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.4

SPACE :&$,&; TIME

IN CONTEMPOKARY PHYSICS

AN INTRODUCTION TO

THE THEORY OF
RELATIVITY AND GRAVITATION

BY

MOBJTZ SCHLICK

PROFESSOR OF PHILOSOPHY AT ROSTOCK UNIVERSITY

RENDERED INTO ENGLISH BY

HENRY L. BROSE
WITH AN INTRODUCTION BY

F. A. LINDEMANN

PROFESSOR OF EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF OXFORD

NEW YORK
OXFORD UNIVERSITY PRESS

AMERICAN BRANCH: 85 WEST S2ND STREET
LONDON. TORONTO. MELBOURNE, AND BOMBAY

1920

COPYRIGHT, 1920

BY THE

OXFORD UNIVERSITY PRESS

AMERICAN BRANCH

Printed in U. S. A.

INTRODUCTION

PBOBABLY no physical theory in recent times has given
rise to more discussion amongst philosophers than the
principle of relativity. One school of thought agrees that
physicists may well be led to recast their notions of space
and time in the light of experimental results. Another
school, however, is of opinion that these questions are no
concern of the physicists, who should make their theories fit
the philosophers' conceptions of these fundamental units.

The theory of relativity consists of two parts, the old
special theory, and the more recent general theory.

The main philosophic achievement of the special theory
of relativity is probably the recognition that the description
of an event, which is admittedly only perfect if both the
space and time co-ordinates are specified, will vary accord-
ing to the relative motion of the observer ; that it is impos-
sible to say, for instance, whether the interval separating
two events is so many centimetres and so many seconds, but
that this interval may be split up into length and time in
different ways, which depend upon the observer who is
describing it.

The reasons which force this conclusion upon the physi-
cist may be made clear by considering what will be the im-
pression of two observers passing one another who send out
a flash of light at the moment at which they are close to-
gether. The light spreads out in a spherical shell, and
it might seem obvious, since the observers are moving
relatively to one another, that they cannot both remain at
the centre of this shell. The celebrated Michelson-Morley
experiment proves that each observer will conclude that he

iv Introduction

does remain at the centre of the shell. The only explana-
tion for this is that the ideas of length and time of the one
observer differ from those of the other. It is not difficult to
find out exactly how much they differ, and it may be shown
that there is only one set of transformations, the Lorentz-
Einstein transformations, which account for the fact that
each observer believes himself to be at the centre of the
spherical shell. It is further a simple matter of geometry
to show that these transformations are equivalent to a
rotation about the axis at right angles to the relative veloc-
ity and the time. In other words, if the world is regarded
as a four-dimensional space-time-manifold^ the Lorentz-
Einstein equations imply that each observer regards sec-
tions at right angles to his own world-line as instantaneous
times. He is quite justified in doing so since the principle
of relativity asserts that the space-time-manifold is homa-
loidal There is no more intrinsic difference between length
and time than there is between length and breadth.

The main achievement of the general theory of relativity
has caused almost more difficulty to the school of philoso-
phers, who would like to save absolute space and time, than
the welding of space and time itself. Briefly this may be
stated as the recognition of the fact that it is impossible to
distinguish between a universal force and a curvature of
the space-time-manifold, and that it is more logical to say
the space-time-manifold is non-Euclidean than to assert
that it is Euclidean, but that all our measurements will
prove that it is not, on account of some hypothetical force.
Perhaps a simple analogy may make this clearer. Suppose
a golfer had always been told that all the greens were level,
and had always found that a putt on a level green proceeded
in a straight line. Now suppose he were playing on a
strange course and found that a ball placed on the green
rolled into the hole, that any putt ran in a spiral and finally

Introduction v

reached the hole. If tie were sufficiently imbued with the
conviction that all greens are and must be level, he might
conclude that there was some force attracting the ball to the
Jiole. If he were of an inquiring turn of mind the golfer
might try another make of ball, and possibly quite different
types of balls such as tennis balls or cricket balls. If he
found them all to behave in exactly the same way, though
one was made of rubber, another of leather, and another
filled with air, he might reasonably begin to doubt the as-
sumption that there was a mysterious force acting on all
these balls alike and begin to suspect the putting-green,

In gravitational phenomena we are confronted with an
analogous case. Anywhere at a distance from matter a
body set in motion continues on a straight course. In the
neighbourhood of matter, however, this course is deflected.
All bodies, whether large or small, dense or gaseous, behave
in exactly the same way and are deflected by the same
amount. Even light, which is certainly as different from
matter as two things can well be, obeys the universal law.
Are we not therefore bound to consider whether our space-
time-manifold may not be curved rather than flat, non-
Euclidean rather than Euclidean?

At first sight it might appear that there must be an easy
way to settle the question. The golfer has only to fix three
points on his putting-green, join them by straight lines, and
measure the sum of the three angles between these lines.
If the sum is two right angles the green is flat, if not, it is
curved. The difficulty, of course, is to define a straight line.
If we accept the definition of the shortest line, we have
carried out the experiment, for the path of a ray of light is
the shortest line and the experiment which determines its
deflection may be read as showing that the three angles
of the triangle star comparison star telescope are not
equal to two right angles when the line star-telescope

vi Introduction

passes near tlie sun. But some philosophers appear not to
accept the shortest line as the straight line. What defini-
tion they put in its place is not clear, and until they make it
clear their position is evidently a weak one. It is to be
hoped they will endeavour to do this, and to explain the ob-
served phenomena rather than adopt a merely negative
attitude.

This translation of Schlick's book should interest a wide
circle, especially amongst those who are concerned with the
general conceptions rather than the details. It would jus-
tify all, and more than all, the trouble that has been ex-
pended on it, if it served to render philosophers more con-
versant with the physicist's point of view and to enlist their
co-operation in the serious difficulties in modern physics,
which yet await solution.

F. A. LINDEMANN.
CLAEENDOIST LABOBATOEY,

OXFOBD.
March, 1920.

AUTHOR'S PREFACE TO SECOND

EDITION

THE second edition of this book differs from the first
chiefly in Chapters II and IX, which are entirely new
additions. The second chapter gives a brief account of the
< special ' theory of relativity. It will probably be welcome
to many readers. It seemed advisable not to assume the
reader to be acquainted with the earlier theory since it has
appeared that many have acquired the book, who are quite
unfamiliar with the subject. The book itself gains con-
siderably in completeness by this addition, as it now repre-
sents an introduction to the whole set of ideas contained in
the theory of relativity, i.e. to the special theory as well as
to the general theory. The beginner need not seek an
entrance to the rudiments of the former from other sources.

Chapter IX of the present edition is also quite new, and
cannot be omitted in a description of the fundamental no-
tions of the theory of relativity. It develops the highly sig-
nificant ideas of Einstein concerning the construction of the
cosmos as a whole, by which he crowned his theory about
two years ago, and which are of paramount importance for
natural philosophy and for our world-view. The essential
purpose of the book is to describe the physical doctrines
under consideration with particular reference to their im-
portance for our knowledge, i.e. their philosophic signifi-
cance, in order that the relativity and gravitation theory of
Einstein may exert the influence, to which it is justly en-
titled, upon contemporary thought. The fact that the sec-
ond edition has rapidly succeeded the first is welcomed as
an indication of a general wish to imbibe the new ideas and
to strive to digest them. The book again offers its help in
this endeavour. May it be of service in bringing this goal
ever nearer.

I owe Professor Einstein my hearty thanks for giving me
many useful hints as in the first edition.

MOEITZ SCHLICK.

ROSTOCK, January 1919.

PREFACE TO THE THIRD EDITION

SINCE the appearance of the second edition the physical
theory which is expounded in the book has been brilliantly
confirmed by astronomical observations (v. page 65). Gen-
eral interest has been excited to a high degree, and the
name of its creator shines with still greater lustre than be-
fore. The fundamental importance of the theory of rela-
tivity is beginning to be recognized more and more on all
sides, and there is no doubt but that, before long, it will be-
come an accepted constituent of the scientific world- view.
The number of those who are filled with wonder at this
achievement of genius has increased much more rapidly
than the number of those who thoroughly understand it.
For this reason, the demand for explanations of the under-
lying principles of the theory has not decreased but, on the
contrary, is growing. This is shown by the fact that the
second edition, although more numerous than the first, be-
came exhausted more rapidly.

The present edition varies from the previous one only in
small additions and other slight improvements. I have
endeavoured to meet the wishes which observant readers
have expressed to me personally or in writing. I hope that
the book will now somewhat better fulfil its good purpose
of leading as far as possible into the wonderful thought-
world of the theory of relativity. Among those to whom I
am indebted for suggestions, I wish to express my special
thanks to Professor E. Cohn, of Strassburg (now at Eos-
tock). MORITZ SCHLICK

BOSTOQK, January 1920,

BIBLIOGRAPHICAL NOTE

EEFERENCE may be made to the following books dealing
with the general theory of relativity :

A. S. Eddington. Report on the Relativity Theory of

Gravitation. Fleetway Press.

A. S. Eddington, Space, Time, and Gravitation. Camb.
Univ. Press. (In the Press.)

An elementary account is given in :

Erwin Freundlich. The Foundations of Einstem's

Theory of Gravitation (trans, by Henry L. Brose).

Camb. Univ. Press.
Albert Einstein. The Special and General Theory of

Relativity (trans, by R. W. Lawson) . Messrs. Methuen.
Henry L. Brose. The Theory of Relativity. An Essay.

B. H. Blackwell, Oxford.

The most important German book on the subject is :
Hermann Weyl. Raum, Zeit und Materie. Jul Springer,

Berlin. This gives all the details of the mathematical

reasoning.

Einstein's epoch-making papers are:

'Grundlagen der allgemeinen Kelativitatstheorie.' Ann.

d. Physik, 4. Folge. Bd. 49, S. 769.
Grundlagen des allgemeinen Relativitdtsprincips. J. A.

Barth. Leipzig. 1916.
'Erklarung der Perihelbewegnng des Merknr aus der

allgemeinen Eelativitatstheorie. ' Sitzungsberichte der

Jconigl. preuss. AJcad. der Wissenschaften, Nov. 1915.

Bd. xlvii

x* Bibliographical Note

The evolution of the ideas which are discussed in Chapter
X of the present book may be traced in the following works,
in addition to those mentioned in the text :
Jevons. The Principles of Science. Macmillan & Co.
H. Poincare. La Valeur de la Science. Paris. La

Science et I' Hypothese. Paris.
Ernest Mach. Erkenntnis und Irrtum. Leipzig. Die

Analyse der Empfindungen.
Jos. Petzoldt. Das Weltproblem. Leipzig.
Aloys Miiller. Das Problem des absoluten Raumes und

seine Bezieliung zum allegemeinen Rawnproblem.

Vieweg, Braunschweig.
Moritz Schlick. Allegemeine Erkenntnislehre. JnL

Springer, Berlin.

I wish to take this opportunity of thanking Mr. J. W. N.
Smith, M.A., of Christ Church (now at Rugby) for the
great care he has taken in revising the proof-sheets. Pro-
fessor Schlick and Dr. Wichmann kindly compared the
translation with the original, and made a number of help-
ful suggestions. I am indebted to Miss Olwen Joergens for
the English rendering of the quotation from Giordano
Bruno. Professor Mitchell, Vice- Chancellor of Adelaide
University, kindly verified the philosophical terminology
of the last chapter.

HENRY L. BROSE.

CHEIST CHTTBOH,
March, 1920.

CONTENTS

PAGE

I. From Newton to Einstein 1

II, The Special Principle of Relativity . . . 7

III. The Geometrical Eelativity of Space . . 22

IV. The Mathematical Formulation of Spatial

Eelativity 28

V. The Inseparability of Geometry and Physics

in Experience 32

VI. The Eelativity of Motions and its Connexion

with Inertia and Gravitation .... 37

VII. The General Postulate of Eelativity and the
Measure-Determinations of the Space-time
Continuum .... ... 46

VIIL Enunciation and Significance of the Funda-
mental Law of the New Theory ... 57

IX. The Finitude of the Universe .... 67

X. Eelations to Philosophy 76

INDEX 89

FKOM NEWTON TO EINSTEIN

AT the present day physical research has reached such a
degree of generalization of its first principles, and its stand-
point has attained to such truly philosophic heights, that
all previous achievements of scientific thought are left far
behind. Physics has ascended to summits hitherto visible
only to philosophers, whose gaze has, however, not always
been free from metaphysical haziness. Albert Einstein is
the guide who has directed us along a practicable path lead-
ing to these summits. Employing an astoundingly ingeni-
ous analysis, he has purged the most fundamental concep-
tions of natural science by removing all the prejudices
which have for centuries past remained undetected in them :
thus revealing entirely new points of view, and building up
a physical theory upon a basis which can be verified by
actual observation. The fact that the refinement of the con-
ceptions, by a critical examination of them from the view-
point of the theory of knowledge, is simultaneously com-
bined with the physical application which immediately
made his ideas experimentally verifiable, is perhaps the
most noteworthy feature of his achievement : and it would
be remarkable, even if the problem with which he was able
to grapple by using these weapons had not happened to be
gravitation that riddle of physics which so obstinately re-
sisted all efforts to read it, and the solution of which must
of necessity afford us glimpses into the inner structure of
the universe.

i

2 From Newton to Einstein

The most fundamental conceptions in physics are those
of Space and Time. The unrivalled achievements in re-
search, which in past centuries have enriched our knowl-
edge of physical nature, left these underlying conceptions
untouched until the year 1905. The efforts of physicists
had always been directed solely at the substratum which
occupied space and time : they had taught us to know, more
and more accurately, the constitution of matter and the law
of events which occurred m vacuo, or as it had, till recently,
been expressed, in the * aether'. Space and Time were re-
garded, so to speak, as vessels containing this substratum
and furnishing fixed systems of reference, with the help of
which the mutual relations between bodies and events had
to be determined: in short, they actually played the part
which Newton had set down for them in the well-known
words: ' Absolute, true and mathematical time flows in vir-
tue of its own nature uniformly and without reference to
any external object'; and * absolute space, by virtue of its
own nature and without reference to any external object,
always remains the same and is immovable 7 .

From the standpoint of the theory of knowledge, the
objection was quite early raised against Newton, that there
was no meaning in the terms Space and Time as used with-
out ' reference to an object' ; but, for the time being, physics
had no cause to trouble about these questions: it merely
sought to explain observed phenomena in the usual way, by
refining and modifying its ideas of the constitution and con-
sistent behaviour of matter and the 'aether'.

An example of this method is the hypothesis which was
put forward by H. A. Lorentz and Fitzgerald, that every
body which is in motion relatively to the aether is subject to
a definite contraction along the direction of motion (the so-
called Lorentz-contraction), which depends upon the veloc-
ity of the body. This hypothesis was set up in order to ex-

From Newton to Emstem 3

plain why It seemed impossible to detect c absolute' rectilin-
ear motion of our instruments by means of the experiment
of Michelson and Morley (which will be discussed below),
whereas, according to the prevalent physical ideas of the
time, this should have been possible. The whole trend of
physical discovery made it evident that this hypothesis
would not be permanently satisfactory (as we shall see im-
mediately), and this meant that the time was come when the
consideration of motion in physics had to be founded on re-
flections of a philosophic nature. For Einstein recognized
that there is a much simpler way of explaining from first '
principles the negative result of Michelson and Morley's
experiment. No special physical hypothesis at all is re-
quired. It is only necessary to recognize the principle of
relativity, according to which a rectilinear uniform ' abso-
lute' motion can never be detected, and the fact that the con-
ception of motion has only a physical meaning when re-
ferred to a material body of reference. He saw also that a
critical examination of the assumptions upon which our
space- and time-measurements have hitherto been tacitly
founded is necessary. Amongst these unnecessary and
unwarrantable assumptions were found, e.g. those which
concerned the absolute significance of such space- and time-
conceptions as/ length', ' simultaneity', &c. If these assump-
tions are dropped, the result of Michelson and Morley 's
experiment appears self-evident, and on the ground thus
cleared is constructed a physical theory of wonderful com-
pleteness, which develops the consequences of the above
fundamental principle; it is called the 'special theory of
relativity, because, according to it, the relativity of motions
is valid only for the special case of uniform rectilinear
motion.

The special principle of relativity indeed takes one con-
siderably beyond the Newtonian conceptions of Space and

4 From Newton to Einstem

Time (as will be seen from the short account in the next
chapter), but does not fully satisfy the philosophic mind,
inasmuch as this restricted theory is only valid for uniform
rectilinear motions. From the philosophic standpoint it is
desirable to be able to affirm that every motion is relative,
i.e. not the particular class of uniform translations only.
According to the special theory, irregular motions would
still be absolute in character; in discussing them we could
not avoid speaking of Space and Time ' without reference
to an object 7 .

But since the year 1905, when Einstein set up the special
principle of relativity for the whole realm of physics, and
not for mechanics alone, he has striven to formulate a gen-
eralized principle which is valid not only for uniform recti-
linear motions, but also for any arbitrary motion whatso-
ever. These endeavours were brought to a happy conclusion
in 1915, being crowned with complete success. They led to
such an extreme degree of relativization of all space- and
time-determinations that it seems impossible to extend it
any further; these space- and time-determinations will
henceforth be inseparably connected with matter, and will
have meaning only when referred to it. Moreover, they
lead to a new theory of gravitational phenomena which
takes physics very far beyond that of Newton. Space,
time, and gravitation play in Einstein's physics a part
fundamentally different from that assigned to them by
Newton.

