of the new theory are to be written in an
explicit form, and (2) the transition to Newton's
fundamental equations is to be performed.

i. From the equation of variation 8{S^s} = o
where

4

ds2 = Zg^dXpdXv,
i

we have, after carrying out the operation of varia-
tion, the four total differential equations :

These are the equations of motion of a material point
in the gravitational field defined by the g^'s.
The symbol Y^v here denotes the expression

W , Sfr. S

The symbol gffa denotes the minor of gu in the
determinant

ten, ...,..., £14]

divided by the determinant itself.

The ten differential equations for the " gravita-
tional potentials " g^ are :

The quantities T^ and T are expressions which
are related in a simple manner to the components
of the stress-energy-tensor (which plays the part

100 THE FOUNDATIONS OF EINSTEIN'S

of the quantity exciting the field in the new theory
in place of the density of mass). K is essentially
equal to the gravitational constant of Newton's
theory.

The differential equations (i) and (2) are the funda-
mental equations of the new theory. The derivation
of these equations is carried out in detail in the
tract by A. Einstein, " The Foundations of the
General Principle of Relativity," J. A. Barth,
Leipzig, 1916.

2. In order to obtain a connection between these
equations and Newton's theory, we must make
several simplifying assumptions. We shall first
assume that the g^'s differ only by quantities which
are small compared with unity from the values
given by the scheme :

l #12 #13 £l4"

1 £22 £23 £24
£31 £32 £33 £34
^£41 £42 £43 £44-

These values for the g^'s characterize the case of
the special theory of relativity, i.e. the case of the
condition free of gravitation. We shall also assume
that, at infinite distances, the gMI/s tend to, and
do finally, assume the above values ; that is, that
matter does not extend into infinite space.

Secondly, we shall assume that the velocities of
matter are small compared with the velocity of
light, and can be regarded as small quantities of
the first order. The quantities

dxlt dx2 dx$
~Js~' ~ds' ~fc'

will then be infinitely small quantities of the first

order, and -^i will equal i, except for quantities of
as

THEORY OF GRAVITATION 101

the second order. From the equations defining
the T*v it will then be seen that these quantities will
be infinitely small, of the first order. If we neglect
quantities of the second order, and finally assume
that, for small velocities of matter, the changes of
the gravitational field with respect to time are small
(i.e. that the derivatives of the gMI/s with respect
to time may be neglected in comparison with the
derivatives taken with regard to the space-co-
ordinates) the system of equations (i) assumes the
form :

This would be the equation of motion of a point-
mass as already given by Newton's mechanics, if
i#44 be taken as representing the ordinary gravita-
tional potential. It still remains to be seen what the
differential equation for g44 becomes in the new theory
under the simplifying assumptions we have chosen.
The stress-energy tensor, which excites the field,
degenerates, as a result of our quite special assump-
tions, into the density of mass p :

In the differential equations (2) the second term
on the left-hand side is the product of two magni-
tudes, which, according to the above arguments,
are to be regarded as infinitely small quantities of
the first order. Thus the second term, being of
the second order of small quantities, may be dis-
missed. The first term, on the other hand, if we
omit the terms differentiated with respect to time,
as above (i.e. if we regard the gravitational field
as " stationary "), reduces to :

= v = 4.

102 THE FOUNDATIONS OF EINSTEIN'S

The differential equation for g44 thus degenerates
into Poisson's equation :

(2,0) A#44 = Kp.

Thus, to a first approximation (i.e. if one regards the
velocity of light as infinitely great, and this is a
characteristic feature of the classical theory, as
was explained in detail in § 3 (b) : if certain simple
assumptions are made about the behaviour of the
gMI/s at infinity ; and if the time-changes of the
gravitational field are neglected) the well-known
equations of Newtonian mechanics emerge out of
the differential equations of Einstein's theory,
which were obtained from perfectly general begin-
nings.

Note 27 (p. 53). The theory of surfaces, i.e.
the study of geometry upon surfaces, makes it
immediately apparent that the theorems, which
have been established for any surface, also hold
for any surface which can be generated by dis-
torting the first without tearing. For if two surfaces
have a point-to-point correspondence, such that
the line-elements are equal at corresponding points,
then corresponding finite arcs, angles, and areas,
etc., will be equal. One thus arrives at the same
planimetrical theorems for the two surfaces. Such
surfaces are called " deformable " surfaces. The
necessary and sufficient condition that surfaces be
continuously deformable is that the expression for
the line-element of the one surface

ds2 = gudxS + g^Xjdxi +
can be transformed into that for the other,

According to Gauss, it is necessary that both sur-
faces have equal measures of curvature. If the latter

THEORY OF GRAVITATION 103

is constant over the whole surface, as e.g. in the
case of a cylinder or a plane, all conditions for the
deformability of the surfaces are fulfilled. In other
cases, special equations offer a criterion as to whether
surfaces, or portions of surfaces, are deformable
into one another. The numerous subsidiary prob-
lems, which result out of these questions, are
discussed at length in every book dealing with
differential geometry (e.g. Bianchi-Lukat).* This
branch of training, which was hitherto of interest
only to mathematicians, now assumes very con-
siderable importance for the physicist too.

Note 28 (p. 61). One must avoid being de-
ceived into the belief that Newton's fundamental
law is in any way to be regarded as an explanation
of gravitation. The conception of attractive force
is borrowed from our muscular sensations, and has
therefore no meaning when applied to dead matter.
C. Neumann, who took great pains to place Newton's
mechanics on a solid basis, glosses upon this point
himself in a drastic fashion, in the following narra-
tive, which shows up the weaknesses of the former
view :

" Let us suppose an explorer to narrate to us his
experiences in yonder mysterious ocean. He had
succeeded in gaining access to it, and a remarkable
sight had greeted his eyes. In the middle of the
sea he had observed two floating icebergs, a larger
and a smaller one, at a considerable distance from
one another. Out of the interior of the larger one,
a voice had resounded, issuing the following com-
mand in a peremptory tone : ' Ten feet nearer ! '
The little iceberg had immediately carried out the
order, approaching ten feet nearer the larger
one. Again, the larger gave out the order : ' Six
feet nearer ! ' The other had again immediately

* Forsyth's " Differential Geometry."— H. L. B.

104 THE FOUNDATIONS OF EINSTEIN'S

i

executed it. And in this manner order after order
had echoed out : and the little iceberg had con-
tinually been in motion, eager to put every command
immediately and implicitly into action.

" We should certainly consign such a report to
the realm of fables. But let us not scoff too soon !
The ideas, which appear so extraordinary to us in
this case, are exactly the same as those which lie
at the base of the most complete branch of natural
science, and to which the most famous of physicists
owes the glory attached to his name.

" For in cosmic space such commands are con-
tinually resounding, proceeding from each of the
heavenly bodies — from the sun, planets, moons, and
comets. Every single body in space hearkens to the
orders which the other bodies give it, always striving
to carry them out punctiliously. Our earth would
dash through space in a straight line, if she were
not controlled and guided by the voice of command,
issuing from moment to moment, from the sun, in
which the instructions of the remaining cosmic
bodies are less audibly mingled.

" These commands are certainly given just as
silently as they are obeyed ; and Newton has
denominated this play of interchange between com-
manding and obeying by another name. He talks
quite briefly of a mutual attractive force, which
exists between cosmic bodies. But the fact remains
the same. For this mutual influence consists in
one body dealing out orders, and the other obeying
them."

