NSTEIN'S THEORY OBRELATWITY
MAX BORN
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BOSTON UNIVERSITY
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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, M.A.
CHRIST CHURCH, OXFORD
WITH I35 DIAGRAMS AND A PORTRAIT
NEW YORK
E. P. DUTTON AND COMPANY
PUBLISHERS
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FROM THE PREFACE TO THE FIRST EDITION
THIS book is an elaboration of certain lectures which were given last winter to a somewhat con- siderable 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 em- phasised 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 throughout. I believe I am 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
§ 2. Units of Length and Time .
§ 3. Origin and Co-ordinate System .
§ 4. The Axioms of Geometry .
§ 5. The Ptolemaic System
§ 6. The Copernican System
§ 7. The Elaboration of the Copernican Doctrine
7 7 8
9 10 11 13
CHAPTER II The Fundamental Laws of Classical Mechanics
§ 1. Equilibrium and the Conception of Force .
§ 2. The Study of Motions — Rectilinear Motion
§ 3. Motion in a Plane
§ 4. Circular Motion
§ 5. Motion in Space
§ 6. Dynamics — The Law of Inertia
§ 7. Impulses ....
§ 8. The Law of Impulses .
§ 9. Mass ....
§ 10. Force and Acceleration §11. Example — Elastic Vibrations §12. Weight and Mass §13. Analytical Mechanics . § 14. The Law of Energy . § 15. Dynamical Units of Force and Mass
15 16
23 24 26 27 28 29 30 32 34 36 39 4i 4S
CHAPTER III
The Newtonian World-System
§ 1. Absolute Space and Absolute Time . § 2. Newton's Law of Attraction
General Gravitation .....
Celest'al Mechanics ..... § 5. The Relativity Principle of Classical Mechanics § 6. Limited Absolute Space ....
Galilei Transformations ....
Inertial Forces ......
Centrifugal Forces and Absolute Space
b ix
§4-
§7- §8. §9-
4S 5* 53 56.
59' 61 62 67 69
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
75 76 79 83 90
93 102 104 no 120 122
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 § 3. A 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 .....
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 ....
§ 6. Mathematics and Reality
§ 7. The Measure-determination of the Space-time Continuum
§ 8. The Fundamental Laws of the New Mechanics .
§ 9. Mechanical Consequences and Confirmations § 10. Optical Consequences and Confirmations
§11. Macrocosm and Microcosm
§ 12. Conclusion ......
PAOI
2 62 264 268
272
275 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, indeed, signify a "world-view" (Weltanschauung) ? We 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, acoustics, optics, and theory of heat. We 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 wave- length of one hundred millionth of a millimetre) over the Ront- gen 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 concep- tions 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 in- dividual 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 estab- lishes 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 the ways of doing this are different. We 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 perception. We spoke of the inaudible tones, of the invisible light, of physics. We 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 a truly wonderful way. We shall have to speak of this in detail in the sequel.
Thus the achievement of Einstein's theory is the relativiza- tion 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 dis- closed 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 re- gard 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 mathe- maticians 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 de- terminations 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 co- ordinate 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, figures in space, and the laws governing these questions. We 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 relativiza- tion. Again centuries have passed since that first turning- point, 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 circum- navigated 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 im- portance 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 immeasur- able 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 world- view 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 re- ligious 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 de- scribe 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 construc- tions, such as the epicycles of the Ptolemaic system or an im- provement in the calculations of the orbits could be success- fully 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 elementary text-books. We 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 rigid bodies, such as the laws of levers. We 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 formulae to express this. We 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.
t. \ O T 2 ^ A ^ 6 7 8...
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
2 3
Fig. 7.
>>A
draw a single figure in which the x's and the t's occur as co- ordinates (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 proportion- ally), 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 RELATIVITY
TV
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 vx (say, 3 cms. per sec), and afterwards vt (say, 5 cms. per sec), then the increase of velocity is v2 — vx (that is, 5 — 3 = 2 cms. per sec, added in each sec). If v 2 is less than vx (say, v1=i cm. per sec), then vz — vt 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
->x
fc-
FlQ. 10.
Fig. 11.
and are sufficiently small, the polygon will no longer be dis- tinguishable 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,
1 say, t = - 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)
nw = — = b
t
Cf. Fig. 12.
Here
_ i . i
20* 10
*>i =
_ i
2
3 5
2 2
T W
b=~= 10.
w
This quantity b is the measure of the acceleration. Its dimensions are clearly [&]== — = _, and its unit is that
1
6
10
k 1
to I
't
£
7
El
. . . 1 . . . .
i
i .... i
i
i . . .
1
w
L
£
0
&
i
1
i
14
/ ^
*
Fig. 12.
acceleration which causes unit velocity to increase by one unit in the unit of time, that is, referred to the physical system of measure cm. /sec.
If we wish to know how far a movable point moves forward during a uniformly accelerated motion in any time t, we imagine the time t divided into n equal parts,* and suppose the point to receive a sudden increase of velocity w at the end of each
small interval of time -. This little increase is connected with
n
the acceleration b by the formula (2), if we replace the small interval of time t in it by -, thus : w = b- .
* In this case any arbitrary length of time t, and not as before, the unit of time, 1 sec, is divided into n parts.
LAWS OF CLASSICAL MECHANICS 21
Then the velocity
after the first interval of time is : vx = w,
,, second ,, ,, v2 = vx -f- w = 2w
„ third ,, ,, v3 = v2 + w = 31V,
and so forth.
t
The point advances after the first interval of time to : x, = v-
n
second ,, ,, x2 = xx -f v2- = (vx + v2) -,
n n
third „ „ x9=xt+vr-=(v1+vt+v9)-»
n 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 V1-\- V2-\- . . . Vn = IW + 2Z£> + Zw + • • • 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
n t . x bt bt , , x vx + v2 + . . . vn = - (» + 1) - = - (n + 1),
2 w 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=- bt2. 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 . 22 = 5 . 4 = 20 metres in the second sec, 5 . 32 = 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
t
^^-~-
^^-
E
z
0
5 10 2
0 30 W
50 60 70 6
0 90 100 HO W
130
Fig. 13.
intervals, for which the quantity under consideration may be regarded as constant, is carefully studied. This leads us to the conception of the limiting value which forms the starting- point of the differential calculus. Historically, the doctrine of motion was actually the problem for the solution of which Newton invented the differential calculus and its converse, the integral calculus.
The theory of motion (kinematics, phoronomy) is the fore- runner of the proper mechanics of forces, or dynamics. It it evidently a sort of geometry of motion. As a matter of fact, in our graphical representation each motion is represented by a geometric configuration in the plane, with the co-ordinates x, t. In this we are concerned with more than a mere analogy. It is just the principle of relativity that attaches fundamental importance to the introduction of time as a co-ordinate in conjunction with the spatial dimensions.
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 arbi- trary curve. The projection of this curve into the ^y-plane is the orbit in the plane (or plane- orbit). The velocity and the ac- celeration are again calculated by 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 the velocity alters. A 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
-*~JT
Fig. 16.
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 projected motions. We 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 magni- tude 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 into the second. We 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 point traverses, at the centre of the circle. For the velocities at A and B are perpendicular to the radii MA and MB, and hence include the same angle. Consequently the two isosceles triangles MAB and CDE are similar, and we get the proportion
Fig. 18.
DE CD
AB MA
Now DE = w, CD = v, and further, MA is equal to the radius r of the circle, and AB 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 sv
— = - or w = — . v r r
S IS)
We now divide by t and notice that - = v, — = 0. Hence
t t
the acceleration
b =
• (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
parallel to the *-axis during the circular motion. We 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 un- fortunately 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 dimen- sions. 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 de- termined 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. Min- kowski 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 all. A 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 by philosophy. We 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 on the duration of the action, even if this is very short. We therefore define the intensity of a blow or impulse as follows :
n impulses J, each of which consists of the force K acting
during the time t = — sees., will, if they follow each other with-
n
out appreciable pauses, have exactly the same effect as if the force K were to continue to act throughout the whole second. Thus we should have
J = ,JJ = K,
t
or, J = ?K = <K . . . . (5)
n
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 " Im- pulse Balance " enables us to measure the intensity of blows even when we cannot ascertain the duration and the force of each one singly. We need only find the force K 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 velo- city, 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
Hitherto we have dealt with a single sphere. We 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 horizontal force. We 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 a difference of resistance of the spheres to impacts. We 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 in- crease 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, if the bodies of mass mx and m2 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, Jx = — J2, or their sum is zero :
Ji + J 2 = mfvx + m2w2 = o . . . (8)
32 THE THEORY OF RELATIVITY
From this it follows that
w2= — — m,
that is, when one sphere loses velocity (w1 negative), the other gains velocity (w2 positive), and vice versa.
If we introduce the velocities of the two spheres before and after the impact, namely, vlf v^ for the first sphere, and v2, v2' for the second, then the changes of velocity are
wx = vj — vx w2 = v2 — v2
and we may also write the equation (8) thus :
tn^Vi — vj + m2(v2 — v2) = 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
m1v1 + m2v2 = m^v-l + m2v2 . . . (9)
and this equation may be interpreted as follows :
To bring a body of mass m 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 carried along by the two spheres before the impact is m1v1 + m2v2. 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 velocity alters continuously. We 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
each of them has the value J = - K or = /K, where t is the short
n
interval occupied by each blow. At each impulse a change of velocity w occurs which, according to (7), is determined by
w
w
b, thus we get
10)
mw = J = tK, or m- = K. But, by (2), t
mb = K .
This is the law of motion of dynamics for forces that act continuously. It states in words that a force produces an acceleration that is proportioned to it ; the constant ratio K : 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
Fig. 21.
along in the time t required to effect it. Accordingly we may express the fundamental law ex- pressed in formula (10) thus :
If a force K 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 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 bear the relation mb — K 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 under the action of elastic forces. We take, say, a straight broad steel spring and fasten it at one end so that it lies hori- zontally 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.
In many cases the restoring force K 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
(where tt == 3*14159 . . .), in the time T = - 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 fastened to the spring. This point A 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
tf
P
7 4
\
?
1 y
/a
i i
1 x
1 0
x A
1
Fig. 22.
Fig. 23.
and s are the side and the hypotenuse of a small right-angled triangle, which is clearly similar to the large right-angled triangle OAP. Hence we have the proportion
s a a
Hence the velocity of A becomes
f
2-nvy.
Now, the point of projection B of the point P executes exactly the same pendulum motion on the v-axis. During the small displacement s of P the point B moves forward a distance 77, and just as for f, we have
x
- or 77
a
36 THE THEORY OF RELATIVITY
This change 77 of y corresponds to a change in the velocity v = 2-nvy of the point A which is given by
w
27TV7) = 27TVS-,
a
and hence to an acceleration of A,
w
s x
b = — = 2ttv- . - = (2ttv)2X
t a
The acceleration in this vibrational motion of the point A is thus actually at every moment proportional to the deflection x. We 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 thus determine the mass m 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 = 0 the sphere is moving through the middle position x = 0 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 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.
Fig. 24.