The importance of these results, in their bearing upon the
underlying principles of natural philosophy, is so stupen-
dous that even those who have only a modest interest in
physics or the theory of knowledge cannot afford to pass
them by. One has to delve deep into the history of science
to discover theoretical achievements worthy to rank with
them. The discovery of Copernicus might suggest itself to

From Newton to Einstem 5

the mind; and if Einstein ? s results do not exert as great an
influence on the world-view of people in general as the
Copernican revolution, their importance as affecting the
purely theoretical picture of the world is correspondingly
greater, inasmuch as the deepest foundations of our knowl-
edge concerning physical nature have to be remod-
elled much more radically than after the discovery of
Copernicus.

It is therefore easy to understand, and gratifying to note,
that there is a general desire to penetrate into this new
field of thought. Many are, it is true, repelled by the exter-
nal form of the theory, because they cannot acquire the
highly* complicated mathematical technique which is neces-
sary for an understanding of Einstein's researches : but the
wish to be initiated into these new views, even without this
technical help, must be satisfied, if the theory is to exercise
its rightful influence in forming the modern view of the
world. And it can be satisfied without difficulty, for the
principles are as simple as they are profound. The concep-
tions of Space and Time were not in the first place evolved
by a complicated process of scientific thinking, but we are
compelled to use them incessantly in our daily life. Start-
ing from the most familiar conceptions of everyday life, we
can proceed step by step to exclude all arbitrary and un-
justified assumptions, until we are finally left with Space
and Time in the simple form in which they play their part
in Einstein's physics. We shall adopt this plan here, in
order to crystallize the fundamental ideas in particular of
the new theory of Space. We get them without any effort,
by merely expelling from the traditional notion of Space all
ambiguities and unnecessary thought-elements. We shall
clear a way leading to the general theory of relativity, if we
get our ideas of Space and Time precise by subjecting them
to a critical -examination, inasmuch as they serve as a

6 From Newton to Emstem

foundation for the new doctrine and make it intelligible.
We shall prepare ourselves for this task by considering
first the thoughts underlying the * special' theory of rela-
tivity.

n

THE SPECIAL PRINCIPLE OF EELATIVITY

MICHELSON and Morley ? s experiment forms the best intro-
duction to this principle, both historically and for its own
sake. Historically, because it gave the first impulse towards
setting up the relativity-theory; and in itself, because the
suggested explanations of the experiment bring the old and
new currents of thought into strongest relief with one an-
other.

The condition of affairs was as follows. The electro-
magnetic waves, of which light is composed, and which,
propagate themselves with a velocity c equal to 300,000 kilo-
metres per second (186,000 miles per sec.), were regarded
by the older physicists as changes of state, transmitted as a
wave-disturbance in a substance called f aether 5 , which com-
pletely filled all empty space, including even that between
the smallest particles of material bodies. Accordingly, light
would be transmitted relatively to the aether with the above
velocity c (i.e. one would obtain the value 300,000 kilometres
per second) if the velocity were measured in a co-ordinate
system, fixed in the aether. If, however, the velocity of light
were to be measured from a body which was moving rela-
tively to the aether with the velocity q in the direction of the
light-rays, the observed velocity of the light-rays should be
c q, for the light waves would hurry past the observer
more slowly since he is moving with them in their direction.
If he were moving directly towards the waves of light, he
should get c+q for its velocity by measurement.

7

8 The Special Principle of Relativity

But, so the argument continues, we on the earth are
exactly in the position of the observer moving relatively to
the aether : for numerous observations had compelled us to
assume that the aether does not partake of the motion of
bodies moving through it, but preserves its state of undis-
turbed rest. ,; This means that our planet, our measuring in-
struments, and all other things on it, rush through the
aether, without in the slightest dragging it along with them ;
it slips through all bodies with infinitely greater ease
than the air between the planes of a flying machine. Since
the aether is nowhere in the world to take part in any mo-
tion of such bodies, a co-ordinate-system which is station-
ary in it fulfils the function of a system which is * absolutely
at rest'; and there would thus be meaning in the phrase
'absolute motion 7 in physics. This would indeed not be ab-
solute motion in the strictly philosophical sense, for we
should understand it as a motion relative to the aether, and
we could still ascribe to the aether and the cosmos embedded
in it any arbitrary motion or rest in * space 5 but the pos-
sibility is quite devoid of meaning, as we should no longer
be dealing with observable quantities. If there is an aether,
the system of reference which is fixed, i.e. at rest, in it must
be unique amongst all others. The proof of theVphysical
reality of the aether would necessarily, and could only, con-
sist in discovering this -unique system of reference. For
example, we might show that only with reference to this sys-
tem is the velocity of transmission of light the same in all
directions, viz. c, and that this velocity is different when
measured relatively to other bodies. After what has been
said, it is clear that this unique system, which is absolutely
at rest, could not be moving with the earth, for the earth
traverses about 30 kilometres per second in its course round
the sun. Our instruments thus move with this velocity rela-
tive to the aether (if we neglect the velocity of the solar sys-

The Special Principle of Relativity 9

tern, which would have to be added to this). This velocity
of 30 kilometres per second for a first approximation we
may suppose it to be uniform and rectilinear is indeed
small in comparison with. c\ but, with the help of a suffi-
ciently delicately arranged experiment, it should be -pos-
sible to measure a change of this order in the velocity of
light, without difficulty. Such an experiment was devised by
Michelson and Morley. It was carefully arranged in such a
way that even the hundredth part of the expected
amount could not have escaped detection if it had been
present.

But no trace of a change was to be found. The principle
of the experiment consisted In a ray of light being reflected
to and fro between two fixed mirrors placed opposite to one
another, the line joining the centres of the mirrors being in
one case parallel to the earth's motion, and in another per-
pendicular to it. An easy calculation shows that the time
taken by the light to traverse the space between the two
mirrors (once to and fro) is in the second case only
VI g z /c 2 of the value obtained in the first case, if q de-
notes the velocity of the earth relatively to the aether. The
absence of any change, in the initial interference fringes,
proves with great accuracy that the time taken is exactly
the same in both cases.

Hence the experiment teaches us that light also propa-
gates itself in reference to the earth with equal velocity
in all directions, and that we cannot detect ' absolute '
motion (i.e. motion with respect to the aether) by this
means.

The same result holds for other methods; for, besides
Michelson and Morley 5 s attempt, other experiments (for
instance, that of Trouton and Noble concerning the be-
haviour of a charged condenser) have led to the conclusion
that absolute motion (we are throughout these remarks

10 The Special Principle of Relativity

only speaking of uniform rectilinear motion) cannot be
established in any way.

This fact seemed new as far as optical and other electro-
magnetic experiments were concerned. It had long been
known, on the other hand, that it was impossible to detect
any absolute rectilinear uniform motion by means of me-
chanical experiments. This principle had been clearly stated
in Newtonian mechanics. It is a matter of everyday ex-
perience that all mechanical events take place in a system
which is moving uniformly and rectilinearly (e.g. in a mov-
ing ship or train) exactly in the same way as in a system
which is at rest relatively to the earth. But for the inevi-
table occurrence of jerks and rocking (which are non-wii-
form motions) an observer enclosed in a moving air-ship
or train could in no wise establish that his vehicle was
moving.

To this old theorem of mechanics there was now to be
added the corollary that electrodynamical experiments
(which include optical ones) give an observer no indication
as to whether he and his apparatus are at rest or moving
uniformly and rectilinearly.

In other words, experience teaches us that the following
theorem holds for all physics: 'All laws of physical nature
which have been formulated with reference to a definite co-
ordinate system are valid, in precisely the same form, when
referred to another co-ordinate system which is in uniform
rectilinear motion with respect to the first.' This empirical
law is called the < special theory of relativity', because it af-
firms the relativity of uniform translations only, i.e. of a
very special class of motions. All physical events take place
in any system in just the same way, whether the system is at
rest or whether it is moving uniformly and rectilinearly.
There is no absolute difference between these two states;
I may regard the second equally well as being that of rest.

The Special Principle of Relativity 11

The empirical fact of the validity of the special principle
of relativity, however, entirely contradicts the considera-
tions made above concerning the phenomenon of light-
transmission, as founded upon the aether-theory. For, ac-
cording to the latter, there should be one unique system
of reference (that which is fixed with reference to the
' aether ')? aM the value obtained for the velocity of light
should have been dependent upon the motion of the sys-
tem of reference used by the observer. Physicists were
confronted with the difficult problem of explaining and
disposing of this fundamental contradiction; this is the
point of divergence of the old and the new physics.

H. A. Lorentz and Fitzgerald removed the difficulty by
making a new physical hypothesis. They assumed that all
bodies, which are put in motion with reference to the aether,
suffer a contraction to VI cf/c 2 of their length in the
direction of their motion. Hereby the negative result of
Michelson and Morley's experiment would in fact be com-
pletely explained; for, if the line between the two mirrors
used for the purpose were to shorten of its own accord as
soon as it is turned so as to be in the direction of the earth's
motion, light would take less time to traverse it, and indeed,
the reductions would be exactly the amount given above
(viz. that by which the time of passage should have been
greater than in the position perpendicular to the earth 's
motion). The effect of the absolute motion would thus be
exactly counterbalanced by this Lorentz-Fitzgerald con-
traction ; and, by means of similar hypotheses, it would also
be possible to give a satisfactory account of Trouton and
Noble's condenser experiment and other experimental
facts.

We thus see that, according to the point of view just
described, there is actually to be an absolute motion in the
physical sense of the term (viz. with reference to a material

12 The Special Principle of Relativity

aether) ; but, since sncli a motion cannot be observed in any
way, special hypotheses are devised to explain why it
always eludes our perception. In other words ? according
to this view the principle of relativity does not hold, and the
physicist is obliged to explain, by means of special hy-
potheses, why all physical phenomena in spite of this take
place actually as if it did hold. An aether is really to exist,
although a unique body of reference of this Mnd nowhere
manifests itself.

In opposition to this view, modern physics, following
Einstein, asserts that, since experience teaches us that the
special principle of relativity actually holds, it is to be re-
garded as a real physical law; since, furthermore, the aether
as a substance obstinately evades all our attempts at observ-
ing it, and all phenomena occur as if it did not exist, the word
' aether' lacks physical meaning, and therefore aether does
not exist. If the principle of relativity and the non-exist-
ence of the aether cannot be brought to harmonize with our
previous arguments about the transmission of light, these
arguments must clearly be reconsidered and revised. It is
to Einstein that the credit falls of discovering that such a
revision is possible, viz. that these arguments are based on
assumptions concerning the measurement of space and time
which have not been tested, and which we only require to
discard in order to do away with the contradiction between
the principle of relativity and our notions about the trans-
mission of light.

Thus, if an event propagates itself, with respect to a co-
ordinate system K, in any direction with the velocity c, and
if a second system K l move relative to K in the same direc-
tion with the velocity q, the velocity of transmission of the
event as viewed from the system K 1 is of course only equal
to c q, if it is assumed that distances and times are meas-
ured in the two systems with the same measuring units. This

Tltie Special Principle of Relativity 13

assumption had hitherto been tacitly used as a basis.
Einstein showed that it is in no wise self-evident : that one
could with equal right (indeed with greater right, as the
results will show) put the value for the velocity of trans-
mission in both systems equal to c; and that the lengths of
distances and of times then have different values for differ-
ent systems of reference moving with reference to one an-
other. The length of a rod, the duration of an event, are not
absolute quantities, as was always assumed in physics be-
fore the advent of Einstein, but are dependent on the state
of motion of the co-ordinate system in which they are
measured. The methods which are at our disposal for
measuring distances and times yield different values in
systems which are in motion relatively to one another. We
shall now proceed to explain this more clearly.

For the purpose of ' measurement ', i.e. for the quantitative
comparison of lengths and times, we require measuring-
rods and clocks. Bigid bodies, the size of which we assume
to be independent of their position, serve as measuring-rods ;
the term clock need not necessarily be confined to the
familiar mechanical object, but may denote any physical con-
trivance which exactly repeats the same event periodically;
e.g. light-vibrations may serve as a clock (this was the case
in Michelson and Morley's experiment).

No essential difficulty arises in determining a moment
or the duration of an event, if a clock is at our disposal at
the place where the event is happening; for we need only
note the reading of the clock at the moment the event under
observation begins, and again at the moment it ceases. The
sole assumption we make is that the conception of the
1 simultaneity (time-coincidence) of two events occurring' at
the same place' (viz. the reading of the clock and the begin-
ning of the event) has an absolutely definite meaning. We
may make the assumption, although, we cannot define the

14 The Special Principle of Relativity

conception or express its content more clearly; it belongs to
those ultimate data, which become directly known to us as
an experience of our consciousness.

The position is different, however, when we are dealing
with two events which occur at different places. To compare
these events in point of time, we must erect a clock at each
place, and bring these two clocks into agreement with one
another, viz. regulate them so that they beat synchronously,
i.e. give the same reading at the 'same moment'. This
regulation, which is equivalent to establishing the conception
of simultaneity for different places, requires a special proc-
ess. We are obliged to resort to the following method. We
send a light-signal from the one clock placed at A (let us
say) to the second at B, and reflect it thence back to A. Sup-
pose that, from the moment of sending to that of receiving
the signal, the clock A has run on for two seconds, then this
is the time which the light has required to traverse the dis-
tance AB twice. Now since (according to our postulate)
light propagates itself in all directions with the same veloc-
ity c, it takes Justus long for the initial as for the return
journey, i.e. one second for each. If we now emit a light-
signal in A at precisely twelve o'clock, after having ar-
ranged with an observer in B to set his clock at one second
past 12 o'clock when he receives the signal, then we shall
rightly consider that we have solved the problem of syn-
chronizing the two clocks. If there are other clocks at other
places, and if we bring them all into agreement with the one
at A according to the method described for B, then they will
agree amongst themselves if compared by the same process.
Experience teaches us that the only time-data which do not
lead to contradictions are those which are got by using
signals which are independent of matter, i.e. are transmitted
with the same velocity through a vacuum. Electro-magnetic
waves travelling with the speed of light fulfil this condition.

The Special Principle of Relativity 15

If we were to use sound-signals In the air, for instance, the
direction of tlie wind would have to be taken into account.
The velocity of light c thus plays a unique part in Na-
ture.

Hitherto we have assumed that the clocks are at rest
relatively to one another and to a fixed body of reference K
(as the earth). We shall now suppose a system of reference
K 1 (e.g. a railway train travelling at an enormous rate)
moving relatively to K with the velocity q in the direction
of A to B. The clocks at different points in K 1 are to be
supposed regulated with one another in exactly the same
way as was just described for those in K. K 1 may for this
purpose be considered to be at rest equally well as K, when
its clocks were regulated. What happens when observers
in K and K* attempt to get into communication with one
another?

Suppose a clock -4 1 at rest in ? to be in immediate proxi-
mity to the clock A at rest in K, at precisely the moment
at which both clocks A and A ^ indicate 12; and suppose
a second clock B l at rest in -B? to be at the place B, whilst
the corresponding clock at rest in K at the same place
indicates 12. Asa observer on K will then say that A 1 coin-
cides with A at the same moment, i.e. simultaneously (at
exactly 12 o'clock) when B 1 coincides with B. At the
moment when the coincident clocks A and A^ both indicate
12, let a light-signal flash out from their common position.
The rays reach B when the clock at B indicates one second
past 12 ; but the clock B 1 , being on the moving body JBT 1 , has
moved away from B a distance q, and will have moved
slightly further away before it is reached by the light-signal.
This means that, for an observer at rest on K, the light takes
longer than one second to travel from A 1 to B 1 . It will now
be reflected at B\ and will arrive back at A 1 in less than
one second, since A*, according to the observer in K, moves

16 The Special Principle of Relativity

towards tlie light. This observer will therefore conclude that
the light takes longer to traverse the distance from A 1 to B 1
than that from B l to A 1 : since in the first case B 1 hastens
away from the light-ray, whereas in the second case A 1
goes to meet it. An observer in K 1 , however, judges other-
wise. Since he is at rest relatively to A 1 and J? 1 , the times
taken by the signal to travel from A 1 to B 1 , and thence back
from B 1 to A\ are exactly the same : for, with reference to
his system 5?, light propagates itself with equal velocity c
in both directions (according to the postulate we have
established on the basis of Michelson and Morley's re-
sult).

We thus arrive at the conclusion that two events, which
are of equal duration in the system J5T 1 , occupy different
lengths of time when measured from the system K . Both
systems accordingly use a different time-measure ; the con-
ception of duration has become relative, being dependent on
the system of reference, in which it is measured. The same
holds true, as immediately follows, of the conception of
simultaneity: two events, which, viewed from one system,
occur simultaneously, happen for an observer in another
system at different times. In our example, when A coin-
cides with A 1 in position, the two clocks at the common
point indicate the same time as the clock B when B coincides
with B 1 ; but the clock B\ belonging to the system K\
indicates a different time at this place. The former two co-
incidences are thus only simultaneous in K but not in the
system K 1 .