THEORY OF GRAVITATION 105

ON THE THEORY OF RELATIVITY
By Henry L. Brose, M.A.
INTRODUCTORY

PHYSICS, being a science of observation
which seeks to arrange natural phenomena
into a consistent scheme by using the
methods and language of mathematics, has to
inquire whether the assumptions implied in any
branch of mathematics used for this purpose are
legitimate in its sphere, or whether they are merely
the outcome of convention, or have been built up
from abstract notions containing foreign elements.
The use of a unit length as an unalterable measure,
or of a time-division, has been accepted in tradi-
tional mechanics without inconsistency manifesting
itself in general until the field of electrodynamics
became accessible to investigators and rendered a
re-examination of the foundations of our modes of
measurement necessary. It is upon these that the
whole science of mathematical physics rests. The
road of advance of all science is in like manner
conditioned by the inter-play of observations and
notions, each assisting the other in giving us a
clearer view of Nature regarded purely as a physical
reality. The discovery of additional phenomena
presages a still greater unification, revealing new
relations and exposing new differences ; the ultimate
aim of physics would seem to consist in reaching

106 THE FOUNDATIONS OF EINSTEIN'S

perfect separation and distinctness of detail simul-
taneously with perfect co-ordination of the whole.
" The all-embracing harmony of the world is the
true source of beauty and is the real truth," as
Poincare has expressed it. The noblest task of
co-ordinating all knowledge falls to the lot of phil-
osophy.

A principle which has proved fruitful in one
sphere of physics suggests that its range may be
extended into others ; nowhere has this led to more
successful results than in the increasing generaliza-
tion which has characterized the advance of the
principle of relativity. This advance is marked by
three stages, quite distinct, indeed, in the nucleus
of their growth, yet each succeeding stage including
the results of the earlier.

Relativity first makes its appearance as a govern-
ing principle in Newtonian or Galilean mechanics ;
difficulties arising out of the study of the phenomena
of radiations led to a new enunciation of the principle
upon another basis by Einstein in 1905, an enuncia-
tion which comprised the phenomena of both me-
chanics and radiation: this will be referred to as
the " special " principle of relativity to distinguish it
from the " general" principle of relativity enunciated
by Einstein in 1915, and which applied to all physical
phenomena and every kind of motion. The latter
theory also led to a new theory of gravitation.

I. THE MECHANICAL THEORY OF
RELATIVITY

In order to arrive at the precise significance of
the principle of relativity in the form in which it
held sway in classical mechanics, we must briefly
analyse the terms which will be used to express it.

THEORY OF GRAVITATION 107

Mechanics is usually denned as the science which
describes how the " position of bodies in ' space '
alters with the ' time.' ' We shall for the present
discuss only the term " position," which also in-
volves " distance/' leaving time and space to be
dealt with later when we have to consider the mean-
ing of physical simultaneity. Modern pure geometry
starts out from certain conceptions such as " point,"
" straight line," and " plane," which were originally
abstracted from natural objects and which are
implicitly denned by a number of irreducible and
independent axioms ; from these a series of pro-
positions is deduced by the application of logical
rules which we feel compelled to regard as legitimate.
The great similarity which exists between geometri-
cally constructed figures and objects in Nature has
led people erroneously to regard these propositions
as true : but the truth of the propositions depends
on the truth of the axioms from which the proposi-
tions were logically derived. Now empirical truth
implies exact correspondence with reality. But
pure geometry by the very nature of its genesis
excludes the test of truth. There are no geometrical
points or straight lines in Nature, nor geometrical
surfaces ; we only find coarse approximations which
are helpful in representing these mathematically
conceived elements.

If, however, certain principles of mechanics are
conjoined with the axioms of geometry, we leave
the realm of pure geometry and obtain a set of
propositions which may be verified by comparison
with experience, but only within limits, viz. in
respect to numerical relations, for again no exact
correspondence is possible, merely a superposition of
geometrical points with places occupied by matter.
Our idea of the form of space is derived from the
behaviour of matter, which, indeed, conditions it.

108 THE FOUNDATIONS OF EINSTEIN'S

Space itself is amorphous, and we are at liberty to
build up any geometry we choose for the purpose
of making empirical content fit into it. Neither
Euclidean, nor any of the forms of meta-geometry,
has any claim to precedence. We may select for
a consistent description of physical phenomena
whichever is the more convenient, and requires a
minimum of auxiliary hypotheses to express the
laws of physical nature.

Applied geometry is thus to be treated as a branch
of physics. We are accustomed to associate two
points on a straight line with two marks on a
(practically) rigid body : when once we have chosen
an arbitrary, rigid body of reference, we can discuss
motions or events mechanically by using the body
as the seat of a set of axes of co-ordinates. The
use of the rule and compasses gives us a physical
interpretation of the distance between points, and
enables us to state this distance by measurement
numerically, inasmuch as we may fix upon an
arbitrary unit of length and count how often it has
to be applied end to end to occupy the distance
between the points. Every description in space
of the scene of an event or of the position of a body
consists in designating a point or points on a rigid
body imagined for the purpose, which coincides with
the spot at which the event takes place or the object
is situated. We ordinarily choose as our rigid
body a portion of the earth or a set of axes attached
to it.

Now Newton's (or Galilei's) law of motion states
that a body which is sufficiently far removed from
all other bodies continues in its state of rest or
uniform motion in a straight line. This holds very
approximately for the fixed stars. If, however, we
refer the motion of the stars to a set of axes fixed
to the earth, the stars describe circles of immense

THEORY OF GRAVITATION 109

radius ; that is, for such a system of reference the
law of inertia only holds approximately. Hence
we are led to the definition of Galilean systems of
co-ordinates. A Galilean system is one, the
state of motion of which is such that the law
of inertia holds for it. It follows naturally that
Newtonian or Galilean mechanics is valid only for
such Galilean or inertial systems of co-ordinates, i.e.
in formulating expressions for the motion of bodies
we must choose some such system at an immense
distance where the Newtonian law would hold. It
will be noticed that this is an abstraction, and that
such a system is merely postulated by the law of
motion. It is the foundation of classical mechanics,
and hence also of the first or " mechanical " principle
of relativity.

If we suppose a crow flying in a straight line
at uniform velocity with respect to the earth dia-
gonally over a train likewise moving uniformly and
rectilinearly with respect to the earth (since motion
is change of position we must specify our rigid body
of reference, viz. the earth), then an observer in
the train would also see the crow flying in a straight
line, but with a different uniform velocity, judged
from a system of co-ordinates attached to the train.
We may consider both the train and the earth to be
carriers of inertial systems as we are only dealing
with small distances. We can then formulate the
mechanical principle of relativity as follows : —

If a body be moving uniformly and rectilinearly
with respect to a co-ordinate system K then it will
likewise move uniformly and rectilinearly with re-
spect to a second co-ordinate system K', provided
that the latter be moving uniformly and rectilinearly
with respect to the first system K.

In our illustration, the crow represents the body,
K is the earth, and K' is carried by the train.