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 the velocity attained. We 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 have twice the weight ; or, more generally, the masses m 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
— = g or G = mg . . . (12)
m
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 body roll or fall down equally fast. We 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 = mgt
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 distri- bution 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.
Fig. 26 exhibits the bob at the deflection x. We 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 of which are G± and G2, respectively :
(27TV)2M1
_G.
(27Tv)2m2 = -p
t
thus
— 2 = (27TV) n,
m.
that is, the ratio of the weight to the mass is the same for both pendulums. We 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
the motion when the forces K 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 vary- ing position of the moving point, is a problem of the integral calculus that may be very difficult if the force alters in a com- plicated 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
V 5
f>
7
3^
y,
)
< \
/
'
r \
i8
/
>
\
0
Fig. 27.
an arbitrary initial velocity it arrives after t sees, at a point 1 where the first blow strikes it. From this point it pursues its course in another direction with a different velocity for t sees, until at a point 2 it is struck by the second blow, which again deflects it, and so forth. Each individual deflection may be determined from the law of impulses. Accordingly, we may draw the whole motion, and we see that the initial point, the initial direction, and the initial velocity completely determine the subsequent course of the motion. This jerky motion gives us a rough picture of the motion of a sphere on an inclined plane. The graph coincides the more closely with the event, which is continuous in reality, the smaller we choose the time interval between the blows.
What is here achieved by direct construction is usually done in the case of forces acting continuously by means of the integral calculus. In this case, too, the point of departure and the initial velocity remain quite arbitrary as regards
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 under- stand 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.
tance for the development of the physical sciences. We cannot, of course, enunciate it generally here nor prove it. We 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 orbit from A 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 velocity on the height. We assert that it states the following :
Let x be the path fallen through, measured upwards (Fig 30), v the velocity, m 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
is, of course, zero, and vice versa. We 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, (v -f w)2 = V2 -f- w2 -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
m
e = YJ — E = — w2 -{- mvw — Gvt,
2
or, since w — bt,
e = ? h2t2 + mvbt - Gvt.
2
Hence the rate of change becomes
m
b2t -\- mvb — Gv.
w
0
Fig. 30.
v.w.
«2
V2
v.w.
V
Fig. 31.
W
The term involving t may be neglected since it can be made vanishingly small by making the time interval vanishingly small. Hence we get finally for the rate of change of E,
= v(mb — G).
But on the basis of the laws of mechanics this expression has the value zero, for by (13) we have mb = mg = G. Hence we have proved that the quantity E (15) remains unchanged with time. If the initial point and the initial velocity of the motion are given, that is, the values of x and v for t = 0, then the expression E, according to (15), acquires a definite value. It then retains this value during the whole motion.
From this it follows that if the body rises, that is, if x increases, v must decrease, and vice versa. Either of the two terms of the expression E can increase only at the expense of
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 against the force of gravitation. We have special names for these terms.
T = — v2 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 de- pending 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 (i7)
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 ex- periments 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 usage applies to the small globe of the earth. We 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 gravitational acceleration g. We 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 de- flections. 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 points of the earth. We 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«] = [^]
2
and its unit is grm. cm.2/sec.
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 experi- mented 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 so that it can be turned by means of a handle. A 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 experi- ments 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 dis- covered 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 ma- terial 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 starting- point 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 im- movable 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. " Hypo- theses non fingo," is his brief and definite expression. But what exists " without reference to any external object what- soever " is not ascertainable, and is not a fact. Here we have clearly a case in which the ideas of unanalysed consciousness are applied without reflection to the objective world. We 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 hypo- theses, but by pursuing the syste- matic and exact path of analysing the known facts of planetary mo- 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 between the acceleration b of the motion, and the force K that
Fig. 33.
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 astro- nomical 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 /tQ\
v = — . . . . (18)
so that 0 = ^-^ = =Vr
ri2 1-
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 :
Ti = C or T* = ;t ■ ■ ■ (I9)
If we insert this in the value for b above, we get
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 like- wise 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 inertia that here expresses itself as a centrifugal force. We 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
47T2C
where C has the same value as for the moon, namely, by (19),
r*
L - f"»"
If we insert this value in that for g, we get
g
-W . . . (22:
TV
Now, the " sidereal " period of revolution of the moon, that is, the time between two positions in which the line con- necting 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 . io7)th part of the earth's circumference 2na
100 = , or a = 6-37 . io8 cms. . . (21)
4 . io7 v J/
If we insert all these values in (22) we get
att2 . 6o-i3 . 6-37 . io8 0 , 2 , .
g = -1 _ 4Z == q8i cm. /sec.2 . (24)
6 2-36* . io12 J l 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 Newton's law its final form. We saw that a planet situated at a distance r from the sun experiences from it an attraction of the value (21)
ir Att 2C
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
attract the sun. If M 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 of mechanics. If we apply it here, we must set K = K', or
47T2C
nF—— =
r2
r2
From this it follows
that
mC =
Mc,
or
C M
c m
that is, this ratio has the same value for both bodies (sun and
planets), and hence also for any body whatsoever. If we call
k this value — , then we may write
47T2
477-2C = kM 4tt2c = km . . . (25)
The factor of proportionality k is called the gravitational con- stant.
The Newtonian law of general gravitation then assumes the symmetrical form
K = k~^r ■ ■ ■ ■ (26)
In words it states :
Two bodies attract each other with a force that is proportional to the mass of each body and is inversely proportional to the square of their distance apart.
4. Celestial Mechanics
It is only in this general form that the Newtonian law denotes a real advance in the calculation of the planetary orbits. For in the original form it was deduced from Kepler's laws by calculation and denoted no more than a very short and striking resume of these laws. It is also possible to prove conversely that the motion of a body about a central body that is at rest and that attracts it according to Newton's law
THE NEWTONIAN WORLD-SYSTEM 57
is necessarily a Kepler elliptic motion. A 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. Ac- cordingly, 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 as deviations from the Kepler laws. We 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 calcu- late 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 peri- helion (Fig. 35), does not remain fixed after all the above cor- rections 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 attrac- tion 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 con- firmed 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 laws. We 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 in its orbit about the sun with stupendous speed. How 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 only in space absolutely at rest ? We 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, v1 and v2, 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 vx = 5 cms. per sec, then an observer moving with the velocity a = 2 cms. per sec. in the same direction would measure only the relative velocity v^ = v± — a= 5 — 2 = 3. If the body now experiences a blow in the direction of motion which magnifies its velocity to v2 = 7 cms. per sec, then the moving observer would measure the final velocity as v2' = v2 — a = 7 — 2 = 5. Thus the change of velocity produced by the blow is w — v2 — v1 = 7 — 5 = 2 in absolute space. On the other hand the moving observer notes the increase of velocity as
w' = v2 — Vi = (v2 — a) — {v1—a) = v2 — v± = w = 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 to which a motion of translation occurs when no forces act. A 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 the terms " real " in the sense of physics. A 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
preference over the other. A 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 considerable part of its weird existence. A 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 physical laws determine space. We 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 velo- city 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 to another, which is moving relatively to the former. We 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 co- ordinates of the points of the other body fix the relative position
Y
S'
> X
© —
s
*-x
>x
Fig. 36.
Fig. 37.
completely. It is, of course, immaterial whether the co- ordinate system is chosen as rectangular, oblique, polar or still more general. It is also immaterial how it is orientated with respect to the first body, except that either this orientation must be maintained, or, if it is changed, we must specify how the co-ordinate system alters its position with respect to the body. If, for example, we operate with rectangular co- ordinates in a plane, then in place of the system S first chosen we may select a second, S', which is displaced (Fig. 36) or turned (Fig. 37) with respect to S. But we must specify ex- actly how great is the displacement and the turning. From these data we can then calculate what the co-ordinates of a point P that had the values x, y in the old system S are in the new system S'. If we call them %' , y' we get formulae that allow us to calculate x' ,y' from x, y. We shall do this for the
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 hold in the other case. We 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-transfor-
mation concerned. Let us consider, as an example, the transformation (27)
>*' above, that expresses a
~x~ displacement along the
Fig. 38- #-axis. It is clear that
the difference of the ^-co- ordinates of two points P and Q, namely, x2 — xlf 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 accelera- tions 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 remains free. We may take a displaced or a rotated system, only it must have the same state of motion. In the fore- going 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
*-J?
Fig. 39.
Fig. 40.
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 = 0 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 in which we have added the unchanged
z' = z . . (29)
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 (Fig. 41). A 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 first ? First of all, the time-measure of the second
Fig. 41.
system S' is exactly the same as that of the first, namely, the one absolute time t — t' ; thus the #-axis, on which t = 0 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 " abso- lutely 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, or equivalent among themselves. We 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
where K is the true, and — mk the apparent or inertial force.
Now, if there is no true force acting, that is, if K = o, then the total force becomes equal to the force of inertia
K' = — mk .... (30)
Thus this force acts on a body left to itself. We may recog- nize its action from the following considerations. We 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
v2 value b = — , 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
_ A^r
k =
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 A77'"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, the decrease of gravity from the pole towards the equator. We 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
o
o
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 to- wards 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 centri- fugal force that keeps equilibrium with the force K, and we get
y r2
This formula is, of course, nothing other than Kepler's third law. For if we cancel the mass m 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
r3 &M _ r
f2 " £T* " U
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 in favour of absolute time. A 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 imponder- able 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 particu- larly 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 con- temporary, 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 pro- pagation 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 = o, 1, 2, 3 . . . 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.
UO
t--1
t--2
t--3
t=b «i
p ^
p X
V P
P X
ft >\
^ P
P >K
0 >X
^ P
P *V
V P
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 sur- face 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 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.
Fig. 48.
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 sound- waves, 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 indepen- dently 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 • • .«' • * •
0 4 i ♦ • '• > *. '» '
6 7 8 9 10 71 12 13 K 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, and he calculated the velocity of light on this basis. We shall in future call this velocity c. Its exact value, to which Romer approximated very closely, is
c = 300,000 km./sec. = 3 . io10 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 of view of the emission theory. We 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 deflec- tion is determined by the ratio d : /, and is evidently inde- pendent 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 £ :
fl = v .... (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 velo- city 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
FUNDAMENTAL LAWS OF OPTICS 83
light is projected from the source Q on to a distant mirror S, where it is reflected and returns to its starting-point (Fig. 54). It traverses the same path twice, and hence it is only the mean velocity during the motion there and back that is measured. The following result, which is important for later considerations, arises from this circumstance : if we suppose that the velocity of light is not the same in both directions, because the earth itself is in motion — we shall discuss this point later (IV, 9, p. no) — then this influence will be wholly or partially cancelled in the motion to and fro. Therefore, in view of the smallness of the velocity of the earth in comparison with that of light, we need take no account of the earth's motion in these measure- ments in practice.
The measurements of the velocity of light were later repeated with improved apparatus, and a considerable degree of accuracy was obtained. Nowadays they can be carried out in a room of moderate length. The result is the value (32) given above. Foucault's method also allowed the velocity of light to be measured in water. It was found to be smaller than that obtained for air. This gave a definite decision on one of the most important points under dispute between the emission and the undulatory theory in favour of the latter. This occurred, indeed, at a time when the triumph of the wave- theory had already long been assured on other grounds.