All this arises, as we see, as a necessary consequence of
the regulation of clocks, which was founded upon the
principle that light always transmits itself with constant
velocity: no other means of regulation is possible without
introducing arbitrary assumptions.

We also obtain different values for the lengths of bodies

The Special Principle of Relativity 17

taken along the direction of motion, if they are measured
from different systems. This is immediately evident from
the following. If I happen to be at rest in a system K, and
wish to measure the length of a rod AB which is moving
with reference to K in the direction of its own axis, I must
either note the time that the rod takes to move past a fixed
point in K, and multiply this time by the velocity of the rod
relative to K (by doing which we should find the length to be
dependent on the velocity, on account of the relativity of
duration) ; or I could proceed to mark on K at a definite
moment two points P and Q, which are occupied by the two
ends A and B respectively at that precise moment, and then
measure the length of PQ in K. Since simultaneity is
a relative conception, the coincidence of A with P, if I
make observations from a system moving with the rod, will
not be simultaneous with the coincidence of B with Q : but
at the time that A coincides with P, the point B will, for me,
be at a point Q 1 slightly removed from Q, and I shall regard
the distance PQ 1 as the true length of the rod. Calculation
shows that the length of a rod, which has a value a in
a system with reference to which it is at rest, assumes the
value a VI g*/>c* for a system which is moving relatively
to it with the velocity q. This is precisely the Lorentz-con-
traction. It no longer appears as a physical effect brought
about by the influence of ' absolute motion', as was the case
according to Lorentz and Fitzgerald, but is merely the result
of our methods of measuring length and times. The ques-
tion which is often put forward by the beginner, as to what
the 'reaP length of a rodas, and whether it i really ' contracts
on being moved, or whether the change in length is only an
apparent one is suggested by a misunderstanding. The
diverse lengths, which are measured in various systems
moving with uniform motion relatively to one another, all
'really* belong to the rod equally; for all such systems are

18 The Special Principle of Relativity

equivalent. No contradiction is contained in this, since
1 length' is only a relative conception.

The conceptions 'more slowly' and 'more quickly' (not
only 'slowly ' and * quickly ') are, according to the new theory,
relative. For, if an observer in K always compares his
clock with the one in JS?, which he just happens to be passing,
he will find that these clocks lag more and more behind his
own : he will hence declare the rate of the clocks in K 1 to be
slower than his own. Exactly the same, moreover, happens
to the observer in K 1 , if he compares his clock with the
successive clocks of K which he happens to encounter. He
will assert that the clocks fixed in his own system are going
at a faster rate ; and this indeed with just as much right as
the other had in affirming the contrary.

All these connected results can be most easily followed if
they are expressed mathematically; we can then grasp
them as a whole. For this purpose we only require to set
up the equations, which enable us to express the time and
place of an event, referred to one system by corresponding
quantities referred to the other system. If x, x* 9 x 3 are the
space-co-ordinates of an event happening at the time t in
the system K\ and if #\, x\, a?*, i 1 are the corresponding
quantities referred to K 1 ; then these equations of transfor-
mation (they are termed the 'Lorentz-transformation')
enable us to calculate the quantities x\, %\, o^ 3? t\ if o?i, x 29
o? 3 , t are given and vice versa. (For further details see
the references at the end of this book.)

Such are, in a few words, the main features of the kine-
matics of the special theory of relativity. Its great impor-
tance in physics is derived from the electro-dynamics and
mechanics which correspond to this type of kinematics.
But for our present purpose it is not necessary to go into

greater detail We shall only mention one extraordinary

result.

The Special Principle of Relativity 19

Whereas in the older physics the law of Conservation of
Energy and that of Conservation of Mass existed entirely
unrelated, it has been shown that the second law is no longer
strictly in agreement with the former, and must therefore be
abandoned. Theory leads to the following view. If a body
take up an amount of energy E (measured in a system which
is at rest with reference to that of the body), the body be-

E

haves as if its mass were increased by the amount . That

<?

is, we cannot say that each body has a constant factor w
which has the significance of a mass independent of its veloc-

E
ity. If, now, the quantity is to be regarded as an actual

c 2

increase of mass, Le. if energy has the property of inertia,
it is an obvious step not only to trace the increase of mass
back to an increase of energy but also to regard the inertial
mass m as being dependent upon a quantity of energy
E = me 2 contained by the body. This amount is very great
owing to the enormous value of c, the velocity of light. This
assumption is in very good agreement with the enormous
store of internal energy of the atom, as deduced from recent
researches. Physics, therefore, no longer recognizes both
of the above laws, but only that of the Conservation of
Energy. The Principle of Conservation of Mass, which has
hitherto been regarded as a distinct fundamental law of
natural science, has been traced back to the Principle of
Energy, and has been recognized as being only approxi-
mately true. It is found to be nearly true, inasmuch as all
increases of energy which are experimentally possible are
in general negligible compared with the enormous store of
internal energy me 2 , so that these changes of mass are
scarcely observable.
That which particularly interests us here is that the

20 The Special Principle of Relativity

theory of relativity entirely does away with the traditional
conceptions of space and time, and banishes i aether 5 as a
substance out of physics. We saw earlier that the existence
of such an aether implied in physical terms that a definite
co-ordinate system (that which is at rest relatively to the
aether) would have to be unique amongst all others, i.e.
with reference to this system physical laws would assume a
particular form. As our theory allows no such unique
system, and since, on the contrary, all systems which
have a uniform translation with regard to one another
are equivalent, the belief in a material aether is incom-
patible with the principle of relativity. We may no longer re-
gard light- waves as a change in the condition of a substance,
in which they are propagated with the velocity c m , for then
this substance would have to be at rest in all equivalent
systems, and that of course entails a contradiction. The
electromagnetic field is, on the contrary, to be regarded as
being independent and not requiring a * carrier'. Since we
are free to use words at pleasure, there is no objection to
using the word * aether' in future to represent the vacuum
(empty space) with its electromagnetic field, or as endowed
with the metrical properties which are to be discussed be-
low; we must be very cautious, however, not to picture it as
matter.

We thus see that, in addition to the conceptions of space
and time, that of substance is crystallized in a purified form
by the critical application of the special theory of relativity.
This process only reaches completion, however, in the
general theory of relativity. However great the revolution
wrought by the special theory may have seemed, the claim
that all motions without exception should be of a relative
character (Le. that only motions of bodies relatively to one
another are to enter into physical laws) brings about such
a strange world-picture and leads to such bold conclusions

The Special Principle of Relativity 21

that, in comparison with it, the reconstruction of conceptions
imposed upon us by the special theory of relativity seems-
modest and incomplete.

To gain an easy approach to the formidable structure of
ideas contained in the general theory of relativity, we shall
start afresh with quite elementary reflections and simple
questionings.

in

THE GEOMETRICAL EELATIVITY OF SPACE

THE most fundamental question which may be asked con-
cerning Space and Time is, to express it in familiar language
for the present: are Space and Time actually real?

From the earliest times an inconclusive controversy was
waged by the philosophers as to whether empty space, the
Hsvovy were real, or merely identical with nothingness. But
even at the present day not every one, be he scientist, philo-
sopher, or general reader, would straightway answer this
question by a simple negative or affirmative. No one, in-
deed, regards Space and Time as real in quite the
same sense as the chair on which I sit, or the air which
I breathe. I cannot deal with space as with material objects
or with energy, which I can transport from one place to
another, manipulate at will, buy and sell. Every one feels
that there is some difference between them; Space and Time
are, in some sense or other, less independent than the things
which exist in them; and philosophers have often emphasized
this lack of independence by stating that neither exists in
itself. We could not speak of Space if there were no
material bodies ; and the conception of Time would likewise
be devoid of meaning if no events or changes took place in
the world. But, even for the popular mind, Space and Time
are not merely nothmg ; for are there not great departments
of engineering which are wholly devoted to overcoming
them?

Of course the decision of this question depends upon
what is understood by 'Beality*. Now, even if this concep-

The Geometrical Relativity of Space 23

tion is difficult, perhaps even impossible, to define, yet tlie
physicist is in the happy position of being able to satisfy
himself with a definition which allows him to fix the limits
of his realm with absolute certainty. * Whatever can be
measured is real.' The physicist may use this sentence of
Planck's as a general criterion, and say that only that which
is measurable possesses indisputable reality, or, to define it
more carefully, physical objectivity.

Are Space and Time measurable? The answer seems
obvious. What would indeed be 'measurable if it were not
Space and Time? Do not our clocks and measuring-scales
serve just this purpose? Is there not even a special science
which is concerned with nothing else than with the measure-
ment of space, without reference to any bodies, viz. metrical
geometry?

But let us be cautious ! It is known that there is differ-
ence of opinion about the nature of geometrical objects
even if this were not the case, we have recently learnt to
look searchingly into the fundamental conceptions of the
sciences above all for concealed or unproved premises.. We
shall thus have to investigate whether the current view of
geometry, as a doctrine of the properties of space, is not
influenced by certain unjustified notions, from which it must
be released.* In fact, philosophic criticism has for some time
affirmed the necessity for doing so, and busied itself with
the task, and has thereby already developed ideas about the
relativity of all spatial relations. We may regard the space-
time-view of Einstein's theory as the logical shaping and
application of these ideas; a continuous path leads from
them to the theory, along which the meaning of the question
of the reality of Space and Time becomes ever clearer. We
shall use this road as a means of access to the new ideas.

Let us begin by reflecting on a simple imaginary experi-
ment, which almost every one who has thought about these

24 The Geometrical Relativity of Space

matters has performed mentally and which, is particularly
well described by Henri Poincare. Let us suppose that all
material bodies in the world increase enormously in size
over-night to a hundred times their original dimensions ; my
room, which is to-day six metres long, would to-morrow have
a length of 600 metres. I myself should be a Goliath 180
metres high, and should be inscribing letters a metre high on
paper with a pen 15 metres long; and similarly all other
dimensions of the universe are to be supposed altered to
a like degree, so that the new world, although a hundred
times increased, would still be geometrically similar to the
old one. '"What would my impressions be in the morning,'
Poincare asks, ' after this astonishing change?' And he
answers : *I should not observe the slightest difference. For
since, according to our assumption, all objects, including my
own body, all measuring-scales and instruments, have shared
in this hundredfold magnification, every means of detecting
this change would be wanting; I should call the length of
my room 6 metres as before, since my metre-scale would
divide into it six times, and so on. 5 What is still more im-
portant, this whole alteration would exist only for those who
erroneously argue that Space is absolute. Truth compels us
to say that, since space is relative, no change has taken place,
and that this is the reason why we were unable to notice
anything. Thus, the universe, which we imagined magnified
a hundredfold, is not only indistinguishable from the orig-
inal one ; it is simply the same universe. There is no mean-
ing in talking of a difference, because the absolute size of a
body is not 'real 7 .

The exposition of Poincare must be carried a little further
to be quite convincing. The fiction of a universal alteration
in the size of the world, or a part of it, is devoid of any ap-
preciable meaning from the very outset, unless definite as-
sumptions are made as to how the physical constants are to

The Geometrical Relativity of Space 25

behave in this deformation. For natural bodies have not
only a geometrical shape, but they also possess physical
properties, e. g. mass. If, after a hundredfold linear mag-
nification of the world, we substitute the former values for
the mass of the earth and the objects it contains, in New-
ton's attraction formula, we shall only get a 10,000th of the
previous value for the weight of a body on the earth's
surface, since this weight is inversely proportional to the
square of the distance from the earth's centre. Can we not
establish this change in weight, and thus arrive indirectly
at the absolute increase in size? We might think that this
would be possible by observations of a pendulum, for the
time of vibration (period) of a pendulum would be just
1,000 times slower on account of the decrease in weight and
increase in length. But would this retardation be observ-
able? Would it possess physical reality? The question is
again unanswerable, unless it is stated how the rotational
velocity of the earth is affected by the deformation ; for our
time-measures are based upon comparison with the former.

The attempt to observe the decrease in weight by means
of a spring-balance (say) would likewise be in vain; for
special assumptions about the behaviour of the coefficient of
elasticity of the spring would again be necessary in this
supposed magnification.

The fiction of a purely geometrical deformation of all
bodies is therefore entirely without significance; it has
no definite physical meaning. If one fine day we were
to observe a slowing down of all our pendulum-clocks,
we could not infer that the universe had been magnified
during the night, but the remarkable phenomenon could be
explained by means of other physical hypotheses. Inversely,
if I assert that all linear dimensions have been lengthened
a hundredfold since yesterday, no experience could prove
the contrary; I should only have to affirm at the same time

26 The Geometrical Relativity of Space

that while all masses had increased a hundredfold in value,
the rate of the earth's spin and of other events had, on the
other hand, decreased to a hundredth of their former value.
It is easily seen from the elementary formulae of Newtonian
Mechanics that, with these assumptions, exactly the same
numbers result from the calculations as before for all
observable quantities (at least as far as gravitational and
inertial effects are concerned). The change has thus no
physical meaning.

From reflections of this kind, which may be multiplied at
pleasure, and which are still based on Newtonian mechanics,
it is already clear that space-time considerations are in-
separably bound up with other physical quantities; and
if we abstract some from the rest, we must by careful com-
parison with experience try to discover in what sense a real
meaning is to be attached to the abstraction.

The reflections of Poincare, supplemented in the manner
indicated, teach us beyond doubt that we can imagine the
world transformed by means of far-reaching geometrical-
physical changes into a new one, which is completely
indistinguishable from the first, and which is completely
identical with it physically, so that the transformation
would not actually signify a real happening. "We started
by considering the case in which the imaginary transformed
world is geometrically similar to the original one; the
conclusions drawn are not in the slightest affected by
dropping this assumption. If we, for instance, assumed that
the dimensions of all objects are lengthened or shortened in
one direction only, say that of the earth's axis, we should
again not notice this transformation, although the shape
of bodies would have changed completely, spheres becoming
ellipsoids of rotation, cubes becoming parallelopideds, and
indeed perhaps very elongated ones. But if we wished to
establish, by means of a measuring-scale, the change in.

The Geometrical Relativity of Space 27

length, as compared with, the breadth, our effort would be
In vain ; since onr measuring-rod, when we turned it into the
direction of the earth's axis for the purpose of carrying
out a measurement, would, according to our supposition, be
correspondingly lengthened or shortened. Nor could we be-
come aware of the deformation directly by means of the
senses of sight or touch; for our own body has- likewise be-
come deformed, as well as our eye-balls, and also the wave-
surfaces of light. Again, we must conclude that there is
no 'real' distinction between the two worlds; the imagined
deformation is not ascertainable by any measurement, i.e.
has no physical objectivity. It is easily seen that the argu-
ments just presented may be generalized still further : we
can imagine with Poincare that the objects in the universe
are arbitrarily distorted in arbitrary directions, and the dis-
tortion need not be the same for all points, but may vary
from place to place. As long as we suppose that all meas-
uring instruments, including our own bodies with their
sense-organs, share in the local deformation for each place,
the whole transformation immediately becomes unascer-
tainable; it does not ' really 7 exist for the physicist

IV

THE MATHEMATICAL FOEMTJLATION OF
SPATIAL EELATIVITY

IJST mathematical phraseology we can express this result
by saying : two worlds, which can be transformed into one
another by a perfectly arbitrary (but continuous and one-to-
one) point-transformation, are, with respect to their phys-
ical reality, identical. That is : if the nniverse is deformed in
any way, so that the points of all physical bodies are dis-
placed to new positions, then (taking account of the above
supplementary considerations), no measurable, no 'real'
change has happened at all, if the co-ordinates of a physical
point in the new position are any arbitrary functions what-
soever of the co-ordinates of its old position. Of course,
it will have to be postulated that the points of the
bodies retain their connexion, and that points which were
neighbouring before the deformation remain so after it
(i.e. these functions must be continuous) ; and, moreover,
to every point of the original world only one point of
the new world must correspond, and vice versa (i.e. these
functions must be one-valued).

It is easy to picture the relations described by imagining
space to be divided by three families of planes, respectively
parallel to the co-ordinate planes, into a number of little
cubes. Those points of the world, which lie on such a plane
(e.g. the ceiling of a room) will, after the deformation, form
a more or less bent surface. The second world will thus be
divided by the system of these bent planes into eight-

28

The Mathematical Formulation of Spatial Relativity 29

cornered cells, which will in general be different in size
and form. But in this world, we should, just as before,
denominate these surfaces ' planes' and their curves of inter--
section ' straight lines', and the cells * cubes'; for every
mea^is of proving that they are not 6 really' so would
be lacking. If we suppose the planes numbered in order,
then every physical point of the deformed world is defined
by three numbers, namely the numbers of the three surfaces
which intersect at it; we can thus use these numbers as
co-ordinates of this point, and shall fittingly call them
* Gaussian co-ordinates', since they signify the same for
three-dimensional configurations as the co-ordinates which,
Gauss in his time introduced for the examination of two-
dimensional configurations (surfaces). He supposed two
intersecting families of curves to be drawn on any arbitrar-
ily curved surface in such a way as to lie entirely on the sur-
face. Each surface-point is then defined by specifying the
two curves (one member from each family) which pass
through the point. It is now evident that with these assump-
tions the bounding surfaces of bodies, the path of light-
rays, all motions and all natural laws in the deformed world,
expressed in these new co-ordinates, will be represented by
identically the same equations as the corresponding objects
and events of the original world, referred to ordinary Car-
tesian co-ordinates, provided that the numbering of the sur-
faces is carried out correctly. A difference between the two
worlds exists, -as we have said, only so long a)S one erro-
neously supposes that planes and lines can be defined in
space at all without reference to bodies in it, as if it were
endowed with * absolute' properties.

