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NSTEIN'S THEORY OBRELATWITY
MAX BORN
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BOSTON UNIVERSITY COLLEGE OF LIBERAL ARTS
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CHESTER C. CORBIN LIBRARY FUND
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Einstein's theory of relativity

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EINSTEIN'S THEORY
OF RELATIVITY
BY
MAX BORN
PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF GOTT1NGEN

TRANSLATED BY
HENRY L. BROSE,
CHRIST CHURCH, OXFORD

M.A.

WITH I35 DIAGRAMS AND A PORTRAIT

NEW YORK E. P. DUTTON AND COMPANY
PUBLISHERS
Printed in Great Britain
I

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2 2.
FROM THE PREFACE TO THE
FIRST EDITION
THIS book is an elaboration of certain lectures which were given last winter to a somewhat considerable audience. The difficulty which persons not conversant with mathematics and physics experience
in understanding the theory of relativity seems to me to
be due for the most part to the circumstance that they are not familiar with the fundamental conceptions and facts of physics, in particular of mechanics. During the lectures I therefore showed some quite simple qualitative experiments to serve as an introduction to such concep-
tions as velocity, acceleration, mass, force, intensity of field,
and so forth. In my endeavour to find a similar means,
adapted to book purposes, the semi-historical method of representation here chosen occurred to me, and I hope I have succeeded in avoiding the uninspiring method of the elementary text books of physics. But it must be emphasised that the historical arrangement has been selected only as a cloak which is to bring into stronger relief the outline of the main theme, the logical relationship. Having once started this process I found myself compelled to con-
tinue, and in this way my undertaking increased to the
dimensions of this book.
The reader is assumed to have but little mathematical
knowledge. I have attempted to avoid not only the higher mathematics but even the use f of elementary functions, such as logarithms, trigonometrical functions, and so forth. Nevertheless, proportions, linear equations, and occasionally squares and square roots had to be intro-
duced. I advise the reader who is troubled with the formulae to pass them by on the first reading and to seek
to arrive at an understanding of the mathematical symbols

vi THE THEORY OF RELATIVITY

from the text itself. I have made abundant use of figures and graphical representations. Even those who are un-

practised in the use of co-ordinates will learn to read the

curves easily.
The philosophical questions to which the theory of

relativity gives rise will only be touched on in this book.

Nevertheless a definite logical point of view is maintained

am throughout. I believe I

right in asserting that this

view agrees in the main with Einstein's own opinion.

Moritz Schlick takes up a similar view in his valuable work "Allgemeine Erkenntislehre " (The General Theory

of Knowledge).

Of the other books which I have used I should like
to quote, above all, Ernst Mach's classical ''Mechanics"

(which has appeared in English), and then the very lucidly

written volume by E. T. Whittaker, "A History of the

Theories of Aether and Electricity" (London, Longmans,

& Green

Co., 19 10), and the comprehensive account of

the Theory of Relativity given by Hermann Weyl in his

" Space, Time, Matter " (English translation published
& by Messrs. Methuen Co., Ltd., 1922). Anyone who
wishes to penetrate further into Einstein's doctrines must
study the latter work. It is impossible to enumerate the countless books and essays from which I have drawn more

or less directly. In conformity with the character of the

book I have refrained from giving references.

MAX BORN

Frankfurt on the Main
June, 1920

PREFACE TO THE THIRD EDITION

APART from a number of minor alterations, this edition differs from its two predecessors in that the

chapter on Einsteinian dynamics has been revised.

Previously, in forming the acceleration, we did not dis-

tinguish sharply between time and proper time, and we

used Minkowski's covariant force-vector in place of ordin-

ary force ;

this of

course increased the

difficulty of

under-

standing a chapter which was, from the outset, not easy.
Dr. W. Pauli, jun., called my attention to a method of

deriving the relativistic formula of mass proposed by Lewis

and Tolman, which fitted in admirably with the scheme of

this book, as it linked up with the conception of momentum

in the same way as the account of mechanics here chosen. The chapter on Einsteinian dynamics was revised in con-

formity with this point of view this also entailed some ;
alterations in the manner of presenting ordinary mechanics.

It is hoped that these changes will simplify the reading.

I should not like to lose this opportunity of thanking
Dr. W. Pauli for his advice. His great work on the

theory of relativity which has appeared as Article 19 in

the fifth volume of the " Enzyklopadie der mathematischen

Wissenschaften," which appeared recently, has been of

great service to me. It is to be recommended foremost

of all to those who wish to become intimately acquainted

with the theory of relativity.

MAX BORN

GOTTINGEN 6th March, 1922

.

CONTENTS

CHAPTER I
Geometry and Cosmology

§ i. The Origin of the Art of Measuring Space and Time

7

§ 2. Units of Length and Time .

7

§ 3. Origin and Co-ordinate System .

8

§ 4. The Axioms of Geometry .

9

§ 5. The Ptolemaic System

10

§ 6. The Copernican System

11

§ 7. The Elaboration of the Copernican Doctrine

13

CHAPTER II
The Fundamental Laws of Classical Mechanics

§ 1. Equilibrium and the Conception of Force .

15

— § 2. The Study of Motions Rectilinear Motion

16

§ 3. Motion in a Plane

23

§ 4. Circular Motion

24

—.... § 5. Motion in Space
§ 6. Dynamics The Law of Inertia
§ 7. Impulses

26 27 28

.... § 8. The Law of Impulses .
§ 9. Mass

29 30

§ 10. Force and Acceleration
— §11. Example Elastic Vibrations

32 34

§12. Weight and Mass

36

§13. Analytical Mechanics

39

§ 14. The Law of Energy .

4i

§ 15. Dynamical Units of Force and Mass

4S

CHAPTER III
The Newtonian World-System

§ 1. Absolute Space and Absolute Time .

4S

§ 2. Newton's Law of Attraction
..... General Gravitation ..... §4- Celest'al Mechanics

5*
53 56.

§ 5. The Relativity Principle of Classical Mechanics
.... § 6. Limited Absolute Space .... §7- Galilei Transformations
...... §8. Inertial Forces

59'
61 62 67

§9- Centrifugal Forces and Absolute Space

69

b

ix

THE THEORY OF RELATIVITY

CHAPTER IV

.... The Fundamental Laws of Optics

§ i. The Ether

.

§ 2. The Corpuscular and the Undulatory Theory

§ 3. The Velocity of Light § 4. Fundamental Conceptions of the Wave Theory Interference

§ 5. Polarisation and Transversality of Light \\ aves

§ 6. The Ether as an Elastic Solid .

§ 7. The Optics of Moving Bodies

§ 8. The Doppler Effect .
.... § 9. The Convection of Light by Matter
§ 10. Aberration

§11. Retrospect and Future Prospects

CHAPTER V
The Fundamental Laws of Electrodynamics
§ 1. Electrostatics and Magnetostatics § 2. Voltaic Electricity and Electrolysis § 3. Resistance and Heating due to Currents § 4. Electromagnetism § 5. Faraday's Lines of Force § 6. Magnetic Induction . § 7. Maxwell's Contact Theory § 8. Displacement Currents § 9. The Electromagnetic Theory of Light § 10. The Luminiferous Ether . § 1 1. Hertz' Theory of Moving Bodies §12. Lorentz' Theory of Electrons § 13. Electromagnetic Mass § 14. Michelson and Morley's Experiment . §15. The Contraction Hypothesis

CHAPTER VI
Einstein's Special Principle of Relativity
§ 1. The Conception of Simultaneity . § 2. Einstein's Kinematics and Lorentz' Transformations
A § 3. Geometrical Representation of Einstein's Kinematics
§ 4. Moving Measuring-rods and Clocks § 5. Appearance and Reality § 6. The Addition of Velocities . § 7. Einstein's Dynamics . § 8. The Inertia of Energy § 9. The Optics of Moving Bodies § 10. Minkowski's Absolute World

CHAPTER VII
.... Einstein's General Theory of Relativity
§1. The Relativity of Arbitrary Motions § 2. The Principle of Equivalence
.... §3. The Failure of Euclidean Geometry
..... § 4. Geometry on Curved Surfaces

75 76 79 83 90 93 102 104
no
120 122
125 134 137 139 142 147 149 153 155 160 162 168 175 180 184
192 198 200 206 210 217 221 230 237 242
247 250 254 257

CONTENTS

XI

§ 5. The Two-dimensional Continuum

....

PAOI 2 62

§ 6. Mathematics and Reality

264

§ 7. The Measure-determination of the Space-time Continuum

268

§ 8. The Fundamental Laws of the New Mechanics .

272

§ 9. Mechanical Consequences and Confirmations

275

§ 10. Optical Consequences and Confirmations
...... §11. Macrocosm and Microcosm
§ 12. Conclusion

2 80 286 289

[SDKS.
591

EINSTEIN'S
THEORY OF RELATIVITY

INTRODUCTION

Das schonste Gliick des denkenden Menschen ist, das Erforschliche
— erforscht zu haben und das Unerforschliche ruhig zu verehren. Goethe.

THE world is not presented to the reflective mind as a finished product. The mind has to form its picture

from innumerable sensations, experiences, communica-

tions, memories, perceptions. Hence there are probably not

two thinking people whose picture of the world coincides in

every respect.
When an idea in its main lines becomes the common property

of large numbers of people, the movements of spirit that are called religious creeds, philosophic schools, and scientific systems

arise ; they present the aspect of a chaos of opinions, of articles of faith, of convictions, that resist all efforts to disentangle

them. It seems a sheer impossibility to find a thread that

will guide us along a definite path through these widely ramified

doctrines that branch off perchance to recombine at other points.

What place are we to assign to Einstein's theory of rela-

tivity, of which this book seeks to give an account ? Is it

only a special part of physics or astronomy, interesting in

itself but of no great importance for the development of the

human spirit ? Or is it at least a symbol of a particular trend

of thought characteristic of our times ? Or does it itself,

We indeed, signify a "world-view" (Weltanschauung) ?

shall

be able to answer these questions with confidence only when we

have become acquainted with the content of Einstein's doctrine.

But we may be allowed to present here a point of view which,

even if only roughly, classifies the totality of all world-views and ascribes to Einstein's theory a definite position within

a uniform view of the world as a whole. The world is composed of the ego and the non-ego, the inner
world and the outer world. The relations of these two poles

2 THE THEORY OF RELATIVITY

are the object of every religion, of every philosophy. But the

part that each doctrine assigns to the ego in the world is different.
The importance of the ego in the world-picture seems to me a measure according to which we may order confessions of

faith, philosophic systems, world-views rooted in art or science,
like pearls on a string. However enticing it may be to pursue

this idea through the history of thought, we must not diverge

too far from our theme, and we shall apply it only to that

special realm of human thought to which Einstein's theory
— belongs to natural science.

Natural science is situated at the end of this series, at the

point where the ego, the subject, plays only an insignificant

part ; every advance in the mouldings of the conceptions of

physics, astronomy, and chemistry denotes a further step

towards the goal of excluding the ego. This does not, of course,

deal with the act of knowing, which is bound to the subject,

but with the finished picture of Nature, the basis of which is

the idea that the ordinary world exists independently of and

uninfluenced by the process of knowing.

The doors through which Nature imposes her presence on

us are the senses. Their properties determine the extent of

what is accessible to sensation or to intuitive perception.

The further we go back in the history of the sciences, the more

we find the natural picture of the world determined by the

qualities of sense. Older physics was subdivided into mechanics,

We acoustics, optics, and theory of heat.

see the connexions

with the organs of sense, the perceptions of motion, impressions

of sound, light, and heat. Here the qualities of the subject

are still decisive for the formation of conceptions. The deve-

lopment of the exact sciences leads along a definite path from

this state to a goal which, even if far from being attained,

yet lies clearly exposed before us : it is that of creating a picture

of nature which, confined within no limits of possible perception

or intuition, represents a pure structure of conception, con-

ceived for the purpose of depicting the sum of all experiences

uniformly and without inconsistencies.

Nowadays mechanical force is an abstraction which has

only its name in common with the subjective feeling of force.

Mechanical mass is no longer an attribute of tangible bodies

but is also possessed by empty spaces filled only by ether

radiation. The realm of audible tones has become a small

province in the world of inaudible vibrations, distinguishable

physically from these solely by the accidental property of the

human ear which makes it react only to a definite interval of

frequency numbers. Modern optics is a special chapter out

of the theory of electricity and magnetism, and it treats of the

INTRODUCTION

8

electro-magnetic vibrations of all wave-lengths, passing from the shortest 7-rays of radioactive substances (having a wavelength of one hundred millionth of a millimetre) over the Rontgen rays, the ultraviolet, visible light, the infra-red, to the longest
wireless (Hertzian) waves (which have a wave-length of many
kilometres). In the flood of invisible light that is accessible to the mental eye of the physicist, the material eye is almost blind, so small is the interval of vibrations which it converts
into sensations. The theory of heat, too, is but, a special part of mechanics and electro-dynamics. Its fundamental conceptions of absolute temperature, of energy, and of entropy belong to the most subtle logical configurations of exact science, and, again, only their name still carries a memory of the subjec-
tive impression of heat or cold.
Inaudible tones, invisible light, imperceptible heat, these
constitute the world of physics, cold and dead for him who
wishes to experience living Nature, to grasp its relationships as a harmony, to marvel at her greatness in reverential awe. Goethe abhorred this motionless world. His bitter polemic
against Newton, whom he regarded as the personification of a
hostile view of Nature, proves that it was not merely a question of an isolated struggle between two investigators about individual questions of the theory of colour. Goethe is the representative of a world-view which is situated somewhere near the opposite end of the scale suggested above (constructed according to the relative importance of the ego), that is, the end opposite to that occupied by the world-picture of the exact sciences. The essence of poetry is inspiration, intuition, the visionary comprehension of the world of sense in symbolic forms. But the source of poetic power is experience, whether it be the clearly conscious perception of a sense-stimulus, or
the powerfully represented idea of a relationship or connexion.
What is logically formal and rational plays no part in the world-
picture of such a type of gifted or indeed heaven-blessed
spirit. The world as the sum of abstractions that are connected
only indirectly with experience is a province that is foreign to it. Only what is directly presented to the ego, only what can be felt or at least represented as a possible experience is real to
it and has significance for it. Thus to later readers, who survey the development of exact methods during the centurv after Goethe's time and who measure the power and significance of Goethe's works on the history of natural science by their
fruits, these works appear as documents of a visionary mind,
as the expression of a marvellous sense of one-ness with (Ein-
fuhlung) the natural relationships, but his physical assertions will seem to such a reader as misunderstandings and fruit]- SS

4 THE THEORY OF RELATIVITY

rebellions against a greater power, whose victory was assured

even at that time.
Now in what does this power consist, what is its aim and

device ?
It both takes and renounces. The exact sciences presume to aim at making objective statements, but they surrender their

absolute validity. This formula is to bring out the following

contrast.
All direct experiences lead to statements which must be

allowed a certain degree of absolute validity. If I see a red flower, if I experience pleasure or pain, I experience events which
it is meaningless to doubt. They are indubitably valid, but only for me. They are absolute, but they are subjective. All seekers after human knowledge aim at taking us out of the narrow circle of the ego, out of the still narrower circle of the ego that is bound to a moment of time, and at establishing common ground with other thinking creatures. It first establishes a link with the ego as it is at another moment, and then with other human beings or gods. All religions, philosophies,

and sciences have been evolved for the purpose of expanding

the ego to the wider community that " we " represent. But

We the ways of doing this are different.

are again confronted

by the chaos of contradictory doctrines and opinions. Yet we

no longer feel consternation, but order them according to the

importance that is given to the subject in the mode of com-

prehension aimed at. This brings us back to our initial prin-

ciple, for the completed process of comprehension is the

world-picture. Here again the opposite poles appear.

' ,

The minds of one group do not wish to deny or to sacrifice

the absolute, and they therefore remain clinging to the ego.

They create a world-picture that can be produced by no sys-

tematic process, but by the unfathomable action of religious,

artistic, or poetic means of expression in other souls. Here

faith, pious ardour, love of brotherly communion, but often

also fanaticism, intolerance, intellectual suppression hold sway.

— — The minds of the opposite group sacrifice the absolute.
They discover often with feelings of terror the fact that

inner experiences cannot be communicated. They no longer

fight for what cannot be attained, and they resign themselves.

But they wish to reach agreement at least in the sphere of the

attainable. They therefore seek to discover what is common

in their ego and in that of the other egos ; and the best that

was there found was not the experiences of the soul itself, not

sensations, ideas, or feelings, but abstract conceptions of the
— simplest kind numbers, logical forms ; in short, the means of
expression of the exact sciences. Here we are no longer con-

INTRODUCTION

cerncd with what is absolute. The height of a cathedral does

not, in the special sphere of the scientist, inspire reverence, but

is measured in metres and centimetres. The course of life is

no longer experienced as the running out of the sands of time,

but is counted in years and days. Relative measures take tin-
place of absolute impressions. And we get a world, narrow,

one-sided, with sharp edges, bare of all sensual attraction, of

all colours and tones. But in one respect it is superior to other

world-pictures : the fact that it establishes a bridge from mind

to mind cannot be doubted. It is possible to agree as to

whether iron has a specific gravity greater than wood,

whether water freezes more readily than mercury, whether
Sirius is a planet or a star. There may be dissensions, it may sometimes seem as if a new doctrine upsets all the old " facts," yet he who has not shrunk from the effort of penetrating into

the interior of this world will feel that the regions known with

certainty are growing, and this feeling relieves the pain which

arises from solitude of the spirit, and the bridge to kindred

spirits becomes built.
We have endeavoured in this way to express the nature of
scientific research, and now we can assign Einstein's theory of

relativity to its category.