4. Fundamental Conceptions of the Wave Theory Interference
Newton's greatest achievement in optics was the resolution of white light into its coloured constituents by means of a prism and the exact examination of the spectrum, which led him to the conviction that the individual spectral colours were the indivisible constituents of light. He is the founder of the theory of colour, the physical content of which is still fully valid to-day — in spite of Goethe's attacks. The power of Newton's discoveries paralysed the free thought of the succeed- ing generations. His refusal to accept the undulatory theory blocked the road to its acceptance for well nigh a century. Nevertheless it found isolated supporters such as, for example, the great mathematician Leonhard Euler in the 18th century.
The revival of the wave-theory is due to the works of Thomas Young (1802), who adduced the principle of interfer- ence to explain the coloured rings and fringes which even Newton had observed in thin layers of transparent substances. We shall at this stage deal somewhat in detail with the pheno- menon of interference because it plays a decisive part in all
84 THE THEORY OF RELATIVITY
finer optical measurements, particularly in researches that constitute the foundations of the theory of relativity.
We explained the nature of waves above : it consists in the individual particles of a body executing periodic oscillations about their positions of equilibrium, whereby the momentary position or the phase of the motion is different for neighbouring particles and moves forward with constant velocity. The time that a definite particle requires for one vibration, to and fro, is called the time of vibration or the period, and is denoted by T. The number of vibrations in one second or the frequency is designated by v. Since the time of a vibration multiplied by their number per second must give exactly one second, we must have vT = i, thus
„ = * or T = T . . . (34)
T v
Fig. 55.
Instead of vibration number or frequency we often say " colour," because a light wave of a definite frequency pro- duces a definite sensation of colour in the eye. We shall not enter into the complicated question as to how " physical colours," as we may call the great manifold of psychological impressions of colour, come about through the conjoined action of simple periodic vibrations. The waves that start out from a small source of light have the form of spheres. This means that particles on a sphere drawn around the source as centre are always in the same state of vibration or they are of equal " phase " (Fig. 55). By means of refraction or other influences a part of such a spherical wave may be deformed so that the surfaces of equal phase or the wave surfaces have some other
FUNDAMENTAL LAWS OF OPTICS 85
form. The simplest wave-surface is evidently the plane, and it is clear that a sufficiently small piece of any arbitrary wave surface, hence even of a spherical surface, may always be regarded in approximations as plane. Hence we consider in particular the propagation of plane waves (Fig. 56). The direction that is perpendicular to the planes of the waves, that is, the normal to the waves, is at the same time the direction of propagation. It is clearly sufficient to consider the state of vibration along a straight line parallel to this direction.
Whether the vibration of the individual particle occurs parallel or perpendicularly to the direction of propagation, whether it is longitudinal or transverse, will be left quite open at this stage. In the figures we shall always draw wave lines and call the greatest displacements upwards and downwards crests and hollows.
The distance from one crest to the next is called wave- length and is designated by A. The distance between the suc- cessive or any two consecutive planes in the same phase is obviously exactly the same amount.
Fig. 56.
During a vibration of a definite particle to and fro, the duration of which is T, the whole wave moves forward exactly a wave-length A (Fig. 47, p. 77). Since the velocity in every motion is equal to the ratio of the path traversed to the time required to do so, the wave velocity c is equal to the ratio of the wave-length to the time of vibration :
A T or
Xv
(35)
If a wave enters from one medium into another, say, from air into glass, the time rhythm of the vibrations is, of course, carried over the bounding surface, that is, T (or v) remains the same. On the other hand, the velocity c and hence, on account of formula (35), also the wave-length A changes. Thus all methods of measuring A may serve to compare the velocity of light in various substances or under various circumstances. We shall make use of this fact later.
86
THE THEORY OF RELATIVITY
We are now in a position to understand the nature of inter- ference phenomena, the discovery of which helped the wave- theory to prevail. Interference may be described by the paradoxical words : light added to light does not necessarily give intensified light, but may become extinguished.
The reason for this is that, according to the wave-theory, light is no stream of material particles but a state of motion. Two vibration impulses that occur together may, however, destroy the motion just like two people who wish to do contrary things impede each other and produce nothing. Let us imagine two trains of waves that intersect. This phenomenon can be conveniently observed if we look from a hillock down into a lake in which the waves caused by two ships meet (Fig. 57). These two wave-systems interpenetrate without dis- turbing each other. In the region at which both exist simul- taneously a complicated motion arises, but so soon as the one
Fig. 57.
Fig. 59.
wave has passed through the other, it continues further as if nothing had happened to it. If we fix our attention on one particular vibrating particle, we see that it experiences inde- pendent impulses from both waves. Hence its displacement at any point is simply the sum of the displacements that it would have under the influence of the individual waves. Two wave-motions are said to superpose without disturbing each other. From this it follows that at points where crest and crest and also at points where hollow and hollow meet, where two equal waves encounter each other, the elevations and the depressions are twice as great (Fig. 58). But at points where crest and hollow meet the impulses destroy each other and no displacement occurs at all (Fig. 59).
If we wish to observe interference of light it will not do simply to take two sources of light and to allow the trains of waves emerging from them to interpenetrate. No observable inter-
FUNDAMENTAL LAWS OF OPTICS 87
ference phenomenon occurs through this, because the actual light waves are no absolutely regular waves. Rather, the state of vibration suddenly alters in a striking way after a series of regular vibrations have occurred, corresponding to the accidental phenomena that occur during the emission of light in the source. These irregular changes effect a corresponding fluctuation of the interference phenomena, which occurs much too quickly for the eye to follow it, and hence we see only uni- form light.
To obtain observable interferences we must resolve a ray of light by artificial means, by reflection or refraction, into two rays, and afterwards make them come together again. Then the irregularities of the vibrations in both rays occur in exactly the same time rhythm, and hence it follows that the inter- ference phenomena do not fluctuate in space, but remain fixed. Wherever the waves strengthen or extinguish each other at a certain moment, they do so at every moment. If we direct the eye, armed with a magnifying glass or a telescope, at such a point we see fringes or rings, provided we use light of one colour (monochromatic light), such as is approximately emitted by a Bunsen flame coloured yellow by common salt. In ordinary light which is composed of many colours the interference spots corresponding to the various wave-lengths do not exactly coincide. At one point red is intensified, say, and blue is extinguished, at other points other colours occur, and hence spots and fringes arise with wonderful colourings. It would, however, take us away from our path of enquiry to pursue these interesting phenomena further.
The simplest arrangements for producing interferences were given by Fresnel (1822), an investigator whose works have furnished the foundation for the theory of light which has remained unattacked up to the present day. We shall often meet with his name in the sequel. That time, the first de- cades of the nineteenth century, must in many respects have resembled our own. Just as nowadays through the discovery of radioactivity and the associated phenomena of radiation, through the enunciation of the physical principle of relativity and of the doctrine of quanta, our knowledge of physical nature is undergoing a stupendous process of deepening and enlargement, which seems to the beholders a complete revolution of all conceptions, so, a hundred years ago, the thousands of individual observations, theoretical experiments, physical or metaphycisal speculations coalesced for the first time into complete and uniform ideas and theories, the application of which at once suggested an undreamed-of abundance of new observations and experiments. At that time Lagrange's
88
THE THEORY OF RELATIVITY
" Analytical Mechanics " and Laplace's " Celestial Mechanics " appeared, the two works that brought Newton's ideas to their conclusion. From them there was developed on the one hand, by Navier, Poisson, Cauchy, and Green, the mechanics of deformable bodies and the theory of fluids and elastic sub- stances ; on the other hand, by the works of Young, Fresnel, Arago, Malus, and Brewster, the theory of light. At the same time began the era of electromagnetic discoveries, of which we shall speak later. At that time physical, research was almost entirely in the hands of the French, Italians, and English. Nowadays all educated nations participate, and the authors of the great revolutionary theories of relativity and quanta, Einstein and Planck, are German.
Fresnel allowed a ray of light to be reflected at two mirrors,
Fig. 60.
Fig. 6i.
Sj and S2 (Fig. 60), slightly inclined to each other. At the points where they meet, these two reflected rays give inter- ference fringes that can be seen with a magnifying glass. Similar arrangements of apparatus have been given in great numbers. We shall here enter only into a field which is im- portant for our purpose, namely, that of the experimental methods of measuring minute changes in the velocity of light. The apparatus used is called an interferometer. It depends on the fact that the wave-length alters proportionately with the velocity of the light, and hence the interferences are dis- placed. An example of an apparatus of this kind is the inter- ferometer of Michelson. It consists in the main (Fig. 61) of a glass plate P that is slightly silvered so as to allow one half of the light from the source Q to pass through while the other half is reflected.
FUNDAMENTAL LAWS OF OPTICS 80
These two component rays travel on to two mirrors Sj and S2, where they are reflected and again encounter the semi- transparent glass plate P, which again resolves them, sending one half of each ray into the observing telescope F. If the two paths PSj and PS2 are exactly equal the two component rays arrive at the telescope in the same phase of vibration and recombine to form the original light again. But if the path of the first ray be lengthened by displacing the mirror S2, then the crests and hollows of the two trains of waves no longer coincide when the rays are recombined at F, but are displaced with respect to each other and weaken each other more or less. If the mirror Si is moved slowly we see alternate patches of light and darkness in the telescope F. The distance of the positions of Sx for two successive dark fields is exactly equal to the wave-length of the light. In this way Michelson has made measurements of wave-length that exceed almost all other physical measurements in accuracy. This is done by counting the changes of light and darkness during a considerable shift of the mirror Slf which comprises many thousands of wave- lengths. The error of observation of an individual wave- length then becomes just as many thousand times smaller.
We have here to give several numerical data. By the above method it is found that the wave-length of the yellow light that is sent out by a Bunsen flame coloured with common salt (NaCl), and the source of which is sodium atoms, is about
%mm. = 6'io ~ 5 cms. in vacuo. All visible light lies within
10,000
the small region of wave-lengths stretching from about 4.10 ~5
(violet) to 8.10 ~ 5 cms. (red). Thus in the language of acoustics
this comprises one octave ; that is, it is the region between
one wave and another that is twice as long. From formula (35)
there then follows for the vibration number of yellow sodium
c 'Vio*^
light the stupendous number i> = - = _- = 5'io14 or 500
r A 610-5
billion vibrations per second. The most rapid acoustic vibra- tions that are still audible vibrate only about 50,000 times per second.
The astonishing accuracy of optical methods of measurement rests on the multiplication of the individual wave-lengths used in interferometric measurements. For example, it allows us to ascertain that the velocity of light in a gas likewise alters if there is a very small change of pressure or temperature (due, say, to the apparatus being touched by the hand). To show this the gas is passed into a cylinder between the glass plate P and the mirror Sr It is then seen that for even the slightest increase of pressure the interference changes, light fields being converted into darkness and vice versa.
90 THE THEORY OF RELATIVITY
For the rest, we must remark that in the interferometer we do not simply see a light on a dark field in the telescope, but a system of light and dark rings. This is due to the fact that the two rays are not exactly parallel and the waves are not exactly plane. The separate parts of the two rays have thus to traverse paths of different length. We shall not, however, enter into the geometric details, but mention this circumstance only because it is customary to speak of interference bands or fringes.