But, if we regard the old co-ordinates, i.e. the system of
perpendicularly intersecting planes, from the point of view
of the new universe, these planes will now reciprocally
seem to be an entirely curved and distorted system; and geo-

30 The Mathematical Formulation of Spatial Relativity

metrical forms and physical laws, when referred to this sys-
tem, assume an entirely new appearance. Thus, instead of
saying that I deform the world in a certain way, I can
equally well say that I am describing the unchanged world
by means of new co-ordinates, the plane-system of which is
deformed in some definite way as compared with the first.
Both processes are truly the same ; and these imaginary de-
formations would not signify any real alteration of the
world, but merely a reference to other co-ordinates.

We may therefore also regard the world in which we
live as the distorted one, and say that the surfaces of
bodies (e.g. the ceiling of a room), which we call planes, are
not 'really' such; our straight lines (light-rays) are 'in
reality' curved lines, &c. We could, without any contra-
diction manifesting itself, assume that a cube which is taken
into another room alters its shape and size considerably on
the way ; we should not be aware of the change, because we
ourselves, with all measuring instruments and the whole
surroundings, suffer analogous changes ; certain curved lines
would have to be considered as the 'true' straight lines.
The angles of our cubes, which we call right angles, would,
'in reality 5 , not be so yet we could not establish this : since
the measure by means of which we have determined the
arms of the angles would correspondingly change in length,
when we turned it round to measure the circular arc belong-
ing to the angle. The sum of the angles of our square would
'in reality' not amount to four right angles in short, it
would be as if we used a geometry other than Euclidean.
The whole assumption would be tantamount to maintaining
that certain surfaces and lines, that appear curved to us,
are really 'true' planes and straight lines, and that we
should have to use them as co-ordinates.

Why do we not actually suppose anything of the sort,
although it would be theoretically possible, and although all

The Mathematical Formulation of Spatial Relativity 31

our observations could be explained by this means? Simply
because this explanation could be given only in a very
complicated way, viz. by assuming extremely intricate
physical laws. The shape of a body would be dependent
upon its position; it would, if sufficiently far removed from
the influence of external forces, describe a curved line, &c. ;
in a word, we should arrive at a very involved system of
physics, and most important of all it would be quite arbi-
trary ; for there would be an unlimited number of similarly
complicated systems of physics, which would all serve
equally well for describing Experience. Compared with
these, the usual system, which applies Euclidean geometry,
distinguishes itself as the simplest, as far as can be judged
up to the present. The lines which we call ' straight' play a
special role in physics ; they are, as Poincare expresses it,
more ' important ' than other lines. A co-ordinate system
founded on these lines therefore leads to the simplest for-
mulae for physical laws.

THE INSEPARABILITY OF GEOMETRY AND
PHYSICS IN EXPERIENCE

THE reasons for preferring the usual system of geometry
and physics to all other possible ones, and for considering it
to be the only 'true' one, are exactly the same as those
which make the Copernican view of the world superior to
that of Ptolemy; the former leads to a much simpler system
of celestial mechanics. The formulation of the laws of plane-
tary motions become excessively complicated, if we refer
them, as Ptolemy did, to a co-ordinate system rigidly at-
tached to the earth; on the other hand, the process becomes
quite simple, if a co-ordinate system which is at rest with
respect to the fixed stars be chosen.

We thus see that experience in no wise compels us to make
use of an absolute geometry, e.g. that of Euclid, for the
physical description of nature. It teaches us only what
geometry we must use, if we wish to arrive at the simplest
formulae to express the laws of physics. From this it
immediately follows that there is no meaning in talking of
an absolute geometry of * space', omitting all reference to
physics and the behaviour of physical bodies; for, since
experience leads us to choose only a certain geometry, in
that it shows us in what way the behaviour of bodies can be
described most simply in mathematical language, it is mean-
ingless to attempt to assign a distinctive position to any one
geometry, as long as we leave material bodies out of account.
Poincare has expressed this tersely in the words : ' Space

32

Inseparability of Geometry <md Physics m Experience 33

itself is amorphous; only the things in it give it a form.'
I shall just recall a few remarks of Helmholtz, in which
he expresses the same truth. At the conclusion of his
lecture on the Origin and Significance of the Axioms of
Geometry, he says: 'If for some particular reason we were
to find it expedient, we could quite logically consider the
space in which we live to be like the apparent space as
pictured in a convex mirror, wherein lines converge and the
background is contracted; or, we could take a limited spheri-
cal portion of our space, beyond the boundaries of which our
perceptions do not extend, and regard it as boundless
pseudo-spherical space. We should, in that case, have to
ascribe to bodies which appear rigid to us, and to our own
bodies at the same time, only the corresponding extensions
and contractions ; and we should, of course, have to alter our
system of mechanical principles entirely. For even the
simple theorem that every point which is in motion and is not
acted on by any forces continues to move in a straight line
with invariable velocity, no longer holds true for the world
which is represented in a convex mirror. . . . Geometrical
axioms are in no way confined to relations in space alone,
but also make assertions about the mechanical behaviour of
our most rigid bodies when in motion/ !

Since the time of Riemann and Helmholtz we have been ac-
customed to talk of plane, spherical, pseudo-spherical and
other spaces, and discriminate from our observations to
which of these classes our 'reaP space belongs. We now un-
derstand how to interpret this : viz. not as if one of these can
be predicated of space, without taking account of objects in
it ; but in the sense that experience teaches us only whether it
is more practical to use Euclidean or non-Euclidean geom-
etry for the physical description of nature. Eiemann him-
self, and likewise Helmholtz, was quite clear about the ques-
tion; but the results of faoth these investigators have often

34 Inseparability of Geometry and Physics in Experience

been misinterpreted, so that they have occasionally even
been used to strengthen the belief that absolute space has a
particular form of its own ascertainable from experience.
We must be on our guard against assuming that space has
any ' physical reality' in this sense. It is well known that
Gauss tried to measure directly, by means of theodolites,
whether the sum of the angles of a very large triangle
amount to two right angles or not. That is, he measured the
angles which three light-rays, emitted from three fixed
points (The Brocken, Hoher Hagen, and Inselberg), made
with each other. Supposing that a deviation from two right
angles had manifested itself, we could either regard the
light-rays as curved and still use Euclidean geometry, or
we could still call the path of a light-ray straight, but we
should then have to introduce a non-Euclidean geometry.
It is therefore not correct to say that experience could ever
prove space to be ' non-Euclidean in structure 7 , i.e. could
ever compel us to adopt the second of these alternatives. On
the other hand, Poincare also errs when he somewhere ex-
presses the opinion that the physicist would actually always
choose the first assumption. For no one was able to predict
whether it might not some time be necessary to depart from
Euclidean measure-determinations in order to be able to de-
scribe the physical behaviour of bodies most simply.

All that could be affirmed at that time was that we should
never find occasion to depart from Euclidean geometry to
any considerable degree, since otherwise our observations,
particularly in astronomy, would long ago have called our
attention to this fact. Hitherto, however, by using
Euclidean geometry as a foundation, we have admirably
succeeded in arriving at simple physical principles. From
this we may conclude that it will always be suited for at least
an approximate description of physical events. If, therefore,
to attain simplicity of expression, it should prove convenient

Inseparability of Geometry and Physics in Experience 35

to give up Euclidean measure-determinations In physics,
such resulting deviations could only be slight, and would
show differences only in regions on the outskirts of our field
of observation. The essential significance of these devia-
tions, whether great or small, naturally remains the same.
This case, hitherto only a theoretical possibility, has now
presented itself. Einstein shows that non-Euclidean rela-
tions must actually be used in representing spatial condi-
tions in physics so that it may be possible to maintain the
extraordinary simplification of the principles underlying our
view of physical nature, as embodied in the general theory
of relativity. We shall return to this point presently. Mean-
while, we shall accept the result that space itself in no wise
has a form of its own ; it is neither Euclidean nor non-Euclid-
ean in constitution, just as it is not a peculiarity of distance
to be measured in kilometres and not in miles. In the same
way as a distance only acquires a definite length when we
have chosen a particular measure as unit, and in addition set
out the mode of measurement, so a definite geometry can be
applied to physical reality only when a definite method has
been fixed upon, according to which spatial conditions are to
be abstracted from physical conditions. Every measurement
of spatial distances, when reduced to the essentials, is per-
formed by placing one body against another ; if such a com-
parison between two bodies is to become a measurement,
it must be interpreted by taking due account of certain
principles (e.g. one must assume that certain bodies are
to be regarded as rigid, i.e. endure a translation without
change of form) . Precisely similar reflections may be made
mutatis mutandis for time. Experience cannot compel us
to found our description of physical nature upon a definite
measure and rate of time ; we choose just that measure and
rate which enable us to formulate physical laws most simply.
All time-determinations are just as indissolubly associated

36 Inseparability of Geometry and Physics in Experience

with physical occurrences as spatial ones are with physical
bodies. Quantitative observations of any physical occur-
rence, such as e.g. the propagation of light from one point
to another, imply that readings must be taken from a clock,
and thus assume a method according to which clocks in
different localities are to be regulated with one another.
Without this means, the conceptions of simultaneity and
equal duration have no definite meaning. These are matters
to which we called attention earlier, when we were discussing
the special theory of relativity. All time-measurements are
undertaken by comparing two events, and if they are to have
the significance of a true measurement, some convention or
principle must be assumed, the choice of which will again
be determined by the endeavour to obtain physical laws in
the simplest form.

We thus see: Time and Space can be dissociated from
physical things and events only in abstraction, i.e. mentally.
The combination or oneness of space, time, and things is
alone reality; each by itself is an abstraction. Whenever
we make an abstraction, we must always ask whether it
has a physical meaning, i.e. whether the products of ab-
straction are actually independent of one another.

VI

THE RELATIVITY OF MOTIONS AND ITS
CONNEXION WITH INERTIA AND GRAVITATION

IF one had not lost sight of this last truth, the celebrated
controversy, whch was always being renewed, about so-
called absolute motion would from the very outset have
assumed a different aspect. The conception of motion has,
in the first place, a real meaning only in dynamics, as the
change of position of material bodies with time; so-called
pure kinematics (known as 'phoronomy' in Kant's time)
arises out of dynamics by abstracting from mass, and is thus
the time-change of the position of mere mathematical points.
How far this product of abstraction may serve for describing
physical nature can be decided only by experience. Before
the time of Einstein, the opponents of absolute motion
(e.g. Mach) always argued thus: Every deterfnination of
position, being only defined for a definite system of refer-
ence, is, as regards its conception, relative, and therefore
also every change of position. Hence only relative motion
exists, ie. there can be no unique system of reference;
for, since the conception of rest is only relative, I must
be able to regard every system of reference as being at
rest. This method of proof, however, overlooks the fact that
the definition of motion as being merely change of position
applies to motion only in the kinematical sense. For real
motions, i.e. for mechanics or dynamics, this conclusion
need not be regarded as final; experience must prove
whether it is justified. From the purely kinematical point

37

38 Relativity of Motions and its Connexion

of view, it is, of course, the same to say that the earth
rotates as that the stellar heavens are rotating around
the earth. It does not follow, however, that both state-
ments are indistinguishable dynamically. Newton, as is
known, assumed the contrary. He believed apparently in
perfect agreement with experience that a rotating body
could be distinguished from one at rest by the appearance
of centrifugal forces (with resultant flattening) ; and abso-
lute rest (leaving out of account any motion of uniform
translation) would be defined by the absence of centrifugal
forces. In realizable experience, every accelerated change
of position is accompanied by the appearance of inertial re-
sistances (e.g. centrifugal forces) ; and it is quite arbitrary
to declare one of these factors, which both belong equally to
physical motion, and are only separable in abstraction, to be
the cause of the other, viz. to regard the inertial resistances
as the effect of the acceleration. It cannot therefore be
proved out of the mere conception of motion (as Mach en-
deavoured to do) that there can be no unique system of ref-
erence, i.e. that there can be no absolute motion; the de-
cision can only be left to observation.

Newton certainly erred in believing that observation had
already decided this question, viz. in the sense that two uni-
form rectilinear motions were in fact relative (i.e. that the
laws of dynamics are exactly the same for two systems of
reference which are moving uniformly and rectilinearly
with regard to one another), but that this was not true for
accelerated motions (e.g. rotations). Accelerations, he
thought, were of an absolute nature ; certain systems of ref-
erence were unique in that the Law of Inertia held for them
alone. They were therefore called Inertial Systems. Ac-
cording to Newton, an Inertial System would thus be de-
fined and recognizable as one in which a body, upon which
no forces act, would move uniformly and rectilinearly (or

with Inertia and Gravitation 39

remain at rest) ; and consequently centrifugal forces (or
flattening) would only fail to manifest themselves in or on a
body if .t were not rotating with reference to the inertial
system. Newton used these views as a foundation for me-
chanics unjustifiably ; for actually they are rot sufficiently
founded on experience. No j>bs<^

wj^jdij^ and no experience has yet

proved whether a body which is at rest in an inertial system
might not be subject to centrifugal forces if an extraordi-
narily great mass were to rotate near it, i.e. whether these
forces are not, after all, only peculiarities of relative
rotation.

The state of affairs was in fact as follows* On the one
hand, the experiences so far known did not suffice to prove
the correctness of Newton's assumption that absolute ac-
celerations existed (i.e. unique systems of reference) ; on
the other hand, the general arguments in favour of the rela-
tivity of all accelerations, e.g. Mach's, were not, as we have
just shown, conclusive. From the standpoint of actual ex-
perience, both points of view had for the time being to be
considered admissible. But, regarded philosophically, the
standpoint which denied the existence of unique systems of
reference, thus affirming all motions to be relative, is very
attractive, and possesses great advantages over the New-
tonian view; for, if it were realizable, it would signify an
extraordinary simplification of our picture of the world.
It would be exceedingly satisfactory to be able to say that
not only uniform, but indeed all, motions are relative. The
Mnematical and dynamical conception of motion would then
become identical in essence. To determine the character of
motion, purely Mnematical observations would suffice. It
would not be necessary to add observations about centrifu-

i Mach and Pearson called particular attention to this. Karl Pearson,
Grammar of Science, Chap. VIII, 4.

40 Relativity of Motions and its Comiexion

gal forces, as It was for the Newtonian view. A system of
mechanics built iip on relative motions would thus result in
a much more compact and complete view of the world than
that of Newton. It would not indeed (as was apparently
the opinion of Mach) be proved to be the only correct view
of the universe; but (as Einstein points out) it would
recommend itself from the very outset by its imposing sim-
plicity and finish. 1

Up to the time of Einstein, however, such a world-view,
i.e. the idea, of a system of mechanics founded on relative
motions, had been only a desire, an alluring goal; such a
system of mechanics had never been enunciated, nor had a
possible way to it even been pointed out. There was no
means of knowing whether, and under what conditions, it
was possible at all or compatible with empirical facts. In-
deed, science seemed to be constrained to develop in the
contrary direction; for, whereas in classical mechanics all
systems moving uniformly and rectilinearly with respect
to one inertial system were likewise inertial systems (so
that at least all uniform motions of translation preserved

i Einstein adds tliat Newton's mechanics only seemingly satisfies the
demands of causality, e.g. in the case of bodies which are rotating and suffer
a flattening. But this mode of expression does not appear to me to be quite
free from objection. We need not look upon the Newtonian doctrine as
making Galilean space, which is of course not an observable thing, the cause
of centrifugal forces; but we can also consider the expression 'absolute
space* to be a paraphrase of the mere fact that these forces exist, They
would then simply be immediate data; and the question why they arise in
certain bodies and are wanting in others would be on the same level with
the question why a body is present at one place in the world and not at
another. Absolute rotation need not be regarded as the cause of the flatten-
ing, but we can say that the former is defined by the latter. In this way
I believe that Newton's dynamics is quite in order as regards the principle
of causality. It would be easy to defend it against the objection that purely
fictitious causes are introduced into it, although Newton's own formulation
was incorrect.

with Inertia and Gravitation 41

the character of being relative ), in the case of electromag-
netic and optical phenomena, even this no longer seemed to
hold; in Lorentz's Electrodynamics there was only one
unique system of reference (the one which is < at rest in
the aether '). Only after Einstein had succeeded in ex-
tending the special principle of relativity, which was valid
in classical mechanics, to all physical phenomena, could the
idea of the entirely general relativity of any arbitrary
motions again be taken up on the ground thus prepared;
and again it was in the hands of Einstein that it bore
fruit. He transplanted it as it were from regions of phil-
osophy to those of physics, and thereby brought it within
the range of scientific research.