110 THE FOUNDATIONS OF EINSTEIN'S

Or, we may say that if K be an inertial system
then K', which moves uniformly and rectilinearly
with respect to K, is also an inertial system. Hence,
since the laws of Newtonian mechanics are based
on inertial systems, it follows that all such systems
are equivalent for the description of the laws of
mechanics : no one system amongst them is unique,
and we cannot define absolute motion or rest ; any
systems moving with mutual rectilinear uniform
motion may be regarded as being at rest. Mathe-
matically, this means that the laws of mechanics
remain unchanged in form for any transformation
from one set of inertial axes to another.

The development of electrodynamics and the
phenomena of radiation generally showed, however,
that the laws of radiation in one inertial system did
not preserve their form when referred to another
inertial system : K and K' were no longer equivalent
for the description of phenomena such as that of
light passing through a moving medium. This meant
that either there was a unique inertial system
enabling us to define absolute motion and rest in
nature, or that we would have to build up a theory
of relativity, not on the inertial law and inertial
systems, but on some new foundation which would
definitely ensure that the form of all physical laws
would be preserved in passing from one system of
reference to another.

This dilemma arose out of the conflicting results
of two experiments, viz. Fizeau's and Michelson
and Morley's.

Fizeau's experiment was designed to determine
whether the velocity of light through moving
liquid media was different from that through a
stationary medium, i.e. whether the motion of the
liquid caused a drag on the aether, which it would
do if the mechanical law of relativity held for light

THEORY OF GRAVITATION 111

phenomena, for then the light ray would be in the
same position as a swimmer travelling upstream or
downstream respectively.*

No "ether-drag" was, however, detected; only a
fraction of the velocity of the liquid seemed to be
added to the velocity of light (c) under ordinary
conditions, and this fraction depends on the refractive
index of the liquid, and had previously been calcu-
lated by Fresnel : for a vacuum this fraction
vanishes.

This result seemed to favour the hypothesis of a
fixed ether, as was supported by Fresnel and
Lorentz. But a fixed ether implies that we should
be able to detect absolute motion, that is, motion
with respect to the ether.

Arguing from this, let us consider an observer in
the liquid moving with it. // there is a fixed
ether, he should find a lesser value for the velocity
of light (i.e. < c) owing to his own velocity in the
same direction, or vice versa in the opposite
direction.

But we on the earth are in the position of the
observer in the liquid since we revolve around the
sun at the rate of, approximately, 30 kms. per second

(i.e. — - — ), and we are subject to a translatory
io,ooo;

motion of about the same magnitude : hence we
should be able to detect a change in the velocity
of light due to our change of motion through the
ether. These considerations give rise to Michelson
and Morley's experiment.

Michelson and Morley attempted to detect motion
relative to the supposedly fixed ether by the inter-
ference of two rays of light, one travelling in the

* It is well known that it takes a swimmer longer to travel a
certain distance up and down stream than to swim across the stream
and back an equal distance.

THE FOUNDATIONS OF EINSTEIN'S

direction of motion of the earth's velocity, the other
travelling across this direction of motion.

No change in the initial interference bands was,
however, observed when the position of the instru-
ment was changed, although such an effect was
easily within the limits of accuracy of the experi-
ment. Many modifications of the experiment like-
wise failed to demonstrate the presence of an
" ether-wind."

To account for these negative results as con-
tradicting deductions from Fizeau's experiment,
Fitzgerald and, later, independently, Lorentz sug-
gested the theory that bodies automatically contract
when moving through the ether, and since our
measuring scales contract in the same ratio, we are
unable to detect this alteration in length ; this
effect would lead us always to get the same result
for the velocity of light. This contraction-hypoth-
esis agrees well with the electrical theory of matter
and may be attributed to changes in the electro-
magnetic forces, acting between particles, which
determine the equilibrium of a so-called rigid body.

Thus Michelson and Morley's experiment seems to
prove that the principle of relativity of mechanics
also holds for radiation effects, that is, it is impossible
to determine absolute motion through the ether or
space : this implies that there is no unique system
of co-ordinates. It disagrees with Fizeau's result
and seems to indicate the existence of a " moving
ether," i.e. an ether which is carried along by
moving bodies, as was upheld by Stokes and Hertz.
Lord Rayleigh pointed out that if the contraction-
hypothesis of Lorentz and Fitzgerald were true,
isotropic bodies ought to become anisotropic on
account of the motion of the earth, and that conse-
quently, phenomena of double refraction should
make their appearance. Experiments which he

THEORY OF GRAVITATION 113

himself conducted with carbon bisulphide and
others carried out by Bruce with water and glass
produced a negative result.

II. THE " SPECIAL " THEORY OF
RELATIVITY

Einstein, in the special theory of relativity,
surmounts these difficulties by doing away with the
ether (as a substance) and assumes that light-
signals project themselves as such through space.
Faraday had already long ago expressed the opinion
that the field in which radiations take place must
not be founded upon considerations of matter, but
rather that matter should be regarded as singu-
larities or places of a singular character in the
field. We may retain the name " ether " for the
field as long as we do not regard it as composed of
matter of the kind we know. Einstein arrives at
these conclusions by critically examining our notions
of space and time or of distance and simultaneity.

We know what simultaneity (time-coincidence of
two events) means for our consciousness, but in
making use of the idea of simultaneity in physics,
we must be able to prove by actual experiment
or observation that two events are simultaneous
according to some definition of simultaneity. A
conception only has meaning for the physicist
if the possibility of verifying that it agrees
with actual experience is given. In other words,
we must have a definition of simultaneity which
gives us an immediate means of proving by experi-
ment whether, e.g. two lightning-strokes at different
places occur simultaneously for an observer situated
somewher ebetween them or not. Whenever measure-
ments are undertaken in physics two points are made
to coincide, whether they be marks on a scale and on
an object, or whether they be cross-wires in a tele-
8

114 THE FOUNDATIONS OF EINSTEIN'S

scope which have been made to coincide with a distant
object to allow angular measurements to be made ;
coincidence is the only exact mode of observation, and
lies at the bottom of all physical measurements. The
same importance attaches to simultaneity, which is
coincidence in time. It is to be noted that no
definition will be made for simultaneity occurring
at (practically) one point : for this case psychological
simultaneity must be accepted as the basis : the
necessity for a physical definition arises only when
two events happening at great distances apart are
to be compared as regards the moment of their
happening. We cannot do more than reduce the
simultaneity of two events happening a great dis-
tance apart to simultaneity referred to a single
observer at one point : this would satisfy the re-
quirements of physics.

Einstein, accepting Michelson and Morley's result,
introduced the convention in 1905 that light is
propagated with a constant velocity ( = c, i.e.
300,000 kms. per sec. approximately) in vacuo in all
directions, and he then makes use of light-signals
to connect up two events in time.

He illustrates his line of argument roughly by
assuming two points, A and B, very far apart on a
railway embankment and an observer at M midway
between A and B, provided with a contrivance
such as two mirrors inclined at 90° and adjusted so
that light from A and B would be reflected into his
vertical line of sight (Fig. 3).

Two events such as lightning-strokes are then to
be defined as simultaneous for the observer at M
if rays of light from them reach the observer at the
same moment (psychologically) : i.e. if he sees the
strokes in his mirror-contrivance simultaneously.

Next suppose that a very long train is moving
with very great uniform velocity along the embank-

THEORY OF GRAVITATION 115

ment, and that the lightning-strokes pass through the
two corresponding points A1 and B1 of the train thus :

Train

MM//w/Mf/M

/ Embankment /

FIG. 3.