We shall meet with Michelson's interferometer again when we have to decide the question as to whether the earth's motion influences the velocity of light.
5. Polarisation and Transversality of Light Waves
Although interference phenomena allow scarcely any inter- pretation other than that of the wave-theory, its general recognition was impeded by two difficulties which, as we saw above, were regarded by Newton as being decisive contradictions to it : firstly, the general rectilinear propagation of light (that is, except for trifling diffraction phenomena) ; secondly, the ex- planation of polarization phenomena. The first difficulty became removed when the wave theory itself was worked out more exactly ; for it was found that waves do, indeed, " bend round corners," but only in regions that are of the order of magnitude of the wave-length. As this is very small in the case of light, our ordinary unrefined vision receives the impression of sharp shadows and rectilinearly bounded rays. Only minute observa- tion is able to detect the interference fringes of diffracted light along the edges of the shadow. The merit of elaborating the theory of diffraction is due to Fresnel, later Kirchhoff, (1882), and, more recently, Sommerfeld (1895). They have deduced the finer phenomena mathematically and have defined the limits within which the conception ray of light may be applied.
The second difficulty concerned the phenomena due to the polarization of light.
When we earlier spoke of waves we always had in mind longitudinal waves such as are known in the case of sound. For a sound wave consists of rhythmical condensations and rarefactions, during which the individual particles of air move to and fro in the direction of propagation of the wave. Trans- versal waves were, indeed, also known ; for example, the waves on a surface of water, or the vibrations of a stretched string, in which the particles vibrate perpendicularly to the direction of propagation of the wave. But in this case we are dealing not with waves that advance in the interior of a substance
FUNDAMENTAL LAWS OF OPTICS 91
but in part with phenomena on the upper surface (water waves) and in part with motions of whole configurations (vibration of strings). Observations or theories about the propagation of waves in elastic solid bodies were not yet known. This accounts for the circumstance, which appears strange to us, that it was so long before optical waves were recognized as transverse vibrations. In fact, the remarkable instance occurred that the impulse for the development of the mechanics of coarse-grained solid elastic bodies was given by experiments and conceptions derived from the dynamics of the imponderable and intangible ether.
We explained above (p. 79) what constitutes the nature of polarization. The two rays that emerge out of a doubly refracting crystal of calcite do not behave like ordinary light when they pass through a second such crystal, that is, they
Fig. 62.
Fig. 63.
do not again resolve into the equally intense rays, but into two of unequal intensity, one of which may under certain circum- stances vanish entirely.
In ordinary " white " light the various directions within the plane of a wave are of equal value or equivalent (Fig. 62). In polarized light this is obviously no longer the case. Malus discovered (1808) that polarization is not a peculiarity of the light that has passed through a doubly refracting crystal, but may also be produced by simple reflection. He showed that light which has been reflected from a mirror at a definite angle is reflected by a second mirror to a varying degree, if the latter mirror is turned about the incident ray (Fig. 63). The plane perpendicular to the surface of the mirror and con- taining the incident and the reflected ray is called the incident plane. The reflected ray is then said to be polarized in the incident plane ; this implies no more than that it behaves
92 THE THEORY OF RELATIVITY
differently towards a second mirror according to the position of the second incident plane to the first. If these mirrors are perpendicular to each other no reflection at all occurs at the second mirror.
The two rays that emerge out of a crystal of calcite are polarized perpendicularly to each other. If we allow them both to fall on to a mirror at an appropriate angle the one is completely extinguished just when the other is reflected to its full amount.
Fresnel and Arago made the decisive experiment (1816) when they attempted to make two such rays, polarized per- pendicularly to each other, interfere. They did not succeed. Fresnel and also Young then drew the inference (1817) that light vibrations must be transversal.
As a matter of fact this deduction makes the peculiar behaviour of polarized light intelligible at once. The vibra- tions of the ether particles do not occur in the direction of
Fig. 64. Fig. 65.
propagation but perpendicularly to it, that is, in the plane of the wave (Fig. 62). But every motion of a point in a plane may be regarded as composed of two motions in two directions perpendicular to each other. In dealing with the kinematics of a point we saw that its motion is determined uniquely when its rectangular co-ordinates, which vary with the time, are given. Now a doubly refracting crystal clearly has the property of transmitting the vibrations of light in it at different velocities in two mutually perpendicular directions. Hence, by Huyghens' principle, when these vibrations enter the crystal they will be deflected to different extents, or refracted differently, that is, they will be separated in space. Each of the emergent rays then consists only of vibrations that take place in a certain plane that passes through the direction of the ray, and the planes belonging to these two rays are mutually perpendicular (Fig. 64). Two such vibrations clearly cannot influence each other : they cannot interfere. Now, if a polarized ray enters into a second crystal it is transmitted without being weakened
FUNDAMENTAL LAWS OF OPTICS 93
only if its direction of vibration is in just the right position with respect to the crystal, being just that in which this vibra- tion can propagate itself. In all other positions the ray is weakened : in the perpendicular position it is not transmitted at all.
Similar conditions obtain in reflection. If this occurs at the appropriate angle, then only one of the two vibrations, the one parallel and perpendicular to the incident plane, is reflected ; the other penetrates into the mirror and is absorbed (Fig. 65). Whether the reflected vibration is that which takes place in the incident plane or perpendicularly to it cannot, of course, be ascertained. (In Fig. 65 the latter is assumed to be the case.) But this question of the position of vibration with respect to the plane or incidence or the direction of polarization has given rise to elaborate researches, theories and discussions, as we shall presently see.
6. The Ether as an Elastic Solid
After the transversality of light waves had been proved in this way and by numerous experiments, there arose in Fresnel's mind the vision of a future dynamical theory of light, which was to derive optical phenomena from the properties of the ether and the forces acting in it, in conformity with the method of mechanics. The ether was necessarily a kind of elastic solid, for it is only in such a substance that mechanical transverse waves can occur. But in Fresnel's time the mathe- matical theory of elasticity of solid bodies had not yet been developed. Possibly he also thought from the outset that the analogy of the ether with material substances was not to be carried too far. At any rate he preferred to investigate the laws of the propagation of light empirically and to interpret them by means of the idea of transversal waves. Above all it was to be expected that the optical phenomena in crystals would shed light on the behaviour of the ether. Fresnel's work in this field is to be ranked among the most beautiful achievements of systematic physics, both in experimental as well as in theoretical respects. Yet we must not digress too far in pursuing details, but must keep in view our problem : how is the ether constituted ?
Fresnel's results appeared to confirm the analogy of light waves with elastic waves. This gave a powerful stimulus to the working out of the theory of elasticity, which had already been begun by Navier (1821) and Cauchy (1822), and to which Poisson (1828) devoted his attention. Cauchy then at once applied the laws derived from elastic waves to optics (1829). We shall try to give an idea of the content of this ether theory,
94 THE THEORY OF RELATIVITY
The difficulty involved is that the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations. Since we do not wish to take these as known, all that can be done is to illustrate them by a simple example, and then to add at the end that in the general case the same holds, but in a more complicated way. The non-mathematical reader may perhaps then get a rough idea of what is involved. It will not, however, give him a real estimate of the power of achievement expressed in the physical pictures and in the mathematical methods used. We are fully conscious of the impossibility of entirely satisfying the non-mathematician, but we cannot refrain from attempting to illustrate the mechanics of continua, because all subsequent theories, not only of the elastic ether, but also electrodynamics in all its ramifications, and, above all, Einstein's theory of gravi- tation, are built up on these conceptions.
A very thin stretched string is in a certain sense a one- dimensional elastic configuration. We shall use it to develop the theory of elasticity. To link up with ordinary mechanics, which deals only with individual rigid solids, we suppose the string to be not continuous but of an atomistic structure, as it were. Let it consist of a series of equal small bodies that are arranged in a line at equal distances from each other (Fig. 66). The particles are to possess inertial mass and each is to exert forces on its two neighbours : these forces are to be such that they resist both an increase and a decrease of the distance between these particles. If we wish to have a concrete picture of such forces, we need only think of small spiral springs that are fixed between the particles. These resist compression as well as extension. But such a representation must not be taken literally. Forces of this kind, indeed, constitute just the essential phenomena of elasticity.
Now if the first particle is displaced a little in the longi- tudinal or in the transverse direction, it immediately acts on the second particle ; the latter in its turn passes the action on to the next, and so forth. The disturbance of the equili- brium of the first particle thus passes along the whole series like a short wave and finally also reaches the last particle. This process does not, however, occur infinitely quickly. At every particle a small fraction of time is lost because the particle, owing to its inertia, does not instantaneously respond to the impulse. For the force does not produce an instantane- ous displacement but an acceleration, that is, a change of velocity during a small interval of time, and the change of velocity again requires time to produce its displacement. Only when this displacement has reached its full value does
FUNDAMENTAL LAWS OF OPTICS 95
the force act to its full extent on the next particle, and from then onwards the process repeats itself with a loss of time that is dependent on the mass of the particles. If the force that arises through the displacement of the first particle were to influence the last particle of the series directly, the action would occur instantaneously, According to Newton's theory of gravitation this is actually supposed to be the case in the mutual attraction of the heavenly bodies. The force with whi h one acts on the other is always directed at the point momentarily occupied by the other and is determined by the distance separating these points at that moment. Newtonian gravi- tation is said to be an action at a distance, for it acts between points at a distance although there is no intervening medium to convey this action.
In contrast with this our series of equidistant points is the simplest model of contiguous action or action by contact. For the action exerted by the first point on the last is transferred by the intervening masses, and hence does not occur instan- taneously but with a loss of time. The force exerted by a
-e — g
Fig. 66.
a Q a P a R a Fig. 67.
particle on its neighbours is certainly still imagined as an action at a distance, although only at a small distance. We may, however, suppose these distances between the particles to grow smaller and smaller, their number becoming correspondingly greater and greater, but in such a way that their total mass remains the same. The chain or particles then passes over into the limiting conception of a material continuum. The forces act between infinitely near particles, and the laws of motion assume the form of differential equations. They ex- press mathematically the physical conception of contiguous action.
We shall pursue this limiting process of the laws of motion a little further for the case of our chain of mass-particles. Let us consider purely transverse displacements (Fig. 67). In the theory of elasticity it is assumed that a particle P is pulled back by its neighbour Q more strongly in proportion to the amount that Q is displaced transversely beyond P. If u is the excess of the transverse displacement of P beyond that of Q, and if a is the original distance between the particles
96 THE THEORY OF RELATIVITY
along the straight line, then the restoring force is to be pro- portional to the ratio - = d, which is called the deformation.
u
a We set
K = p . u = pd
where p is a constant number which is clearly equal to the force if the deformation d be chosen equal to i. ^> is called the elastic constant.
Now the same particle likewise experiences a force K' =
P— == pd' from its other neighbour R. But except in the
a singular case, when the deflection of P is exactly a maximum, the particle R will be more strongly displaced than P, and hence will not pull back the latter, but will tend to increase its displacement. Thus K' will work against K.