Although the philosophical arguments were so powerful
in themselves, Einstein gave them additional weight by add-
ing to them the physical argument that all motions were
most probably endowed with a relative character. This
physical argument is built on the equality of inertial and
gravitational mass. We can see it more clearly in the fol-
lowing way. If we assume all accelerations to be relative,
then all centrifugal forces, or other inertial resistances
which we observe, must depend on motion relative to other
bodies ; we must therefore seek the cause of these inertial
resistances in the presence of those other bodies. If, for
example, there were no other body present in the heavens
except the earth, we could not speak of a rotation of the
earth, and the earth could not be flattened at the poles. The
centrifugal forces, as a consequence of which the earth's
flattening comes about, must thus owe their existence to the
action on the earth of the heavenly bodies. Now, as a matter
of fact, classical mechanics is acquainted with an action
which all bodies exert on one another, viz. Gravitation.
Does experience lend any support to the suggestion that this
gravitational influence might be made answerable for the

42 Relativity of Motions and its Connexion

Inertial effects! This support is actually to be found, and
is very remarkable ; it consists in the circumstance that one
and the same constant plays the determining role for both
iuertial and gravitational effects, viz, the quantity known as
mass. If, for instance, a body describes a circular path
relatively to an inertial system, the necessary central force
Is, according to classical mechanics, proportional to a factor
m which is a characteristic for the body; but if the body is
attracted by another body (e.g. the earth) in virtue of gravi-
tation, the force acting on it (e.g. its weight) is proportional
to this same factor m. It is on account of this that, at the
same place in a gravitational field, all bodies without excep-
tion suffer the same acceleration; for the mass of a body
eliminates itself, since it occurs as a factor of proportion-
ality both in the expression for the inertial resistance and
in that for the attraction.

Einstein has made the connexion between gravitation and
inertia extraordinarily clear by the following reflection. If
a physicist, enclosed in a box somewhere out in space, were
to observe that all objects left to themselves in the box ac-
quired a certain acceleration, e.g. fell to the bottom with
constant acceleration, he could interpret this phenomenon,
in two ways : in the first place, he could assume that his
box was resting on the surface of some heavenly body, ancf
he would then ascribe the falling of the objects to the gravi-
tational influence of the heavenly body; or, he could assume
instead that the box was moving 'upwards' with constant
acceleration, and then the behaviour of the 'falling' bodies
would be explained by their inertia. Both explanations are
equally possible, and the enclosed physicist would have no
means of discriminating between them. If we now assume
that all accelerations are relative, and that a means of dis-
crimination is essentially wanting, this may be generalized.
We may consider the observed acceleration of any body left

with Inertia and Gravitation 43

to Itself, at any point in the universe, to be due to the effect
either of inertia or of gravitation, i.e. we may either say
'the system of reference, from which I am observing this
event, is accelerated' or 'the event is taking place in a
gravitational field'. We shall follow Einstein, and call
the statement that both interpretations are equally jus-
tifiable the Principle of Equivalence. It is founded, as we
have seen, on the identity of inertial and gravitational
mass.

The circumstance of the identity of these two factors is
very striking, and when we get to realize its full import, it
seems astonishing that it did not occur to any one before
Einstein to bring gravitation and inertia into closer con-
nexion with one another. If something analogous had been
observed in another branch of physics (e.g. if an effect had
been found which was proportional to the quantity of elec-
tricity associated with a body) we should immediately have
brought it into relationship with the remaining electrical
phenomena ; we should have regarded electrical forces, and
the supposed new effect, as different manifestations of one
and the same governing principle. In classical mechanics,
however, not the slightest connexion was introduced "be-
tween gravitational and inertial phenomena; they were not
comprised under one sole principle, but existed side by side
totally unrelated. The fact that one and the same factor
mass played a similar part in each seemed mere chance to
Newton. Is it really only chance? This seems improbable
in the highest degree.

The identity of inertial and gravitational mass is thus the
real ground of experience which gives us the right to assume
or assert that the inertial effects which we observe in bodies
are to be traced back to the influence which is exerted
upon them by other bodies. (This influence is, of course,
in accordance with modern views, to be conceived not as

44 Relativity of Motions and its Connexion

an action at a distance, but as being transmitted through.
a field.)

The above assertion (of identity) implies the postulate of
an unlimited relativity of motions ; for, since all phenomena
are to depend only on the mutual position and motion of
bodies, reference to any particular co-ordinate system no
longer occurs. The expression of physical laws, with refer-
ence to a co-ordinate system attached to any arbitrary body
(e.g. the sun), must be the same as with reference to one
attached to any other body whatsoever (e.g. a merry-go-
round on the earth) ; we should be able to look upon both
with equal right as being 'at rest'. The laws of Newtonian
mechanics had to be referred to a perfectly definite system
(an Inertial System) which was quite independent of the
mutual position of bodies ; for the Law of Inertia held for
these only. In the new mechanics, on the other hand, which
has to look upon inertial and gravitational forces as the ex-
pression of a single fundamental law, not only gravitational
phenomena, but also inertial phenomena, are to depend ex-
clusively on the position and motion of bodies relative to
one another. The expression for this fundamental law
must accordingly be such that no co-ordinate system plays
a unique part compared with the others, but that all remain
valid for any arbitrary system. It is evident that the old
Newtonian dynamics can signify only a first approximation
to the new mechanics ; for the latter demands, in contradis-
tinction to the former, that centrifugal accelerations, for
example, must be induced in a body if large masses rotate
around it; and the contradiction between the new theory
and classical mechanics does not come into evidence in this
particular case, merely because these forces are so small,
even for the greatest available masses in the experiment,
that they escape our observation,

Einstein has actually succeeded in establishing a funda-

with Inertia and Gravitation 45

mental law which, comprises inertial and gravitational
phenomena alike. We are now better prepared to follow
the line of argument by which Einstein arrived at this

result

VII

THE GENERAL POSTULATE OF EELATIVITY
AND THE MEASURE-DETERMINATIONS OF
THE SPACE-TIME CONTINUUM

THE idea of relativity has only been applied in the preced-
ing pages to physical thought in so far as it beaxs on mo-
tions. If these are really relative without exception, any co-
ordinate systems moving arbitrarily with reference to one
another are equivalent, and space loses its objectivity, in so
far as it is not possible to define any motions or accelera-
tions with respect to it. Yet it still preserves a certain ob-
jectivity, so long as we tacitly imagine it to be provided with
absolutely definite metrical properties. In the older physics
every process of measurement was unhesitatingly founded
on the notion of a rigid rod, which preserved the same
length at all times, no matter what its position and sur-
roundings might be ; and proceeding from this, all measure-
ments were determined according to the rules of Euclidean
geometry. This process was not changed in any way in the
new physics which is based on the special theory of relativ-
ity, provided that the condition was fulfilled that the meas-
urements were all carried out within the same co-ordinate
system, by means of a rod respectively arrest with regard
to each system in question. In this way space was still en-
dowed with the independent property, as it were, of being
' Euclidean* in * structure', since the results of these meas-
urcf-determiations were regarded as being entirely inde-

46 .

Measure-Determinations of the Space-time Contimmm 47

pendent of the physical conditions prevailing in space, e.g.
of the distribution of bodies and their gravitational fields.
Now we have seen that it is always possible to fix the posi-
tion- and magnitude-relations of bodies and events accord-
ing to the ordinary Euclidean rule, e.g. by means of Car-
tesian co-ordinates, so long as the laws of physics have been
correspondingly formulated. But we are subject to a limi-
tation : we had set out to determine them, if possible, in such
a manner that the general postulate of relativity would be
fulfilled. Now it by no means follows that we shall succeed
in fulfilling this condition if we use Euclidean geometry.
We have to take into account the possibility that this may
not be so. Just in the same way as we found that the postu-
late of special relativity could be satisfied only if the con-
ception of time which had previously prevailed in physics
was modified, it is likewise quite possible that the general-
ized principle of relativity might compel us to depart from
ordinary Euclidean geometry.

Einstein, by considering a very simple example, comes to
the conclusion that we are actually compelled to make this
departure. If we fix our attention upon two rotating co-
ordinate systems, and assume that in one of them, say K,
the positional relations of the bodies at rest (in K) can be
determined by means of Euclidean geometry (at least in
a certain domain of K), then this is certainly not possible
for the second system K 1 . This is easily seen as follows.
Let the origin of co-ordinates and the -axis of the two
systems coincide, and let the one system rotate relatively to
the other about this common axis. We shall suppose a
circle described about the origin as centre in the ^-^-plane
of jST; for reasons of symmetry this is also a circle in K 1 .
If Euclidean geometry holds in K, then the ratio of the cir-
cumference to the diameter is in this system n ; but if we
determine this same ratio by means of measurements with

4:8 General Postulate of Relativity ancl

rods which are at rest in K l , we obtain a value greater than
a. For, if we regard this process of measurement from the
system K, the measuring-rod has the same length in meas-
uring the diameter as if it were at rest in K: whereas in
measuring the circumference it is shortened, owing to the
Lorentz-Fitzgerald contraction ; the ratio of these numbers
thus becomes greater than n and the geometry which holds
in K 1 is not Euclidean. Now, the centrifugal forces with
respect to -5?, which are due to inertial effects (on the old
theory), may, however, be regarded at every point, accord-
ing to the Principle of Equivalence, as gravitational effects.
From this it can be seen that the existence of a gravita-
tional field demands that non-Euclidean measure-determina-
tions be used. Strictly speaking, there is, however, no finite
domain which is entirely free from gravitational effects ; so
that, if we wish to maintain the postulate of general rela-
tivity, we must refrain from describing metrical and posi-
tional relations of bodies by Euclidean methods. This does
not mean that in place of Euclidean geometry we are now to
use some other definite geometry, such as that of Lobats-
chewsky or Eiemann, for the whole of space (cf. Section IX
below) , but that all types of measure-determination are to
be used : in general, a different sort at every place. Which
it is to be, depends upon the gravitational field at the place.
There is not the slightest difficulty in thinking of space in
this way ; for we fully convinced ourselves above that it is
only the things in space which give it a definite structure
or constitution ; and now we have only to assign this role
as we shall immediately see to gravitational masses or
their gravitational fields respectively. It becomes impossible
to define and measure lengths and times (as may likewise
easily be shown) in a gravitational field in the simple man-
ner described in Section II, by means of clocks and measur-
ing-rods. Since gravitational fields are nowhere absent, the

Measure-Determinations of tJie Space-time Continuum 49

special theory of relativity nowhere holds accurately; the
velocity of light, for instance, is never in truth absolutely
constant. It would, however, he quite wrong to say that the
special theory had been proved to be false, and had been
overthrown by the general theory. It has really only been
assimilated in the latter. It represents the special case
into which the general theory resolves when gravitational
effects become negligible.

It follows, then, from the general theory of relativity that
it is quite impossible to ascribe any properties to space with-
out taking into account the things in it. The relativization
of space has thus been carried out completely in physics, as
was shown by the above general considerations to be the
most likely result. Space and Time are never objects of
measurement in themselves; only conjointly do they consti-
tute a four-dimensional scheme, into which we arrange
physical objects and processes by the aid of our observa-
tions and measurements. "We choose this scheme in such a
way that the resultant system of physics assumes as simple
a form as possible. (We are free to choose, since we are
dealing with a product of abstraction.)

How is this arrangement to be fitted into the scheme?
What is it that we really observe and measure?

It is easily seen that the possibility of observing accur-
ately depends upon noting identically the same physical
points at various times and in various places ; and that all
measuring reduces itself to establishing that two such
points, upon which we have fixed, coincide at the same place
and at the same time. A length is measured by applying a
unit measure to a body, and observing the coincidence of its
ends with definite points on the body. With our apparatus
the measurement of all physical quantities resolves finally
into the measurement of a length. The adjustment and
reading of all measuring instruments of whatsoever va-

50 General Postulate of Relativity and

riety whether they he provided with pointers or scales,
angular-diversions, water-levels, mercury columns, or any
other means are always accomplished by observing the
space-time-coincidence of two or more points. This is also
true above all of apparatus used to measure time, familiarly
termed clocks. Such coincidences are therefore, strictly
speaMng, alone capable of being observed ; and the whole of
physics may be regarded as a quintessence of laws, accord-
ing to which the occurrence of these space-time-coincidences
takes place. Everything else in our world-picture which can
not be reduced to such coincidences is devoid of physical ob-
jectivity, and may just as well be replaced by something
else. All world pictures which lead to the same laws for
these point-coincidences are, from the point of view of
physics, in every way equivalent. We saw earlier that it
signifies no observable, physically real, change at all, if we
imagine the whole world deformed in any arbitrary manner,
provided that after the deformation the co-ordinates of
every physical point are continuous, single-valued, but
otherwise quite arbitrary, functions of its co-ordinates be-
fore the deformation. Now, such a point-transformation
actually leaves all spatial coincidences totally unaffected;
they are not changed by the distortion, however much all
distances and positions may be altered by them. For, if two
points A and B, which coincide before the deformation (i.e.
are infinitely near one another), are at a point the co-ordi-
nates of which are #i, x 2 , %*, and if A arrives at the point
&i', o/ 2 ', &* 9 as a result of the deformation, then, since by
hypothesis the #' ? s are continuous single-valued functions
of the #'s, B must also have the co-ordinates #/, # 2 ' ? # 3 ',
after the deformation i.e. must be at the same point (or
infinitely near) A. Consequently, all coincidences remain
undisturbed by the deformation.
Earlier, we had only, for the sake of clearness, investi-

Measure-Determinations of tJie Space-time Contmuum 51

gated these effects in the case of space ; we may now gen-
eralize by adding the time t as a fourth co-ordinate. Better
still, we may choose as our fourth co-ordinate the product ct
(= # 4 ) in which c denotes the velocity of light. These are
conventions which simplify the mathematical formulation
and our calculations, and have a merely formal significance
for the present. It would therefore be wrong to associate
any metaphysical speculations with the introduction of the
four-dimensional point of view.

Over and above its convenience for this formulation, we
can see other advantages which accrue from our regarding
time as a fourth co-ordinate, and recognize therein an essen-
tial justification for this mathematical view. To show this
clearly, let us suppose a point to move in any way in a plane
(that of &i-x 2 may be chosen). It describes some curve in
this plane. If we draw this curve, we can, by looking at
it, get an impression of the shape of its path, but not of any
other data of its motion, e.g. the velocity which it has at
different points of its path, or the time at which it passes
through these points. But if we add time x as a third co-
ordinate, the same motion will be represented by a three-
dimensional curve, the form of which immediately gives us
information about the character of the motion ; for we can
recognize directly from it which # 4 belongs to any point x x z
of the path, and we can also read off the velocity at any mo-
ment from the inclination of the curve to the #i-a? 2 -plane.
We shall follow Minkowski by appropriately calling this
curve the world-lme of the point. A circular motion in the
Oa-o^-plane would be represented by a helical world-line in
the av^-a^-manifold. This trajectory of the point only
arbitrarily expresses, as it were, one aspect of its motion,
viz. the projection of the three-dimensional world-line on the
av^vplane. Now, if the motion of the point itself takes
place in three-dimensional space, we obtain for its world-

52 General Postulate of Relativity of

line a curve in the four-dimensional manifold of the x^ #2?
#s, # 4 , and from this line all characteristics of the motion of
the point can be studied with the greatest ease. The path of
the point in space is the projection of the world-line on the
manifold of the ah. 9 x 2 , # 8 , and thus gives an arbitrary and
one-sided view of a few properties only of the motion:
whereas the world-line expresses them all in their en-
tirety.

Our considerations about the general relativity of space
may immediately be extended to the four-dimensional space-
time manifold ; they apply here also, for to increase the num-
ber of co-ordinates by one does not alter the underlying
principle. The system of world-lines in this x^-x 2 -Xy-x^
manifold represents the happening in time of all events in
the world. "Whereas a point transformation m space alone
represented a deformation of the world, i.e. a change of
position and a distortion of bodies, a point-transformation
in the four-dimensional universe also signifies a change in
the state of motion of the three-dimensional world of
bodies: since the time co-ordinate is also affected by the
transformation. "We can always imagine the results which
arise from the four-dimensional forms, by picturing them
as motions of three-dimensional configurations. If we sup-
pose a complete change of this sort to take place, by which
every physical point is transferred to another space-time
point in such a way that its new co-ordinates, x\, x' 2y a/' 3 , x'*,
are quite arbitrary (but continuous and single-valued) func-
tions of its previous co-ordinates x l9 x 2 , x 3 , x : then the new
world is, as in previous cases, not in the slightest degree
different from the old one physically, and the whole change
is only a transformation to other co-ordinates. For that
which we can alone observe by means of our instruments,
viz* space-time-coincidences, remains unaltered. Hence
points which coincided at the world-point x^ x 2 , x*, x in the

Measure-Determinations of the Space-time Continuum 53

one universe would again coincide in the other at the world-
point <x>\ 9 x' 2 , XB, x*. Their coincidence and this is all that
we can observe takes place in the second world precisely
as in the first.