The question now arises : Are the two lightning-
strokes at A and B, which are simultaneous with
respect to the embankment also simultaneous with
respect to the moving train ? It is quite clear that
as M1 is moving towards B1 and away from A1, the
observer at M1 (mid-point of A1B1) will receive the
ray emitted from B1 sooner than that emitted from
A1 and he would say that the lightning-stroke at
B or B1 occurred earlier than the one at A or A1.
Hence our condition of simultaneity is not satisfied
and we are forced to the conclusion that events
which are simultaneous for one rigid body of refer-
ence (the embankment) are not simultaneous for
another body of reference (the train) which is in
motion with regard to the first rigid body of reference.
This establishes the relativity of simultaneity.

This is, of course, only an elementary example of
a very special case of the regulation of clocks by
light-signals. It may be asked how the mid-point
M is found : one might simply fix mirrors at A and
B and by flashing light-signals from points between
A and B ascertain by trial the point (M) at which
the return-flashes are observed simultaneously : this
makes M the mid-point between the " time "-distance
from A and B on the embankment.

The relativity of simultaneity states that every
rigid body of reference (co-ordinate system) has its
own time : a time-datum only has meaning when the

116 THE FOUNDATIONS OF EINSTEIN'S

body of reference is specified, or we may say that
simultaneity is dependent on the state of motion of
the body of reference.

Similar reasoning applies in the case of the distance
between two points on a rigid body. The length
of a rod is defined as the distance, measured
by (say) a metre-rule, between the two points
which are occupied simultaneously by the two
ends. Since simultaneity, as we have just seen, is
relative, the distance between two points, since they
depend on a simultaneous reading of two events, is
also relative, and length only has a meaning if the
body of reference is likewise specified : any change
of motion entails a corresponding change of length :
we cannot detect the change since our measures
alter in the same ratio. Length is thus a relative
conception, and only reveals a relation between
the observer and an object: the "actual" length
of a body in the sense we usually understand it
does not exist : there is no meaning in the term.
The length of a body measured parallel to its direc-
tion of motion will always yield a greater result
when judged from a system attached to it than
from any other system. These few remarks may
suffice to indicate the relativity of distance.

In classical mechanics it had always been assumed
that the time which elapsed between the happening
of two events, and also the distance between two
points of a rigid body were independent of the state
of motion of the body of reference : these hypoth-
eses must, as a result of the relativity of simul-
taneity and distance, be rejected. We may now ask
whether a mathematical relation between the place
and time of occurrence of various events is possible,
such that every ray of light travels with the same
constant velocity c whichever rigid body of reference
be chosen, e.g. such that the rays measured by an

THEORY OF GRAVITATION 117

observer either in the train or on the embankment
travel with the same apparent velocity.

In other words, if we assume the constancy of
propagation of light in vacuo for two systems, K
and K1 moving uniformly and rectilinearly with
respect to one another, what are the values of the
co-ordinates x1, y1, z1, tl of an event with respect to
K1, if the values x, y, z, t of the same event with
respect to K are given ?

It is easy to arrive at this so-called Lorentz-
Einstein transformation, e.g. in the case where K1 is
moving relative to K parallel to K's x axis with
uniform velocity v we get :

x — vt , ,

X1 = ______ y1 = y z1 = Z.

If we put x — ct, we find that — reduces to c.

i.e. c = - = ^- is the same for both systems, and

the condition of the constancy of c, the velocity
of light in vacuo, is preserved.

If N/I — v 2/c2 is to be real, then v cannot be
greater than c, i.e. c is the limiting or maximum
velocity in nature and has thus a universal signifi-
cance.

If we imagine c to be infinitely great in com-
parison with v (and this will be the case for all
ordinary velocities, such as those which occur in
mechanics), the equations of transformation de-
generate into :

x1 = x — vt y1 = y z1 = z t1 = t.

This is the familiar Galilean transformation which
holds for the " mechanical " principle of relativity.

118 THE FOUNDATIONS OF EINSTEIN'S

We see that the Lorentz-Einstein transformation
covers both mechanical and radiational phenomena.
The special theory of relativity may now be
enunciated as follows : All systems of reference
which are in uniform rectilinear motion with
regard to one another can be used for the
description of physical events with equal justi-
fication. That is, if physical laws assume a
particularly simple form when referred to any
particular system of reference, they will preserve
this form when they are transformed to any other
co-ordinate system which is in uniform rectilinear
motion relatively to the first system. The
mathematical significance of the Lorentz-Einstein
equations of transformation is that the expression
for the infinitesimal length of arc ds

(viz. ds* = dx* + dy* + dz* - c* dt2)

in the space-time * manifold x, y, z, t, preserves its
form for all systems moving uniformly and rec-
tilinearly with respect to one another.

Interpreted geometrically this means that the
transformation is conformal in imaginary space of
four dimensions. Moreover, the time-co-ordinate
enters into physical laws in exactly the same way
as the three space-co-ordinates, i.e. we may regard
time spatially as a fourth dimension of space. This
has been very beautifully worked out by Minkowski,
whose premature loss is deeply to be regretted. It
may be fitting here to recall some remarks of Berg-
son in his " Time and Free Will." He there states
that " time is the medium in which conscious states
form discrete series : this time is nothing but space,
and pure duration is something different." Again,

* A continuous manifold may be defined as any continuum of
elements such that a single element is defined by n continuously
variable magnitudes.

THEORY OF GRAVITATION 119

" what we call measuring time is nothing but counting
simultaneities ; owing to the fact that our con-
sciousness has organized the oscillations of a pendu-
lum as a whole in memory, they are first preserved
and afterwards disposed in a series : in a word, we
create for them a fourth dimension of space, which
we call homogeneous time, and which enables the
movement of the pendulum, although taking place
at one spot, to be continually set in juxta-posi-
tion to itself. Duration thus assumes the illusory
form of a homogeneous medium and the connecting
link between these two terms, space and duration,
is simultaneity, which might be defined as the
intersection of time and space." Minkowski calls
the space-time-manifold " world " and each point
(event) " world-point."

The results achieved by the special theory of
relativity may be tabulated as follows : —

(1) It gives a consistent explanation of Fizeau's

and Michelson and Morley's experiment.

(2) It leads mathematically at once to the value

suggested by Fresnel and experimentally
verified by Fizeau for the velocity of a beam
of light through a moving refracting medium
without making any hypothesis about the
physical nature of the liquid.

(3) It gives the contraction in the direction of

motion for electrons moving with high speed,
without requiring any artificial hypothesis
such as that of Lorentz and Fitzgerald to
explain it.

(4) It satisfactorily explains aberration, i.e. the

influence of the relative motion of the earth
to the fixed stars upon the direction of motion
of the light which reaches us.

(5) It accounts for the influence of the radial

component of the motion of the stars, as

120 THE FOUNDATIONS OF EINSTEIN'S

shown by a slight displacement of the spectral
lines of the light which reaches us from the
stars when compared with the position of the
same lines as produced by an earth source.

(6) It accounts for the "fine structure" of the

spectral lines emitted by the atom.*

(7) It gives the expression for the increase of

inertia, owing to the addition of (apparent)
electromagnetic inertia of a charged body in
motion.