The resultant force on the particle P is the difference of these forces
K-K' = p(d-d').
And this determines the motion of P according to the funda- mental dynamical formula : mass times acceleration equals force
= mb = K- K' =p{d-df).
Now let us suppose the number of particles to be increased more and more, but their mass to be decreased in the same ratio so that the mass per unit of length always retains the same value. Let there be n particles in each unit of length,
so that n . a = I, that is, n = -. The mass per unit of length
a
is mn = — . This linear quantity is called the density of mass,
a and is designated by p. By dividing the above equation by a, we get
w, , K - K' J~d'
— b = pb= = p
a a a
and here we have configurations quite similar to those which occurred in the definitions of the conceptions, velocity and acceleration. For just as the velocity was the ratio of the path
x to the time t, v = -, wherein the time t is to be considered t
quite short for an accelerated motion, so we have here the
FUNDAMENTAL LAWS OF OPTICS 97
deformation d = -, the ratio of the relative displacement a
to the original distance, wherein the latter is to be regarded as
extremely small. Just as the acceleration was before defined
IS)
as the ratio of the change of velocity to the time, b = - =
t
— — — , so we have here the quantity / = , which mea- sures in a fully analogous manner the change of the deforma- tion from point to point.
Exactly as the velocity v and the acceleration b retain their sense and their finite values for time intervals that are arbi- trarily small, so the quantities d and / retain their meaning and finite values no matter how small the distance a becomes.
All these are so-called differential co-efficients, v = - and d = -
being such of the first order, b = — — — and / = such
of the second order.
Thus the equation of motion becomes a differential equation of the second order,
pb = Pf ■ ■ . • (36)
both with respect to the time change as well as to the space change of the event . A 11 laws of contiguous action in theoretical physics are of this type. If, for example, we are dealing with elastic bodies that are extended in all directions, we get two analogously formed members for the other two space dimensions. Moreover, precisely similar laws hold in the theory of electric and magnetic events. Finally, the gravitational theory of Einstein has also been brought into such a form.
We have here yet to remark that laws of action at a distance may be written in a form similar to that of formulas for con- tiguous action. For instance, if we strike out the member pb in our equation (36), that is, if we assume that the density of mass is extremely small, then a displacement of the first particle will at the same moment call up a force acting on the last particle, because the inertia of the intervening members has dropped out. Thus we really have the transmission of a force with infinite velocity, a true action at a distance. Nevertheless the law pf = 0 appears in the form of a differential equation, as a contiguous action. Such laws of pseudo-contiguous action will be met with in the theory of electricity and magnetism, where they have really prepared the way for the true laws of contiguous action. The essential factor in the latter is the 7
98 THE THEORY OF RELATIVITY
inertial member that is responsible for the finite velocity of transmission of disturbances of equilibrium, that is, the genera- tion of waves.
Two quantities occur in the law (36) that determine the physical character of the substance : the mass per unit of volume
or the density p, and the elastic constant p. If we write b = ±~f,
P we see that for a given deformation, that is for a given /, the ac- celeration becomes greater in proportion as p becomes greater and p becomes smaller. Thus p is just a measure of the elastic rigidity of the substance, and p is a measure of the inertial mass, and it is clear that an increase of the rigidity accelerates the motion, an increase of the inertia retards it. Accordingly
the velocity c of a wave will depend only on the ratio ?. For the
P more quickly the wave travels the greater are the accelerations of the individual particles of the substance. The exact law for this relationship is found by the following considerations.
Each individual point-mass executes a simple periodic motion of the kind which we investigated earlier (II, 11, p. 34). We showed there that in it the acceleration is connected with the deflection x according to formula (n)
b = (2ttv)2x
where v is the number of vibrations per second. If we insert in place of v the time of vibration according to formula (34),
p. 84, T = -, we get v
-(¥)
The same argument that has here been used for succession in time may also be applied for succession in space, and must lead to relations that correspond entirely. We have simply to replace the quantity / (the second space-coefficient) and the time of vibration T (the period in time succession) by the wave-length A (the " space-period "). We thus get the formula
/=(?)'*
If we form the quotient of the two expressions for b and / the factor (277) 2x cancels out, and there remains
f T2'
FUNDAMENTAL LAWS OF OPTICS 90
Now, on the one hand we have by formula (35), p. 85,
that - = c, on the other hand bv (36), p. 97, that - = £.. 1 / P
Hence it follows that
1 P c = £ or c
-4.
• (37)
This relation holds for all bodies, no matter whether they be gaseous, liquid, or solid. But there is the following differ- ence.
In liquids and gases there is no elastic resistance to the lateral displacement of the particles, but only to the change of volume. Hence only longitudinal waves can propagate themselves in such substances, their velocities being determined according to formula (37), by the elastic constant p which is decisive in such changes of volume.
On the other hand, in solid bodies, on account of the elastic rigidity which opposes lateral displacements, three waves, one longitudinal and two transverse, with differ- ent velocities, can transmit themselves in each direc- tion. This is due to the fact that the compressions and rarefactions of the longitudinal waves involve an elastic constant p, which is different from that which comes into action for the lateral dis- tortions due to the transverse vibrations.
Moreover, in non-crystalline bodies the two transverse waves have, indeed, different directions of vibration, perpendicular to each other, but they have the same velocity Ct ; the longi- tudinal wave has a different velocity C, (Fig. 68).
All these facts allow themselves to be confirmed by experi- ments on acoustic waves in solid bodies.
We now return to the starting-point of our reflections, namely, to the elastic theory of light.
This consists in identifying the ether as the carrier of light vibrations with a solid elastic body. The light waves are then, as it were, to be sound waves in this hypothetical medium.
Now, what properties are to be ascribed to this elastic ether ? In the first place the enormous velocity of propagation c
Fig. 68.
100 THE THEORY OF RELATIVITY
requires either that the elastic ridigity p be very great or that the density of mass p be very small, or that both conditions hold simultaneously. But since the velocity of light is different in different substances, either the ether within a material body must be condensed or its elasticity must be changed or again both may be true simultaneously. We see that different courses are here open to us. The number of possibilities is still further increased by the fact that, as we saw above (IV, 5, P- 93) > experiment cannot decide whether the vibrations of polarized light are parallel or perpendicular to the plane of polarization (the incident plane of the polarizing mirror).
Corresponding to the indefinite nature of the problem we also find an innumerable number of different theories of the elastic ether in history. We have already mentioned the most important authors ; to the mathematicians Poisson, Fresnel, Cauchy, and Green there is to be added for the first time a German physicist of note, Franz Neumann, who became the teacher of the generation of physicists of his own country characterized by Helmholtz, Kirchhoff and Clausius.
Nowadays we feel surprise at the amount of ingenuity and labour that was expended on the problem of comprehending optical phenomena in their totality as motions of an elastic ether having the same properties as those possessed by material elastic solids. It seems to us that the principle which defines explanations as the reduction of unknown things to known things was overstrained. For we now know that the nature of elastic solids is by no means simple and is certainly not to be regarded as known. The physics of the ether has shown itself to be simpler and more easily intelligible than the physics of matter, and modern research is directed at tracing back the constitution of matter, as a secondary phenomenon, to the properties of the fields of force that represent the remains of the ether of the older theory. This change in the programme of science is not least due to the failures attending the attempts to build up a logical theory of the elastic ether.
One objection to the latter theory, which seems of impor- tance, is that an all-pervading ether (which fills astronomic space) of great rigidity, which it must have as the carrier of the rapid vibrations of light, would necessarily offer resistance to the motion of heavenly bodies, in particular, to that of the planets. Astronomy has never detected departures from Newton's laws of motion that would point to such a resistance. Stokes (1845) partly disposed of this objection by remarking that the conception of solidity of a body is in itself really some- thing relative and depends on the relation of the deforming forces to time. A piece of pitch— sealing-wax and glass behave
FUNDAMENTAL LAWS OF OPTICS 101
similarly — when struck with a hammer splits cleanly. But if it is loaded with a weight, the latter sinks gradually, although perhaps only slowly, into the pitch as if pitch were a very viscous fluid. Now, the forces that occur in light vibrations change stupendously quickly (600 billion times per sec.) com- pared with the relatively slow processes that occur in planetary motions in the course of time ; the ratio of these forces is much more extreme than that ot the hammer blow to the superimposed weight. Therefore the ether may function for light as an elastic solid and yet give way completely to the motion of the planets.
Now, even if we wish to content ourselves with this astro- nomic space filled with pitch, serious difficulties arise out of the laws of the propagation of light themselves. Above all, we have to take into account that in elastic solids a longitudinal wave always occurs conjointly with the two transversal waves. If we follow out the refraction of a wave at the boundary of two media, and if we assume that the wave vibrates purely transversally in the first medium, then a longitudinal wave must arise in the second medium together with the transverse wave. All attempts to escape this consequence of the theory by making more or less arbitrary changes have been doomed to failure. Extraordinary hypotheses were suggested, such as that the ether opposes to compression an infinitely small or an infinitely great resistance compared with its rigidity towards transversal distortions. In the former case the longitudinal waves would travel forward infinitely slowly, in the latter infinitely quickly, and would at any rate not manifest them- selves as light. A physicist, MacCullagh (1839), went so far as to construct an ether that departed altogether from the model of elastic bodies. For whereas in these the particles oppose a resistance to every change of their distance from each other, but follow pure twists without resistance, MacCullagh's ether was to behave in just the contrary way. We cannot here enter into the theory. However strange it may appear, it is nevertheless of importance as the fore-runner of the electro- magnetic theory of light. It leads to almost the same formulae as the latter, and is actually able to give an account of optical phenomena that is to a considerable degree correct. But its weakness is that it disclosed no relationship between optical phenomena and other physical phenomena. It is clear that by means of arbitrary constructions ether models can be found that allow a certain region of phenomena to be repre- sented. Such inventions acquire a value as contributions to our knowledge only when they lead to a fusion of two formerly unconnected physical regions. This is the great
102 THE THEORY OF RELATIVITY
advance achieved by Maxwell when he fitted optics into the scheme of electromagnetic phenomena.
7. The Optics of Moving Bodies
Before we pursue this development further we wish to pause and to ask how the doctrine of the elastic ether behaves towards the space-time problem and relativity. Whereas in our optical investigations so far we have taken no account of the position or motion of the bodies that emit, receive, or allow the passage of light, we shall now concentrate our at- tention on just these conditions.
The space of mechanics is regarded as empty wherever there are no material bodies present. The space of optics is rilled with ether. But here the ether is for us actually a kind of matter that has a certain mass, density, and elasticity. Accordingly we can immediately apply Newtonian mechanics with its doctrine of space and time to the universe full of ether. This universe then no longer consists of isolated masses that are separated by empty spaces but is completely filled with the thin mass of the ether, in which the coarse masses of matter are floating. The ether and matter act on each other with mechanical forces and move according to the Newtonian laws. Thus Newton's standpoint is logically applicable to optics. The question is only whether observation is in agreement with it.
But this question cannot be answered simply by unam- biguous experiments. For the state of motion of the ether outside and inside matter is not known, and we are free to think out hypotheses about it. Thus we must put the question in the form : is it possible to make assumptions about the mutual actions of the motions of the ether and of matter such that all optical phenomena are thereby explained ?