The desire to include, in our expression for physical laws,
;>nly what we physically observe leads to the postulate that
the equations of physics do not alter their form in the above
arbitrary transformation, i.e. that they are valid for any
space-time co-ordinate systems whatever. In short, ex-
pressed mathematically, they are *covariant ? for all substi-
tutions. This postulate contains our general postulate of
relativity; for, of course, the term 'all substitutions 5 in-
cludes those which represent transformations of entirely
arbitrary three-dimensional systems in motion. But it goes
further than this, inasmuch as it allows the relativity of
space, in the most general sense discussed above, to be valid
even within these co-ordinate systems. In this way Space
and Time are deprived of the 'last vestige of physical ob-
jectivity', to use Einstein's words.

As explained above, 1 we may determine the position of
a point by supposing three families of surfaces to be drawn
through space, and then, after assigning a definite number,
a parametric value, to each successive surface of each fam-
ily, we may regard the numbers of the three surfaces which
intersect at the point as its co-ordinates. (Each family
must be numbered independently of the others.) Of course,
the relations between co-ordinates which are defined in this
way (Gaussian co-ordinates) will not in general be the same
as those which hold between the ordinary Cartesian co-ordi-
nates of Euclidean geometry. The Cartesian #-co-ordinate
of a point, for example, is ascertained by marking off the dis-
tance from the beginning- of the #-axis by means of a rigid
unit measure; the number of times this measure has to be

a Page 29.

54 General Postulate of Relativity and

applied end to end gives the desired co-ordinate number. In
tlie case of the new co-ordinates other conditions hold (cf.
page 48 above), since the value of a parameter is not im-
mediately obtainable as a number by applying the unit
measure. "We must consequently regard the x^ x 2 ? # 3 , %* of
the four-dimensional world as parameters, each of which
represents a family of three-dimensional manifolds; the
space-time continuum is partitioned by four such famil-
ies, and four three-dimensional continua intersect at
each world-point, their parameters thus being its co-or-
dinates.

If we now consider that the principle by which the co-
ordinates are to be feed consists in a perfectly arbitrary
partition of the continuum by means of families of surfaces
for, physical laws are to remain invariant for arbitrary
transformation it seems at first sight as if we no longer
had any firm footing or raeans of orientation. "We do not
immediately see how measurements are possible at all, and
how we can succeed in ascribing definite number values to
the new co-ordinates, even if these are no longer directly
results of measurement. Comparing measuring-rods and
observing coincidences result in a measurement, as we have
seen, only if they are founded on some idea, or some physi-
cal assumption or, rather, convention ; the choice of which,
strictly speaking, is essentially of an arbitrary nature, even
if experience points so unmistakably to it as being the sim-
plest that we do not waver in our selection. We therefore
find it necessary to make some convention, and we arrive at
this by a sort of principle of continuation, as follows. In
ordinary physics we are accustomed to assume without
argument that we may speak of rigid systems of reference,
and can realize them to a certain degree of approximation ;
length may then be regarded as being one and the same
quantity at every arbitrary point, in every position and

Measure-Determinations of the Space-time Continuum 55

state of motion. This assumption had already been modi-
fied to a certain extent in the special theory of relativity.
According to the latter, the length of a rod is in general de-
pendent npon its velocity relative to the observer ; and the
same holds of the indications of a clock. The connexion
with the older physics, and, as it were, the continuous transi-
tion to it, are due to the circumstance that the alterations in
the length- and time-data become imperceptibly small, if the
velocity is not great; for small speeds (compared with those
of light) we may regard the assumptions of the old theory
as being allowable. The special theory of relativity so ad-
justs its equations that they degenerate into the equations
of ordinary physics for small velocities.

In the general theory, the relativity of lengths and time
goes much further still; the length of a rod, according to it,
can also depend on its place and its position. To gain a
starting point at all, a Ao$ poi TTOV <rrc5, we shall of course
maintain continuity with the physics which has hitherto
proved its worth, and accordingly assume that this relativity
vanishes for extremely small changes. We shall thus con-
sider the length of a rod to remain constant as long as its
place, its position, and its velocity change only slightly in
other words, we shall adopt the convention that, for infi-
nitely small domains, and for systems of reference, in which
the bodies under consideration possess no acceleration, the
special theory of relativity holds. Since the special theory
uses Euclidean measure-determinations, this includes the
assumption that, for the systems designated above, Eucli-
dean geometry is to remain valid for infinitely small por-
tions. (Such an infinitely small domain may still be large
compared with the dimensions which are used elsewhere in
physics.) The equations of the general theory of relativity
must be, in the special case mentioned, transformed into
those of the special theory. We have now founded our

56 Measure-Determinations of the Space-time Contmuum

theory on an idea which makes measurement possible, and
we have reviewed the assumptions by means of which we
can successfully solve the problem proposed by the postu-
late of general relativity.

vni

ENUNCIATION AND SIGNIFICANCE OF THE
FUNDAMENTAL LAW OF THE NEW THEOEY

IN accordance with tlie last remarks, we shall turn our
attention to the realm of the infinitely small, and in it choose
a three-dimensional Euclidean system of co-ordinates, in
such a way that the bodies which are to be considered have
no perceptible acceleration with respect to it. This choice
is equivalent to the introduction of a definite four-dimen-
sional co-ordinate system for the domain in question. Let
us fix any point-event in this domain, i.e. a world-point A in
the space-time-continuum, the co-ordinates of which we
shall assume to be Xi, J 2 , X^ X*, in our local system; of
these Xi, X 2 , X s are measured by applying a small measur-
ing rule of unit length end to end, and the value of X* is
determined by the reading of a clock. B is to represent a
space-time point-event infinitely near A] its co-ordinates
differ, by the values dXi, dZ 2 , dZ 3 , cLX 4 , from those of A.
The ' distance' of these two world-points is then given by
the well-known simple formula

This ' distance', the line-element of the world-line, connect-
ing A and J?, is, of course, not in general a space-distance
[length] , but, since it is a combination of space- and time-
quantities, has the physical significance of a motional event,
as we clearly pointed out in introducing the notion of world-
lines, The numerical value of ds is always the same, what-
ever orientation the chosen local co-ordinate system may
have,

67

58 Enwciation and Significance of the

(The special theory of relativity throws a clearer light on
ds. If, for example, ds 2 is negative, it states that we can, by
appropriately choosing co-ordinate directions, obtain ds 2 =
dXi, whilst the other three cLX y s vanish. There is then no
difference between the space co-ordinates of the two world-
points ; the events corresponding to them thus occur in this
system at the same place, but with a time-difference dX*.
In this case ds is said to belong to the * time-class' of events ;
on the other hand it is assigned to the ' space-class ' of
events if 'ds 2 ' is positive; for in the latter case the co-
ordinate directions may be so chosen that &K vanishes.
The two point-events then take place simultaneously for
this system, and ds gives a measure of the distance which
separates them. Finally ds = signifies a motion which
takes place with the velocity of light, as is easily seen if we
substitute for dX 4 its value c-di.)

We shall now introduce any new co-ordinates o?i, x 2 , X B) x* 9
which are quite arbitrary functions of X ly X 2 , X 3 , X 4 , i.e. we
shall pass from our local system to any other arbitrary sys-
tem. Certain co-ordinate differences d# 1? cb 2 , d# 3 , d% 4 , cor-
respond to the ' distance! ' between the points A and B in
this new system, and the old co-ordinate difference dX can
be expressed in terms of the new dr's by using elementary
formulae of the differential calculus. 1 If we insert the ex-
pressions thus obtained for the dJX's in the above formula
for the line-element, we obtain its value expressed in the
new co-ordinates in the following form :
ds 2 = ucte

Fundamental Law of the New Theory 59

i.e. as a sum of ten terms, in which the ten quantities g
are certain functions of the co-ordinates X 1 They do not
depend on the particular choice of the local system, for
the value of ds 2 was itself independent thereof.

"When Riemann and Helmholtz examined three-dimen-
sional non-Euclidean continua, they regarded the factors g,
which occur above in the expression for the line-element, as
purely geometrical quantities, by which the metrical prop-
erties of space were determined. They were perfectly
aware, however, that we could not well speak of measure-
ments and space without making some physical assump-
tions. Helmholtz 's words were quoted above ; here we need
only allude to Eiemann ? s remarks at the close of his in-
augural dissertation (p. 268 of his Gesammelte Werke).
He there states that, in the case of a continuous manifold,
the principle of its measure-relations is not already con-
tained in the conception of the manifold, but must 'come
from elsewhere'; it is to be sought in 'binding forces', i.e.
the ground of these measure-relations must be physical in
nature. We know that reflections in the realm of metrical
geometry acquire a meaning only when its relationship to
physics is borne in mind. The above # ? s do not therefore
merely allow a physical interpretation, but indeed demand
it. Einstein's general theory of relativity gives them such
an interpretation directly. For, to recognize the signifi-
cance of the g 's, we need only call to mind the physical mean-
ing of the transformation from a local system to the general
system, as was discussed just above. The former was de~

performing the operations indicated we easily find that:
,AXA fXXjf /*Xj\* fiXtf*

*~Cs5?) + Gsf) + fe?) -fe?) &c -

60 Enunciation and Significance of the

fined by the property that a material point, left to itself in
the space of the X 1? X 2 , Xs, moves rectilinearly and uni-
formly in this space; its world-line, 1 i.e. the law of its
motion, is consequently a four-dimensional straight line, the
line element of which is given by :

ds 2 = dX 2 + dJXt '+] d! - dXi

If we transform to the new co-ordinates a*, o? 2 , ^ &*, this
means that we are viewing the same event, the same motion
of the point, from some other arbitrary system, with respect
to which the local system is of course moving with accelera-
tion in some way. Therefore, in the space of the o?i, o? 2 , o? 8 ,
the point moves curvilinearly and non-uniformly. The
equation of its world-line, i.e. its law of motion, alters, in-
asmuch as its line-element, expressed in the new co-ordi-
nates, is now given by:

ds 2 = ffu cte| + . . . . rj rg^ cfoi d# 2 + . - -

We now recall the i Principle of Equivalence' (p. 41). Ac-
cording to this, the statement that 'a point left to itself
moves with certain accelerations ? is identical with the state-
ment that 'the point is in motion under the influence of a
gravitational field \ The equation of the world-line ex-
pressed in the new co-ordinates thus represents the motion
of a point in the gravitational field. The factors g are hence
the quantities which determine the field. "We see that their
part in the new theory is analogous to that played by the
gravitational potential in the Newtonian theory. We may,
therefore, term them the 10 components of the gravitational
potential.

The world-line of the point, which was a straight line for
the local system, i.e. the shortest connecting line between

ilts equation, expressed in the form of tlie shortest (geodetic) line, is

Fundamental Law of the New Theory 61

two world-points, likewise represents a shortest line in the
new system of x^ # 2 , % 3 , # 4 , for the definition of a geodetic
line is independent of the co-ordinate system. If we could
now regard the domains of the *locaP system as being- in-
finitesimal, the whole world-line in it would shrink to an
element ds. The reflection made above would become mean-
ingless, and we conld draw no further inferences. Since the
Law of Inertia and the Special Theory of Relativity have,
however, been so widely confirmed by experience, it is clear
that there must in reality be finite regions, for which, if
we choose a suitable system of reference, ds 2 = d#i 2 + cb 2 2
F +'d#! dn?: viz. those parts of the world in which, with
this chosen system, no perceptible influence of gravitating
matter exists. In it the world-line is for this system a
straight line, and consequently for arbitrary systems a geo-
detic line. We now again recall our Principle of Continuity
(according to which the new laws are to be assumed, in such
a way that the old laws are contained in them unchanged as
nearly a$ possible, and the new ones resolve into the latter
for the limiting case) ; and we then make the hypothesis that
the relation obtained in this way is valid quite generally for
every motion of a point under the influence of inertia and
gravitation, i.e. that the world-line of the point is always a
geodetic even when matter is present. This gives us the
desired fundamental law. Whereas the Law of Inertia of
Newton and Galilei states : A point under no forces moves
uniformly and rectilinearly', the Einstein Law, which com-
prises both inertial and gravitational effects, asserts : The
world-line of a material point is a geodetic line m the space-
time continuum. This laws fulfils the condition of relativ-
ity ; for it is an invariant for any arbitrary transformations,
since the geodetic line is defined independently of the sys-
tem of reference.
We must again emphasize that the co-ordinates 2* . . OR*

62 Enunciation and Significance of the

are number-values, wMcli fix the time and place of an event,
but have not the significance of distances and times as
measured in the ordinary way. The 'line-element' As, on
the other hand, has a direct physical meaning, and can
be ascertained by means of measuring-scales and clocks.
It is, by definition, independent of the system of co-ordi-
nates; hence we need only betake ourselves to the local
system of Xi . . X* 9 and the value which we there obtain
for ds is valid generally.

Those steps have now been taken which are of general
philosophic importance, and fundamental for the view of
space and time according to the new doctrine: it is in
these that we are here primarily interested. For Einstein
they were merely the preliminary stage for the physical
problem of getting at the actual values of the quantities
ff, i.e. of discovering how they depend upon the distribution
and motion of the gravitating masses. In accordance with
the Principle of Continuity, Einstein starts here again by
working from the results of the special theory of relativity.
The latter had taught us that not only matter in the ordi-
nary sense, but also every kind of energy, has gravitational
mass, and that inertial mass is altogether identical with
energy. This implies that not the 'masses ' but the ener-
gies * should figure in the differential equations giving the
ff's. The equations must of course remain covariant for
any arbitrary substitutions. In addition to these initial
assumptions which, from the point of view of the theory,
are quite obvious, Einstein makes the further assumption
that the differential equations are of the second order ; he
was guided by the fact that the old Newtonian potential sat-
isfied a differential equation of just this type. In this way
we arrive at perfectly definite -equations for the g's, and

iThey are represented in the special theory of relativity by the com-
ponents ol a four-dimensional 'tensor', the Impulse-energy tensor.

Fundamental Law of the New Theory 63

thus the problem of establishing them is (theoretically)
solved.

So we see that, except for the last-mentioned purely for-
mal analogy, the entire theory is built on foundations which
have absolutely nothing in common with Newton's old
theory of potential ; it is, on the contrary, developed purely
from the postulate of general relativity, and from well-
known results of physics (as given by the special principle
of relativity). It is so much the more surprising that the
new equations, which have been obtained by such different
means, actually degenerate into the Newtonian formula for
general mass-attraction for a first approximation. This is
in itself such an excellent confirmation of the lines of argu-
ment that it must inspire very considerable confidence in
their correctness. But, as we know, the achievements of the
new theory do not end here. For, if we work out the equa-
tions to a second approximation, there immediately
emerges, without the help of any auxiliary assumptions, a
quantitatively exact explanation of the anomalous motion
of Mercury's perihelion, a phenomenon, which the Newto-
nian Theory could account for only by introducing special
hypotheses of a rather arbitrary nature. These are aston-
ishing results, the scope of which cannot easily be over-
estimated : and we must agree with Einstein when he says
at the conclusion of 14 of his essay Die Grwidlage der all-
gemeinen Relativitatstheorie' : 'The fact that the equa-
tions deduced from the postulate of general relativity by
purely mathematical processes . . . give us to a first ap-
proximation the Newtonian law of attraction, and to a sec-
ond approximation the motion of Mercury's perihelion . . .
discovered by Leverrier, is a convincing proof that the
theory is physically correct'. The new fundamental law
has an additional advantage over the Newtonian attraction
formula, inasmuch as it is expressed as a differential law;

64 Emmciation and Significance of the

I.e. according to it, events at one point in the space-time
manifold depend only upon the events of points infinitely
near it on all sides, whereas in Newton's attraction formula
gravitation occurs as a force acting at a distance. This
means that we have considerably simplified the physical
picture of the world, and consequently have now advanced
another step in the theory of knowledge, by banishing
gravitation, the last force acting at a distance, out of
physics, and expressing all the laws underlying physical
events solely by differential equations.

All the other laws must, of course, also be formulated in
such a way that they remain invariant after any arbitrary
transformations. The method of doing this is prescribed
by the special principle of relativity and the principle of
continuity, and has already been applied by Einstein and
others. Chief interest circles around electrodynamics, from
which it is to be hoped that, by combining it with the new
theory of gravitation, it will be possible to build up a flaw-
less system of physics. It is the great problem for physi-
cists of the future to bring electrodynamics and gravita-
tional theory under a common law, and thus embrace both
realms in one theory. The endeavours which have been
carried out in this direction have so far been unavailing;
probably this is due, above all, to the absence of further
data of experience, in which gravitational and electrical
phenomena occur simultaneously.

In addition to the astronomical confirmation mentioned
above, there are still other possibilities of verifying the
theory by observation ; for, according to it, there should be
a still perceptible lengthening of the time of oscillation of
light in a very strong gravitational field, and a curvature of
the light rays should manifest itself. (The path of the lat-
ter being the geodetic lines ds = 0.) The presence of the
first effect, which consists in a displacement of the spectral

Fundamental Laiv of the New Theory 65

lines towards the red end, has not yet been definitely estab-
lished. Whereas the efforts to detect this shift in the gravi-
tational field of the sun have so far been fruitless, observa-
tions of the spectra of other fixed stars seem to indicate
with great probability that it actually exists. The second
effect, however, viz. the deflection of light by gravitation
was established beyond doubt on May 29, 1919, on the oc-
casion of the total eclipse of the sun. The light from a star
which, on its way to the earth, passes close by the sun, is
attracted by the latter 7 s intense gravitational field. This
should, according to theory, express itself in an apparent
displacement of the star. Since these stars which happen
to be near the sun (as projected on the celestial sphere) are
only visible to the eye or a photographic plate during a total
eclipse of the sun, this inference from theory can only be
tested upon such occasions. Two expeditions were sent out
from England to observe the above eclipse. They succeeded
in finding that the displacement of the apparent position of
these stars was actually such as had been prophesied by
Einstein, and, indeed, to the exact amount he had previ-
ously calculated. This confirmation is doubtless one of the
most brilliant achievements of human thought, and, in its
theoretical significance, even surpasses the famous discov-
ery of the planet Neptune from the calculation of Leverrier
and Adams. The general theory of relativity has in this
way successfully undergone the severest tests. The world
of science pays homage to the triumphant power with which
the correctness of the physical content of the theory and
the truth of its philosophical foundations are confirmed by
experience.