In the first place, it is a pure product of the striving after

the liberation of the ego, after the release from sensation and

We perception.

spoke of the inaudible tones, of the invisible

We light, of physics.

find similar conditions in related sciences,

in chemistry, which asserts the existence of certain (radioactive)

substances, of which no one has ever perceived the smallest
— trace with any sense directly or in astronomy, to which we
refer below. These " extensions of the world," as we might

call them, essentially concern sense-qualities. But everything

takes place in the space and the time which was presented to

mechanics by its founder, Newton. Now, Einstein's discovery

is that this space and this time are still entirely embedded in

the ego, and that the world-picture of natural science becomes

more beautiful and grander if these fundamental conceptions

are also subjected to relativization. Whereas, before, space

was closely associated with the subjective, absolute sensation

of extension, and time with that of the course of life, they are
now purely conceptual schemes, just as far removed from direct

perception as entities, as the whole region of wave-lengths of

present-day optics is inaccessible to the sensation of light except

for a very small interval. But just as in the latter case, the

space and time of perception allow themselves to be ordered

without giving rise to difficulties, into the system of physical

conceptions. Thus an objectivation is attained, which has

6 THE THEORY OF RELATIVITY

manifested its power by predicting natural phenomena in

We a truly wonderful way.

shall have to speak of this in

detail in the sequel.
Thus the achievement of Einstein's theory is the relativization and objectivation of the conceptions of space and time. At the present day it is the final picture of the world as

presented by science.

CHAPTER I GEOMETRY AND COSMOLOGY

i. The Origin of the Art of Measuring Space and Time

THE physical problem presented by space and time is nothing more than the familiar task of fixing numerically a place and a point of time for every phy event, thus enabling us to single it out, as it were, from the chaos of the co-existence and succession of things.
The first problem of Man was to find his way about on the
earth. Hence the art of measuring the earth (geodesy) became the source of the doctrine of space, which derived its name " geometry " from the Greek word for earth. From the v outset, however, the measure of time arose from the regular change of night and day, of the phases of the moon and of the seasons. These phenomena forced themselves on Man's attention and first moved him to direct his gaze to the stars, which were the source of the doctrine of the anivc cosmology. Astronomic science applied the teachings of g metry that had been tested on the earth to the heavenly regions, allowing distances and orbits to be defined. For this purpose it gave the inhabitants of the earth the celestial (astronomic)
measure of time which taught Man to distinguish between
Past, Present, and Future, and to assign to each thing its place
in the realm of Time.

2. Units of Length and Time

The foundation of every space- and time-measurement is
laid by fixing the unit. A datum of length, " so and so many

metres."

denotes

the

ratio

of

the

length

to

be

measured

to

the "

length of a metre. A time-datum of " so and so many seconds

denotes the ratio of the time to be measured to the duration

of a second. Thus we are always dealing with ratio-nunV:

relative data concerning the units. The latter themselves

are to a high degree arbitrary, and are chosen for reasons of

their being capable of easy reproduction, of being easily

transportable, durable, and so forth.

.
8 THE THEORY OF RELATIVITY
In physics the measure of length is the centimetre (cm.), the hundredth part of a metre rod that is preserved in Paris. This was originally intended to bear a simple ratio to the circumference of the earth, namely, to be the ten-millionth part of a quadrant, but more recent measurements have disclosed that this is not accurately true.
The unit of time in physics is the second (sec), which bears the well-known relation to the time of rotation of the earth on its axis.
3. Origin and Co-ordinate System
But if we wish not only to determine lengths and periods of time, but also to designate places and points of time, further
conventions must be made. In the case of time, which we regard as a one-dimensional con-
figuration, it is sufficient to
specify an origin (or zero-point) Historians reckon dates by counting the years from the birth of Christ. Astronomers choose other origins or initial points, according to the objects of their researches ; these they call epochs. If the unit and the origin are fixed, every event
may be singled out by assigning
a number-datum to it. In geometry in the narrower
sense, the determination of position on the earth, two data
must be given to fix a point. To say " My house is in Baker
Street," is not sufficient to fix it. The number of the house must also be given. In many American towns the streets themselves are numbered. The address No. 25, 13th Street, thus consists of two number-data. It is exactly what mathematicians call a " co-ordinate determination." The earth's surface is covered with a network of intersecting lines, which are numbered, or whose position is determined by a number, distance, or angle (made with respect to a fixed initial or zero-
line).
Geographers generally use geographic longitude (east of Greenwich) and latitude (north or south) (Fig. 1). These determinations at the same time fix the zero-lines from which the co-ordinates are to be counted, namely, for geographical longitude the meridian of Greenwich, and for the latitude the

GEOMETRY AND COSMOLOGY 9
equator. In investigations of plane geometry we generally use rectilinear [Cartesian) co-ordinates (Fig. 2), x, y, which signify the distances from two mutually perpendicular coordinate axes ; or, occasionally, we also use oblique co-ordinates
(Fig. 3), polar co-ordinates (Fig. 4), and others. When the

Fig. 2.

Fig. 3.

co-ordinate system has been specified, we can seek out each

point or place if two numbers are given.

In precisely the same way we require three co-ordinates

to fix points in space. It is simplest to choose mutually per-

pendicular

rectilinear

co-ordinates

again ;

we

denote

them

by

x,y,z (Fig. 5).

Fig. 4.

Fig. 5.

4. The Axioms of Geometry
Ancient geometry, regarded as a science, was less concerned with the question of determining positions on the earth's surface, than with determining the size and form of areas,
We figures in space, and the laws governing these questions.
see traces of the origin of this geometry in the art of surveying
and of architecture. That is also the reason why it managed
without the conception of co-ordinates. First and foremost,

10 THE THEORY OF RELATIVITY
geometric theorems assert properties of things that are called points, straight lines, planes. In the classic canon of Greek geometry, the work of Euclid (300 B.C.), these things are not defined further but are only denominated or described. Thus
we here recognize an appeal to intuition. You must already know what is a straight line if you wish to take up the study
of geometry. Picture the edge of a house, or the stretched cable of your surveying instruments, form an abstraction from what is material and you will get your straight line. Next,
laws are set up that are to hold between these configurations of abstraction, and it is to the credit of the Greeks to have made the great discovery that we need assume only a small number of these theorems to make all others come out of them correctly with logical inevitableness. These theorems, which are used as the foundation, are the axioms. Their correctness cannot be proved. They do not arise from logic but from other
sources of knowledge. What these sources are has formed the
subject of the theories of all the philosophies of the succeeding
centuries. Scientific geometry itself, up to the end of the 18th century, accepted these axioms as given, and built up its purely deductive system of theorems on them.
We shall not be able to avoid discussing in detail the question
of the meaning of the elementary configurations called point, straight line, and so forth, and the grounds of our knowledge of the geometric axioms. For the present, however, we shall adopt the standpoint that we are clear about these things, and we shall thus operate with the geometric conceptions in the
way we learned (or should have learned) at school, and in the way numberless generations of people have done, without scruples. The intuitional truth of numerous geometric theo-
rems, and the utility of the whole system in giving us bearings in our ordinary real world is to suffice for the present as our
justification for using them.
5. The Ptolemaic System
To the eye the heavens appear as a more or less flat dome to which the stars are attached. But in the course of a day the whole dome turns about an axis whose position in the heavens is denoted by the pole-star. So long as this visual appearance was regarded as reality an application of geometry from the earth to astronomic space was superfluous, and was, as a matter of fact, not carried out. For lengths and distances measurable with earthly units were not present. To denote
the positions of the stars only the apparent angle that the line of vision from the observer to the star formed with the

GEOMETRY AND COSMOLOGY 11
horizon and with another appropriately chosen plane had to be known. At this stage of knowledge the earth's surface wa considered at rest and was the eternal basis of the universe. The words " above " and " below " had an absolute meaning and when poetic fancy or philosophic speculation undertook to estimate the height of the heavens or the depth of Tartarus, the meaning of these terms required no word of elucidation. At this stage scientific concepts were still being drawn from the abundance of subjective data. The world-system called after Ptolemy (150 a.d.) is the scientific formulation of this mental attitude. It was already aware of a number of detailed facts concerning the motion of the sun, the moon, and the planets, and it had a considerable theoretical grasp of them, but it retained the notion that the earth is at rest and that the stars are revolving about it at immeasurable distances. Its orbits were determined as circles and epi-cycles according to the laws of earthly geometry, yet astronomic space was not actually through this subjected to geometry. For the orbits were fastened like rings to the crystal shells, which, arranged in strata, signified the heavens.
6. The Copernican System
It is known that Greek thinkers had already discovered the spherical shape of the earth and ventured to take the first
steps from the geometric world-systems of Ptolemy to higher abstractions. But only long after Greek civilization and culture had died, did the peoples of other countries accept the spherical shape of the earth as a physical reality. This is the first truly great departure from the evidence of our eyes, and at the same time the first truly great step towards relativization. Again centuries have passed since that first turningpoint, and what was at that time an unprecedented discovery
has now become a platitude for school-children. This makes it difficult to convey an impression of what it signified to thinkers
to see the conceptions " above " and " below " lose their absolute meaning, and to recognize the right of the inhabitants of the antipodes to call " above " in their regions what we call " below " in ours. But after the earth had once been circumnavigated all dissentient voices became silent. For this reason, too, the discovery of the sphericity of the earth offered no reason for strife between the objective and the subjective view of the world, between scientific research and the church. This strife broke out only after Copernicus (1543) displaced the earth from its central position in the universe and created the helio-
centric world-system.

12 THE THEORY OF RELATIVITY
In itself the process of relativization was hardly advanced by this, but the importance of the discovery for the develop-
ment of the human spirit consisted in the fact that the earth, mankind, the individual ego, became dethroned. The earth became a satellite of the sun and carried around in space
the peoples swarming on it. Similar planets of equal importance accompany it in describing orbits about the sun.
Man is no longer important in astronomy, except for himself.
But still more, none of these amazing facts arise from ordinary observation (such as is the case with a circumnavigation of the globe), but from observations which were, for the time in question, very delicate and subtle, from different calculations of planetary orbits. The evidence was at any rate such as
was neither accessible to all men nor of importance for everyday
life. Ocular evidence, intuitive perceptions, sacred and pagan tradition alike speak against the new doctrine. In place of the visible disc of the sun it puts a ball of fire, gigantic beyond
imagination; in place of the friendly lights of the heavens, similar balls of fire at inconceivable distances, or spheres like the
earth, that reflect light from other sources; and all visible measures are to be regarded as deception, whereas immeasurable distances and incredible velocities are to represent the true state of affairs. Yet this new doctrine was destined to be victorious. For it drew its power from the burning wish of all thinking minds to comprehend all things of the material
— world, be they ever so unimportant for human existence,
as a co-ordinate unity to make them a permanent possession of the intellect and communicable to others. In this process, which constitutes the essence of scientific research, the human spirit neither hesitates nor fears to doubt the most striking facts of visual perception, and to declare them to be illusions, but prefers to resort to the most extreme abstractions rather
than exclude from the scientific description of Nature one
established fact, be it ever so insignificant. That, too, is why
the church, at that time the carrier of the subjective worldview then dominant, had to persecute the followers of the
Copernican doctrine, and that is why Galilei had to be brought
before the inquisitorial tribunal as a heretic. It was not so
much the contradictions to traditional dogmas as the changed
attitude towards spiritual events that called this struggle into being. If the experience of the soul, the direct perception of things, was no longer to have significance in Nature, then religious experience might also one day be subjected to doubt.
However far even the boldest thinkers of those times were removed from feelings of religious scepticism, the church
scented the enemy.

GEOMETRY AND COSMOLOGY 18
The great relativizing achievement of Copernicus was the root of all the innumerable similar but lesser relativizations of growing natural science until the time when Einstein's discovery ranged itself as a worthy result alongside that of
its great predecessor.
But now we must sketch in a few words the cosmos as mapped out by Copernicus.
We have first to remark that the conceptions and laws
of earthly geometry can be directly applied to astronomic space. In place of the cycles of the Ptolemaic world, which
were supposed to occur on surfaces, we now have real orbits in space, the planes of which may have different positions. The centre of the world-system is the sun. The planets describe their circles about it, and one of them is the earth, which rotates about its own axis, and the moon in its turn revolves in its orbit about the earth. But beyond, at enormous dis-
tances, the fixed stars are suns like our own, at rest in space. Copernicus' constructive achievement consists in the fact that with this assumption the heavens must exhibit all these pheno-
mena which the traditional world-system was able to explain only by means of complicated and artificial hypotheses. The alternation of day and night, the seasons, the phenomena
of the moon's phases, the winding planetary orbits, all these things become at one stroke clear, intelligible, and accessible to
simple calculations.
7. The Elaboration of the Copernican Doctrine
The circular orbits of Copernicus soon no longer sufficed to account for the observations. The real orbits were evidently considerably more complicated. Now, an important point for the new view of the world was whether artificial constructions, such as the epicycles of the Ptolemaic system or an improvement in the calculations of the orbits could be successfully carried out without introducing complications. It was the immortal achievement of Kepler (1618) to discover the simple and striking laws of the planetary orbits, and hence to save the Copernican system at a critical period. The
orbits are not, indeed, circles about the sun, but curves closely related to circles, namely, ellipses, in one focus of which the sun is situated. Just as this law determines the form of the orbits in a very simple manner, so the other two laws of Kepler determine the sizes of the orbits and the velocities with
which they are traversed. Kepler's contemporary, Galilei (1610), directed a telescope,
which had just then been invented, at the heavens and

14 THE THEORY OF RELATIVITY

discovered the moons of Jupiter. In them he recognized a

microscopic model of the planetary system and saw Coper-

nicus' ideas as optical realities. But it is Galilei's greater merit

to have developed the principles of mechanics, the application of

which to planetary orbits by Newton (1867) brought about the

completion of the Copernican world-system.

Copernicus' circles and Kepler's ellipses are what modern

science calls a kinematic or phoronomic description of the orbits,

namely, a mathematical formulation of the motions which does

not contain the causes and relationships that bring about these

same motions. The causal expression of the laws of motion

is the content of dynamics or kinetics, founded by Galilei.

Newton has applied this doctrine to the motions of the heavenly

bodies, and by interpreting Kepler's laws in a very ingenious way he introduced the causal conception of mechanical force

into astronomy. Newton's law of gravitation proved its

superiority over the older theories by accounting for all the

deviations from Kepler's laws, the so-called perturbations of

orbits, which refinements in the methods of observation had

in the meantime brought to light.

This dynamical view of the phenomena of motion in astro-

nomic space, however, at the same time demanded a more

precise formulation of the assumptions concerning space and

time. These axioms occur in Newton's work for the first time

as explicit definitions. It is therefore justifiable to regard the

theorems that held up to the advent of Einstein's theory as

expressions of Newton's doctrine of space and time. To under-

stand them it is absolutely necessary to have a clear survey

of the fundamental laws of mechanics, and that, indeed, from

a point of view which places the question of relativity in the

foreground, a standpoint that is usually neglected in the

We elementary text-books.

shall therefore next have to

discuss the simplest facts, definitions, and laws of mechanics.

CHAPTER II
THE FUNDAMENTAL LAWS OF CLASSICAL MECHANICS
i. Equilibrium and the Conception of Force
HISTORICALLY, mechanics took its start from the
doctrine of equilibrium or statics ; logically, too, the
development from this point is the most natural one. The fundamental conception of statics is force. It is derived from the subjective feeling of exertion experienced when we
perform work with our bodies. Of two men he is the stronger who can lift the heavier stone or stretch the stiffer bow. This
measure of force, with which Ulysses established his right
among the suitors, and which, indeed, plays a great part in the stories of ancient heroes, already contains the germ of the objectivation of the subjective feeling of exertion. The next step was the choice of a unit of force and the measurement
of all forces in terms of their ratios to the unit of force, that is, the relativization of the conception of force. Weight, being
the most evident manifestation of force, and making all things tend downwards, offered the unit of force in a convenient form, namely, a piece of metal which was chosen as the unit of weight through some decree of the state or of the church. Nowadays it is an international congress that fixes the units. The unit
of weight in technical matters is the weight of a definite piece
of platinum in Paris. This unit, called the gramme (grm.) will be used in the sequel till otherwise stated. The instrument used to compare the weights of different bodies is the balance.
Two bodies have the same weight, or are equally heavy,
when, on being placed in the two scales of the balance, they do not disturb its equilibrium. If we place two bodies found to be equally heavy in this manner in one pan of the balance, but, in the other, a body such that the equilibrium is again not disturbed, then this new body has twice the weight of
either of the other two. Continuing in this way we get,
starting from the unit of weight, a set of weights with the help
of which the weight of every body may be conveniently deter-
mined.
15

16 THE THEORY OF RELATIVITY

It is not our task here to show how these means enabled

man to find and interpret the simple laws of the statics of

We rigid bodies, such as the laws of levers.

here introduce

only just those conceptions that are indispensable for an

understanding of the theory of relativity.
Besides the forces that occur in man's body or in that

of his domestic pets he encounters others, above all in the
events that we nowadays call elastic. The force necessary to stretch a cross-bow or any other bow belongs to this category.