We now call to mind the doctrine of the principle of rela- tivity of classical mechanics. According to it absolute space exists only in a restricted sense ; for all inertial systems that move rectilinearly and uniformly with respect to each other may be regarded with equal right as being at rest in space. The first hypothesis that suggests itself to us concerning the luminiferous ether is the following
The ether in astronomic space far removed from material bodies is at rest in an inertial system.
For if this were not the case parts of the ether would be accelerated. Centrifugal forces would arise in it and would bring about changes of density and elasticity, and we should expect that the light from stars would have given us indications of this.
FUNDAMENTAL LAWS OF OPTICS 103
In form this hypothesis satisfies the classical principle of relativity. If the ether is counted among material bodies, then motions of translations of bodies with respect to the ether are just as much relative motions as those of two bodies with respect to each other, and a common motion of translation of the ether and all matter should be capable of detection either mechanically or optically.
But the physics of material bodies alone, without the ether, need no longer satisfy the principle of relativity. A common translation of all matter in which the ether does not participate, that is, a relative motion with respect to the latter, could very well be ascertained by optical experiments. Then the ether would practically define a system of reference that is absolutely at rest. The question which is important above all else for the sequel is whether the observable optical phenomena depend only on the relative motions of material bodies or whether the motion in the sea of ether makes itself remarked.
A light wave has three characteristics :
1. The vibration
number or fre- quency.
2. The velocity.
3. The direction of
propagation.
We shall now investi- gate systematically what influence relative motions of the bodies emitting and receiving light with respect to each other and to the transmitting medium, be it the ether in free astronomic space or be it a transparent substance, have on these three characteristics.
We shall apply the following method. We consider a train of waves, which leaves the zero-point o in any direction at the time t = 0, and we count the individual waves that pass over any point P up to the time t. This number is evidently quite independent of the co-ordinate system in which the co-ordinates P are measured, whether this system be at rest or moving. We determine this number thus :
The first wave that leaves the zero-point t = 0 has to advance along a certain distance s (Fig. 69) until it reaches the point P,
and it takes the time S to do this. From this moment onwards
104 THE THEORY OF RELATIVITY
we count the waves that pass over P up to the moment t, that is, during the time t — -. Now since the light executes v
vibrations in one second, and every wave that passes by cor- responds exactly to one vibration, v waves pass by in one
second, and hence vlt — - J waves pass over the point P in the
time t — - sees. c
Thus the wave number vft — - j is dependent only on how
the two points O and P are situated with respect to each other and to the train of waves, and on how great the interval of time t is between the departure of the first wave at O and the arrival of the last at P. This number has nothing to do with the system of reference. Thus it is an invariant in the sense that we have attached to this word above.
This comes out most clearly if we use Minkowski's mode of expression. According to this the departure of the first wave from the zero point at the time / = o is an event, a world-point ; the arrival of the last wave at the time t at the point P is another event, a second world-point. But world-points exist without relation to definite co-ordinate systems. And since the wave- number v(t — - j is determined by the two world-points, only
it is independent of the system of reference, or is invariant.
From this there easily follow, either by intuition or by apply- ing Galilei-transformations, all theorems about the behaviour of the three characteristics of the wave, the frequency, direction, and velocity, when the system of reference is changed. We shall deduce these theorems in order and shall compare them with experience.
8. The Doppler Effect
The fact that the observed frequency of a wave depends on the motion both of the source of light and of the observer, each with respect to the intervening medium, was discovered by Christian Doppler (1842). The phenomenon may easily be observed in the case of sound waves. The whistle of a locomotive seems higher when it is approaching the observer and becomes deeper at the moment of passing. The rapidly approaching source of sound carries the impulses forwards so that they succeed each other more rapidly. The motion of an observer moving towards the source has a similar effect ; he then receives the waves in more rapid succession. So the
FUNDAMENTAL LAWS OF OPTICS 105
same must hold in the case of light. Now the frequency of the light determines its colour ; the rapid vibrations correspond to the violet end of the spectrum, the slower vibrations to the red. Hence when a light source is approaching an observer or vice versa the colour of the light inclines a little towards violet ; in the case when either is receding, a little towards red. This phenomenon has actually been observed. Now the light which comes from luminescent gases does not consist of all possible vibrations but of a number of separate frequencies. The spectrum that a prism or a spectral apparatus depending on interference exhibits is no continuous band of colour like the rainbow but separate sharp-coloured lines. The frequency of these spectral lines is characteristic of the chemical elements that are emitting light in the flame (spectral analysis by Bunsen and Kirchhoff 1859). The stars, too, have such line spectra, whose lines for the most part coincide with those of the earth's elements. From this it is to be inferred that the matter in the furthermost depths of astronomic space is composed of the same primary constituents. The lines of the stars do not, however, exactly coincide with the corre- sponding lines on the earth but show small displacements towards the one side for one half of the year, and towards the other during the other half. These changes of frequency are the results of the Doppler effect of the earth's motion about the sun. During the one half of the year the earth moves towards a definite star, and hence the frequency of all the light waves coming from this star are magnified and the spectral lines of the star appear shifted towards the side of rapid frequencies (the violet end), whereas during the second half of the year the earth moves away from the star, and hence the spectral lines are then displaced towards the other side (the red end).
This wonderful picture of the earth's motion in the spectrum of the stars does not, indeed, present itself in an unadulterated form. For it is clear that there will be superposed on it the Doppler effect due to the emission of the light by a moving source. Now, if the stars are not all at rest in the ether, their motion must again manifest itself in a displacement of the spectral lines. This becomes added to that due to the earth's motion, but does not show the annual change, and hence may easily be distinguished and separated from the former. Astro- nomically this phenomenon is much more important still, for it gives us information about the velocities of even the most distant stars so far as the motion entails an approach towards or a recession from the earth. It is not our object, however, to enter more closely into these investigations.
106
THE THEORY OF RELATIVITY
We are interested above all in the question as to what happens when the observer and the source of light move in the same direction with the same velocity. Does the Doppler effect then vanish, does it depend only on the relative motion of the material bodies, or does it not vanish and thereby betray the motion of bodies through the ether ? In the former case the principle of relativity would be fulfilled for the optical phenomena that occur between material bodies.
The ether theory gives the following answer to this question. The Doppler effect does not only depend on the relative motion of the source of light and of the observer, but also to a slight extent on the motions of both with respect to the ether. But this influence is so small that it escapes observation; moreover, in the case of a common translation of the source of
light and of the observer it is rigorously equal to zero.
The latter point is so clear intuitively that it need hardly be emphasized. It is only neces- sary to reflect that the waves pass by any two points at rest relatively to each other in the same rhythm, irrespective of whether the two points are at rest in the ether or move with a common motion. Nevertheless the principle of relativity does not hold rigorously, but only ap- proximately, for the bodies emit- ting and absorbing the light. We shall prove this. For this purpose we make use of the theorem above de- rived concerning the invariance of the wave-number.
Let us allow a train of waves to start out in the ^-direction from the zero-point of the system S which is at rest in the ether, and let us count the waves that pass over a point P until the time t has elapsed (Fig. 70). The path which the waves traverse in this time is equal to the ^-co-ordinate of the point P. Thus we must set s = x, and the wave-number
y
) —
p
<
X >
X
Fig. 70.
amounts to
<--i)
Let us next consider a system S' moving in the ^-direction with the velocity v, and let the observer be at rest in it at a point with the co-ordinate %' . At the time t == 0 S and S' are
FUNDAMENTAL LAWS OF OPTICS 107
to coincide, and at the time t the observer is just to have reach. <1 the point P. Then the same wave-number in the system S' is equal to
"-?)■
where v and c' denote the frequency and velocity as measured by the moving observer. We thus have
"(<-;) ="'('-?) • • • (38)
where the co-ordinates are connected by the Galilei trans- formation (29) on p. 65.
%' = x — vt or x = %' -f- vt
If we insert this we get
%' + vt
v[t
H"-?) • • (39)
and this must of course hold for all values of x' and t. If we choose, in particular, t = I, %' = 0, we get
-:)
(40)
That is the desired law. It expresses that an observer moving in the same direction as the light waves measures a
frequency v that is reduced in the ratio (1 — - J : 1.
Conversely we now consider a light wave source that vibrates with the frequency vQ, and moves in the direction of the #-axis with the velocity v0. Let an observer at rest in the ether measure the frequency v. This case is immediately reducible to the preceding one. For it is quite immaterial for our argument whether it is the light source or the observer that is moving, it only depends on the rhythm with which the waves impinge on a moving point. The moving point is now the source of light. We thus get the formula for this case from the preceding case if we replace v in it by v0 and v by y0 :
•(-?)—■
But here v0 is given as the frequency of the source of light, and
108 THE THEORY OF RELATIVITY
v, the observed frequency, is being sought. Thus we must solve for v and we get
*-;v • • • (4i)
c
The observed frequency, therefore, appears magnified, since the denominator is less than i, in the ratio I : ( i — -?Y
We see at once that it is not immaterial whether the observer moves in the one direction or the source in the opposite direction with the same velocity.
For if, in formula (41), we set v0 = — - v, it becomes
c
and this is different from (40) . In all practical cases the differ- ence is certainly very small. We saw earlier (IV, 3, p. 82) that the ratio of the velocity of the earth in its orbit around the
sun compared with that of light is p = - = 1 : 10,000, and
c
similar small values of /3 hold for all cosmic motions. But we
may then write as a very close approximation
for if we neglect B2 = = io-8 compared with 1,
100,000,000
we have (1 + j8) (1 - j8) = 1 - £2 = 1.
This rejection of the square of /? = - will play an important
c
part in the sequel. It is almost always permissible because
such exceedingly small quantities as jS2 = io-8 are accessible
to observation in only a few cases. The phenomena of the
optics (and electrodynamics) of moving bodies are nowadays,
indeed, classified according to whether they are of the order
P or £2. The former quantities are said to be of the first order,
and the latter of the second order in jS. In this sense we may
assert the following :
The Doppler effect depends only on the relative motion of
the source of light and of the observer if the quantities of the
second order are neglected.
FUNDAMENTAL LAWS OF OPTICS 100
We see this, too, if we assume a simultaneous motion of the source of light (velocity v0) and the observer (velocity v). We then clearly obtain the observed frequency v if we insert v from (41) in (40) :
-,(,-!)
V
I — c
= vt
I-VJ
c
7
If the source of light and the observer have the same velocity v0 = v, the fraction becomes equal to 1 and we get v = v0. Thus the observer notices nothing of a common motion with the source relative to the ether. But as soon as v differs from v0, a Doppler effect comes about, the amount of which depends not only on the difference of the velocities v — v0. This would allow the motion relative to the ether to be ascertained if the difference were not of the second order and hence much too small to be observed.