The assertion that all motions and accelerations are rela-
tive is equivalent to the assertion that space and time have
no physical objectivity. One statement comprehends the
other. Space and time are not measurable in themselves :

66 Fundamental Law of the New Theory

they only form a framework into which we arrange physical
events. As a matter of principle, we can choose this frame-
work at pleasure ; but actually we do so in such a way that It
conforms most closely to observed events (e.g. so that the
'geodetic lines 3 of the framework assume a distinctive
physical role) ; we thus arrive at the simplest formulation
of physical laws. An order has no independent existence,
but manifests itself only in ordered things. Minkowski had
as a result of the special theory of relativity enunciated
the proposition In terse language (perhaps not wholly free
from criticism) that space and time in themselves are re-
duced to the status of mere shadows, and only an indis-
soluble synthesis of both has an independent existence. So,
on the basis of the general theory of relativity, we may now
say that this synthesis itself has become a mere shadow, an
abstraction; and that only the oneness of space, time, and
things has an independent existence.

IX

THE TINITUDE OF THE UNIVEBSE

IN Newton's mechanics, and, indeed, in pre-Einsteinian
physics altogether, space played a part which was alto-
gether independent of any considerations about matter.
Jnst as a vessel can exist free of content and preserve its
form, space was to preserve its properties, whether * occu-
pied' by matter or not. The general theory of relativity
has taught ns that this view is groundless and misleading.
'Space', according to it, is possible only when matter is
present, which then determines its physical properties.

This standpoint, which arises out of the general theory of
relativity, is proved to be the only justifiable one, when we
approach the cosmological question of the structure of the
universe as a whole. Certain difficulties had already been
encountered earlier, which clearly showed that Newton's
cosmology was untenable; but it never suggested itself to
anyone that Newton's doctrine of space might be partly re-
sponsible for these difficulties. The relativity theory yields
an unexpected and wondrous solution of the discrepancies,
which is of exceeding importance for our picture of the
world.

It was generally believed by the ancients that the cosmos
was bounded by a mighty sphere, to the inner surface of
which the fixed stars were thought to be attached in some
way. Even Copernicus did not succeed in destroying this
belief. He had placed the sun in the middle of the planets
moving around it, and recognized the earth as one planet

67

68 The Pinitude of the Universe

amongst many others, but not yet the sun as one of many
fixed stars. In comparison with this naive view, the picture
of the world must have seemed to become both enriched and
exalted when Giordano Bruno propounded the doctrine of
the infinity of the worlds in space. It was alluring to the
imagination to think of the innumerable stars as being also
suns similar to our own, and poised in space, and of space
as extending to infinity, not limited by any rigid sphere, nor
enclosed by any 'crystal dome'. Bruno glorifies the free-
dom of spirit which emanates from this extension of the
world system in rapturous lines :

Now unconfmed the wings stretch ont to heaven,
Nor shrink beneath a crystal firmament
Aloft into the aether's fragrant deeps,
Leaving below the earth-world with its pain,
And all the passions of mortality.

Up to the present day the conception of the world as a
whole described in these lines has had complete sway. It
was certainly, from an aesthetic standpoint, most attractive
and most satisfactory for the philosopher to picture the
cosmos as composed of the world of matter infinitely ex-
tended into infinite space; a traveller on the way to infi-
nitely distant regions meets with ever new stars, even if
he continue through all eternity, without reaching the limits
of their realms or exhausting their number. It is true that
the stars have "been sown with great scarcity in the heav-
enly regions ; a comparatively small amount only of matter
is scattered over a great volume of space; but its mean
density is to be the same everywhere, and is not to become
zero even at infinity. So that, if we fix upon a certain
amount of mass in*some great volume of celestial space, and
divide it by the size of this volume, we should by choosing
a continually larger volume arrive at a constant finite value

The Finitude of the Universe 69

for tlie mean density. From the point of view of natural
philosophy, such a picture of the world would be highly sat-
isfactory. It would have neither beginning nor end, neither
a centre nor boundaries, and space would nowhere be
empty.

But the celestial mechanics of Newton is incompatible
with this view. For, if we assume the strict validity of
Newton's gravitational formula, according to which masses
exert a mutually attractive force varying inversely as the
square of the distance, calculation shows that the effects at
a certain point of an infinite number of masses present at
infinite distances, according to the above view, do not sum
up to a certain finite gravitational force at the point, but
that only infinite and indeterminate values are obtained.

Einstein proves this in an elementary way as follows : If
p is the average density of matter in the universe, then the
amount of matter contained in a large sphere of radius R is
4/3 Ttp R*. The same expression (by a familiar theorem of
the theory of potential) gives the number of 'lines of force',
due to gravitation which pass through the surface of the
sphere. The extent of this surface is 4:rtR 2 , so that there are
ipR lines of force to every unit area of surface. But this
latter number expresses the intensity of the force which is
exerted by the gravitational effect of the contents of the
sphere at a point on the surface : it clearly becomes infinity
if R increases beyond all limits.

As this is impossible, the universe cannot, on Newton's
theory, be constituted as was just portrayed; gravi-
tational potential must become zero at infinity, and
the cosmos must present the picture of an island of finite
extent surrounded on all sides by infinite 'empty space':
and the mean density of matter would be infinitely small.

But such a picture of the universe would be unsatisfactory
to the highest degree. The energy of the cosmos would con-

70 The Finitude of the Universe

stantly decrease, as radiation would disappear into infinite
space; and matter, too, would gradually disperse. After a
certain time the world would have died an inglorious death.

Now these exceedingly awkward consequences are insep-
arably connected with Newton's theory. The astronomer
Seeliger,-who laid bare these shortcomings to their full ex-
tent, sought to escape them by assuming that the attractive
force between two masses decreased more rapidly than
Newton's law demands. With the help of this hypothesis,
he actually succeeds in maintaining without contradiction
this idea of a world infinitely extended, filling all space with
matter of a mean density. An unsatisfactory feature of this
theory is, however, contained in the fact that the hypothesis
is invented ad hoc, and is not occasiqied or supported by
any other experience.

Great interest thus circles round the question whether
it is not possible to solve the cosmological problem by some
new theory which is entirely satisfactory in every way. The
suggestion forces itself upon us that the general theory of
relativity might be able to do this ; for, in the first place, it
gives us information about the nature of gravitation towards
which the Newtonian law represents only an approximation ;
secondly, it sheds an entirely new light on the problem of
space. We have therefore reason for hoping that it will give
us important disclosures about the question of the finitude
of the world in space.

When Einstein investigated whether his theory was to be
brought into closer harmony with the assumption of an in-
finite world with an average uniform density of distribution
of stars than had been possible for Newton's theory, he first
met with disappointment. For it appeared that a universe
constructed in accordance with the hopes expressed above
was just as little compatible with the new mechanics as with
that of Newton.

The Fimtude of the Universe 71

As we know, the space of the new theory of gravitation
Is not Euclidean in structure, but departs somewhat from
this shape, conforming in its measure-relations to the distri-
bution of matter. Now if it were possible that, correspond-
ing to the world-picture of Giordano Bruno, a uniform dis-
tribution of stars on the average existed for infinite space,
then, in spite of deviations in particular places, space could
still roughly be called Euclidean as a whole : just as I might
call the ceiling of my room plane, by forming an abstraction
which neglects the little roughnesses of its surface. Calcula-
tion, however, shows that such a structure of space Ein-
stein calls it quasi-Euclidean is not possible in the general
theory of relativity. On the contrary, according to this
theory, the mean density of matter must necessarily be zero
in infinite non-Euclidean space ; i.e. we are again driven to
the world-system which was discussed above, which would
consist of a finite aggregation of matter in otherwise empty
space of infinite dimensions.

This view, which was unsatisfactory for Newton's theory,
is still more so for the general theory of relativity. Not only
do the objections which were pointed out above apply in this
case also, but new ones arise in addition. For, if we seek to
find the mathematical boundary conditions for the quantities
g at infinity, which correspond to this case, Einstein shows
that we may attempt it in two ways. We might, in the first
place, think of assigning to the g ? s the same boundary values
which are allotted to them at infinity in the mathe-
matical treatment of planetary motions. For the planetary
system certain limiting values (gi = g 22 = g BZ = 1, g^ =
+' 1, the other # ? s 0) are permissible, since we have still to
take into account the presence of the stellar system at great
distances; but the extension of this method to the whole
universe is incompatible with the fundamental ideas of the
relativity theory in two respects. First, a perfectly definite

72 The Finitude of the Universe

choice of co-ordinate systems would be imperative for tMs ;
and second, the inertial mass of a, body would, contrary to
our hypotheses, no longer be solely due to the presence of
other bodies ; but a material point would still possess inertial
mass if it were at an infinite distance from other bodies, or
even if it were entirely isolated and left alone in the world-
space. This is contrary to the trend of thought of the
general principle of relativity ; and we see that only those
solutions come into consideration in which the inertia of a
body vanishes at infinity.

Einstein now showed (and this appeared to be the second
way) that one might indeed assume boundary conditions for
the g 's at infinity, which would fulfil the latter demand ; and
that a world-picture drawn in this way would even have an
advantage over the Newtonian one, inasmuch as no star and
no ray of light, according to it, could disappear in infinite
space, but would finally have to return into the system. But
he also showed that such boundary conditions would be in
absolute disagreement with the actual state of the stellar
system, as experience presents it to us. The gravitational
potentials would have to increase at infinity beyond all
limits, and very great relative velocities of the stars would
necessarily occur, whereas, in fact, we observe that the mo-
tions of the stars take place extremely slowly compared with
the velocity of light. The fact of the small velocity of the
stars is indeed one of the most striking peculiarities, com-
mon to all members of the stellar system, which offer them-
selves to observation, and can be used as a basis for cos-
mological speculations. In virtue of this property, we can
unhesitatingly regard the matter in the cosmos as at rest to
a first approximation (if we choose an appropriate system
of reference) ; we consequently base our calculations on this
assumption.

We thus find that the second method likewise does not

The Finitude of the Universe 73

lead to the goal. The inference is that, according to the
relativity theory, the universe cannot be a finite complex of
stars existing in infinite space; this, after the above re-
marks, means that we cannot regard space as quasi-
Euclidean. What possibility now remains?

At first it seemed as if no reply was forthcoming from
the theory; but Einstein soon discovered that it was still
possible to generalize his original gravitational equations
slightly further. After this small extension of the formulae,
the general theory of relativity has the inestimable advan-
tage of giving us an unmistakable answer, whereas the pre-
vious Newtonian theory left us in total uncertainty, and
could only rescue us from forming a highly undesirable pic-
ture of the universe by making new and unconfirmed hy-
potheses.

If we again suppose the matter of the universe to be
distributed with absolutely uniform density and to be at rest,
the calculation leaves no doubt but that space is spherical
in structure (there is the additional possibility that it might
be * elliptical' in constitution, but we may neglect this case,
which seems to be of mathematical rather than of physical
interest) . Since matter does not actually occupy space uni-
formly and is not at rest, but only shows the same density
of distribution as a mean, we must regard space as quasi-
spherical (i.e. on the whole it is spherical, but departs from
this form in its smaller parts, just as the earth is only an
ellipsoid as a whole, but is, when considered in smaller
portions, possessed of an irregularly formed surface).

"What the term * spherical space' is intended to convey is
probably known to the reader through Helmholtz's popular
essays. He, as we know, describes the three-dimensional
analogy to a spherical surface ; the former has, like the lat-
ter, the property of being circumscribed, i.e. it is unlimited
and yet finite. The comparison with the surface of a sphere

The Fmitude of the Universe

nust not mislead one to confuse in one's mind ' spherical '
vith sphere-shaped. A sphere is bounded by its surface,
;he latter cutting it out of space as a part of it ; spherical
space, however, is not a part of infinite space, but has sim-
ply no limits. If I start out from a point of our spherical
jvorld and continually proceed along a ' straight line', I shall
ciever reach a limiting surface; the * crystal dome', which
according to the ancients was supposed to encompass the
universe, exists just as little for Einstein as it did for
Giordano Bruno. There is no space outside the world;
space exists only in so far as matter exists, for space in
itself is merely a product of abstraction. If, from any
point, we draw the straightest lines in all directions, these
at first, of course, diverge from one another, but then
approach again, in order finally to meet at one point as
before. ' The totality of such lines fills the world-space
entirely, and the volume of the latter is finite. Einstein's
theory even enables us to calculate its numerical value for
a given density of distribution ; we thus obtain the volume

7.10* 1
y = , - cubic centimetres an enormously high figure ;

for p, the mean density of matter, has an exceedingly small
value. The structure of the universe, which the general
theory of relativity unveils to us, is astounding in its logical
consistency, imposing in its grandeur, and equally satisfying
for the physicist as for the philosopher. All the difficulties
which arose from Newton's theory are overcome ; yet all the
advantages which the modern picture of the world presents,
and which elevate it above the view of the ancients, shine
with a clearer lustre than before. The world is not confined
by any boundaries, and is yet harmoniously complete in
itself. It is saved from the danger of becoming desolate, for
no energy or matter 'can wander off to infinity, because

The Finitude of the Universe 75

space Is not infinite. The infinite space of the cosmos has
certainly had to "be rejected; but this does not signify such
sacrifice as to reduce the sublimity of the picture of the
world. For that which causes the idea of the infinite to in-
spire sublime feelings is beyond doubt the idea of the end-
lessness of space (actual infinity could not in any case be
imagined) ; and this absence of any barrier, which excited
Giordano Bruno to such ecstasy, is not infringed in any way.
By a combination of physical, mathematical, and philo-
sophic thought genius has made it possible to answer, by
means of exact methods, questions concerning the universe
which seemed doomed for ever to remain the objects of
vague speculation. Once again we recognize the power of
the theory of relativity in emancipating human thought,
which it endows with a freedom and a sense of power such as
has been scarcely attained through any other feat of science.

BELATIONS TO PHILOSOPHY

IT is scarcely necessary to mention that the words space
and time in the preceding chapters have been used only in
the L objective' sense in which these conceptions occur in nat-
ural science. 'Subjective' psychological experience of exten-
sion in space and order in time is quite distinct from these.

Ordinarily there is nothing to induce us to analyse this
difference in detail; the physicist does not need to concern
himself in the slightest with the investigations of the psy-
chologist into spatial perception. But when we wish to
form a clear picture of the ultimate epistemolqgical f ounda.-
tions of natural science, it becomes necessary to give an
adequate account of the relationship between these two
points of view. This is the task of the philosopher ; for it is
generally accepted that it is for philosophy to reveal the
fundamental assumptions of the separate sciences, and bring
them into harmony with one another.

What leads us to speak of space and time at all? "What
is the psychological source of these notions? There is no
doubt that all our perceptions of space, and the conclusions
resulting therefrom, emanate from certain properties of our
sense-impressions, viz. from those properties which we term
* spatial' and which do not allow of closer definition: for we
get our knowledge of them only from direct experience.
Just as it is impossible for me to explain to a person who
has been born blind, by means of a definition in words, what
I experience when I see a green surface, so it is impossible

76

Relations to Philosophy 77

for me to describe wliat Is meant when I ascribe to this green
appearance a definite extension and position in tlie field of
vision. In order to know what is meant, we must be able
to 'behold' it: we must have visual perceptions or Impres-
sions. This spatial quality, which is an essential accompani-
ment of visual impressions, is thus intuitive (' anschaulicJi') .
We assign the term in an extended sense to all the other
data of our world of presentations and perceptions, not
only the visual ones. The perceptions of the other senses,
more particularly the tactual and kinaesthetie (muscular
and articular) presentations, have properties which we
likewise term 'spatial'. In fact the intuition which the
blind have of space consists, exclusively, of such data. A
sphere feels different from a cube to the touch: I experience
different muscular sensations in the arm, according as
I describe with my hand a long or a short, a gently curved or
a zigzag line. These differences constitute the space quality
(RaumlicTikeit) of the tactual and muscular perceptions: it
is these that the person born blind has In his mind when he
hears of different localities or dimensions.

The data, however, of the various realms of perception
cannot be compared with one another (e.g. the space arising
out of tactual presentations Is entirely dissimilar in Mnd
from that of the optical presentations : a man bora blind,
who has a knowledge of the first only, cannot, from it, form
any notion of the latter). Tactual space has so far not the
slightest resemblance to visual space, and the psychologist
finds himself obliged to say that there are just as many
spaces for our intuition as we have senses.