The last result, however, introduces an anomaly
inasmuch as the inertial mass of a quickly-moving
body increases, but not the gravitational mass, i.e.
there is an increase of inertia without a corresponding
increase of weight asserting itself. One of the most
firmly established facts in all physics is hereby
transgressed. This result of the theory suggested
a new basis for a more general theory of relativity,
viz. that proposed by Einstein in 1915. As the
special theory of relativity deals only with uniform,
rectilinear motions, its structure is not affected by
any alteration of the ideas underlying gravitation.

III. THE GENERAL THEORY OF RELATIVITY

We have seen that the first or " mechanical "
theory of relativity was built up on the notion of
inertial systems as deduced from the law of inertia ;
the " special " theory of relativity was built up on
the universal significance and invariance of c, the
velocity of light in vacuo ; the third or general form
of relativity is to be founded on the principle of the
equality of inertial and gravitational mass and in
contradistinction to the other two is to hold not only
for systems moving uniformly and rectilinearly with
respect to one another, but for all systems whatever

* See Sommerfeld, " Atomic Structure and Spectral Lines," p. 474.

THEORY OF GRAVITATION 121

their motion ; i.e. physical laws are to preserve
their form for any arbitrary transformation of the
variables from one system to another.

Mass enters into the formulae of the older physics
in two forms : (i) Force = inertial mass multiplied
by the acceleration. (2) Force — gravitational mass
multiplied by the intensity of the field of gravitation ;
or,

p = m . a p = ml.g

m1

«~w*

Observation tells us that for a given field of gravi-
tation the acceleration is independent of the nature
and state of a body ; this means that the propor-
tionality between the two characteristic masses
(inertial and gravitational) must be the same for all
bodies. By a suitable choice of units we can make
the factor of proportionality unity, i.e. m = m1.

This fact had been noticed in classical mechanics,
but not interpreted.

Eotvos in 1891 devised an experiment to test the
law of the equality of inertial and gravitational mass :
he argued that if the centre of inertia of a hetero-
geneous body did not coincide with the centre of
gravity of the same body, the centrifugal forces act-
ing on the body due to the earth's rotation acting at
the centre of inertia would not, when combined with
the gravitational forces acting at the centre of gravi-
tational mass, resolve into a single resultant, but that
a torque or turning couple would exist which would
manifest itself, if the body were suspended by a very
delicate torsionless thread or filament. His experi-
ment disclosed that the law of proportionality of
inertial and gravitational mass is obeyed with
extreme accuracy : fluctuations in the ratio could
only be less than a twenty-millionth.

THE FOUNDATIONS OF EINSTEIN'S

Einstein hence assumes the exact validity of the
law, and asserts that inertia and gravitation are
merely manifestations of the same quality of a
body according to circumstances. As an illustra-
tion of the purport of this equivalence he takes the
case of an observer enclosed in a box in free space
(i.e. gravitation is absent) to the top of which a
hook is fastened. Some agency or other pulls this
hook (and together with it the box) with a constant
force. To an observer outside, not being pulled, the
box will appear to move with constant acceleration
upwards, and finally acquire an enormous velocity.
But how would the observer in the box interpret the
state of affairs ? He would have to use his legs to
support himself and this would give him the sensa-
tion of weight. Objects which he is holding in his
hands and releases will fall relatively to the floor
with acceleration, for the acceleration of the box will
no longer be communicated to them by the hand ;
moreover, all bodies will " fall " to the floor with the
same acceleration. The observer in the box, whom
we suppose to be familiar with gravitational fields,
will conclude that he is situated in a uniform field of
gravitation : the hook in the ceiling will lead him to
suppose that the box is suspended at rest in the
field and will account for the box not falling in the
field. Now the interpretation of the observer in the
box and the observer outside, who is not being pulled,
are equally justifiable and valid, as long as the
equality of inertial and gravitational mass is main-
tained.

We may now enunciate Einstein's Principle of
Equivalence : Any change which an observer per-
ceives in the passage of an event to be due to a
gravitational field would be perceived by him ex-
actly in the same way, if the gravitational field
were not present and provided that he — the ob-

THEORY OF GRAVITATION

server — make his system of reference move with
the acceleration which was characteristic of the
gravitation at his point of observation.

It might be concluded from this that one can
always choose a rigid body of reference such that,
with respect to it, no gravitational field exists,
i.e. the gravitational field may be eliminated ; this,
however, only holds for particular cases. It would
be impossible, for example, to choose a rigid body
of reference such that the gravitational field of the
earth with respect to it vanishes entirely.

The principle of equivalence enables us theoretic-
ally to deduce the influence of a gravitational field
on events, the laws of which are known for the
special case in which the gravitational field is absent.

We are familiar with space- time-domains, which
are approximately Galilean when referred to an
appropriate rigid body of reference. If we refer
such a domain to a rigid body of reference K1 moving
irregularly in any arbitrary fashion, we may assume
that a gravitational field varying both with respect
to time and to space is present for K1 : the nature
of this field depends on the choice of the motion of
K1. This enabled Einstein to discover the laws
which a gravitational field itself satisfies. It is im-.
portant to notice that Einstein does not seek to
build up a model to explain gravitation but merely
proposes a theory of motions. His equations describe
the motion of any body in terms of co-ordinates of
the space-time manifold, making use of the inter-
changeability and equivalence implied in relativity.
He does not discuss forces as such ; they are, after
all, as Karl Pearson states, " arbitrary conceptual
measures of motion without any perceptual equiva-
lent." They are simply intermediaries which have
been inserted between matter and motion from
analogy with our muscular sense.

THE FOUNDATIONS OF EINSTEIN'S

A direct consequence of the application of the
Principle of Equivalence in its general form is that
the velocity of light varies for different gravitational
fields, and is constant only for uniform fields (this
does not contradict the special theory of relativity,
which was built up for uniform fields, and only
makes it a special case of this much more general
theory of relativity) . But change of velocity implies
refraction, i.e. a ray of light must have a curved
path in passing through a variable field of gravita-
tion. This affords a very valuable test of the truth
of the theory, since a star, the rays from which pass
very near the sun before reaching us, would have to
appear displaced (owing to the stronger gravitational
field around the sun), in comparison with its relative
position when the sun is in another part of the
heavens : this effect can only be investigated during
a total eclipse of the sun, when its light does not
overpower the rays passing close to it from the star
in question.* The calculated curvature is, of course,
exceedingly small (1-7 seconds of arc), but, never-
theless, should be observable.

The motion of the perihelion of Mercury, discovered
by Leverrier, which long proved an insuperable
obstacle regarded in the light of Newtonian
mechanics, is immediately accounted for by the
general theory of relativity ; this is a very remark-
able confirmation of the theory.

Before we finally enunciate the general theory of
relativity, it is necessary to consider a special form
of acceleration, viz. rotation. Let us suppose a
space-time-domain (referred to a rigid body K) in
which the first Newtonian Law holds, i.e. a Galilean
field : we shall suppose a second rigid body of refer-
ence K1 to be rotating uniformly with respect to K,

* We shall return to this test at the conclusion of the chapter.