Now, these can easily be compared with weights. If, for

example, we wish to measure the force that is necessary to stretch a spiral spring a certain distance (Fig. 6), then we find

by trial what weight must be suspended from it to effect equilibrium for just this extension. Then the force of the

spring is equal to that of the weight, except that the former exerts a pull upwards but the latter

X~

? downwards. The principle that

* action and reaction are equal and

opposite in the condition of equilib-

rium has tacitly been applied.

If such a state of equilibrium

be disturbed by weakening or re-

moving one of the forces, motion

occurs. The raised weight falls

when it is released by the hand sup-

porting it and thus furnishing the

reacting force. The arrow shoots

forth when the archer releases the

FlG 6

string of the stretched bow. Force tends to produce motion. This is

the starting-point of dynamics, which seeks to discover the

laws of this process.

— 2. The Study of Motions Rectilinear Motion

It is first necessary to subject the conception of motion

itself to analysis. The exact mathematical description of the

motion of a point consists in specifying at what place relative

to the previously selected co-ordinate system the point is

situated from moment to moment. Mathematicians use

We formulae to express this.

shall as much as possible avoid

this method of representing laws and relationships, which is

not familiar to everyone, and shall instead make use of a graphi-

cal method of representation. Let us illustrate this for the

simplest case, the motion of a point in a straight line. Let the

unit of length be the centimetre, as usual in physics, and let the

;

LAWS OF CLASSICAL MECHANICS 17

= moving point be at the distance x i cm. from the zero point
or origin at the moment at which we start our considerations
= and which we call the moment t o. In the course of i sec.
suppose the point to have moved a distance of \ cm. to the
= right, so that for t i the distance from the origin amounts
to i-5 cms. In the next second let it move by the same amount
= to x 2 cms., and so forth. The following small table gives
the distances x corresponding to the times t.

8... t.

\

O

T

2

^

A

^

6

7

x

i i-5 2 2-5 3 3*5 4 4-55...

\

We see the same relationship pictured in the successive
lines of Fig. 7, in which the moving point is indicated as a small circle on the scale of distances. Now, instead of drawing a
number of small diagrams, one above the other, we may also

t'6
US

t'-<+

t*3

t=2

t-1

t'O

>>A

2

3

Fig. 7.

draw a single figure in which the x's and the t's occur as coordinates (Fig. 8). In addition, this has the advantage of allowing the place of the point to be depicted not only at the
beginning of each full second but also at all intermediate times,
We need only connect the positions marked in Fig. 7 by a
continuous curve. In our case this is obviously a straight line. For the point advances equal distances in equal times the co-ordinates x, t thus change in the same ratio (or proportionally), and it is evident that the graph of this law is a straight
line. Such a motion is called uniform. The name velocity
v of the motion designates the ratio of the path traversed
to the time required in doing so, or in symbols :

v=

)

18 THE THEORY OF RELATIVITTYV

In our example the point traverses \ cm. of path in each

second. The velocity remains the same throughout and

amounts to J cm. per sec.

The

unit

of

velocity

is

already

fixed

by

this

definition ;

it is the velocity which the point would have if it traversed

i cm. per sec. It is said to be a derived unit, and, without

introducing a new value, we call it cm. per sec. or cm. /sec. To express that the measurement of velocities may be referred

back to measurements of lengths and times in accordance with

formula (i) we also say that velocity has the dimensions length

= divided by time, written thus : [v]

[_yj

or

[L.T""

1 ].

In the

same way we assign definite dimensions to every quantity that allows itself to be built up of the fundamental quantities,

>x
length /, time t, and weight G. When the latter are known the unit of the quantity may at once be expressed by means of
those of length, time, and weight, say, cm., sec. and grm. In the case of great velocities the path % traversed in the
time t is great, thus the graph line has only a small inclination to the x- axis : the smaller the velocity, the steeper the graph.
A point that is at rest has zero velocity and is represented
in our diagram by a straight line parallel to the i!-axis, for the points of this straight line have the same value of % for all
times t (Fig. 9 a) .
If a point is firstly at rest and then at a certain moment suddenly acquires a velocity and moves on with this velocity, we get as the graph a straight line one part of which is bent,
the other being vertical (Fig. 9 b). Similarly broken lines

LAWS OF CLASSICAL MECHANICS 19

represent the cases when a point that is initially moving uni-

formly for a while to the right or to the left suddenly changes

its velocity (Figs. 9 c and 9 d).

If

the velocity before

the

sudden

change is

v x

(say,

3

cms.

— — = per sec), and afterwards v t (say, 5 cms. per sec), then the

increase of velocity is v 2

v x (that is, 5

3

2 cms. per sec,

=i added in each sec). If v 2 is less than v x (say, v 1

cm. per

— — = — sec), then v z v t is negative (namely, 1 3

2 cms. per

sec), and this clearly denotes that the moving point is suddenly

retarded.
If a point experiences a series of sudden changes of velocity then the graph of its motion is a succession of straight lines

joined together (polygon) as in Fig. 10.
If the changes of velocity occur more and more frequently

t

kt

FlQ. 10.

->x

fc-

Fig. 11.

and are sufficiently small, the polygon will no longer be distinguishable from a curved line. It then represents a motion whose velocity is continually changing, that is, one which is
non-uniform, accelerated or retarded (Fig. 11).
An exact measure of the velocity and its change, accelera-
tion, can be obtained in this case only with the aid of the methods of infinitesimal geometry. It suffices for us to imagine the continuous curve replaced by a polygon whose straight sides represent uniform motions with definite velocities. The bends
of the polygon, that is, the sudden changes of velocity, may
be supposed to succeed each other at equal intervals of time,

= say, t

1
- sees.

If, in addition these changes are equally great, the motion is said to be " uniformly accelerated." Let each such change

20 THE THEORY OF RELATIVITY
of velocity have the value w, then if there are n per sec. the total change of velocity per sec. is
(2)

LAWS OF CLASSICAL MECHANICS 21

Then the velocity

after the first interval of time is :

,, second ,,

,,

„ third

,,

,,

= v x

w,

= = v 2

w v x -f-

2w

= + = v3

v2

w

31V,

and so forth.

The point advances

= after the first interval of time to : x,

t v-

n

second ,,

,,

= = + x 2

x x

-f

v

2

n

(v x

n v 2 ) -,

third „

„

x9

=x

t

+vr-=(v n

1

+v

t

+v9)-n»

and so forth. After the nth. interval of time, that is, at the end of the time /, the point will have arrived at

x= + + (v ±

v2

.

.

.

vn )~.

n

But

V V -\1

-\-

2

.

.

.

= + + w + Vn

IW

2Z£>

Z

•

•

•

Htf'

= + + + (1

2

3

. . . n)w.

The sum of the numbers from 1 to n can be calculated quite simply by adding the first and the last ; the second and the second to last ; and so forth ; in each case we get for the sum
+ of the two numbers n 1, and altogether we have — of such

+ + = sums or pairs. Thus we get 1

2

n .

.

.

- (n -\- 1).

If,

further, we replace w by b -, we get
11

+ + = + - = - + v x

v2

vn .

.

.

nt

.

x bt

- (» 1)

2

w

bt

(n ,

,

x

1),

2

thus

— = + = + — ^

(n

1) -

(1

-).

2

«2

n

Here we may choose n to be as great as we please. Then becomes arbitrarily small and we get

x=- bt 2.
2
This signifies that in equal times the paths traversed are proportional to the squares of the times. If, for example,

22 THE THEORY OF RELATIVITY

= the acceleration b 10 metres per sec, then the point traverses

= = 5 metres in the first sec, 5 . 2 2 5 . 4 20 metres in the second

= sec,

5

.

2
3

45 in the third sec, and so forth. This relationship

is represented by a curved line, called a parabola, in the xt

plane (Fig. 13). If we compare the figure with Fig. 12 we see how the polygon approximately represents the continuously
= curved parabola. In both figures the acceleration b 10

has been chosen, and this determines the appearance of the

curves, whereas the units of length and time are unessential.
We may also apply the conception of acceleration to non-

uniformly accelerated motions, by using instead of 1 sec. a

time of observation which is so small that, during it, the motion

may be regarded as uniformly accelerated. The acceleration

itself then becomes continuously variable.

All these definitions become rigorous and at the same time

convenient to handle if the process of sub-division into small

l
6 5
V 3 Z
1

LAWS OF CLASSICAL MECHANICS 23
3. Motion in a Plane
If we wish to study the motion of a point in a plane, our method of representation at once allows itself to be extended
to this case. We take in the plane an ^-co-ordinate system and
erect a rf-axis perpendicular to it (Fig. 14). Then a straight line in the #y/-space corresponds to a rectilinear and uniform motion

Fig. 14.

Fig. 15.

in the #y-plane. For if we project the points of the straight
= line that correspond to the points of time t o, I, 2, 3, . . .
on to the %y-plane, we see that the positional displacement takes place along a straight line and at equal intervals.
Every non-rectilinear but uniform motion is said to be accelerated even if, for example, a curved path is traversed with constant velocity. For in this

case the direction of the velocity

changes although its numerical
value remains constant. An ac-

celerated motion is represented in

the #v/-plane (Fig. 15) by an arbitrary curve. The projection of

this curve into the ^y-plane is

the orbit in the plane (or plane-
orbit). The velocity and the ac-

-*~JT

celeration are again calculated by

Fig. 16.

supposing the curve replaced by

a polygon closely wrapped round the curve. At each corner

of this polygon not only the amount but also the direction of

A the velocity alters.

more exact analysis of the conception

of acceleration would take us too far. It is sufficient to

mention that it is best to project the graph of the moving point

on to the co-ordinate axes x, y, and to follow out the rectilinear motion of these two points, or what is the same, the change

24 THE THEORY OF RELATIVITY

in time of the co-ordinates x, y. The conceptions denned for

rectilinear motions as given above may now be applied to these

We projected motions.

thus get two components of velocity

vx , vy , and two components of acceleration bx , by , that together

fix the velocity or the acceleration of the moving point at

a given instant.

In the case of a plane motion (and also in one that occurs

in space) velocity and acceleration are thus directed magnitudes

(vectors). They have a definite direction and a definite magni-

tude. The latter can be calculated from the components.

For example, we get the direction and magnitude of the velocity

from the diagonal of the rectangle with the sides vx and vy (Fig. 16). Thus, by Pythagoras' theorem, its magnitude is

v = VV + V

•

•

•

(3)

An exactly corresponding result holds for the acceleration.

4. Circular Motion
There is only one case which we wish to consider in greater
detail, namely, the motion of a point in a circular orbit with

Fig. 17.
constant speed (Fig. 17). According to what was said above, it is an accelerated motion, since the direction of the velocity constantly alters. If the motion were unaccelerated the moving
point would move forward from A in a straight line with the
uniform velocity v. But in reality the point is to remain on the circle, and hence it must have a supplementary velocity or acceleration that is directed to the central point M. This
is called the centripetal acceleration. It causes the velocity at a neighbouring point B, which is reached after a short interval t, to have a direction different from that at the point A.
From a point c we next draw the velocities at A and B in a

.

LAWS OF CLASSICAL MECHANICS 25

separate diagram (Fig. 17), paying due regard to their magnitude and direction. Their magnitude will be the same, namely

v, since the circle is to be traversed

with constant speed, but their
direction is different. If we con-
nect the end-points D and E of

the two velocity lines, then the

connecting line is clearly the

supplementary velocity w, which

transforms the first velocity state

We into the second.

thus get an

isosceles triangle CED, having the

base w and the sides v, and we at

once see that the angle a at the

vertex is equal to the angle sub-

tended by the arc AB, which the

Fig. 18.

point traverses, at the centre of

A the circle. For the velocities at and B are perpendicular

MA to the radii

and MB, and hence include the same angle.

MAB Consequently the two isosceles triangles

and CDE are

similar, and we get the proportion

DE AB CD MA

= = MA Now DE w, CD v, and further,

is equal to the

AB radius r of the circle, and

is equal to the arc s except for

a small error that can be made as small as we please by choosing

the time-interval t sufficiently small.
Hence we have

w—

=

s -

or

w

=

s—v

v

r

r

= — = We

now

divide

by t

and

notice that

S -

IS)

v,

0.

t

t

the acceleration

b=

Hence
• (4)

that is, the centripetal acceleration is equal to the square of the velocity in the circle divided by the radius.
This theorem, as we shall see, is the basis of one of the first and most important empirical proofs of Newton's theory of
gravitation.
Perhaps it is not superfluous to have a clear idea of what this uniform circular motion looks like in the graphical representation in the *y£-space. This is obviously produced by allowing the moving point to move upwards regularly

26 THE THEORY OF RELATIVITY

We parallel to the *-axis during the circular motion.

thus get

a helix (screw line), which now represents the orbit and the course

of the motion in time completely. In Fig. 1 8 it is drawn on the

surface of a cylinder that has its base on the #y-plane.

5. Motion in Space
Our graphical method of representation fails for motions in space, for in this case we have three space co-ordinates x, y, z, and time has to be added as a fourth co-ordinate. But unfortunately our visual powers are confined to three-dimensional space. The symbolic language of mathematics must now lend us a helping hand. For the methods of analytical geometry allow us to treat the properties and relationships of spatial configurations as pure matters of calculation without requiring us to use our visual power or to sketch figures. Indeed, this process is much more powerful than geometric construction. Above all, it is not bound to the dimensional number three but is immediately applicable to spaces of four or more dimensions. In the language of mathematics the conception of a space of more than three dimensions is not at all mystical but is simply an abbreviated expression of the fact that we are dealing with things that allow themselves to be fully determined by more than three number data. Thus the position of a point at a given moment of time can be fixed only by specifying four number data, the three space-co-ordinates x, y, z and the time t. After we have learned to deal with the xyt-spa.ce as a means of depicting plane motion it will not be difficult also to regard the motions in three-
dimensional space in the light of curves in the xyzt-spa.ce. This view of kinematics as geometry in a four-dimensional xyzt-spa,ce has the advantage of allowing us to apply the well-known laws of geometry to the study of motions.
But it has a still deeper significance that will become clearly apparent in Einstein's theory. It will be shown that the conceptions space and time, which are contents of experience of quite different kinds, cannot be sharply differentiated at all as objects of physical measurement. If physics is to retain
its maxim of recognizing as real only what is physically observ-
able it must combine the conceptions space and time to a higher unity, namely, the four-dimensional xyzt-spa.ee. Minkowski called this the " world " (1908), by which he wished to express that the element of all order of real things is not place nor point of time but the " event " or the " world-point/' that
is, a place at a definite time. He called the graphical picture of a moving point " world-line," an expression that we shall

LAWS OF CLASSICAL MECHANICS 27

continue to use in the sequel. Rectilinear uniform motion thus corresponds to a straight world-line, accelerated motion to one that is curved.

— 6. Dynamics The Law of Inertia

After these preliminaries we revert to the question with

which we started, namely, as to how forces generate motions.

The simplest case is that in which no forces are present at

A all.

body at rest will then certainly not be set into motion.

The ancients had already made this discovery, but, above this,

they also believed the converse to be true, namely, that wherever

there is motion there must be forces that maintain them. This
view at once leads to difficulties if we reflect on why a stone or a spear that has been thrown continues to move when it

has been released from the hand. It is clearly the latter that

has set it into motion, but its influence is at an end so soon as
the motion has actually begun. Ancient thinkers were much

troubled in trying to discover what forces actually maintain the

motion of the thrown stone. Galilei was the first to find the right

point of view. He observed that it is a prejudiced idea to as-

sume that wherever there is motion there must always be force.

Rather it must be asked what quantitative property of motion

has a regular relationship with force, whether it be the place

of the moving body, its velocity, its acceleration, or some
composite quantity dependent on all of these. No amount

of reflection will allow us to evolve an answer to these questions

We by philosophy.

must address ourselves directly to nature.

The question which she gives is, firstly, that force has an influence

in effecting changes of velocity. No force is necessary to

maintain a motion in which the magnitude and the direction

of the velocity remain unaltered. And conversely, where

there are no forces, the magnitude and direction of the velocity

remain unaltered ; thus a body which is at rest remains at rest,

and one that is moving uniformly and rectilinearly continues to

move uniformly and rectilinearly.

This law of inertia (or of persistence) is by no means so

obvious as its simple expression might lead us to surmise.

For in our experience we do not know of bodies that are really

withdrawn from all influences from without, and if we use our

imaginations to picture how they travel on in their solitary

rectilinear paths with constant velocity throughout astronomic

space, we are at once confronted with the problem of the

absolutely straight path in space absolutely at rest, with which

we shall have to deal in detail later on. For the present, then,

we shall interpret the law of inertia in the restricted sense in

which Galilei meant it.