We see that the Doppler effect gives no useful practical
method of establishing motions
with respect to the ether in f Iff *^\
astronomic space. t"_~~~====|3 \-j- +
We must further add that \
the Doppler effect has been de- tected with sources of light on the earth. This required sources of light moving with extremely great speed in order that
the ratio jS = - might attain a perceptible value. For this 0
purpose J. Stark (1906) used the so-called canal rays. If two electrodes are fixed in an evacuated tube containing hydrogen of very small density, and if one of the electrodes is perforated and made the negative terminal (cathode) of an electric dis- charge (Fig. 71), we get in the first place the so-called cathode rays, and secondly, as Goldstein discovered in 1886, a reddish luminescence penetrates through the hole or holes of the cathode, due to positively charged hydrogen atoms or molecules moving at a great speed. The velocity of these canal rays is of the order v = io8 cms. per sec, thus ]8 has the value
3I010 300
which is fairly high compared with the astronomic values.
Stark investigated the spectrum of canal rays and found that the bright lines of hydrogen exhibited the displacement
110 THE THEORY OF RELATIVITY
that was to be expected on the grounds of the Doppler effect. This discovery became of great importance for atomic physics. But it does not belong to our proper theme.
Finally we have to mention that Beloposki (1895) and Galitzin (1907) proved the existence of a sort of Doppler effect with the help of sources of light on the earth and moving mirrors.
9. The Convection of Light by Matter
We have next to investigate the second characteristic of a source of light, namely, its velocity. According to the ether theory the velocity of light is a quantity that is determined by the density of mass and the elasticity of the ether. Thus it has a fixed value in the ether of astronomic space, but a different value in every material body, which will depend on how the matter influences the ether in its interior and carries it along with itself.
If we first treat the velocity of light in astronomic space we must conclude that an observer moving relatively to the ether will measure a velocity different from that measured by our observer at rest. For here the elementary laws of relative motion clearly hold. If the observer moves in the same direc- tion as the light, its velocity relative to the ether will seem diminished by the amount of the velocity v of the observer. Indeed, beings can be imagined that could overtake light. The same result arises from the formulae above derived that express the general relations between the properties of light as established by two observers moving with translation relatively to each other. If we set t = o, %' = I in formula (39) we get
and if we insert the value for v from (40) in this we get
c c\ cJ
or, since v cancels out,
c' = c(i-V) = c-v . . . (42)
This signifies that the velocity of light in the moving system is determined according to the rules of relative motion.
This may also be interpreted by regarding an observer who is moving through the ether as being in an ether wind that blows away from or against the light waves just like the air
FUNDAMENTAL LAWS OF OPTICS 111
brushes past a quickly moving motor car and carries the sound with it.
Now, this furnishes us with a means of establishing the motion of, say, the earth or the solar system relative to the ether. We have two essentially different methods of measuring the velocity of light, an astronomical and a terrestrial method. The former, the old process of Romer, makes use of the eclipses of Jupiter's satellites ; it measures the velocity of the light that traverses the space between Jupiter and the earth. In the latter method the source of light and the observer partici- pate in the motion of the earth. Do these two methods give exactly the same result or are there deviations that betray motion relative to the ether ?
Maxwell (1879) called attention to the fact that by observing the eclipses of Jupiter's moons it should be possible to ascertain
Jupiter
Fig. 72.
a motion of the whole solar system with respect to the ether. Let us suppose the planet Jupiter at the point A of its orbit (Fig. 72), which is the point nearest to the orbit of the sun in the motion of the solar system in the direction shown. (It has been assumed in the diagram that the orbit of Jupiter intersects the orbit of the solar system at A.) In the course of a year Jupiter moves only a short distance away from A, since its time of revolution in its own orbit is about twelve years. In one year the earth traverses its orbit once, and by observing eclipses it is possible to find the time required by the light to travel across the diameter of the earth's orbit. Now, since the whole solar system moves in the direction of the sun to- wards A the light from Jupiter to the earth runs contrary to this motion, and its velocity appears increased. Let us now wait for six years until Jupiter is situated at the opposite point B of its orbit. The light now runs in the same direction
112 THE THEORY OF RELATIVITY
as the solar system, and thus requires a longer time to cross the earth's orbit ; so its velocity appears smaller.
When Jupiter is at A the eclipses of one of his satellites during half a year (of the earth) must be delayed by the amount
of time U = , where I denotes the diameter of the earth's
c -f v
orbit. When Jupiter is at B the delay amounts to t2 = . If the solar system were at rest in the ether both delays
would be equal to t0 = -. Their actual difference, namely,
c
t - t - if Z T \ - 2^v 2^v
2 1~ \c — v c + v) ~ c2 — V2 ~ c2(i — £2)
for which, by neglecting jS2 in comparison with i, we may write
h — h = -=■ = 2/o0,
c2
allows us to determine jS and hence also the velocity v = pc of the solar system relative to the ether. Now, light takes about eight minutes to travel from the sun to the earth, thus t0 = 16 mins. or = iooo sees, (in round numbers). Thus from
a time-difference t» — t-i = i sec. we should get B =
21 & H 2000
or v = Be = ^ =i^o kms. per sec.
r 2000 J r
The velocities of the stars relative to the solar system, which may be deduced from the Doppler effect, are mostly of the order 20 kms. per sec, but velocities up to 300 kms. per sec. occur in certain clusters of stars and spiral nebulae. The accuracy of the astronomic determinations of time has thus far not sufficed to establish a delay in the eclipses of a satellite of Jupiter to the extent of one sec. or less in the course of half a year. Yet it is not out of the question that refinement of the methods of observation will yet disclose such a delay.
An observer situated on the sun, who happened to know the value of the velocity of light in the ether at rest, would also be able to ascertain the motion of the solar system through the ether by means of the eclipses of Jupiter's satellites. To do this he would have to measure the delay in the eclipses during half a revolution of Jupiter in his orbit. The same formula t2 — tt = 2t0p is valid for this, but now t0 denotes the time that the light requires to traverse the diameter of Jupiter's orbit. This value of t0 is (about 2\ times) greater than the
FUNDAMENTAL LAWS OF OPTICS 113
value used above for the earth's orbit, 16 mins., and the delay t% — tt becomes greater in the same proportion. But for the same reason the time of revolution of Jupiter, during which the eclipses must be observed consecutively, is much greater than (about 12 times as great as) an earth year, so that this method, which could also be applied by an observer on the earth, seems to promise no advantage.
At any rate the fact that the accuracy that is nowadays attainable has brought to light not even a delay of several seconds proves that the velocity of the solar system with respect to the ether is not much greater than the greatest known velo- cities of the stars relative to each other.
We next turn our attention to the terrestrial modes of measuring the velocity of light. Here it is easy to see why they do not allow us to draw conclusions about the motion of the earth through the ether. We have already indicated the ground for this above when mentioning these methods for the first time (IV, 3, p. 82), for the light traverses one and the same path in its journey there and back. It is only a mean velocity during the path to and fro that is actually measured. The deviation of this from the velocity of light c in the ether is, however, a quantity of the second order with respect to j8 and is not accessible to observation. For if / is the length of path then the time that the light requires for the first journey, in the direction of the earth's motion, is equal to
and the time for the return journey is — - — , thus the
c — v whole time is
/I I \ _ 2lc _ 2lc
\c + v c — v) ~ (c + v)(c — v) c2 —
The mean velocity is 2/ divided by this time, thus it is
t =* , e(, _ J)
and hence it differs from c by a quantity of the second order.
Besides the direct measurement of the velocity of light there are numberless other experiments in which the velocity of light comes into play. All interference and diffraction pheno- mena are brought about by making light-waves that travel along different paths meet at the same place and causing them to be superposed on each other. Refraction at the boundary of two bodies arises through light having different velocities in them ; thus this velocity enters into the action of all optical apparatus that contains lenses, prisms, and similar things. Is it not possible
8
114 THE THEORY OF RELATIVITY
to think out arrangements in which the motion of the earth and the " ether wind " produced by it make themselves remarked? Very many experiments have been designed and carried out to discover this motion. The general result of experi- ments with sources of light on the earth teaches us that not the slightest influence of the ether-wind is ever observable. It is true that up to recent times we have been dealing with ex- perimental arrangements that allow only quantities of the first order in jS to be measured. The fact that this must always lead to a negative result easily follows from the circumstance that the true duration of the motion of the light from one place to another is never measured, but only differences of such times for the same light-path or their sum for the motion there and back. For the reason given above we thus see that the quantities of the first order always cancel out.
But we might expect a positive result if we took a source not on the earth but in the heavens. If we direct a telescope
at a star, to which the momentary velocity v of the earth is just directed (Fig. 73), the velocity of light in the lenses of the telescope relative to the substance of the glass will be greater by the amount v Fig. 73. than if the earth were at rest,
and if we look at the same star six months later through the telescope, the velocity of light in the lenses will be smaller by the amount v. Now, since the amount of the refraction in a lens is determined by the velocity of light, we might expect the focus of the lens to have a different position in these two cases. This would be an effect of the first order. For the difference of the velocity of light in the two cases would be 2Vt and its ratio to the velocity
in the ether at rest would be— = 2)3.
c
Arago actually carried out this experiment, but found no change in the position of the focus. How is this to be ex- plained ?
We clearly made the assumption above that the velocity of light in a body that moves in the ether against the ray with the velocity v is greater by just this amount than if the body were at rest in the ether. In other words, we have assumed that material bodies pierce through the ether without carrying it along in the slightest, just like a net that is carried through water by a boat.
FUNDAMENTAL LAWS OF OPTICS 115
The results of experiment teach us that this is manifestly not the case. Rather, the ether must participate in the motion of matter. It is only a question of how much.
Fresnel established that to explain Arago's observation and all other effects of the first order it was sufficient to assume that the ether is only partly carried along by matter. We shall forthwith discuss in detail this theory, which has been brilliantly confirmed by experiment.
It was Stokes (1845) above all others who later adopted the more radical standpoint that the ether in the interior of matter shares completely in its motion. He assumed that the earth carries along with itself the ether which is in its interior, and that this ether motion gradually decreases outwards until the state of rest of the ether in the universe is reached. It is clear that then all optical phenomena on the earth occur exactly as if the earth were at rest. But in order that the light that comes from the stars may not experience deflections and changes of velocity in the transi-
tional stratum between the ,9^/ ^ ^ \^,
ether of space and the ether converted by the earth, special hypotheses concern-
St
r n
l\ \
/
7
u
/*
If
/
F
*' /
ing the motions of the ether ^ _^l ^ J/f^
must be made. Stokes found a hypothesis such as satisfied all optical condi- tions. But it was shown later not to be in agree- Fig. 74.
ment with the laws of
mechanics. Numerous attempts at rescuing Stokes's theory have led to no result, and it would have succumbed to internal difficulties even if Fresnel's theory had not been confirmed by Fizeau's experiment (see p. 119 below).
Fresnel's idea of partial convection cannot easily be deduced from Arago's experiment because refraction in lenses is a com- plicated process which concerns not only the velocity but also the direction of the waves. But there is a fully equivalent experiment that was carried out by Hoek (1868) later and which is much easier to follow.
The principle underlying the arrangement of the apparatus is that of the interferometer (Fig. 74) . The light falls from the source Q on to a half-silvered glass plate P inclined at 450 to the direction of the ray. This glass plate divides the ray into two parts. The reflected ray (ray 1) strikes consecutively the mirrors S2, SB, S8, that form the corners of a square with P, and on its return to P is partly reflected into the telescope F.