THEORY OF GRAVITATION 125

say a plane disc rotating in its plane with constant
angular velocity. An observer situated on the disc
near its periphery will experience a force radially
outwards, which is interpreted by an external
observer at rest relatively to K as centrifugal force,
due to the inertia of the rotating observer. But
according to the principle of equivalence the rotating
observer is justified in assuming himself to be at rest,
i.e. the disc to be at rest. He regards the force
acting on him as an effect of a particular sort of
gravitational field (in which the field vanishes at the
centre and increases as the distance from the centre
outwards). This rotating observer, who considers
himself at rest, now performs experiments with
clocks and measuring-scales in order to be able to
define time- and space-data with reference to K1.
It is easy to show that if, of two clocks which go at
exactly the same rate when relatively at rest in the
Galilean field K, one be placed at the centre of the
rotating disc and one at the circumference, the latter
will continually lose time as compared with the
former.

Secondly, if an observer at rest in K measure the
radius and circumference of the rotating disc, he will
obtain the same value for the radius as when the
disc is at rest, but since, when he measures the cir-
cumference of the disc, the scale lies along the
direction of motion, it suffers contraction, and,
consequently, will divide more often into the circum-
ference than if the scale and the disc were at rest.
(The circumference does not change, of course, in
rotation.) That is, he would get a value greater

,, r ,, ,. circumference. „„ . ,,

than 77 for the ratio — ^ — This means that

diameter

Euclidean geometry does not hold for the observer
making his observations on the disc, and we are
obliged to use co-ordinates which will enable his

126 THE FOUNDATIONS OF EINSTEIN'S

results to be expressed consistently. Gauss in-
vented a method for the mathematical treatment of
any continua whatsoever, in which measure-relations
(" distance " of neighbouring points) are denned.
Just as many numbers (Gaussian or curvilinear
co-ordinates) are assigned to each point as the
continuum has dimensions. The allocation of num-
bers is such that the uniqueness of each point is
preserved and that numbers whose difference is
infinitely small are assigned to infinitely near points.
This Gaussian or curvilinear system of co-ordinates
is a logical generalization of the Cartesian system.
It has the great advantage of also being applicable
to non-Euclidean continua, but only in the cases in
which infinitesimal portions of the continuum con-
sidered are of the Euclidean form. This calls to mind
the remarks made at the commencement of this
sketch about the validity of geometrical theorems.
It seems as though the miniature view that we
can take of straight lines in the immensity of space
led to a firm belief in the universal significance
of Euclidean geometry. When we deal with light
phenomena which range to enormous distances, we
find that we are not justified in confining ourselves
to Euclidean geometry ; the " straightest " line in
the time-space-manifold is " curved." We must
therefore choose that geometry which, expressed
analytically, enables us to describe observed pheno-
mena most simply : it is clear that for even large
finite portions of space the non-Euclidean geometry
chosen must practically coincide with Euclidean
geometry.

We now see that the general theory of relativity
cannot admit that all rigid bodies of reference K, K1,
etc., are equally justifiable for the description of the
general laws underlying the phenomena of physical
nature, since it is, in general, not possible to make

THEORY OF GRAVITATION 127

use of rigid bodies of reference for space-time descrip-
tions of events in the manner of the special theory of
relativity. Using Gaussian co-ordinates, i.e. label-
ling each point in space with four arbitrary numbers
in the way specified above (three of these correspond
to three space dimensions and one to time), the
general principle of relativity may be enunciated
thus :—

All Gaussian four-dimensional co-ordinate-sys-
tems are equally applicable for formulating the
general laws of physics. This carries the principle
of relativity, i.e. of equivalence of systems, to an
extreme limit.

With regard to the relativity of rotations, it may
be briefly mentioned that centrifugal forces can,
according to the general theory of relativity, be due
only to the presence of other bodies. This will be
better understood by imagining an isolated body
poised in space ; there could be no meaning in saying
that it rotated, for there would be nothing to which
such a rotation could be referred : classical mechanics
however, asserts that, in spite of the absence of other
bodies, centrifugal forces would manifest themselves :
this is denied by the general theory of relativity.
No experimental test has hitherto been devised
which could be carried out practically to give a
decision in favour of either theory.

A favourable opportunity for detecting the slight
curvature of light rays (which is predicted by the
general theory) when passing in close vicinity to the
sun occurred during the total eclipse of the 2Qth May,
1919. The results, which were made public at the
meeting of the Royal Society on 6th November
following, were reported as confirming the theory.

In addition to the slight motion of Mercury's peri-
helion, there is still a third test which is based upon
a shift of the spectral line towards the infra-red, as

128 THE FOUNDATIONS OF EINSTEIN'S

a result of an application of Doppler's principle ;
this has not yet led to a conclusive experimental
result.

I. NOTE ON NON-EUCLIDEAN GEOMETRY

In practical geometry we do not actually deal with
straight lines, but only with distances, i.e. with
finite parts of straight lines, yet we feel irresistibly
impelled to form some conception of the parts of a
straight line which vanish into inconceivably distant
regions. We are accustomed to imagining that a
straight line may be produced to an infinite distance
in either direction, yet in our mathematical reasoning
we find that in order to preserve consistency (in
Euclid),* we may only allocate to this straight line
one point at infinity : we say that two straight lines
are parallel when they cut at a point at infinity,
i.e. this point is at an infinite distance from an
arbitrary starting-point on either straight line, and is
reached by moving forwards or backwards on either.

Many attempts have been made, without success,
to deduce Euclid's " axiom of parallels," which
asserts that only one straight line can be drawn
parallel to another straight line through a point
outside the latter, from the other axioms. It finally
came to be recognized that this axiom of parallels
was an unnecessary assumption, and that one could
quite well build up other geometries by making other
equally justified assumptions.

If we consider a point, P, outside a straight line, L
(Fig. 4), to send out rays in all directions, then,
starting from the perpendicular position Palt we
find that the more obliquely the ray falls on L
the farther does the point of intersection an travel

* According to the modern analytical interpretation of Euclid.

THEORY OF GRAVITATION

129

along L to the left (say). Our experience teaches
us that the ray and L have one point in common.
There is no justifiable reason, however, for asserting,
as Euclid's axiom does, that for a final infinitely
small increase of the angle at Pan (i.e. additional turn
of Pan about P), an suddenly bounds off to infinity
along L, i.e. an, the point of intersection leaves finite
regions to disappear into so-called " infinity," and
that, for a further infinitesimal increase, an, reappears
at infinity at the other end of L to the right of 04.

One might equally well assume, as Lobatschewsky
did, that P^QQ and Pa _ ^ form an angle which differs
ever so slightly from two right angles, and that there
are an infinite number of other straight lines included

OC4 CX3 OC2 OC,
FIG. 4.

between these two positions (as indicated by the
dotted lines in the figure), which do not cut L at
all. Lobatschewsky (andalso Bolyai) built up an
entirely consistent geometry on this latter assump-
tion.

Riemann later abolished the assumption of infinite
length of a straight line, and assumed that in travel-
ling along a straight line sufficiently far one finally
arrives at the starting-point again without having
encountered any limit or barrier. This means that
our space is regarded as being finite but unbounded.*

* E.g. the surface of a sphere cuts a finite volume out of space,
but particles sliding on the surface nowhere encounter boundaries
or barriers. This is a three-dimensional analogon to the four-

130 THE FOUNDATIONS OF EINSTEIN S

Thus in Riemann's case there is no parallel line to
L for an never leaves L ; there is no a^ . This
geometry was called by Klein elliptical geometry
(and includes spherical geometry as a special case).
He calls Euclidean geometry parabolic (Fig. 5), for
the branches of a parabola continue to recede from
one to another, and yet in order to obtain consistent
results in its formulae we are obliged only to assign
one point at infinity to it, just as to the Euclidean
straight line. Lobatschewsky's geometry is simi-
larly called hyperbolic (Fig. 5), since a hyperbola has

FIG. 5.

two points at infinity, corresponding in analogy to
the two*points at infinity at which the two parallels
through a point external to a straight line cut the
latter.