28 THE THEORY OF RELATIVITY
Let us picture to ourselves a smooth exactly horizontal table on which a smooth sphere is resting. This is kept pressed
against the table by its own weight, but we ascertain that it requires no appreciable force to move the sphere quite slowly
on the table. Evidently there is no force acting in a horizontal direction on the sphere, otherwise it would not itself remain at rest at any point on the table.
But if we now give the sphere a velocity it will continue to move in a straight line and will lose only very little of its speed. This retardation was called a secondary effect by
Galilei, and it is to be ascribed to the friction of the table and the air, even if the frictional forces cannot be proved to be present by the statical methods with which we started. It is just this depth of vision, which correctly differentiates what is essential in an occurrence from disturbing subsidiary effects,
that characterizes the great investigator.
The law of inertia is at any rate confirmed for motion on the table. It has been established that in the absence of forces the velocity remains constant in direction and magnitude.
Consequently the forces will be associated with the change
of velocity, the acceleration. In what way they are associated can again be decided only by experiment.

7. Impulses

We have presented the acceleration of a non-uniform

motion as a limiting case of sudden changes of velocity of brief

uniform motions. Hence we shall first have to enquire how

a single sudden change of velocity is produced by the application

of a force.

For

this

a

force

must

act

for

only

a

short

time ;

it is then what we call an impulse or a blow. The result of such

a blow depends not only on the magnitude of the force but also
We on the duration of the action, even if this is very short.

therefore define the intensity of a blow or impulse as follows :
K n impulses J, each of which consists of the force acting

= during the time t — sees., will, if they follow each other withn

out appreciable pauses, have exactly the same effect as if the
K force were to continue to act throughout the whole second.

Thus we should have

J

=

J
,J

=

K,

t

or,

J

=

?K
n

=

<K

.

.

.

. (5)

LAWS OF CLASSICAL MECHANICS 29

To visualize this, let us imagine a weight placed on one side

of a lever having equal arms (such as a balance), and suppose

a hammer to tap very quickly and evenly on the other side with

blows just powerful enough to preserve equilibrium except for

inappreciable fluctuations (Fig. 19). It is clear that we may

tap more weakly but more often, or more strongly and less often,

so long as the intensity J of the blow multiplied by the number of blows n, or divided by the time t required by each blow,

always remains exactly equal to the weight K. This " Impulse Balance " enables us to measure the intensity of blows

even when we cannot ascertain the duration and the force of each

We K one singly.

need only find the force that keeps equili-

brium with n such equal blows per second (disregarding the

inappreciable trembling of the arms), then the magnitude of

each blow is the wth part of K.
= The dimensions of impulse are [J] [T . G], where G
denotes weight.

Fig. 19.
8. The Law of Impulses
We again consider the sphere on the table and study the
action of impulses on it. To do this we require a hammer that may be swung, say, about a horizontal axis. Firstly, we calibrate the power of the blows of our hammer for each length of drop by means of our " impulse balance." Then we
allow it to impinge against the sphere resting on the table and observe the velocity that it acquires through the blow by
measuring how many cms. it rolls in 1 sec. (Fig. 20). The result
is very simple.
The more powerful the blow the greater the velocity, the relation being such that twice the blow imparts twice the velocity, three times the blow three times the velocity, and so forth, that is, the velocity and the blow bear a constant ratio to each
other (they are proportional). This is the fundamental law of dynamics, the so-called
law of impulse (or momentum) for the simple case when a body is set into motion from rest. If the sphere already has a velocity initially, the blow will increase or decrease it according as it

30 THE THEORY OF RELATIVITY
strikes the sphere in the rear or in the front. By a strong
counter-blow it is possible to reverse the direction of motion of the sphere.
The law of impulse then states that the sudden changes of
velocity of the body are in the ratio of the impulses or blows that
produce them. The velocities are here considered as positive
or negative according to their direction.

9. Mass

We Hitherto we have dealt with a single sphere.

shall

now perform the same impulse experiment with spheres of

different kinds, say, of different size or of different material,

some being solid and others hollow. Suppose all these spheres

to be set into motion by exactly equal blows or impulses.

Experiment shows that they then acquire quite different

Fig. 20.

velocities, and, indeed, it is at once observed that light spheres

are made to travel at great speed, but heavy ones roll away only

slowly. Thus we find a relationship with weight, into which

we shall enter into detail later, for it is one of the empirical

foundations of the general theory of relativity. But here, on

the contrary, we wish to bring out clearly and prominently

that from the abstract point of view the fact that various

spheres acquire various velocities after equally strong impacts

has nothing to do with weight. Weight acts downwards and

produces the pressure of the sphere on the table, but exerts no

We horizontal force.

now find that one sphere opposes greater

resistance to the blow than another ; if the former is at the same time the heavier, then this is a new fact of experience, but does

not from the point of view here adopted allow itself to be de-

duced from the conception of weight. What we establish is

We a difference of resistance of the spheres to impacts.

call

— .

LAWS OF CLASSICAL MECHANICS 3]

it inertial resistance, and measure it as the ratio of the impulse
or impact J to the velocity v generated. The name mass has been chosen for this ratio, and it is denoted by the symbol w. Thus we set

m=I

(6)

This formula states that for one and the same body an increase of the impulse J calls up a greater velocity v in such a
way that their ratio has always the same value m. When mass has been defined in this way its unit can no longer be
chosen at pleasure, because the units of velocity and of impulse have already been fixed. Rather, mass has the dimensions
[«] - [™]

and its unit in the ordinary system of measures is sec. 2grm./cm. In ordinary language the word mass denotes something like
amount of substance or quantity of matter, these conceptions themselves being no further defined. The concept of substance, as a category of the understanding, is counted among those
— — things that are directly given, i.e. are immediate data. In
physics, however as we must very strongly emphasize the word mass has no meaning other than that given by formula (6)
It is the measure of the resistance to changes of velocity.
We may write the law of impulses more generally thus :

mw = J

(7)

It determines the change of velocity w that a body in motion

experiences as the result of an impulse J. The formula is often interpreted too as follows :

The given impulsive force J of the hammer is transferred to the movable sphere. The hammer " loses " the impulse

J, and this impulse reappears in the motion of the sphere to the same extent mw. This impulsive force carries the sphere along, and when the latter itself impinges on another body, it, in its

turn, gives the latter a blow or impulse, and thereby loses

just as much impulse as the other body gains. For example,
m m if the bodies of mass x and 2 impinge against each other
rectilinearly (that is, whilst moving in the same straight line),

then the impulsive forces which they exert on each other are

= — always equal and opposite, that is, J x

J 2 , or their sum

is zero :

= + = + m Ji

J2

mfv x

w
22

o.

.

. (8)

32 THE THEORY OF RELATIVITY

From this it follows that

w =——

2

m,

that is, when one sphere loses velocity (w 1 negative), the other

gains velocity (w 2 positive), and vice versa.

If we introduce the velocities of the two spheres before and

after the impact,

namely, v lf v^ for the first sphere,

and

v 2,

'
v2

for the second, then the changes of velocity are

= — w x

vj

vx

= — w 2

v2

v2

and we may also write the equation (8) thus :

— + — = tn^Vi

vj

m 2 (v 2

v 2)

o

If we then collect all the quantities referring to the motion before the impact on the one side, and all those referring to the motion after the impact on the other, we get

m + m = + m 1v 1

2v 2 m^v-l

v 2 2 .

.

.

(9)

and this equation may be interpreted as follows :
m To bring a body of mass from a state of rest into one in
which it has the velocity v we require the impulse mv ; it then
carries this impulse along with it. Thus the total impulse
m + carried along by the two spheres before the impact is 1v 1 m v 2 2 . The equation (9) then states that this total impulse
is not changed as a result of the impact. This is the law of
conservation of impulse or momentum.

10. Force and Acceleration

Before pursuing further the striking parallelism between
mass and weight we shall apply the laws so far established to

the case of forces that act continuously. Unfortunately, again,
the theorems can be set up rigorously only with the aid of the

methods of the infinitesimal calculus, yet the following considera-
tions may serve to give an approximate idea of the relationships

involved.

A force that acts continuously generates a motion whose

We velocity alters continuously.

now suppose the force re-

placed by a rapid succession of blows or impulses. Then at each

blow the velocity will suffer a sudden change and a world-line

that is bent many times, as in Fig. 10, will result, and which

will fold closely around the true, uniformly curved, world-line

and will be able to be used in place of the latter in the calcula-
tions. Now if w blows per sec. replace the force K, then by (5)

LAWS OF CLASSICAL MECHANICS 33

= K = each of them has the value J

n

or

/K, where t is the short

interval occupied by each blow. At each impulse a change

of velocity w occurs which, according to (7), is determined by

= = = mw J tK, or m-w K. But, by (2), w

b, thus we get

t

mb = K .

10)

This is the law of motion of dynamics for forces that act

continuously. It states in words that a force produces an

K acceleration that is proportioned to it ; the constant ratio

b :

is the mass.
We may give this law still a different form which is advan-

tageous for many purposes, in particular for the generalization

that is necessary in the dynamics of Einstein (see VI, 7, p. 221).

For if the velocity v alters by the amount w, then the impulse

= carried along by the moving body, namely, J mv, alters by

= —mw

mw. Thus we have mb

, the change of the impulse carried

along in the time t required to
effect it. Accordingly we may

express the fundamental law ex-

pressed in formula (10) thus :

K If a force

acts on a body,

— then the impulse J mv carried

along by the body changes in such

a way that its change per unit of time is equal to the force K.

Expressed in this form the law

holds only for motions which take

place in a straight line and in

Fig. 21.

which the force acts in the same

straight line. If this is not the case, that is, if the force

acts obliquely to the momentary direction of motion the

law must be generalized somewhat. Let us suppose the force

drawn as an arrow which is then projected on to three mutually

perpendicular directions, say, the co-ordinate axes. In Fig.

21 the case is represented in which the force acts in the xy-

plane, and its projections on the x- and the jy-axis have been

drawn. Let us imagine the moving point projected on the

axes in the same way. Then each of the points of projection

executes a motion on its axis of projection. The law of motions

then states that the accelerations of these motions of projection

— K bear the relation mb

to the corresponding components of

force. But we shall not enter more closely into these mathe-

matical generalizations, which involve no new conceptions.

3

34 THE THEORY OF RELATIVITY

— ii. Example Elastic Vibrations

As an example of the relation between force, mass, and

acceleration we consider a body that can execute vibrations

We under the action of elastic forces.

take, say, a straight

broad steel spring and fasten it at one end so that it lies horizontally in its position of rest (and does not hang downwards).

It bears a sphere at the other end (Fig. 22). The sphere can

then swing to and fro in the horizontal plane (that of the page).

Gravity has no influence on its motion, which depends only
on the elastic force of the spring. When the displacements are

small the sphere moves almost in a straight line. Let its

direction of motion be the #-axis.
If we set the sphere into motion, it executes a periodic vibration, the nature of which we can make clear to ourselves

as follows : If we displace the sphere slightly out of the position of equilibrium with our hands, we experience the restoring force of the spring. If we let the sphere go, this force imparts to it an acceleration, which causes it to return to the mean

position with increasing velocity. In this process the restoring

force, and hence also the acceleration, continuously decreases, and becomes zero when passing through the mean position

itself, for here the sphere is in equilibrium and no accelerative

force acts on it. At the place, therefore, at which the velocity

is greatest, the acceleration is least. In consequence of its

inertia the sphere passes rapidly through the position of equili-

brium, and then the force of the spring begins to retard it
and applies a brake, as it were, to the motion. When the orig-

inal deflection has been attained on the other side the velocity

has decreased to zero and the force has reached its highest

value. At the same time the acceleration has reached its

greatest value in reversing the direction of the velocity at this
moment. From this point onwards it repeats the process in

the reverse sense.
If we next replace the sphere by another of different mass

we see that the character of the motion remains the same
but the time of a vibration is changed. When the mass is

greater the motion is retarded, and the acceleration becomes less ; a decrease of mass increases the number of vibrations
per sec.
K In many cases the restoring force may be assumed to be

exactly proportional to the deflection x. The course of the
motion may then be represented geometrically as follows : Consider a movable point P on the circumference of a circle

of radius a, which is being traversed uniformly v times per

sec. by P. It then traverses the circumference, which is 27ra

LAWS OF CLASSICAL MECHANICS 35

T = (where tt == 3*14159 . . .), in the time

- sees., thus its

velocity is

2tt(1

2-nav.

Let us now take the centre O of the circle as the origin of a rectangular set of co-ordinates in which P has the co-ordinates x, y. Then the point of projection A of the point P on the #-axis
will move to and fro during the motion just like the mass
A fastened to the spring. This point is to represent the vibrating mass. If P moves forward along a small arc s, then A moves
=t
along the .r-axis a small distance f, and we have v j- as
the velocity of A. Fig. 23 now shows that the displacements f

^

/

B

36 THE THEORY OF RELATIVITY

This change 77 of y corresponds to a change in the velocity
= A v 2-nvy of the point which is given by

w

= 27TV7)

27TVS-,

a

and hence to an acceleration of A,

= = = b

—w

sx 2ttv- . -

(2ttv) 2X

ta

The acceleration in this vibrational motion of the point A

is thus actually at every moment proportional to the deflection

.... We x.

get for the force

= = K mb m(27Tv) 2x

(n)

By measuring the force corresponding to a deflection x and

by counting the vibrations we can

m thus determine the mass

of the

spring pendulum.

The picture of the world-line of

such a vibration is clearly a wave-

line in the ^-plane, if x is the direc-

tion of vibration (Fig. 24). In the

figure it has been assumed that at

= the time t

the sphere is moving

= through the middle position x

towards the right. We see that when-

ever the sphere passes through the
= £-axis, that is, for x 0, the direction

of the curve is most inclined to the

#-axis, and this indicates the greatest

velocity. Hence the curve is not

Fig. 24.

curved at this point, and the change
of velocity or the acceleration is zero.

The opposite is true of those points that correspond to the

extreme deflections.

12. Weight and Mass
At the beginning when we introduced the conception of mass, we observed immediately that mass and weight exhibit a remarkable parallelism. Heavy bodies offer a stronger resistance to an accelerating force than light bodies. Is this, then, an exact law ? As a matter of fact, it is. To have
the facts quite clear, let us again consider the experiment of setting into motion spheres on a smooth horizontal table by
means of impacts or impulses. We take two spheres A and B,

LAWS OF CLASSICAL MECHANICS 37

of which B is twice as heavy as A, that is, on the impulse balance

B exactly counterpoises two bodies each exactly like A. We

next apply equal blows to A and B on the table and observe

We the velocity attained.

find that A rolls away twice as

quickly as B.
Thus the sphere B, which is twice as heavy as A, opposes
a change of velocity exactly twice as strongly as A. We may
also express this as follows : Bodies having twice the mass
m have twice the weight ; or, more generally, the masses are in
the ratio of the weights G. The ratio of the weight to the mass is a perfectly definite number. It is denoted by g, and we

write

—Q
m

=

g

or

G

=

mg

.

.

. (12)

Of course, the experiment used to illustrate the law is very
rough.* But there are many other phenomena that prove the

same fact ; above all, there is the observed phenomenon that all

bodies fall equally fast. It is hereby assumed, of course, that

no forces other than gravity exert an influence on the motion.

This means that the experiment must be carried out in vacuo so
that the resistance of the air may be eliminated. For pur-

poses of demonstration an inclined plane (Fig. 25) is found

suitable, on which two spheres, similar in appearance but of

different weight, are allowed to roll down. It is observed that

they reach the bottom exactly simultaneously.

The weight is the driving force ; the mass determines the
resistance. If they are proportional to each other, then a

heavy body will indeed be driven forward more strongly than

a lighter one, but to balance this it resists the impelling force

more strongly, and the result is that the heavy and the light

We body roll or fall down equally fast.

also see this from our

formulae. For if in (10) we replace the force by the weight G, and assume the latter, by (12), proportional to the mass, we get

mb = G = mg

t

that is,

b=g

(13)

Thus all bodies have one and the same acceleration verti-

cally downwards, if they move under the influence of gravity

alone, whether they fall freely or are thrown. The quantity g,

the acceleration due to gravity, has the value

= g

981 cm./sec. 2 (or 32 ft. /sec. 2 .).

* For example, we have neglected the circumstance that in producing the rotation
of the rolling sphere a resistance must also be overcome which depends on the distribution of mass in the interior of the sphere (the moment of inertia).

'

38 THE THEORY OF RELATIVITY

The most searching experiments for testing this law may

be carried out successfully with the aid of simple pendulums

with very fine threads. Newton even in his time noticed that

the times of swing are always the same for the same length

of pendulum, whatever the composition of the sphere of the pendulum. The process of vibration is exactly the same as

that described above for the elastic pendulum, except that

now it is not a steel spring but gravity that pulls back the sphere.
We must imagine the force of gravity acting on the sphere

to be resolved into two components, one acting in the direction

of the continuation of the thread, and keeping it stretched,

the other acting in the direction of motion and being the driving

force that acts on the sphere or bob.

We Fig. 26 exhibits the bob at the deflection x.

see at once

Fig. 25.