The space of the physicist, however, which we set up as
objective in opposition to these subjective spaces, is a single
definite one, and we think of it as independent of our sense
impressions (but of course not independent of physical
objects; on the contrary, it is only real in conjunction with

78 Relations to Philosophy

them). It is not identical with, any of the above spaces of
intuition, for it has quite different properties. If we look at
a rigid cube, for instance, we find that its form changes for
our visual sense according to the side at which, or the dis-
tance from which, we view it. The apparent length of its
edges varies, and yet we ascribe to it the same physical
shape. "We get a similar result, in forming a judgment about
a cube, by means of our sense of touch : by which we also re-
ceive entirely different impressions, according as we touch
larger or smaller parts of its surface, or according to
the parts of the skin which come in contact with it ; yet in
spite of these different impressions we pronounce the cubical
form of the object to have remained unaltered. The objects
of physics are therefore not the data of sense : the space of
physics is not in any way given with our perceptions, but is
a product of our conceptions. "We cannot therefore ascribe
to physical objects the space of intuition with which our vis-
ual perceptions have made us acquainted, nor that which we
find present in our tactual presentations, but only a concep-
tual arrangement, which we then term objective space, and
determine by means of a suitably disposed manifold of num-
bers (co-ordinates). Hence we see that the same thing holds
true for intuitional space as for other qualities of the sense-
data such as tones, colours, &c. Physics does not know col-
our as a property of the object with which it is associated;,
but only frequencies of the vibrations of electrons. It has
no knowledge of qualities of heat, but only of kinetic energy
of the molecules.

Similar arguments apply in the consideration of subjec-
tive psychological time. A special psychological time can-
not indeed be claimed for the realm governed by each partic-
ular sense ; for it is one and the same time-character which
permeates all experiences not only those of the senses in
the same way. This direct experience of duration, of earlier

Relations to Philosophy 79

and later, is nevertheless an ever-changing Intuitional fac-
tor, which makes one and the same objective event appear,
according to mood and attentiveness, now long, now short :
a factor which vanishes altogether during sleep, and bears
an entirely different stamp according to the wealth of
associations of the experience. In short, it is easily dis-
tinguishable from physical time, which only signifies an
arrangement having the properties of a one-dimensional
continuum. This objective order or arrangement has just as
little to do with the Intuitional experience of duration as the
three-dimensional order of objective space has to do with the
intuitional experiences of extension, as presented optically
or tactually. In recognizing this, we get the pith of Kant's
doctrine of the * Subjectivity of Time and Space', according
to which both are merely * forms' of our intuition, and
cannot be ascribed to the 'tMngs-in-themselves'. Kant
himself does not give clear expression to this truth; for he
always talks of * space' only, without drawing a dividing
line between the intuitional spaces of the various senses, or
between them and the space of bodies as implied in physics.
Instead of this, he merely opposes the unknowable arrange-
ment of the 'things-in- themselves' to the space and time of
the things as given by the senses. We, on the other hand,
find occasion to distinguish only between the intuitional psy-
chological spaces and non-intuitional physical space. Just
because the latter is purely conceptual, it is quite impossible
contrary to the opinion of many a follower of Kant for
intuition to give us the slightest information as to whether
physical space is Euclidean or not. In conjunction with,
objective time, physical space is designated by the four-
dimensional scheme which we have repeatedly discussed
above, and which in mathematical language can simply be
treated as the manifold of all number quadruples # t , a? 2 , &*, &*
In this objective scheme there is no distinction between a

80 Relations to Philosophy

' time' distance and a ' space' distance. TMs is the point
which receives full recognition for the first time through
the theory of relativity. Both simply appear as one-
dimensional continna; and there is no room left in this
conception for the intuitional difference between duration
(length of time) and extension (length of space). It does
not matter how fundamental a part this difference plays for
consciousness.

It is obvious that in the first instance only the intuitional
psychological spaces and times are given us ; and we must
inquire how we have, by starting from them, arrived at the
construction of the objective space-time manifold. This
construction is not indeed a product of natural science, but
is a necessity of our daily life ; for when we ordinarily talk
of the position and shape of bodies, we are always already
thinking of physical space, which is conceived as independent
of individuals and of the organs of sense. Of course, we
always represent to our consciousness shapes and distances,
about which we are thinking, by visual and tactual means and
Mnaesthetic presentations : because we always strive, as far
as possible, to exhibit non-intuitional conceptual relations in
our thinking by sensory substitutes which may act as their
representatives, but are no more than sense-representatives
of the physical conception of space. The former are not to
be confused with the latter, nor must they lead one to regard
the latter with Kant as likewise intuitional.

The answer to our question, as to the genesis of the
physical conception of space from the intuitional data of
the psychological spaces, is now quite plain. These spaces
are essentially dissimilar and incapable of comparison with
one another; but they have, as our experiences teach us,
a perfectly definite uniform functional relation to one
another. TactuaLperceptions, e.g., correlate themselves with
visual perceptions. A certain correspondence exists between

Relations to\ Philosophy 81

the two spheres; and through this correspondence it is
possible to arrange all spatial perceptions into one scheme,
this being just what we call objective space. If in feeling
over an object my skin nerves receive a perception-complex
of the 'cube form', I can, by adopting proper measures
(lighting a candle, opening my eyes, &c.), receive certain
visual perception complexes, which I likewise designate as
'cube form'. 1 The optical impression is toto caelo different
from the tactual one ; but experience teaches me that they go
hand in hand with one another. In the case of persons born
blind, who acquire the sight of their eyes through an opera-
tion, we have an opportunity of studying their gradual train-
ing in associating the data of the two realms of sight and
touch. 2

Now it is important to understand quite clearly what par-
ticular experiences lead us to connect a perfectly definite
element of optical space with a perfectly definite element of
tactual space, and thereby to form the conception of a
' point ' in objective space. For it is here that experiences
arising out of coincidences come into account. In order
to fix a point in space, we must in some way or other, di-
rectly or indirectly, point to it: we must make the point
of a pair of compasses, or a finger, or the intersection of
cross-wires, coincide with it (i.e. bring about a time-space
coincidence of two elements which are usually apart). Now
these coincidences always occur consistently for all the in-
tuitional spaces of the various senses and for various indi-

1 Vide Locke's Essay on Human Understanding, bk. il, ch. 9, s. 8.

2 This view is familiar to the English reader from Berkeley's New Theory
of Vision. (Fraser, Oxford edition, vol. i.) Cf, Dufaur, Archives des sciences
physiques et naturelles, tome 58, p. 232.

Schopenhauer cited various instances in chap, iv of his Fourfold Root of
the Principle of Sufficient Reason, mentioning in partcular Cheselden's blind,
man, a case recorded in Phil Trans, vol. 35 (Trans.),

82 Relations to Philosophy

viduals. It is just on account of this that a < point ? is
defined which is objective, i.e. independent of individual ex-
periences and valid for all. An extended pair of com-
passes applied to the skin excites two sensations of prick-
ing; but if I bring the two points together so that they
occupy the same spot in optical space, I only get one sen-
sation of pricking, and there is also coincidence in tact-
ual space. Upon close investigation, we find that we arrive
at the construction of physical space and time by just this
method of coincidences and by no other process. The
space-time manifold is neither more nor less than the quin-
tessence of objective elements as defined by this method.
The fact of its being a four-dimensional manifold follows
from experience in the application of the method itself.

This is the outcome of our analysis of the conceptions of
space and time; it is an analysis of psychological data
regarded as our sources of knowledge. We see that we
encounter just that significance of space and time which
Einstein has recognized to be essential and unique for
physics, where he has established it in its full right.
For he rejected Newton's conceptions, which denied
the origin we have assigned to them, and founded physics
on the conception of the coincidence of events. Here we
have the realization of an eminently desirable point of con-
tact between physical theory and the theory of knowledge.

In one matter physical theory goes far beyond the bounds
within which we have psychological data. Physics intro-
duces, as its ultimate indefinable conception, the coincidence
of two events; on the other hand, the psycho-genetic analy-
sis of the idea of objective space ends in the conception of
the space-time coincidence of two elements of perception.
Are they to be regarded simply as. one and the same thing?

Eigorous positivism, such as that of Mach, affirms them
to be so. According to him, the directly experienced ele-

Relations to Philosophy 83

ments such as colours, tones, pressures, warmths, &c., are
the sole reality, and there are no other actual events be-
yond the coming and going of these elements. "Wherever
else in physics other coincidences are mentioned, they are
only abbreviated modes of speech, economical working-
hypotheses, not realities as perceptions are. Looked at
from this point of view, the conception of the physical
world in its objective four-dimensional scheme would
merely be an abridged statement of the correspondence of
the subjective time-space experiences in the realms of the
various senses, and nothmg more.

This view is, however, not the only possible interpreta-
tion of scientific facts. If distinguished investigators in the
domain of the exact sciences do not cease to urge that the
picture of the world as offered by Mach fails to satisfy
them, the ground for it is doubtless to be sought in this,
that the quantities which occur in physical laws do not all
indicate 'elements 7 in Mach's sense. 1 The coincidences
which are expressed by the differential equations of physics
are not immediately accessible to experience. They do not
directly signify a coincidence of sense-data; they denote
non-sensory magnitudes, such as electric and magnetic in-
tensities of field and similar quantities. There is no argu-
ment whatsoever to force us to state that only the in-
tuitional elements, colours, tones, &c., -exist in the world.
We might just as well assume that elements or qualities
which cannot be directly experienced also exist. These can
likewise be termed 'reaP, whether they be comparable with
intuitional ones or not. For example, electric forces can
just as well signify elements of reality as colours and tones,
They are measurable, and there is no reason why episte-
mology should reject the criterion for reality which is used

i The English reader will find the corresponding theory in K. Pearson,
Grammar of Science.

&4 Relations to Philosophy

in physics (v. p. 21). The conception of an electron or an
atom would then not necessarily be a mere working hy-
pothesis, a condensed fiction, but could equally well desig-
nate a real connexion or complex of such objective ele-
ments: just as the conception of the ^ego ? denotes a real
complex of intuitional elements. The picture of the world,
as presented by physics-, would then be a system of symbols
arranged into a four-dimensional scheme, by means of
which we get our knowledge of reality; that is, more than a
mere auxiliary conception, allowing us to find our way
through given intuitional elements.

The two views stand in opposition; and I believe that
there is no rigorous proof of the correctness of the one
and the falseness of the other. The reasons which induce
me to declare myself in favour of the second which may,
in contrast to the strictly positivist view, be called realistic
are as follows :

First, it seems to me to be purely arbitrary, nay, dog-
matic, to allow only the intuitional elements and their rela-
tionships to be valid as realities. "Why should intuitional
experiences be the only ' events' in our world? Why
should there not be other events besides these?

We find that the processes of science do not justify us in
thus narrowing the conception of reality. It was put for-
ward in opposition to certain fallacious metaphysical
views ; but these can be avoided in other ways.

Secondly, the strictly positivist picture of the world
seems to me to be unsatisfactory on account of a certain
lack of continuity. In narrowing down the conception of
reality in the above sense, we tear, as it were, certain holes
in the fabric of reality, which are patched up by mere
auxiliary conceptions. The pencil in my hand is to be re-
garded as real, whereas the molecules which compose it are
to be pure fictions. This antithesis, often uncertain and

Relations to Philosophy 85

fluctuating, between conceptions which denote something
real and those which are only working-hypotheses, finally
becomes unbearable. It is avoided by the assumption,
which is certainly allowable, that every conception which is
actually of use for a description of physical nature can like-
wise be regarded as a sign of something real. I believe
that, in striving to illuminate even -the innermost recesses
of the theory of knowledge, we need never give up this as-
sumption, and that it renders possible a view of the world
harmonious in its last details and perfect in itself, which
also satisfies the demands imposed upon thought by the
realist's attitude of mind, but without making it necessary
to give up any of the advantages of the positivist view of
the world.

One of its chief advantages is that the relation of the
separate theories to one another receives due recognition
and a proper measure of value. We felt ourselves impelled
several times in the course of the discussion to explain
clearly to ourselves that, in many cases, there is no possi-
bility, and no urgent need, to distinguish one point of view
from the others as the only true one. It can never be
proved that Copernicus alone is in the right, and that
Ptolemy is wrong. There is no logical ground which can
compel us to set up the theory of relativity as the only true
one in opposition to the absolute theory, or to declare that
the Euclidean determinations of measure are merely right
or wrong. The most that can be done is to show that, of
these alternatives, the one view is simpler than the other,
and leads to a more finished, more satisfactory picture of
the world.

Every theory is composed of a network of conceptions
and judgments, and is correct or true if the system of judg-
ments indicates the world of facts uniquely. For, if such a
unique correspondence exists between conceptions and

86 Relations to Philosophy

reality, it is possible, with the assistance of the network of
judgments in the theory, to derive the successive steps in
the phenomena of nature, e.g. to predict occurrences in the
future. And the fulfilment of such prophecies, the agree-
ment between calculation and observation, is the only
means of proving that a theory is true. It is, however, pos-
sible to indicate identically the same set of facts by means
of various systems of judgments; and consequently there
can be various theories in which the criterion of truth is
equally well satisfied, and which then do equal justice to
the observed facts, and lead to the same predictions. They
are merely different systems of symbols, which are allo-
cated to the same objective reality: different modes of ex-
pression which reproduce the same set of facts. Amongst
all the possible views which, contain the same nucleus oi
truth in this way, there must be one which is simplest ; and
our reason for preferring just this one is not founded upon
reasons of practical economy, a sort of mental indolence (as
has been held by some) . There is a logical reason for it, in-
asmuch as the simplest theory contains a minimum number
of arbitrary factors. The more complicated views neces-
sarily contain superfluous conceptions, of which I can dis-
pose at pleasure, and which are consequently not condi-
tioned by the facts under consideration; about which, there-
fore, I am right in asserting that nothing real corresponds
to them, regarded apart from the other conceptions. In the
case of the simplest theory, on the other hand, the role of
each particular conception is made imperative by the facts :
such a theory forms a system of symbols, all of them in-
dispensable. .Lorentz's aether-theory (v. p. 10), for
example, declares one co-ordinate system to be unique
among all others, but does not essentially afford the means
of ever actually specifying this system. His theory is thus
encumbered with the conception of absolute motion,

Relations to Philosophy 87

whereas the conception of relative motion suffices for a
unique description of the facts. The former is never
capable of application alone, but only in certain combina-
tions, which are embraced in the conception of relative
motion.

Now, the conceptions of space and time, in the form in
which they have hitherto occurred, in physics are included
among these superfluous factors. This we have recognized
as a result of the general theory of relativity. They, too,
cannot be applied separately; but only in so far as they
enter into the conception of the space-time coincidence of
events. We may therefore reiterate that only in this union
do they indicate something real, but not when taken alone.

We see how stupendous is the theoretical range of these
new views. Einstein's analysis of the conceptions of space
and time belongs to the same category of philosophic evolu-
tion as David Hume's criticism of the ideas of substance
and causality. In what way this development will continue,
we cannot yet say. The method which characterizes it is
the only fruitful one for the theory of knowledge, consist-
ing as it does in a searching criticism of the fundamental
ideas of science, stripping off everything that is superfluous
and with ever-increasing clearness exposing the ultimate
pure content.

INDEX

Adams, 65.

Aether, 8, 20.

Arbitrary transformations, 54.

Berkeley, 81.

Bruno, Giordano, 68, 71.

Centrifugal forces, 39, 40, 44.

Clocks, 50.

Conservation of Energy and Mass, 19.

Continuity, 61, 62.

Copernicus, 5, 67, 85.

Distances, 80.

Einstein, 4, 37, 40-45, 53, 59, 61-64,

72, 73, 87,

Equivalence, Principle of, 43, 60.
Euclid, 34, 35, 46, 55.

Four-dimensional manifold, 52.

Galilei, 61.

Gaussian co-ordinates, 29.
Geometry, Foundations of, 34.
Gravitation, 42, 48.

Helmholtz, 33, 59.
Hume, 87.

Inertial systems, 38.
Kant, 79, 80.

Leverrier, 63, 65.
Line-element, 57.
Lobatschewsky, 48.

89

Locke, 81.

Lorentz and Fitzgerald, 2, 11, 17,

Mach, 37, 39, 82, 83.
Mercury, Perihelion of, 63.
Michelson and Morley, 3, 11, 13, 16.
Minkowski, 51, 66.

Newton, 2, 4, 38, 39, 40, 69, 70, 71,
Non-Euclidean geometry, 48.
Objective and Subjective, 76.

Pearson, 39, 83.
Poincare*, 24-7, 32, 33, 34.
Positivism, 82, 83, 84.
Potentials, gravitational, 60, 71.
Ptolemy, 32.

Reality, 22, 23.
Relativity of length, 17.
Riemann, 33, 48, 59.
Rotation, 39.

Schopenhauer, 81.

Space, 33.

Space, spherical, 73.

Space and Time, 2, 4, 5, 23, 36, 53,

76.

Special Theory of Relativity, 7 et seq.
Straight lines, 30, 31.

Tensor, 62.

Trouton and Noble, 9, 11.

World-line, 51, 60, 61.
World-point, 57.

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