The fact that one is obliged to renounce Euclidean
geometry in the general theory of relativity leads to
the conclusion that our space is to be regarded as
finite but unbounded : it is curved, as Einstein
expresses it, like the faintest of ripples on a surface
of water.

dimensional space-time manifold of Minkowski. It does not mean
that the universe is enclosed by a spherical shell, as was supposed by
the ancients. We cannot form a picture of the corresponding result
in the four-dimensional continuum in which, according to the general
theory of relativity, we live.

THEORY OF GRAVITATION 131

SOME ASPECTS OF RELATIVITY

THE THIRD TEST
BY HENRY L. BROSE, M.A.

UP to the present, three methods of verifying
Einstein's Theory of Relativity have been
suggested.

The first one, which was a direct outcome of the
new gravitational field-equations proposed by Ein-
stein, proved successful. The slow motion of Mer-
cury's perihelion which long mystified astronomers
was immediately accounted for. This result is the
more remarkable as all other explanations of this
phenomena were artificial in origin, consisting of a
hypothesis formulated ad hoc which could not be
verified by observation.

The second method involved the deflection of a
ray of light in its passage through a varying gravita-
tional field. The results of the total eclipse of the
sun which occurred on 2Qth May, 1919, have become
famous and were recorded as confirming Einstein's
prediction. The results of a more recent expedition
have proved finally conclusive.

The third test, the results of which are still in
abeyance, is perhaps the most important of the three,
inasmuch as it depends upon a very simple calcula-
tion from Einstein's Principle of Equivalence, which
asserts that an observer cannot discriminate how

1S2 THE FOUNDATIONS OF EINSTEIN'S

much of his motion is due to a gravitational field and
how much is due to an acceleration of his body of
reference. Einstein illustrates his argument by
supposing an observer situated in a closed box in
free space. The observer has at first no sensation
of weight, and need not support himself upon his
feet. Now suppose an external agent to pull the
box in a definite direction with constant force. The
observer in the box performs experiments with
masses of variable material, and as they all fall to
the " floor " of the box at the same time, he concludes
that he is in a gravitational field. He himself has
acquired the sensation of weight. This result led
Einstein to propound the equivalence of gravitational
and accelerational fields. An immediate consequence
of this principle is that the duration of an event
depends upon the gravitational conditions at the
place of the event.

If we consider the light (of frequency j/J which is
emitted by a distant star, and suppose it to traverse
a practically invariable gravitational field in which
bodies are assumed to fall with a constant accelera-
tion g, then an observer at a distance h from the
star will have attained a velocity (= acceleration x

time) = g . - where c is the velocity of light and the

distance h is small in comparison with the distance
traversed by the observer in the time the light takes
to reach him. By Doppler's Principle, the apparent
frequency j>2 is given by

='.(•+ 3 =

Potential of unit mass moved through a distance
h is gh = + (f> (say). This gives the work done in
moving unit mass from the source of light to the

THEORY OF GRAVITATION 133

observer (the source of light is here the point to
which the potential energy is referred in the field) .

Therefore, if we transform the accelerational field
of the observer into the gravitational field, we get
the result :

This means that a spectral line of frequency vl
will appear to a distant observer to be displaced, if
compared with the position of the same line, when
produced by a source at a different point in the
field. Each of these lines, produced by vibrat-
ing electrons, may be regarded as a clock, and this
simple calculation shows how time-measurements
are affected by the state of the gravitational field.
This effect amounts to 0*008 Angstroms, for a wave-
length of 4000 A. The same displacement would be
produced as a Doppler effect by a velocity of 0-6 kms.
per sec. When this test was put into practice, it
was found difficult to discriminate it from the
various superposed effects due to other causes such
as the radial velocities of the stars, proper velocities
of the gaseous envelopes, pressure, etc. The con-
ditions of the emission of light by the sun have not
been fully ascertained, nor is the light of the arc
lamp free from disturbing elements. Dr. Erwin
Freundlich, of the Neubabelsberg Observatory, has
discussed, in conjunction with Professor Einstein,
the possibility of recognizing this effect in spite of
these obscuring influences. He points out three
ways of establishing the result qualitatively. They
may be briefly classified as being based on (i) statis-
tical methods ; (2) nebular spectra ; (3) calcium
lines in the spectra of the atmosphere surrounding
double-stars.

i. If we consider a great number of stars of about

134 THE FOUNDATIONS OF EINSTEIN'S

the same mass evenly distributed over the heavens,
and represent the spectral shift due to radial veloci-
ties (i.e. velocities in the line of sight) graphically,
we should expect these velocities to be distributed
according to the law of probability about the value
zero, i.e. as depicted by Gauss's Error Curve, which
resembles a vertical section of a bell. If, however,
Einstein's gravitational effect really exists, we should
expect these velocities to group themselves sym-
metrically about a positive velocity which would be
that corresponding to this spectral shift. Gauss's
Error Curve would thus appear displaced by pre-
cisely the amount of the radial velocity corresponding
to this shift, as all the radial velocities would be
falsified by just this amount.

The values of the radial velocities have been
plotted in the case of B-stars, called Helium stars
on account of the predominance of helium lines in
their spectra. Other observations have led astro-
nomers to infer that the B-stars have unusually
great masses but small densities. The result has
been distinctly in favour of the Einstein shift on
the basis of the foregoing discussion. The same was
found to hold for the bright K- and M-stars, which
are considered to be at a lower temperature and
possessed of enormous surface extent, which ac-
counts for their brilliance.

If we indicate the mean shift of the lines towards
the red by K, then for

B-stars K = 4-3 kms. + 0-5,
K „ K = 3-5 „ ±0-9,
M „ K = 5-3 „ +2-3.

K is here expressed in terms of a Doppler shift as a
velocity, i.e. as if the Einstein shift were due to an
additional radial motion and hence expressible in
kilometres.

THEORY OF GRAVITATION 135

Alternative ways of accounting for this shift have
been proposed.

(a) It may be regarded as an ordinary Doppler
effect. This would imply that the stars of the B,
K, and M type suffer a general expansion to which
stars of the F and G type (yellow stars like the sun)
and the A-stars are not subject.

This explanation does not seem very probable, as
helium lines were used in determining the shift for
the B-stars, whereas quite different lines were used
for measuring the effect for the K- and M-stars.
It would be a strange coincidence if this shift, to
which all the evidence points as arising from a
common origin, should be manifested just in these
cases which have been made the object of an investi-
gation.

(6) The general shift towards the red might be
ascribed to pressure effects at the surfaces of the
stars or to the presence of other lines which lie on
the red side of the main lines, but which are very
weak or even absent in the comparison spectrum of
the sun. A detailed knowledge of conditions on the
surfaces of stellar bodies could alone give a decision
on this point.