Fig. 26.

the two similar right-angled triangles, the sides of which are

in the same proportion :

K=G

x

I

Accordingly, formula (11) gives for two pendulums, the bobs

G of

which

are

G ±

and

2, respectively :

M _G. (27TV) 2 1

m = (27Tv) 2 2

-p

t

thus

— = 2

(27TV) n,

m.

that is, the ratio of the weight to the mass is the same for both

We pendulums.

called this ratio g in formula (12). Hence

we get the equation

.... = g

(27tv)H,

(14)

LAWS OF CLASSICAL MECHANICS 39
from which we see that g may be determined by measuring
the length / of the pendulum and the vibration number v. The law of the proportionality of weight to mass is often
expressed as follows :
gravitational and inertial mass are equal.
Here gravitational mass simply signifies the weight divided by g, and the proper mass is distinguished by prefixing the word " inertial."
The fact that this law holds very exactly was already known to Newton. Nowadays it has been confirmed by the most delicate measurements known in physics, which were carried out by Eotvos (1890). Hence we are completely justified in using the balance to compare not only weights but also masses.
One might now imagine that such a law is firmly embedded in the foundations of mechanics. Yet this is by no means the case, as is shown by our account, which follows fairly closely the
ideas contained in classical mechanics. Rather, it is attached,
as a sort of curiosity, somewhat loosely to the fabric of the other laws. Probably it has been a source of wonder to many, but no one suspected or sought any deeper relationship that
might be wrapt in it. For there are many kinds of forces
that can act on a mass. Why should there not be one that is exactly proportional to the mass ? A question to which no
answer is expected will receive none. And so the matter rested
for centuries. This was possible only because the successes of the mechanics of Galilei and Newton were overwhelming It controlled not only the motional events on the earth but also those of the stars, and showed itself to be a trustworthy foundation for the whole realm of the exact sciences. For in the middle of the nineteenth century it was looked on as the object of research to interpret all physical events as me-
chanical events in the sense of the Newtonian doctrine. And thus in building up their stately edifice physicists forgot to
ascertain whether the basis was strong enough to support the whole. Einstein was the first to recognize the importance of the law of equality of inertial and gravitational mass for the
foundations of the physical sciences.

13. Analytical Mechanics

The problem of analytical mechanics is to find from the

law of motion

mb = K

K the motion when the forces are given. The formula itself
gives us only the acceleration, that is, the change of velocity.

40 THE THEORY OF RELATIVITY
To get from the latter the velocity, and from this again the varying position of the moving point, is a problem of the integral
calculus that may be very difficult if the force alters in a complicated way with the place and the time. An idea of the
nature of the problem is given by our derivation of the change of position in a uniformly accelerated motion along a straight line (p. 20). The motion is already more complicated when it is in a plane and due to the action of a constant force of definite direction, as in the case of a motion due to falling or
to a throw. Here, too, we may substitute as an approximation
for the continuous course of the motion one consisting of a series of uniform motions, each of which is transformed into the
next by means of impulses. We again call to mind our table
and agree that the sphere rolling on it is to receive a blow of the same size and direction after the same short interval t
(Fig. 27). Now, if the sphere starts off from the point O with

LAWS OF CLASSICAL MECHANICS 41
magnitude and direction. But if these are given, the further course of the motion is fully determined. Thus one and the
same law of force may produce an infinity of motions according
to the choice of the initial conditions. Thus the enormous number of motions due to falling or to throws depends on the same law of force, of gravity that acts vertically downwards.
In mechanical problems we are usually concerned with the motion not of one body but of several that exert forces on one another. The forces are then not themselves given but depend for their part on the unknown motion. It is easy to understand that the problem of determining the motions of several bodies by calculation becomes highly complicated.
14. The Law of Energy
But there is a law which makes these problems much
simpler and affords a survey of the motion. It is the law of the conservation of energy, which has become of very great impor-

Fig. 28.

Fig. 29.

We tance for the development of the physical sciences.

cannot,

We of course, enunciate it generally here nor prove it.

shall

only seek to know its content from simple examples.

A pendulum which is released after the bob has been raised

— to a certain point rises on the opposite side of the mean position
to the same height except for a small error caused by friction

and the resistance of the air (Fig. 28). If we replace the cir-
cular orbit by some other by allowing the sphere to run on rails as in a " toy railway " (Fig. 29), then the same result holds : the sphere always rises to the same height as that from which

it started.
From this it easily follows that the velocity that the sphere
has at any point P of its path depends only on the depth of this point P below the initial point A. To see this we imagine
the piece AP of the orbit changed, the rest PB remaining un-
altered. Now, if the sphere were to arrive at P along the one
A orbit from with a velocity different from that with which it
arrives along the other, then in its further course from P to B it

42 THE THEORY OF RELATIVITY

would not in each case exactly reach its goal B. For, to achieve

this, a uniquely determinate velocity is clearly necessary at P.
Consequently the velocity at P does not depend on the form

of the piece of orbit traversed, and since P is an arbitrary point,

this result holds generally. Hence the velocity v must be

determined by the height of fall h alone. The truth of the law

depends on the circumstance that the path (the rails) as such

opposes no resistance to the motion, that is, exerts no force on

the sphere in its direction of motion, but receives only its

perpendicular pressure. If the rails are not present, we have

the case of a body falling freely or of one that has been thrown,

and the same result holds : the velocity at each point depends

only on the height of fall.

This fact may not only be established experimentally but
may also be derived from our laws of motion. We hereby also

get the form of the law that regulates the dependence of the

We velocity on the height.

assert that it states the following :

Let x be the path fallen through, measured upwards (Fig
m 30), v the velocity, the mass, and G the weight of the body.
Then the quantity

E = -v* + Gx .

.

.

. (15)

has the same value during the whole process of falling.
To prove this we first suppose E to stand for any arbitrary
quantity that depends on the motion and hence alters from
moment to moment. Let E alter by the amount e in a small

interval of time t, then we shall call the ratio ~ the rate of

t

change of E, and, exactly as before in defining the orbital

velocity v and the acceleration b, we suppose that the time

interval t may be taken as small as we please. If the quantity

E does not change in the course of time, then its rate of change

We is, of course, zero, and vice versa.

next form the change

of the above expression E in the time t. During this time

the height of fall x decreases by vt, and the velocity v increases
= by w bt. Hence after the time t the value of E becomes

= + + - E'

-(v

w) 2

G (x

vt).

Now,

= w (v -f w) 2

V 2 -f- 2 -f- 2VW.

This states that the square erected over v and w, joined
together in the same straight line, may be resolved into a square
having the side v, one having the side w, and two equal rectangles
having the sides v and w (Fig. 31).

LAWS OF CLASSICAL MECHANICS 43

Hence we get

= + + — E'

— v2

— w2 -f twvw Ox Gvt.

2

2

If we deduct the old value of E from this, we get as the
change in value

= = — — — e

YJ

E

m w 2 -{- mvw

Gvt,

2

— or, since w

bt,

= ? + - e

h 2t 2

mvbt

Gvt.

2

Hence the rate of change becomes

— m

mvb b

2 t

-\-

Gv.

w

v.w.

Fig. 30.

44 THE THEORY OF RELATIVITY

the other. The first term is characteristic of the state of velo-

city of the body, the second, of the height that it has attained

We against the force of gravitation.

have special names for

these terms.

T = — v 2 is called the vis viva or kinetic energy.

U = Gx is called the capacity for work or the potential

energy.

Their sum,

T +U = E, .

.

.

. (16)

is

called

simply

the

mechanical

energy

of

the

body ;

and the

law which states that it remains constant during the motion

of the body is called the law of conservation of energy.
= The dimensions of energy are [E] [GL]. Its unit is

grm. cm.

The name capacity for doing work is of course derived from

the work done by the human body in lifting a weight. Accord-

ing to the law of conservation of energy this work becomes

transformed into kinetic energy in the process of falling. If,

on the other hand, we give a body kinetic energy by throwing

it upwards, this energy changes into potential energy or capacity

for doing work.

Exactly the same as has been described for falling motions,

holds in the widest sense for systems composed of any number

of bodies, so long as two conditions are fulfilled, namely :

i. External influences must not be involved, that is, the system must be self-contained or isolated.
2. Phenomena must not occur in which mechanical energy
is transformed into heat, electrical tension, chemical affinity, and such like.

If these are fulfilled the law that
E=T+U

always remains constant holds true, the kinetic energy depending on the velocities, the potential energy on the positions of the moving bodies.
In the mechanics of the heavenly bodies this ideal case is
realized very perfectly. Here the ideal dynamics of which we
have developed the principles is valid. But on the earth this is by no means the case. Every
motion is subject to friction, whereby its energy is transformed into heat. The machines by means of which we produce motion, transform thermal, chemical, electric, and magnetic forces into mechanical forces, and hence the law of energy in
its narrow mechanical form does not apply. But it may

LAWS OF CLASSICAL MECHANICS 45

always be maintained in an extended form. If we call the heat

energy Q, the chemical energy C, the electro-magnetic energy
W, and so forth, then the law that for closed systems the sum

E=T+U+Q+C+W

(i 7 )

is always constant holds. It would lead us too far to pursue the discovery and logical
evolution of this fact by Robert Mayer, Joule (1842) and Helmholtz (1847), or to investigate how the non-mechanical forms of energy are determined quantitatively. But we shall use the conception of energy later when we speak of the intimate relationship that the theory of relativity has disclosed between
mass and energy.

15. Dynamical Units of Force and Mass

The validity of the process by which we have derived the

fundamental laws of mechanics is, in a certain sense, restricted

to the surface of our table and its immediate neighbourhood.

For we have abstracted our conceptions and laws from experiments in a very limited space, in the laboratory. The

advantage of this is that we need not trouble our heads about

the assumption concerning space and time. The rectilinear
motions with which the law of inertia deals may be copied

on the table with a ruler. Apparatus and clocks are assumed

to be available for measuring the orbits and the motions.

Our next concern will be to step out of the narrow confines

of our rooms into the wider world of astronomic space. The

first stage will be a " voyage round the world " which idiomatic

We usage applies to the small globe of the earth.

shall pose

the question : do all the laws of mechanics set up apply just as much in a laboratory in Buenos Aires or in Capetown as

here ?

Yes, they do, with one exception, namely, the value of the

We gravitational acceleration g.

have seen that this can be

measured exactly by observations of pendulums. It has been

found that one and the same pendulum swings somewhat more

slowly at the equator than in the more southerly or more nor-

therly regions. Fewer vibrations occur in the course of a day, that is, in the course of one rotation of the earth. From this
it follows that g has a minimum value at the equator and

increases towards the north and the south. This increase is

quite regular as far as the poles, where g has its greatest value.
We shall see later to what this is due. Here we merely take
note of the fact. For the system which we have hitherto used for measuring forces and masses this fact, however, has very

awkward consequences.

46 THE THEORY OF RELATIVITY

So long as weights are compared with each other only by

means of the scale balance, there are no difficulties. But let us

imagine a spring balance here in the laboratory which has been
calibrated with weights. If we then bring this spring balance into more southerly or more northerly regions, we shall find that

when loaded with the same weights it will give different deflections. If, therefore, we identify weight with force as we

have hitherto done, there is nothing left for us but to assert

that the force of the spring has altered and that it depends on the geographical latitude. But this is obviously not the

case. It is not the force of the spring that has altered but the gravitational force. It is, therefore, wrong to take the weight

of one and the same piece of metal as the unit of force at all

We points of the earth.

may choose the weight of a definite

body at a definite point on the earth as the unit of force, and
this may be applied at other points if the acceleration g due to gravity is known by pendulum measurements at
both points. This is, indeed, just what technical science

actually does do. Its unit of force is the weight of a definite
normal body in Paris, the gramme. Hitherto we have always

used this without taking into account its variability with

position. In exact measurements, however, the value must

be reduced to that at the normal place (Paris).

Science has departed from this system of measures, at which

one place on the earth is favoured, and has selected a system

that is less arbitrary.

The fundamental law of mechanics itself offers a suitable

method for doing this. Instead of referring the mass to the

force, we establish the mass as the fundamental quantity of

the independent dimensions [M] and choose its unit arbitrarily :

let a definite piece of metal have the mass I. As a matter of

fact, the same piece of metal that served technical science as

the unit of weight, the Paris gramme, is taken for this purpose, and this unit of mass is likewise called the gramme (grm.).

The fact that the same word is used in technical science

to denote the unit of weight and in physics to denote the unit
of mass may easily lead to error. In the sequel we use the

physical system of measure, the fundamental units of which are : cm. for length, sec. for time, grm. for mass.
Force now has the derived dimensions

[K] = [MB] = [*£]

and the unit, called the dyne, is grm. cm. /sec. 2

= Weight is defined by G

mg thus the unit of mass has ;

= the weight G g dynes, It changes with the geographical

LAWS OF CLASSICAL MECHANICS 17

= latitude, and in our own latitude it has the value g 981 dynes.
This is the technical unit of force. The weight given by a

spring balance,

expressed

in

dynes,

is,

of

course,

a

constant ;

for its power of accelerating a definite mass is independent

of the geographical latitude.
The dimensions of impulse or momentum are now :

[j] = [TK] = pjr]

and its unit is grin. cm. /sec. Finally, the dimensions of energy
are
[^] [E] = [MV«] =

and its unit is grm. cm. 2/sec. 2
Now that we have cleansed the system of measures of all
earthly impurities, we can proceed to the mechanics of the
stars.

CHAPTER III
THE NEWTONIAN WORLD-SYSTEM

i. Absolute Space and Absolute Time

THE principles of mechanics, as here developed, were partly suggested to Newton by Galilei's works and were partly created by himself. To him we owe above all the expression of definitions and laws in such a generalized form that they appear detached from earthly experiments and allow themselves to be applied to events in astronomic

space.

In the first place Newton had to preface the actual mechanical

principles by making definite assertions about space and time.

Without such determinations even the simplest law of mechanics,

that of inertia, has no sense. According to this, a body on which

no force is acting is to move uniformly in a straight line. Let us fix our thoughts on the table with which we first experimented in conjunction with the rolling sphere. If now the sphere rolls on the table in a straight line, an observer who

follows and measures its path from another planet would have

to assert that the path is not a straight line according to his

point of view. For the earth itself is rotating, and it is clear

that a motion that appears rectilinear to the observer travelling

with the earth, because it leaves the trace of a straight line
on his table, must appear curved to another observer who does not participate in the rotation of the earth. This may be

roughly illustrated as follows :

A circular disc of white cardboard is mounted on an axis

A so that it can be turned by means of a handle.

ruler is fixed

in front of the disc. Now turn the disc as uniformly as possible,

and at the same time draw a pencil along the ruler with constant

velocity, so that the pencil marks its course on the disc. This

path will, of course, not be a straight line on the disc, but a

curved line, which will even take the form of a loop if the

rotary motion is sufficiently rapid. Thus, the same motion

which an observer fixed to the ruler would call uniform and

rectilinear, would be called curvilinear (and non-uniform)
by an observer moving with the disc, This motion may be

48

THE NEWTONIAN WORLD-SYSTEM 19
constructed point for point, as is illustrated in the drawing (Fig. 32), which explains itself.
This example shows clearly that the law of inertia has sense, indeed, only when the space, or rather, the system of reference in which the rectilinear character of the motion is
to hold, is exactly specified. It is in conformity with the Copernican world-picture,
of course, not to regard the earth as the system of reference,
for which the law of inertia holds, but one that is somehow
fixed in astronomic space. In experiments on the earth, for example, rolling the sphere on the table, the path of the freely moving body is not then in reality straight but a little curved. The fact that this escapes our primitive type of observation is due only to the shortness of the paths used in the experiments compared with the dimensions of the earth. Here, as has often happened in science, the inaccuracy of observation
Fig. 32.
has led to the discovery of a great relationship. If Galilei
had been able to make observations as refined as those of later centuries the confused mass of phenomena would have made the discovery of the laws much more difficult. Perhaps, too,
Kepler would never have unravelled the motions of the planets, if the orbits had been known to him as accurately as at the present day. For Kepler's ellipses are only approximations from which the real orbits differ considerably in long periods of time. The position was similar, for example, in the case of modern physics with regard to the regularities of spectra
;
the discovery of simple relationships was rendered much more difficult and was considerably delayed by the abundance of
very exact data of observation.
So Newton was confronted with the task of finding the system of reference in which the law of inertia and, further, all the other laws of mechanics were to hold. If he had chosen the sun, the question would not have been solved, but would
4

50 THE THEORY OF RELATIVITY
only have been postponed, for the sun might one day be discovered also to be in motion, as has actually happened in the meantime.
Probably it was for such reasons that Newton gained the conviction that an empirical system of reference fixed by material bodies could, indeed, never be the foundation of a law involving the idea of inertia. But the law itself, through
its close connection with Euclid's doctrine of space, the element of which is the straight line, appears as the natural startingpoint of the dynamics of astronomic space. It is, indeed, in the law of inertia that Euclidean space manifests itself outside the narrow limits of the earth. Similar conditions obtain in the case of time, the flow of which receives expression in the uniform motion due to inertia.
In this way, possibly, Newton came to the conclusion that there is an absolute space and an absolute time. It will be best to give the substance of his own words. Concerning time
he says : I. " Absolute, true and mathematical time flows in itself
and in virtue of its nature uniformly and without reference to any external object whatever. It is also called duration/'
" Relative, apparent, and ordinary time is a perceptible and external, either exact or unequal, measure of duration, which we customarily use instead of true time, such as hour, day, month, year."
" Natural days, which are usually considered as equal measures of time are really unequal. This inequality is some-
what corrected by the astronomers who measure the motion of the heavenly bodies according to the correct time. It may be that there is uniform motion by which time may be measured accurately. All motions may be accelerated or retarded. Only the flow of absolute time cannot be changed. The same duration and the same persistence occurs in the existence
of all things, whether the motions be rapid, slow, or zero."
Concerning space Newton expresses similar opinions. He
says :
II. " Absolute space, in virtue of its nature and without reference to any external object whatsoever, always remains immutable and immovable."
" Relative space is a measure or a movable part of the absolute space. Our senses designate it by its position with respect to other bodies. It is usually mistaken for the immovable space."
"So in human matters we, not inappropriately, make use
of relative places and motions instead of absolute places and motions. In natural science, however, we must abstract from

THE NEWTONIAN WORLD-SYSTEM 51
the data of the senses. For it may be the case that no body that is really at rest exists, with reference to which we may
refer the places and the motion." The definite statement, both in the definition of absolute
time as in that of absolute space, that these two quantities exist " without reference to any external object whatsoever " seems strange in an investigator of Newton's attitude of mind. For he often emphasizes that he wishes to investigate only what is actual, what is ascertainable by observation. " Hypotheses non fingo," is his brief and definite expression. But what exists " without reference to any external object whatsoever " is not ascertainable, and is not a fact. Here we have clearly a case in which the ideas of unanalysed consciousness
We are applied without reflection to the objective world.
shall investigate the question in detail later on.
Our next task is to describe how Newton interpreted the laws of the cosmos and in what the advance due to his doctrine
consisted.

2. Newton's Law of Attraction

Newton's idea consisted in setting up a dynamical idea of planetary orbits, or, as we nowadays express it, in founding Celestial Mechanics. To do this it was necessary to apply Galilei's conception of force to the motions of the stars. Yet Newton did not find the law according to which

the heavenly bodies act on one another by setting up bold hypotheses, but by pursuing the systematic and exact path of analysing
the known facts of planetary mo-

Fig. 33.

tions. These facts were expressed in the three Kepler laws

that compressed all the observations of that period of time
into a wonderfully concise and vivid form. We must here
state Kepler's laws in full. They are :

i. The planets move in ellipses with the sun at one of the

foci (Fig. 33).
2. The radius vector drawn from the sun to a planet de-

scribes equal areas in equal times.
3. The cubes of the major axes of the ellipses are propor-
tional to the squares of the periods of revolution.

Now the fundamental law of mechanics gives a relation
K between the acceleration b of the motion, and the force that

52 THE THEORY OF RELATIVITY
produces it. The acceleration b is completely determined by the course of the motion, and, if this is known, b can be calculated. Newton recognized that the orbit as defined by Kepler's laws just sufficed to allow a calculation of b. Then the law
K = mb
also allows the acting force to be calculated. The ordinary mathematics of Newton's time would not have
enabled him to carry out this calculation. He had first to
invent the mathematical apparatus. Thus there was created in England the Differential and Integral Calculus, the root of the whole of modern mathematics, as a bye-product of astronomical researches, whereas Leibniz (1684) simultaneously invented the same method on the continent by starting from
a totally different point of view.
Since we do not wish to use the infinitesimal calculus in this book we cannot pause to give a picture of the wonderful nature of Newton's inferences. Yet the fundamental idea may be illustrated by a simple example.
The orbits of the planets are slightly eccentric ellipses,
that are almost of a circular shape. It will be permissible to assume approximately that the planets describe circles about
the sun, as was, indeed, supposed by Copernicus. Since circles are special ellipses with the eccentricity zero, this assumption
certainly fulfils Kepler's first law.
The second law next states that every planet traverses its circle with constant speed. Now, by II 4, we know all about
the acceleration in such circular motions. It is directed to-
wards the centre and, by formula (4), p. 25, it has the value

r

= where v the speed in the orbit, and r is the radius of the

circle.

If now T is the period of revolution, the velocity is deter-
= mined as the ratio of the circumference 27Tr(7T 3-14159 . . .)

to the time T, thus

= —27Tr

v

.

.

.

/tQ\ . (18)

so that

=

^-^ ri 2

=

=V1-r

We next direct our attention to the third Kepler law which,
in the case of a circular orbit, clearly states that the ratio of

THE NEWTONIAN WORLD-SYSTEM 58

the cube of the radius, r3 , to the square of the time of revolution

T2 ,

has

the

same

value

C

for

all

planets

:

= = C or

Ti

T* ;t

If we insert this in the value for b above, we get

(I9)

b = *"S .

.

.

. (20)

r2

According to this the value of the centripetal acceleration depends only on the distance of the planet from the sun, being inversely proportional to the square of the distance, but it is quite independent of the properties of the planet, such as its
mass. For the quantity C is, by Kepler's third law, the same for all planets, and can therefore involve at most the nature of the sun and not that of the planets.
— Now, it is a remarkable circumstance that exactly the same
law comes out for elliptic orbits by a rather more laborious calculation, it is true. The acceleration is always directed towards the sun situated at a focus, and has the value given by formula (20).

3. General Gravitation

The law of acceleration thus found has an important property
in common with the gravitational force on the earth (weight) :
it is quite independent of the nature of the moving body. If we calculate the force from the acceleration, we find it likewise directed towards the sun. It is thus an attraction and
has the value

K = mb = m—— .

.

. (21)

r2

It is proportional to the mass of the moving body, just like

the weight

G = mg

of a body on the earth.
This fact suggests to us that both forces may have one and
the same origin. Nowadays, this circumstance, having been handed down to us through the centuries, has become such a truism, as it were, that we can scarcely conceive how bold
and how great was Newton's idea. What a prodigious imagi-
nation it required to conceive the motion of the planets about
the sun or of the moon about the earth as a process of " falling " that takes place according to the same laws and under the action
of the same force as the falling of a stone released by my hand.
The fact that the planets or the moon do not actually rush

54 THE THEORY OF RELATIVITY

into their central bodies of attraction is due to the law of
We inertia that here expresses itself as a centrifugal force.

shall have to deal with this again later.

Newton first tested this idea of general weight or gravitation

in the case of the moon, the distance of which from the earth

was known from angular measurements.

This test is so important that we shall repeat the very

simple calculation here as evidence of the fact that all scien-

tific ideas become valid and of worth only when calculated and

measured numerical values agree.

The

central

body

is

now

the

earth ;

the

moon

takes

the

place of the planet, r denotes the radius of the moon's orbit,

T the period of revolution of the moon. Let the radius of the

earth be a. If the gravitational force on the earth is to have
the same origin as the attraction that the moon experiences

from the earth, then the acceleration g due to gravity must,

by Newton's law (20), have the form

47T 2 C

where C has the same value as for the moon, namely, by (19),

r*
Lf"»"

-W If we insert this value in that for g, we get

g TV

.

.

. (22:

Now, the " sidereal " period of revolution of the moon, that is, the time between two positions in which the line connecting the earth to the moon has the same direction with
respect to the stars, is
= T 27 days 7 hours 43 minutes 12 seconds = 2,360,592 seconds.

In physics it is customary to write down a number to only

so many places as are required for further calculation. So

we write here

T = 2-36 . io6 sees.

The distance of the moon from the centre of the earth is about 60 times the earth's radius, or, more exactly,
= r 6o-ia.

The earth's radius itself is easy to remember because the metric system of measures is simply related to it. For 1 metre
= = 100 cms. one ten-millionth of the earth's quadrant, that

THE NEWTONIAN WORLD-SYSTEM 55

is, the forty-millionth

part

or

(4

.

io 7 th )

part of the earth's

circumference 2na

100 =

= , or a

6-37 . io 8 cms. .

4 . io 7

.

(v 21J)/

If we insert all these values in (22) we get

= _ g6

att 2
-1

.

6o-i 3 2-36*

. .

6-43Z7 io 12

.

io 8

==

qJ8i

cm. /, sec. 2 l

,

.

.

(24) K ^'

This value agrees exactly with that found by pendulum

observations on the earth (see II, 12, p. 37). The great importance of this result is that it represents the

relativization of the force of weight. To the ancients weight denoted a pull towards the absolute " below," which is ex-

perienced by all earthly bodies. The discovery of the spherical

shape of the earth brought with it the relativization of the

direction of earthly weight ; it became a pull towards the

centre of the earth.

And now the identity of earthly weight with the force of attraction that keeps the moon in her orbit is proved, and since

there can be no doubt that the latter is similar in nature to

the force that keeps the earth and the other planets in their

orbits round the sun, we get the idea that bodies are not simply " heavy " but are mutually heavy or heavy relatively to each

other. The earth, being a planet, is attracted towards the sun,

but it itself attracts the moon. Obviously this is only an ap-

proximate description of the true state of affairs, which consists

in the sun, moon, and earth attracting each other. Certainly,

so far as the orbit of the earth round the sun is concerned, the

latter may, to a high degree of approximation, be regarded

as at rest, because its enormous mass hinders the calling up

of appreciable accelerations, and, conversely, the moon, on

account of its size, does not come into account. But an exact

theory will have to take into consideration these influences,
called " perturbations."

Before we begin to consider more closely this view, which

signifies the chief advance of Newton's theory, we shall give

We Newton's law its final form.

saw that a planet situated

at a distance r from the sun experiences from it an attraction

of the value (21)

ir

Att 2 C

r2

where C is a constant depending only on the properties of the sun, not on those of the planet. Now, according to the new
view of mutual or relative weight the planet must likewise

56 THE THEORY OF RELATIVITY

M attract the sun. If

is the mass of the sun, c a constant

dependent only on the nature of the planet, then the force

exerted on the sun by the planet must be expressed by

K' = M^!f.

But earlier, in introducing the conception of force (II, i,

p. 16), we made use of the principle that the reaction equals

the action, which is one of the simplest and most certain laws

K = of mechanics. If we apply it here, we must set

K', or

THE NEWTONIAN WORLD-SYSTEM 57

A is necessarily a Kepler elliptic motion.

new feature aria

only when, firstly, we now regard both bodies as moving and,

secondly, add further bodies in the problem. Then we get the problem of three or more bodies, which cor-
responds exactly to the actual conditions in the planetary
system (Fig. 34). For not only are the planets attracted by the sun and the moons by their planets, but every body, be it sun, planet, moon, or comet, attracts every other body. Accordingly, the Kepler ellipses appear to be only approximately

valid, and they are so only because the sun on account of its

great mass overshadows by far the reciprocal action of all other bodies of the planetary system. But in long periods

of time these reciprocal actions must al?o manifest themselves

We as deviations from the Kepler laws.

speak, as already

remarked, of " perturbations."

In Newton's time such perturbations were already known,

and in the succeeding centuries refinements in the methods of

observation have accumulated an immense number of facts

that had to be accounted for by

Newton's theory. That it suc-

ceeded in doing so is one of the
greatest triumphs of human genius.

It is not our aim here to pursue

the development of mechanics from

Newton's time to the present day,

and to describe the mathematical

methods that were devised to calculate the " perturbed " orbits. The

Fig. 34.

most ingenious mathematicians of all countries have played a part in setting up the " theory of perturbations," and even if no satisfactory solution has yet

been found for the problem of three bodies, it is possible to

calculate with certainty the motions for hundred thousands or millions of years ahead or back. So Newton's theory was
— tested in countless cases in new observations, and it has never
failed except in one case, of which we shall presently speak.

Theoretical astronomy, as founded by Newton, was therefore

long regarded as a model for the exact sciences. It achieved

what had been the longing of mankind since earliest history. It lifts the veil that is spread over the future ; it endows its followers with the gift of prophecy. Even if the subject-

matter of astronomic predictions is unimportant or indifferent

for human life, yet it became a symbol for the liberation of the spirit from the trammels of earthly bonds. We, too,

follow the peoples of earlier times in gazing upwards with rever-

ential awe at the stars, which reveal to us the law of the world.

58 THE THEORY OF RELATIVITY
But the world-law can tolerate no exception. Yet there is one case, as we have already mentioned, in which Newton's theory has failed. Although the error is small, it is not to be denied. It occurs in the case of the planet Mercury, the planet
nearest the sun. The orbit of any planet may be regarded
as a Kepler elliptic motion that is perturbed by the other planets,
that is, the position of the orbital plane, the position of the major axis of the ellipse, its eccentricity, in short, all " elements
of the orbit " undergo gradual changes. If we calculate these according to Newton's law and apply them to the observed orbit, it must become transformed into an exact Kepler orbit, that is, an ellipse in a definite plane at rest, with a major axis of definite direction and length, and so forth. This is so,
indeed, for all planets, except that a little error remains in the
case of Mercury. The direction of the major axis, that is, the line connecting the sun with the nearest focus, the perihelion (Fig. 35), does not remain fixed after all the above corrections have been applied, but executes a very slow motion
Earth
Perihelion
Fig. 35.
— of rotation, advancing 43 sees, of arc every hundred years.
The astronomer Leverrier (1845) the same who predicted the
— existence of the planet Neptune from calculations based on
the perturbations first calculated this motion, and it is fully established. Yet it cannot be explained by the Newtonian attraction of the planetary bodies known to us. Hence recourse has been taken to hypothetical masses whose attraction was to bring about the motion of Mercury's perihelion. Thus, for example, the zodiacal light, which is supposed to emanate from thinly distributed nebulous matter in the neighbourhood of the sun, was brought into relation with the anomaly of Mercury. But this and numerous other hypotheses all suffer from the fault that they have been invented ad hoc and have been confirmed by no other observation.
The fact that the only quite definitely established deviation from Newton's law occurs in the case of Mercury, the planet nearest the sun, indicates that perhaps there is after all some fundamental defect in the law. For the force of attraction

THE NEWTONIAN WORLD-SYSTEM 59

is greatest in the proximity of the sun, and hence deviations

from the law of the inverse square will show themselves first there. Such changes have also been made, but as they are

invented quite arbitrarily and can be tested by no other facts, their correctness is not proved by accounting for the motion

of Mercury's perihelion. If Newton's theory really requires
a refinement we must demand that it emanates, without the

introduction of arbitrary constants, from a principle that is

superior to the existing doctrine in generality and intrinsic

probability.

Einstein was the first to succeed in doing this by making

general relativity the most fundamental postulate of physical

We laws.

shall revert in the last chapter to his explanation

of the motion of Mercury's perihelion.

5. The Relativity Principle of Classical Mechanics

In discussing the great problems of the cosmos we have

almost forgotten the point of departure from the earth. The

laws of dynamics found to hold on the earth were transplanted

to the astronomical space through which the earth rushes

How in its orbit about the sun with stupendous speed.

is it,

then, that we notice so little of this journey through space ?
How is it that Galilei succeeded in finding laws on the moving

earth which, according to Newton, were to be rigorously valid

We only in space absolutely at rest ?

have called attention

to this question above when mentioning Newton's views

about space and time. We stated there that the apparently

straight path of a sphere rolling on the table would, in reality,

owing to the rotation of the earth, be slightly curved, for the

path is straight not with respect to the moving earth but with

respect to absolute space. The fact that we do not notice

this curvature is due to the shortness of the path and of the

time of observation, during which the earth has turned only

slightly. Even if we admit this, we are still left with the

motion of revolution about the sun, which proceeds with the
immense speed of 30 kms. per sec. Why do we notice nothing

of this ?

This motion, due to the revolution, is also, indeed, a rotation,
and this must make itself remarked in earthly motions similarly

to the rotation of the earth on its own axis, only much less,

since the curvature of the earth's orbit is very small. But in our question we do not mean this rotatory motion but the

forward motion, which, in the course of a day, is practically

rectilinear and uniform.

Actuallv, all mechanical events on the earth occur as if this

60 THE THEORY OF RELATIVITY

tremendous forward motion does not exist, and this law holds

quite generally for every system of bodies that executes a

uniform and rectilinear motion through Newton's absolute

space. This is called the relativity principle of classical me-
chanics, and it may be formulated in various ways. For the

present, we shall enunciate it as follows :

Relatively to a co-ordinate system moving rectilinearly and

uniformly through absolute space the laws of mechanics have exactly the same expression as when referred to a co-ordinate system

at rest in space.
To see the truth of this law we need only keep clear in our

minds the fundamental law of mechanics, the law of impulses,
and the conceptions that occur in it. We know that a blow

produces a change of velocity. But such a change is quite

independent of whether the velocities before and after the

blow, v 1 and v 2, are referred to absolute space or to a system of reference which is itself moving with the constant velocity a.

If the moving body is moving before the blow in space with

= the velocity v x 5 cms. per sec, then an observer moving with = the velocity a 2 cms. per sec. in the same direction would

= — a= — = measure only the relative velocity v^ v ±

5

2

3.

If the body now experiences a blow in the direction of motion

= which magnifies its velocity to v 2 7 cms. per sec, then the = — moving observer would measure the final velocity as v 2' v 2 = — = a 7 2 5. Thus the change of velocity produced by the — — = — = blow is w v 2 v 1 7 5 2 in absolute space. On the

other hand the moving observer notes the increase of velocity as

= — = — — = — = = — = w' v 2

Vi

(v 2

a)

{v 1 —a)

w v 2

v±

5

3 2.

Both are of the same value. Exactly the same holds for continuous forces and for the
accelerations produced by them. For the acceleration b was
defined as the ratio of the change of velocity w to the time t required in changing it, and since w is independent of whatever
rectilinear uniform forward motion (motion of translation) the system of reference used for the measurement has, the same
holds for b.
The root of this law is clearly the law of inertia, according
A to which a motion of translation occurs when no forces act.
system of bodies, all of which travel through space with the same constant velocity, is hence not only at rest as regards their geometric configuration, but also no actions of forces manifest themselves on the bodies of the system in consequence of the motion. But if the bodies of the system exert forces on each other, the motions thereby produced will occur relatively
just as if the common motion of translation were not taking

THE NEWTONIAN WORLD-SYSTEM 61

place. Thus, for an observer moving with the system, it would

not be distinguishable from one at rest.