2. It is only possible to prove that the shift K
is not due to a radial velocity if one can measure
the ordinary Doppler effect arising from the radial
velocity separately. Let us consider a single B-star
or group of B-stars which happen to be embedded
in a nebula of great extent which accompanies them
in their motion. The Doppler effect due to the
radial velocity would be the same for the star as the
nebula, but the gravitational effect predicted by
Einstein would not be the same, inasmuch as the
gravitational field at the surface of the star will vary
considerably from that at the outer edge of the
nebula. Hence it would be reasonable to attribute

136 THE FOUNDATIONS OF EINSTEIN'S

any difference in the magnitude of the spectral
shifts in the case of the star and the nebula to the
difference in gravitational fields at each place.

The stars of the nebular group of Orion have
hitherto offered the only possibility of applying this
method. The results have fulfilled Einstein's expec-
tations qualitatively, and it remains to be seen
whether the agreement will hold quantitatively. A
general shift of the star-spectrum as compared with
the corresponding lines of the associated nebula was
observed.

Some very bright B-stars in the constellation of
Orion are considered to form an entity with their
attendant nebula. This conclusion was reached as
the result of independent research.

The radial velocity of the Orion-nebula has been
measured by various observers. The values obtained
are: 17-7 (Wright), 17-4 (Vogel and Eberhardt),
18-5 (Frost and Adams). The mean value is 17-4
kms. per sec. This velocity is derived from the
brightest part of the nebula, the so-called trapezium.
The values obtained in the case of the stars almost
all exceed 20 kms. per sec., and hence it seems likely
that part of this radial velocity, viz. the excess over
that of the nebula, is due to the Einstein effect.
When the difference between the radial velocities of
the stars and the associated nebula are tabulated for
each star, we find that in the case of all members
except two the difference is positive, i.e. indicative
of a shift towards the red end, in agreement with the
statistical investigation applied to the B-, K-, and
M-stars. The difference amounts to 6-0 kms.
+ -i km., and is a little greater than that given by
the statistical method.

The two stars 0 and 36 Orionis give a displace-
ment towards the violet end. It has been suggested
that they do not belong to the more limited group of

THEORY OF GRAVITATION 137

Orion stars, but are only projected into that portion
of the celestial sphere. This is supported by the
fact that both stars have only very small spherical
proper motions, and that the radial velocities
observed for them differ considerably from the mean
of the radial velocities of the others.

This method has not been successfully applied to
other stellar systems inasmuch as the nebulae of
those which are available emit such feeble light that
it has not been possible to establish the displacement
to any degree of accuracy. Eddington recently
pointed out that a very important factor had been
neglected in the fundamental equations of the early
theories concerning the equilibrium of stellar matter,
viz. the pressure due to radiation. According to
his theory, the equilibrium in the interior of the star
(regarded as a gaseous sphere) is determined by three
conditions. These are gaseous pressure, radiational
pressure, and gravitational forces.

Calculation shows that for very great masses the
gravitational pressure is almost entirely balanced by
radiational pressure. This implies that any addi-
tional force such as that due to a centrifugal field of
rotation would lead to an unstable condition.

It can, furthermore, be deduced from Eddington's
theory that only stars whose masses exceed a certain
minimum value can in the course of their evolution
reach the very high surface-temperatures which have
been observed in the case of the O- and B-stars.

It therefore seems likely that the O- and B-stars
have in the process of evolution passed through a
stage of which the radiational pressure has brought
about a condition of unstable equilibrium, and one
might expect them to be surrounded by cosmic dust
which has become dissociated from the nuclei of the
system.

In some cases this dissociated matter may be in
9*

138 THE FOUNDATIONS OF EINSTEIN'S

a very fine state of division, and may extend so far
into space that the absorption lines they produce in
the spectrum of the star they surround may originate
from a gravitational field which differs perceptibly
from that at the surface of the star. There are
definite signs of the existence of such atmospheres.
A high percentage of B-stars are found to be spectro-
scopic double stars, i.e. their spectral lines fluctuate
periodically about some mean position. Hartmann
was the first to notice that in the spectrum of the
B-star S. Orionis the absorption lines K and H of
calcium, viz. 3933-82 and 3968-63 Angstroms, occur,
but that they do not share in the periodic movements
of the other lines. A number of other stars belonging
to early spectral types contain calcium absorption
lines in their spectra, which exhibit a similar anomaly,
inasmuch as they either remain immovable or execute
periodic motions which are of feeble amplitude com-
pared with the proper stellar lines. In view of the
important rdle that calcium plays in the outermost
layers of the gaseous atmosphere encircling the sun,
and in view of the discussion above, the suggestion
forces itself upon one that these calcium lines indicate
the presence of an extensive atmosphere surrounding
the star.

It has often been put forward that these lines are
due to the light from these stars being absorbed by
vast interstellar clouds of calcium. Evershed con-
siders that this is supported by the fact that when
the motion of the solar system is subtracted from
that calculated from the fixed calcium lines (owing
to the ordinary Doppler effect), the remaining motion
is very small. But this argument does not carry
weight inasmuch as it is known that the B-stars, in
the spectrum of which these lines occur, themselves
have very small radial velocities. As Young re-
marked, it seems very strange that these calcium

THEORY OF GRAVITATION 139

clouds should so consistently choose to lie in front of
stars of type B or earlier. An objection against this
hypothesis is to be found in the fact that in the
case of various systems these two calcium lines are
not at rest but move, although with somewhat less
amplitude than the other proper lines of the double
star.

An additional circumstance which lends support
to the theory that calcium lines denote the presence
of an atmosphere around the star is that a great
number of helium-stars are enveloped in a nebulous
atmosphere which is actually visible.

Assuming then that the calcium absorption lines
are due to such atmospheres, we may apply the same
process as in the case of the Orion nebula, i.e. if the
shifts of the spectral-lines of the stars be systematic-
ally falsified by a superposed gravitational effect,
this should be expressed by the lines of the actual
spectrum from a double star being displaced towards
the red as compared with the fixed calcium lines.

This phenomenon has been clearly observed. The
result has not yet been quantitatively fixed, as the
numbers taken are not regarded as final.

All stars in the spectra of which the H and K lines
of calcium occur have been used to test the con-
clusion, and all show a shift to the red end ; the
mean of the shifts corresponds to a velocity of
-j- 6-3 kms. per sec.

The results of this discussion have been formulated
by Dr. Freundlich thus : —

SUMMARY

i. Statistical consideration gives us the means of
separating the mean gravitational effect from the
ordinary Doppler effect in the case of the helium
B-stars and the bright K- and M-stars, which

140 EINSTEIN'S THEORY OF GRAVITATION

astronomical investigations compel us to regard as
being of particularly great mass.

A general shift of the spectra towards the red is
exhibited with considerable certainty.

2. It follows from a comparison of the displacement
of the lines of the star-spectra that the above dis-
placement which was found by a statistical examina-
tion is not an ordinary Doppler effect, but is due to
the conditions of emission of light at the surfaces of
the stars.

3. The close connection of the B- and 0-stars with
nebulous matter in the heavens is a symptom that
these stars are of great mass.

4. If we regard the fixed calcium lines in the
spectra of B- and 0-stars as being caused by absorp-
tion in extended calcium atmospheres moving with
each star in question, the shift towards the red
which manifests itself may be regarded as the effect
predicted by Einstein's theory, i.e. due to the
different gravitational fields from which the absorp-
tion lines and the stellar lines have originated.

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