The experience, repeated daily and thousands of times,

that we observe nothing of the translatory motion of the earth

is a tangible proof of this law. But the same fact manifests

itself in motions on the earth. For when a motion on the earth

is rectilinear and uniform with respect to the earth, it is so

also with respect to space, if we disregard the rotation in the

earth's motion. Everyone knows that in a ship or a railway

carriage moving uniformly mechanical events occur in the

same way as on the earth (considered at rest). On the moving

ship,

too,

for

example,

a

stone

falls

vertically ;

it

falls

along

a vertical that is moving with the ship. If the ship were to

move quite uniformly and without jerks of any kind the

passengers would notice nothing of the motion so long as they

did not observe the apparent movement of the surroundings.

6. Limited Absolute Space

The law of the relativity of mechanical events is the starting-

point of all our later arguments. Its importance rests on the

fact that it is intimately connected with Newton's views on the

absolute space, and essentially limits the physical reality of this

conception from the outset.
We gave as the reason that made it necessary to assume

absolute space and absolute time that without it the law

of inertia would be utterly meaningless. We must now enter

into the question as to how far these conceptions deserve

A the terms " real " in the sense of physics.

conception has

physical reality only when there is something ascertainable

by measurement corresponding to it in the world of phenomena.

This is not the place to enter into a discussion on the philo-

sophic conception of reality. At any rate it is quite certain

that the criterion of reality just given corresponds fully with

the way reality is used in the physical sciences. Every con-

ception that does not satisfy it has gradually been pushed out

of the system of physics.
We see at once that in this sense a definite place in Newton's

absolute space is nothing real. For it is fundamentally im-

possible to find the same place a second time in space. This is clear at once from the principle of relativity. Given

that we had somehow arrived at the assumption that a definite

system of reference is at rest in space, then a system of refer-

ence moving uniformly and rectilinearly with respect to it

may with equal right be regarded as at rest. The mechanical

events in both occur quite similarly and neither system enjoys

62 THE THEORY OF RELATIVITY

A preference over the other.

definite body that seems at rest

in the one system of rest performs a rectilinear and uniform

motion, as seen from the other system, and if anyone were to

assert that this body marks a spot in absolute space, another

may with equal right challenge this and declare the body to

be moving.

In this way the absolute space of Newton already loses

A a considerable part of its weird existence.

space in which

there is no place that can be marked by any physical means

whatsoever, is at any rate a very subtle configuration, and not

simply a box into which material things are crammed.
We must now also alter the terms used in our definition

of the principle of relativity, for in it we still spoke of a co-

ordinate system at rest in absolute space, and this is clearly

without sense physically. To arrive at a definite formulation
= the conception of inertial system {inertia laziness) has been

introduced, and it is taken to signify a co-ordinate system in

which the law of inertia holds in its original form. There is

not only the one system at rest as in Newton's absolute space,

where this is the case, but an infinite number of others that are

all equally justified, and since we cannot well speak of several " spaces " moving with respect to each other, we prefer to avoid the word " space " as much as possible. The principle

of relativity then assumes the following form :

There are an infinite number of equally justifiable systems,

inertial systems, executing a motion of translation with respect

to each other, in which the laws of mechanics hold in their simple

classical form.
We here see clearly how intimately the problem of space

is connected with mechanics. It is not space that is there

and that impresses its form on things, but the things and their

We physical laws determine space.

shall find later how this

view gains more and more ground until it reaches its climax

in the general theory of relativity of Einstein.

7. Galilei Transformations
Although the laws of mechanics are the same in all inertial systems, it does not of course follow that co-ordinates and velocities of bodies with respect to two inertial systems in relative motion are equal. If, for example, a body is at rest in a system S, then it has a constant velocity with respect to the other system S', moving relatively to S. The general laws of mechanics contain only the accelerations, and these, as we saw, are the same for all inertial systems. This is not true of the co-ordinates and the velocities.

THE NEWTONIAN WORLD-SYSTEM 63
Hence the problem arises to find the position and the velocity of a body in an inertial system S' when they are given for another inertial system S.
It is thus a question of passing from one co-ordinate system
We to another, which is moving relatively to the former.
must at this stage interpose a few remarks about equivalent (equally justified) co-ordinate systems in general and about
the laws, the so-called transformation equations, that allow us
to pass from one to the other by calculation. In geometry co-ordinate systems are a means of fixing in
a convenient manner the relative positions of one body with respect to another. For this we suppose the co-ordinate system to be rigidly fixed to the one body. Then the coordinates of the points of the other body fix the relative position

64 THE THEORY OF RELATIVITY

simplest case, namely, that in which the system S' arises from

S as the result of a parallel displacement by the amount a

in the ^-direction (Fig. 38). Then clearly the new co-ordinate x' of a point P will be equal to its old x diminished by the dis-

placement a, whereas the jy-co-ordinate remains unaltered.

Thus we have

= — = %'

x

a, y'

y

.

.

. (27)

Similar, but more complicated, transformation formulae

We hold in the other case.

shall later have to discuss this

more fully. It is important to recognize that every quantity that has a geometric mean-

ly'

ing in itself must be in-

dependent of the choice of

the co-ordinate system and

must hence be expressed in

similar co-ordinate systems

in a similar way. Such

a quantity is said to be

invariant with respect to

y'\jj

the co-ordinate-transformation concerned. Let

us consider, as an example,

~x~

the transformation (27)
>*' above, that expresses a
displacement along the

Fig. 38-

#-axis. It is clear that

the difference of the ^-co-
— ordinates of two points P and Q, namely, x 2 x lf does not
change. As a matter of fact (Fig. 39),

x\ == (x.

= — a)

x2

%

If the two co-ordinate systems S and S' are inclined to each
other, then the distance s of any point P from the origin is an
invariant (Fig. 40). It has the same expression in both systems, for, by Pythagoras' theorem, we have

= + = + / s2

x2 y2

x' 2

2
.

.

. (28)

In the more general case, in which the co-ordinate system
is simultaneously displaced and turned, the distance P, Q of
two points becomes an invariant. The invariants are par-
ticularly important because they represent the geometrical relations in themselves without reference to the accidental
choice of the co-ordinate system. They will play a considerable
part in the sequel.
If we now return after this geometrical digression to our

THE NEWTONIAN WORLD-SYSTEM 65

starting-point, we have to answer the question as to what are the transformation laws that allow us to pass from one inertial

system to another.
We defined the inertial system as a co-ordinate system

in which the law of inertia holds. Only the state of motion is important in this connexion, namely, the absence of accelerations with respect to the absolute space, whereas the nature and position of the co-ordinate system is unessential. If we choose

it to be rectangular, as happens most often, its position still

We remains free.

may take a displaced or a rotated system,

only it must have the same state of motion. In the foregoing we have always spoken of system of reference wherever we were concerned with the state of motion and not with the

+y

fy'

Q

6 *1

L

VA/

*2

Fig. 39.

Fig. 40.

*-J?

nature and position of the co-ordinate system, and we shall

use the expression systematically from now onwards.

If an inertial system S' is moving rectilinearly with respect

to S with the velocity v, we may choose rectangular co-ordinates

in both systems of reference such that the direction of motion

becomes the x- and the #'-axis, respectively. Further, we may

= assume that at the time t

the origin of both systems co-

incides. Then, in the time t the origin of the S'-system will
= have been displaced by the amount a vt in the ^-direction :

thus at this moment the two systems are exactly in the position

that was treated above purely geometrically. Hence the

equations (27) hold, in which a is now to be set equal to vt.

Consequently we get the transformation equations

x = x — vt

y =y

= z'

z

.

. (29)

in which we have added the unchanged co-ordinate. This

66 THE THEORY OF RELATIVITY

law is called a Galilei transformation in honour of the founder

of mechanics.
We may also enunciate the principle of relativity as follows :
The laws of mechanics are invariant with respect to Galilei

transformations.

This is due to the fact that accelerations are invariant, as we have already seen above by considering the change of velocity of a moving body with respect to two inertial systems.
We showed earlier that the theory of motions or kinematics
may be regarded as a geometry in four-dimensional xyzt-sp&ce,
the " world " of Minkowski. In this connexion it is not without interest to consider what the inertial systems and the Galilei

transformations signify in this four-dimensional geometry.
This is by no means difficult, for the y- and the z-co-ordinate

do not enter into the trans-

formation at all. It is thus

sufficient to operate in the

^-plane.
We represent our iner-

tial system S by a rectangu-

lar ^-co-ordinate system

A (Fig. 41).

second inertial

system S' then corresponds

to another co-ordinate sys-

tem x't', and the question

is : what does the second look like and how is it

situated relatively to the

Fig. 41.

first ? First of all, the time-measure of the second

system S' is exactly the same as that of the first, namely, the

— = one absolute time t t' ; thus the #-axis, on which t

lies,

= coincides with the #'-axis, t' 0. Consequently the system S'

can only be an oblique co-ordinate system. The /'-axis is the
= world-line of the point %' 0, that is, of the origin of the
system S'. The ^-co-ordinate of this point which moves with

the velocity v relatively to the system S is equal to vt in this
system at the time t. For any world-point P whatsoever the

figure then at once gives the formula of the Galilei transforma-

= — tion x'

x

vt.

Corresponding to any other inertial system there is another

oblique ^-co-ordinate system with the same #-axis, but a

differently inclined /-axis. The rectangular system from which we started has no favoured position among all these oblique systems. The unit of time is cut off from all the /-axes of the

various co-ordinate systems by the same parallel to the #-axis.

THE NEWTONIAN WORLD-SYSTEM 67
This is in a certain sense the " calibration curve " of the xt-
plane with respect to the time.
We compress the result into the sentence :
In the xt-plane the choice of the direction of the t-axis is quite arbitrary ; in every xt-co-ordinate system having the same x-axis the fundamental laws of mechanics hold.
From the geometric point of view this manifold of equivalent co-ordinate systems is extremely singular and unusual. The
fixed position or the invariance of the %-axis is particularly
remarkable. When we operate in geometry with oblique
co-ordinates there is usually no reason for keeping the position of one axis fixed. But this is required by Newton's fundamental law of absolute time. All events which occur simultaneously, that is for the same value of t, are represented by a parallel to the #-axis. Since, according to Newton, time flows " absolutely and without reference to any object whatsoever," simultaneous events must correspond to the same world-point
in all allowable co-ordinate systems.
We shall see that this unsymmetrical behaviour of the
world-co-ordinates x and t, here only mentioned as an error of style, is actually non-existent. Einstein has eliminated it through his relativization of the conception of time.

8. Inertial Forces

After having recognized that the individual points in

Newton's absolute space have at any rate no physical reality,

we enquire what remains of this conception at all. Well,

it asserts itself quite clearly and emphatically, for the resistance

of all bodies to accelerations must be interpreted in Newton's

sense as the action of absolute space. The locomotive that

sets the train in motion must overcome the inertial resistance.

The shell that demolishes a wall draws its destructive power

from inertia. Inertial actions arise wherever accelerations

occur, and these are nothing more than changes of velocity

in absolute space ; we may use the latter expression, for a

change of velocity has the same value in all inertial systems.

Systems of reference that are themselves accelerated with

respect to inertial systems are thus not equivalent to the latter,

We or equivalent among themselves.

can, of course, also

refer the laws of mechanics to them, but they then assume a
new and more complicated form. Even the path of a body

left to itself is no longer uniform and rectilinear in an accelerated
system (see III, i, p. 48). This may also be expressed by

saying that in an accelerated system apparent forces, inertial
forces, act besides the true forces. A body on which no true

68 THE THEORY OF RELATIVITY

forces act is yet subject to these inertial forces, and its motion is therefore in general neither uniform nor rectilinear. For example, a vehicle when being set into motion or stopped is
such an accelerated system. Railway journeys have made
everyone familiar with the jerk due to the train starting or stopping, and this is nothing other than the inertial force of
which we have spoken.
We shall consider the phenomena individually for a system
S moving rectilinearly, whose acceleration is to be equal to k.
If we now measure the acceleration b of a body with respect to
this moving system S, then the acceleration with respect to absolute space is obviously greater to the extent k. Hence the fundamental dynamical law with respect to space is
= + m(b k) K.

If we write this in the form
mb = K — mk,

we may say that in the accelerated system S a law of motion
of Newtonian form, namely,
mb = K'

again holds, except that now we must write for the force K'
the sum
K' = K - mk

K — where is the true, and mk the apparent or inertial force.

K = Now, if there is no true force acting, that is, if

o,

.... then the total force becomes equal to the force of inertia K' = — mk

(30)

We Thus this force acts on a body left to itself.

may recog-

We nize its action from the following considerations.

know that

the gravitation on the earth, the force of gravity, is determined

= by the formula G mg, where g is the constant acceleration

= — due to gravity. The force of inertia K'

mk thus acts

exactly

like

weight

or

gravity ;

the

minus

sign

denotes

that

the force of acceleration is in a direction opposite to the system

of reference S used as a basis. The value of the apparent

gravitational acceleration k is equal to the acceleration of the

system of reference S. Thus the motion of a body left to itself

in the system S is simply a motion such as that due to falling

or being thrown.

This relationship between the inertial forces in accelerated

systems and the force of gravity still appears quite fortuitous

here. It actually remained unobserved for two hundred years.

THE NEWTONIAN WORLD-SYSTEM 69
But even at this stage we must state that it forms the basis
of Einstein's general theory of relativity.

9. Centrifugal Forces and Absolute Space

In Newton's view the occurrence of inertial forces in acceler-

ated systems proves the existence of absolute space or, rather,

the favoured position of inertial systems. Inertial forces

present themselves particularly clearly in rotating S3'stems
of reference in the form of centrifugal forces. It was from them

that Newton drew his main support for his doctrine of absolute space. Let us give the substance of his own words :
" The effective causes which distinguish absolute and relative

motion from each other are centrifugal forces, the forces tending

to send bodies away from the axis of rotation. In the case

of a motion that is only relatively

circular these forces do not exist, []77

,

__j^

but they are smaller or greater in

proportion to the amount of the

(absolute) motion." " Let us, for example, hang a

vessel by a very long thread and

turn it about its axis until the

thread becomes very stiff through

the torsion (Fig. 42). Then let

us fill it with water and wait till

both vessel and contents are com-
pletely at rest. If it is now made

to rotate in the opposite direction

by a force applied suddenly, and

if this lasts for some time whilst

the thread unwinds itself, the

surface of the water will first be
plane, just as before the vessel began to move, and then when

the force gradually begins to act on the water, the vessel will

make the water participate appreciably in the motion. It (the water) gradually moves away from the middle and mounts up the walls of the vessel, assuming a hollow shape (I have

carried out this experiment personally)."
" At the beginning when the relative motion of the water

in the vessel ( with respect to the walls) was greatest, it displayed no tendency to move away from the axis. The water did not seek to approach the periphery by climbing up the walls, but remained plane, and thus the true circular motion had not yet begun. Later, however, as the relative motion of the water decreased, its ascent up the walls expressed the tendency to

70 THE THEORY OF RELATIVITY

move away from the axis, and this tendency showed the con-

tinually increasing true circular motion of the water, until
this finally reached a maximum, when the water itself was

resting relatively to the vessel." " Moreover, it is very difficult to recognize the true motions
of individual bodies and to distinguish them from the apparent motions, because the parts of that immovable space in which the bodies are truly moving cannot be perceived bjf the senses."
" Yet the position is not quite hopeless. For the necessary auxiliary means are given partly by the apparent motions,

which are the differences of the real ones, and partly by the

forces on which the true motions are founded as working causes. If, for example, two spheres are connected at a given distance apart by means of a thread and thus turned about the usual centre of gravity (Fig. 43), we recognize in the tension
of the thread the tendency of the spheres to move away from the axis of the motion, and from this we can get the magnitude of the circular motion ... In this way we could find both

the magnitude and the direction of this circular motion in

every infinitely great space, even if there were nothing external

and perceptible in it, with which the spheres could be com-

pared."

These words express most clearly the meaning of absolute
space. We have only a few words of explanation to add to

them.

Concerning, firstly, the quantitative conditions in the case
of the centrifugal forces we can at once get a survey of these if we call to mind the magnitude and the direction of the accelera-

tion in the case of circular motions. It was directed towards

the centre and, according to formula (4), p. 25, it had the

= — value b

v2
, where r denotes the circular radius, and v the

r

velocity.

Now, if we have a rotating system of reference S that ro-
tates once in T sees., then the velocity of a point at the distance

r from the axis (see formula (18), p. 52) is

277?'

hence the acceleration relative to the axis, which we denoted by k (see p. 68) is
k =_ A^r
Now, if a body has the acceleration b relatively to S, its
+ absolute acceleration is b k. Just as above in the case of

THE NEWTONIAN WORLD-SYSTEM 71

rectilinear accelerated motion there then results an apparent force of the absolute value

'" '

a

A 77 1

(3i:

which is directed away from the axis. It is the centrifugal
force.
It is well known that the centrifugal force also plays a part
in proving that the earth rotates (Fig. 44). It drives the
masses away from the axis of rotation and through this causes,
firstly, the flattening of the earth at the poles, and, secondly,
We the decrease of gravity from the pole towards the equator.
became acquainted with the latter phenomenon above, when we were dealing with the choice of the unit of force (II, 15, p. 45) without going into its cause. According to Newton it is a proof of the earth's rotation. The centrifugal force, acting outwards, acts against gravity and reduces the weight. The

oo

Fig. 43.

decrease of the acceleration g due to gravity has the value
4*L? at the equator, where a is the earth's radius. If we here

insert for a the value given above (III, 3, (23), p. 55),

= = a 6*37 . 10 cms., and for the time of rotation T 1 day

= = 24 . 60 . 60 sees.

86,400 sees., we get for the difference

of the gravitational acceleration at the pole and at the equator

the value 3*37 cm./sec. 2, which is relatively small compared with 981 ; this value has to be increased slightly, owing to

the flattening of the earth.

According to Newton's doctrine of absolute space these

phenomena are positively to be regarded not as due to motion

relative to other masses, such as the fixed stars, but as due

to absolute rotation in empty space. If the earth were at

rest, and if, instead, the whole stellar system were to rotate

in the opposite sense once around the earth's axis in 24 hours,

72 THE THEORY OF RELATIVITY
then, according to Newton, the centrifugal forces would not occur. The earth would not be flattened and the gravitational force would be just as great at the equator as at the pole. The motion of the heavens, as viewed from the earth, would
be exactly the same in both cases. And yet there is to be a definite difference between them ascertainable physically.
The position is brought out perhaps still more clearly in Foucault's pendulum experiment (1850). According to the laws of Newtonian dynamics a pendulum swinging in a plane must permanently maintain its plane of vibration in absolute space if all deflecting forces are excluded. If the pendulum is suspended at the North Pole, the earth rotates, as it were, below it (Fig. 45). Thus the observer on the earth sees a
rotation of the plane of oscillation in the reverse sense. If the earth were at rest but the stellar system in rotation, then,

^x

Fig. 45.

Fig. 46.

according to Newton, the position of the plane of oscillation should not alter with respect to the earth. The fact that it does so again appears to prove the absolute rotation of the
earth.
— We shall consider a further example the motion of the
moon about the earth (Fig. 46). According to Newton the moon would fall on to the earth if it had not an absolute
rotation about the latter. Let us imagine a co-ordinate system, with its origin at the centre of the earth, and the #y-plane as that of the moon's orbit, the #-axis always passing through the moon. If this system were to be absolutely at rest, then the
moon would be acted on only by the gravitational force towards the centre of the earth, which, by formula (26) on p. 56,
has the value
K = *M?.

THE NEWTONIAN WORLD-SYSTEM 78
Thus it would fall to the earth along the *-axis. The fact that it does not do so apparently proves the absolute rotation of the co-ordinate system xy. For this rotation produces a centrifugal force that keeps equilibrium with the force K, and we get

y

r2

This formula is, of course, nothing other than Kepler's
m third law. For if we cancel the mass of the moon on both
= sides and express v by the period of revolution T, v -=-, we get

or, by (25) on p. 56,

47rV = JM

:

:

T2

r2

fr3 2

"

&M
£T*

_"

rU

An exactly corresponding result holds, of course, for the
rotation of the planets about the sun.
These and many other examples show that Newton's doctrine of absolute space rests on very concrete facts. If we run through the sequence of arguments again, we see the follow-
ing :
The example of the rotating glass of water shows that the relative rotation of the water with respect to the glass is
not responsible for the occurrence of centrifugal forces. It might be that greater masses in the neighbourhood, say the
whole earth, are the cause. The flattening of the earth, the decrease of gravity at the equator, Foucault's pendulum experiment show that the cause is to be sought outside the earth. But the orbits of all moons and planets likewise exist
only through the centrifugal force that maintains equilibrium
with gravitation. Finally, we notice the same phenomena
in the case of the farthermost double stars, the light from which takes thousands of years to reach us. Thus it seems as if the occurrence of centrifugal forces is universal and cannot be due to inter-actions. Hence nothing remains for us but to assume absolute space as their cause.
Such modes of conclusion have been generally current and regarded as valid since the time of Newton. Only few thinkers
have opposed them. We must name among these few above all
Ernst Mach. In his critical account of mechanics he has analysed the Newtonian conceptions and tested their logical
bases. He starts out from the view that mechanical experience
can never teach us anything about absolute space. Relative

74 THE THEORY OF RELATIVITY

positions and relative motions alone may be ascertained and

are hence alone physically real. Hence Newton's proofs of

the existence of absolute space must be illusory. As a matter

of fact, everything depends on whether it is admitted that

if the whole stellar system were to rotate about the earth no

flattening, no decrease of gravity at the equator, and so forth,

would occur. Mach asserts rightly that such statements go far beyond possible experience. He reproaches Newton very

energetically with having become untrue to his principle of
allowing only facts to be considered valid. Mach himself has sought to free mechanics from this grievous blemish. He was

of the opinion that the inertial forces would have to be regarded

as actions of the whole mass of the universe, and sketched the

outlines of an altered system of dynamics in which only relative

quantities occurred. Yet his attempt could not succeed.

In the first place the importance of the relation between

inertia and gravitation that expresses itself in the proportion-

ality of weight to mass escaped him. In the second place he was unacquainted with the relativity theory of optical and

electro-magnetic phenomena which eliminated the prejudice

A in favour of absolute time.

knowledge of both these facts

was necessary to build up the new mechanics, and the dis-

covery of both was the achievement of Einstein.

CHAPTER IV
THE FUNDAMENTAL LAWS OF OPTICS

i. The Ether

MECHANICS is both historically and logically the foun-

dation of physics, but it is nevertheless only a part of

it, and, indeed, a small part. Hitherto to solve the
problem of space and time we have made use only of mechanical
observations and theories. We must now enquire what the

other branches of physical research teach us about it.

It is, above all, the realms of optics, of electricity, and of

magnetism

that

are

connected

with

the

problem

of

space ;

this is due to the circumstance that light and the electric

and magnetic forces traverse empty space. Vessels out of

which the air has been pumped are completely transparent

for light no matter how high the vacuum. Electric and

magnetic forces, too, act across such a vacuum. The light

of the sun and the stars reaches us after its passage through

empty space. The relationships between the sun-spots and

the polar light on the earth and magnetic storms show inde-

pendently of all theory that electromagnetic actions take

place through astronomic space.

The fact that certain physical events propagate themselves

through astronomic space led long ago to the hypothesis that

space is not empty but is filled with an extremely fine imponderable substance, the ether, which is the carrier or medium of

these phenomena. So far as this conception of the ether is still

used nowadays it is taken to mean nothing more than empty
space associated with certain physical states or " fields."

If we were to adopt this abstract conception from the very out-

set, the majority of the problems that are historically connected

with the ether would remain unintelligible. The earlier ether

was indeed regarded as a real substance, not only endowed with

physical states, but also capable of executing motions.
We shall now describe the development, firstly, of the prin-

ciples of optics, and, secondly, of those of electrodynamics.

This will for the present make us digress a little from the problem

75

76 THE THEORY OF RELATIVITY
of space and time, but will then help us to take it up again fortified with new facts and laws.
2. The Corpuscular and the Undulatory Theory
* I say then that pictures of things and thin shapes are emitted from things off their surfaces . . .
Therefore in like manner idols must be able to scour in a moment
of time through space unspeakable . . .
But because we can see with the eyes alone, the consequence is that, to whatever point we turn our sight, there all the several things meet and strike it with their shape and colour . . .
That is what we read in the poem of Titus Lucretius Carus
on the Nature of Things (Book 4), that poetic guide to Epicurean philosophy, which was written in the last century before the
birth of Christ.
The lines quoted contain a sort of corpuscular theory of light which is elaborated by the imaginative power of the poet but at the same time developed in a true scientific spirit. Yet we can no more call this doctrine a scientific doctrine than we can other ancient speculations about light. There is no sign of an attempt to determine the phenomena quantitatively, the first characteristic of objective effort. Moreover it is particularly difficult to dissociate the subjective sensation of light from the physical phenomenon and to render it measurable.
The science of optics maybe dated from the time of Descartes.
His Dioptrics (1638) contains the fundamental laws of the propagation of light, the laws of reflection and refraction.
The former was already known to the ancients, and the latter had been found experimentally shortly before by Snell (about
1618). Descartes evolved the idea of the ether as the carrier of light, and this was the precursor of the undulatory theory. It
was already hinted at by Robert Hooke (1667), and was clearly formulated by Christian Huygens (1678). Their great contemporary, Newton, who was somewhat younger, is regarded
as the author of the opposing doctrine, the corpuscular theory.
Before entering on the struggle between these theories we shall
explain the nature of each in rough outline. The corpuscular theory asserts that luminescent bodies
send out fine particles that move in accordance with the laws of mechanics and that produce the sensation of light when
they strike the eye.
The undulatory theory sets up an analogy between the propagation of light and the motion of waves on the surface of water or sound-waves in air. For this purpose it has to assume the existence of a medium that permeates all trans-
* From Munro's prose translation, published by Deighton, Bell & Co.

FUNDAMENTAL LAWS OF OPTICS 77

parent

bodies

and

that

can

execute

vibrations ;

this

is

the

luminiferous ether. In this process of vibration the individual

particles of this substance move only with a pendulum-like

motion about their positions of equilibrium. That which

moves on as the light-wave is the state of motion of the particles

and not the particles themselves. Fig. 47 illustrates the process

for a series of points that can vibrate up and down. Each of

the diagrams drawn vertically below one another corresponds to

= a moment of time, say, t

3 o, 1, 2,

.

.

.

Each individual

point executes a vibration vertically. The points all taken

together present the aspect of a wave that advances towards

the right from moment to moment.

Now there is a significant objection to the undulatory theory.

p^ UO

pX
VP

PX

ft

>\

t--1
^P

P >K

>X

t--2

^P

t--3

t=b «i

P *V
VP
Fig. 47.

It is known that waves run around obstacles. It is easy to see this on every surface of water, and sound waves also "go around corners." On the other hand, a ray of light travels in a straight line. If we interpose a sharp-edged opaque body in its path we get a shadow with a definite outline.
This fact moved Newton to discard the undulatory theory. He did not himself decide in favour of a definite hypothesis but merely established that light is something that moves away from the luminescent body " like ejected particles." But his
successors interpreted his opinion as being in favour of the
emission theory, and the authority of his name gained the
acceptance of this theory for a whole century. Yet, at that time Grimaldi had already discovered (the result was published posthumously in 1665) that light can also " bend round corners."

78 THE THEORY OF RELATIVITY

At the edges of sharp shadows a weak illumination in successive striae are seen ; this phenomenon is called the diffraction of light. It was this discovery in particular that made Huygens
a zealous pioneer of the undulatory theory. He regarded
as the first and most important argument in favour of it the fact that two rays of light cross each other without interfering with each other, just like two trains of water-waves, whereas bundles of emitted particles would necessarily collide or at least disturb each other. Huygens succeeded in explaining

the reflection and the refraction of light on the basis of the
undulatory theory. He made use of the principle, now called

after his name, according to which every point on which the
light impinges is to be regarded as the source of a new spherical wave of light. This resulted in a fundamental difference between the emission and the undulatory theory, a difference

that later led to the final experi-

mental decision in favour of the

latter.
It is known that a ray of light

which passes through the air and

strikes the plane bounding surface of a denser body such as

glass or water is bent or refracted

so that it is more steeply inclined

to the bounding surface (Fig. 48) The emission theory accounts
for this by assuming that the

corpuscles of light experience an

Fig. 48.

attraction from the denser medium at the moment they enter into it.

Thus they are accelerated by an impulse perpendicular to the

bounding surface and hence deflected towards the normal. It

follows from this that they must move more rapidly in the

denser than in the less dense medium. Huygen's construction

on the wave theory depends on just the opposite assumption

(Fig. 49). When the light wave strikes the bounding surface it

excites elementary waves at every point. If these become

transmitted more slowly in the second, denser, medium, then

the plane that touches all these spherical waves and that represents the refracted wave according to Huygens, is deflected

in the right sense.
Huygens also interpreted the double refraction of Iceland spar, discovered by Erasmus Bartholinus in 1669, on the basis of the wave-theory, by assuming that light can propagate itself in the crystal with two different velocities in such a way that the one elementary wave is a sphere, the other a spheroid.

FUNDAMENTAL LAWS OF OPTICS 70
He discovered the remarkable phenomenon that the two rays
of light that emerge out of such a piece of fluor spar behave quite differently from other light towards a second piece of fluor spar. If the second crystal is turned about a ray that comes out of the first, then two rays arise out of it which are of varying intensity according to the position of the crystal,
and it is possible to make one or other of these rays vanish
Fig. 49.
entirely (Fig. 50). Newton remarked (1717) that it is to be concluded from this that a ray of light corresponds in symmetry
not to a prism with a circular but rather to one with a square
cross-section. He interpreted this as evidence against the
undulatory theory, for at that time, analogously with soundwaves, only waves of compression and rarefaction were thought of, in which the particles swing " longitudinally " in the direction
Fig. 50.
of propagation of the wave (Fig. 51), and it is clear that these must have rotatory symmetry about the direction of pro-
pagation.
3. The Velocity of Light The first determinations of the most important property
of light, that which will form the nucleus of our following

80 THE THEORY OF RELATIVITY
reflections, namely, the velocity of light, were made independently of the controversy between the two hypotheses about the nature of light. The fact that it was enormously great was clear from all observations about the propagation of light. Galilei had endeavoured (1607) to measure it with the aid of
lantern signals but without success, for light traverses earthly distances in extremely short fractions of time. Hence the
measurement succeeded only when the enormous distances
between the heavenly bodies in astronomic space were used.
Olaf Romer observed (1676) that the regular eclipses of
Jupiter's satellites occur earlier or later according as the earth

• • > \ ''•'•»• p •

•

.«'

•

*

•

4

i

> • '•

'»

*.

'

6

7

8

9

K 10 71 12 13

15

Fig. 51.

is nearer to or farther away from Jupiter (Fig. 52). He inter-
preted this phenomenon as being caused by the difference of

time used by the light to traverse the paths of different lengths,

We and he calculated the velocity of light on this basis.

shall

in future call this velocity c. Its exact value, to which Romer

approximated very closely, is

= = c

300,000 km./sec.

3 . io 10 cms. per sec. . (32)

James Bradley discovered (1727) another effect of the

FUNDAMENTAL LAWS OF OPTICS 81

finite velocity of light, namely, that all fixed stars appear to
execute a common annual motion that is evidently a count i r-

part to the rotation of the earth around the sun. It is very

easy to understand how this effect comes about from the point

We of view of the emission theory.

shall give this inter-

pretation here, but we must remark that it is just this pheno-

menon that raises certain difficulties for the wave-theory,
about which we shall yet have much to say. We know

(see III, 7, p. 64) that a motion which is rectilinear and

uniform in our system of reference S is so also in another

system S', if the latter executes a motion of translation with

respect to S. But the magnitude and the direction of the

velocity is different in the two systems. It follows from this

that a stream of light corpuscles which, coming from a fixed

Fig. 52.

Fig. 53.

star, strike the earth, appear to come from another direction.
We shall consider this deflection or aberration for the particular
case when the light impinges perpendicularly to the motion
of the earth (Fig. 53). Let a telescope, on the objective of which a light corpuscle strikes, be in the position 1. Now, whilst the light traverses the length / of the telescope, the earth, and with it the telescope, moves into the position 2 by an amount d. Thus the ray strikes the centre of the eye-piece
only when it comes, not from the direction of the telescopic axis, but from a direction lying somewhat behind the earth's motion. Hence the direction in which the telescope aims
does not point to the true position of the star, but to a point
of the heavens that is displaced forward. The angle of deflection is determined by the ratio d : /, and is evidently independent of the length / of the telescope. For if the latter be
6

82 THE THEORY OF RELATIVITY

increased, so also is the time that the light requires to traverse
it, and hence also the displacement d of the earth is increased in the same ratio. The two paths I and d, traversed in equal times by the light and the earth, must be in the ratio of the

corresponding velocities :

d_v

This ratio, also called the aberration constant, will in future

be denoted by £ :

.... = v
fl

(33)

c

It has a very small numerical value, for the velocity of the
= earth in its orbit about the sun amounts to about v 3°
km. /sec, whereas the velocity of light, as already mentioned, amounts to 300,000 km./sec Hence j3 is of the order 1 : 10,000.
The apparent positions of all the fixed stars are thus always a little displaced in the direction of the earth's motion at that

Fig. 54.
moment, and hence describe a small elliptical figure during the
annual revolution of the earth around the sun. By measuring this ellipse the ratio jS may be found, and since the velocity v of the earth in its orbit is known from astronomic data, the velocity of light c may be determined from it. The result is
in good agreement with Romer's measurement.
We shall next anticipate the historical course of events
and shall give a note on the earthly measurements of the velocity of light. All that was essential for this was a technical device that allowed the extremely short times required by light to traverse earthly distances of a few kilometres or even only a few metres, to be measured with certainty. Fizeau (1849) and Foucault (1865) used two different methods to carry out these measurements, and confirmed the numerical value of c found by the astronomic method. The details of the process need not be discussed here, particularly as they are
to be found in every elementary textbook of physics. We
call attention to only one point : in both processes the ray of