116 THE THEORY OF RELATIVITY
The transmitted ray (ray 2) traverses the same path in the reverse sense and interferes with ray 1 in the field of vision. A transparent body, say a tube W filled with water, is next interposed between Sx and S2, and the whole apparatus is mounted so that the straight line connecting Si with S2 can be placed alternately in the same direction as, and opposite to, the earth's motion about the sun. Let the velocity of light in water that is at rest be cv This value is a little smaller than the
velocity in vacuo and the ratio of the one to the other — = n
ci is called the refractive index of water. The velocity of light in air differs only inappreciably from c, and thus the refractive index of air is almost exactly equal to 1. Now the water is carried along by the earth in its orbit. If the ether in the water were not to participate in this motion at all then the velocity of light in the water relative to the absolute ether (in outside space) would be unaltered ; that is, it would be equal to clt and, for a ray travelling in the direction of the earth's motion, it would be Cj + v, and ct relative to the earth. We shall assume neither of these cases to begin with, but shall leave the amount of convection undetermined. Let the velocity of light in the moving water relative to the absolute ether be a little greater than clt say cx + <j>, and hence cx -f </» — v relative to the earth. We" wish to determine the unknown convection co- efficient <j> from experiment. If it is zero, no convection occurs ; if it is v, complete convection occurs. Its true value must lie between these limits. We shall, however, make one assumption, namely, that the convection in air may be neglected in com- parison with that in water.
Now, let I be the length of the tube of water. Then the ray 1
requires the time = to traverse the tube, if the earth
cx + <t> — v is moving in the direction from Sx to S2. To traverse the cor- responding air distance between S3 and P the same ray requires
the time ■ — — . Thus, on the whole, the time that the ray 1
c -\- v requires to traverse the two equal paths in water and in air is
1 + '
cY -f- (j) — V c -\- V
The ray 2 travels in the reverse direction. It first traverses the air-distance in the time , then the water-distance in
FUNDAMENTAL LAWS OF OPTICS 117
the time , and hence altogether it requires for the
same distances in the air and water the time
1 + l
c — v cY — <f> -\- v
Now, experiment shows that the interferences do not shift in the slightest when the apparatus is turned into the direction opposite to that of the earth's velocity or, indeed, into any other position whatsoever. From this it follows that the rays i and 2 take equal times, independent of the position of the apparatus with respect to the earth's orbit, that is,
cx + </> — v c -f- v c — v cx — <f> -\- v
We can calculate </> from this equation. We shall pass over the somewhat circuitous calculation * and shall give only the result which, if we neglect quantities of the second and higher orders, is :
#-(x-5> • • • (43)
This is the famous convection formula of Fresnel, who, indeed, found it by a different, more speculative, process. Before we mention his assumption let us see what the formula actually asserts. According to it the convection is the greater the more the refractive index exceeds the value i which it has in vacuo. For air cx is almost equal to c, and n almost equal to i, thus <f> is almost zero, as we predicted above. The greater the refractive power, the more complete is the convection of the light. Now, the velocity of light in a moving body, measured
* The steps of the argument are :
(c + v) + fa + <P - ») = fa - <t> + v) + (g - v)f
(*!+<*>- V)(C +V) " (C- »)fa - <t> + V)
[c + ex + <p)(c - »)fa - <P + v) = {c + cy - <t>){c + z^)fa +<t>-v), V* + cl
r-t-
= dm — c
V
and. approximately, we have
(■-*>•
118 THE THEORY OF RELATIVITY
relative to the absolute ether is
'! + *=*!+ (l-i)l/,
and relative to the moving body it is
ct + i> - v = cY + (i - ^)v -v = Cl - J.
This last formula will serve us as a link to Fresnel's inter- pretation. He assumed that the density of the ether in a material body is different from the density in free ether ; let the former be pj and the latter p.
We next imagine the moving body, say, in the form of a beam, whose length is parallel to the direction of motion ; let its basic face be of unit area. In the motion of the beam through the ether the front face advances by the distance v in a unit of
time (Fig. 75), and this sweeps out a
y\ volume v (area of the face multiplied
-i'_ J by the height). This volume con-
] / tains an amount of ether pv. Thus
-* this enters into the beam through
^4
FlG 7- the front face. Here it assumes a
new density and will thus move on
with a different velocity v with respect to the body, since, for
the same reasons as above, its mass must also equal p{ult and
we get
Plv1 = pv
or Vi = -v-
Pi
This is in a certain sense the strength of the ether wind in the beam moving with the velocity v. Light which moves with the velocity cx relatively to the condensed ether moves relatively to the body with the velocity
c _ Vl = cx — £-v. Pi
Now we have seen that according to the result of Hoek's experiment the velocity of light relative to the moving body is
1
Consequently we must have
P. - - L = cj! Pl n2 ' c2'
FUNDAMENTAL LAWS OF OPTICS 119
Thus the condensation £-* is equal to the square of the
coefficient of refraction.
Furthermore, we can conclude from this that the elasticity of the ether must be the same in all bodies. For formula (37)
on p. 99 tells us that in every elastic medium c'1
_#
Thus
%-
/
.\
in ether p — c2p, in matter px = c-fp^ But according to the above result concerning the condensation of ether in matter these two expressions are the same.
This mechanical interpretation of the convection coefficient by Fresnel has exerted a great influence on the elaboration of the elastic theory of light. But we must not disguise from our- selves that it is open to strong objections. As is well known, rays of light of different colour (frequency) have different refractive indices n, that is, different velocities. Hence it follows that the convection ,
coefficient has a different value for each colour. But this is incompatible with Fresnel's interpreta- tion, for then the ether would have to flow with a different velocity in the body according to the colour. Thus there would be just as many ethers as there are colours, and that is surely impossible.
The convection formula (43), however, is founded on the results of experi- ment without regard to the shall see that it is derived
F
Fig. 76.
mechanical interpretations. We in the electro-magnetic theory of light from ideas concerning the atomic structure of matter and electricity.
It is very difficult to test Fresnel's formula by means of experiments on the earth because it requires that transparent substances be moved with extreme rapidity. Fizeau succeeded in carrying out the experiment (185 1) by means of a sensitive interferometer arrangement.
The apparatus used by him is quite similar to that of Hoek, except that both light-paths SXS2 and S3 are furnished with tubes in which the water can circulate ; they are arranged so that the ray 1 flows directly parallel to the water, and the ray 2 directly against it. Fizeau tested whether the water
120 THE THEORY OF RELATIVITY
carries the light along with it by observing whether the inter- ference fringes were displaced when the water was set into rapid motion. This displacement actually occurred, but very much less than to the extent that would correspond to complete convection. Exact measurement disclosed perfect agreement with Fresnel's convection formula (43).
10. Aberration
We shall now discuss the influence of the motion of bodies on the direction of light-rays, in particular the question whether the motion of the earth through the ether can be ascertained by observing any phenomena accompanying changes of direction. Here again we have to distinguish whether we are dealing with an astronomic or an earth source of light.
The apparent deflection of the light that reaches the earth
from the stars is the aberration, which we have already dis- cussed from the point of view of the corpuscular theory (IV, 3, p. 81). Although the ex- planation there given is very simple, it is correspondingly complicated from the point of view of the wave-theory, for it is easy to see that a deflection of the wave-planes does not occur at all. This is seen most readily in the case where the rays fall perpendicularly FlG- 77- to the motion of the observer.
For then the wave-planes of this motion are parallel and are so perceived by the moving observer (Fig. 77). But calculation tells us the same. Let us place a stationary co- ordinate system S and a moving system S' so that the x- and the #'-axis each fall in the direction of motion, and let us count the waves that have passed over any point P from the moment
t = 0 to the moment t. This number is, as we know, v(i — ~\
where s is the path traversed by the waves. In the case of perpendicularly incident waves we clearly have s = y. The in variance of the wave-number requires that
<•- 9 ='(<-9
if the co-ordinates are transformed into each other by the
f p
0 /
+ v
FUNDAMENTAL LAWS OF OPTICS 121
Galilei transformation. In this case the jy-co-ordinate remains unaltered, and hence we must have
v = v and - = — . thus c = c' . c c
Hence the moving observer sees a wave of exactly the same frequency, velocity, and direction. For if this were altered then the wave-number in S', besides depending on y' , would also have to depend on x' .
Thus it seems as if the wave theory is unable to account for the simple phenomenon of aberration, which has been known for almost 200 years.
But the position is not quite so bad as this would indicate. The reason for the failure of the argument given just above is that the optical instruments with which the observations are made, and which include the naked eye, do not establish the position of the wave that arrives, but accomplish something totally different.
The function of the eye or of the telescope is called optical representation, and it consists in combining the rays emitted by a luminescent object into one picture. In this process the vibrational energy of the particles of the object are trans- ported by the light-waves to the corresponding particles of the picture. The paths along which this transference of energy takes place are actually the physical rays. But energy is a quantity which, according to the law of conservation, can move about and be transformed just like a substance, but cannot be created or destroyed. Hence it seems reasonable to apply the laws of the corpuscular theory to the motion of energy. As a matter of fact the simple derivation of the aberration formula given earlier is quite correct if we define the light- rays as the energy-paths of the light-waves and apply the laws of relative motion to them, as if they were streams of projected particles.
But we may also obtain this aberration formula, without applying this conception of rays as energy paths, by following the refraction of waves individually in the lenses or prisms of the optical instrument. For this we require a definite convection theory. Stokes's theory of complete convection can account for aberration only by making assumptions about the motion of the ether which are not admissible. We have already called attention to these difficulties above. Fresnel's theory gives a law of refraction of waves of light at the surface of moving bodies from which the aberration formula follows exactly. The substance of the body through which the light passes does not affect the result, although the value of the convection
122 THE THEORY OF RELATIVITY
coefficient is different in every substance. To test this directly Airy (1871) filled a telescope with water and ascertained that the aberration retained its normal value. The aberration is, of course, no longer an effect of the first order if the light-wave and the observer have no motion relative to each other. From this it also follows that in all optical experiments with sources of light on the earth no deflection of the rays through the ether wind occurs. Fresnel's theory succeeds in representing these facts so that they are in agreement with experiment. It is unnecessary to enter into the details.
11. Retrospect and Further Development
We have treated the luminiferous ether as a substance that obeys the laws of mechanics. Thus it satisfies the law of inertia, and hence where there is no matter, in astronomic space, it will be at rest in an appropriate inertial system. Now if we refer all phenomena to a different inertial system, exactly the same laws hold for the motions of bodies and of the ether, hence also for the propagation of light, but, of course, only in as far as they concern accelerations and mutual force effects. We know that the velocity and the direction of a motion are quite different with respect to different inertial systems ; for we may regard every body moving in a straight line as at rest merely by choosing a suitable system of refer- ence, namely, one that moves with it. Thus in this almost trivial sense the classical principle of relativity must hold for the ether regarded as a mechanical substance.
From this it follows, however, that the velocity and direction of light-rays must appear different in every inertial system. Thus it was to be expected that it would be possible to ascertain the velocity of the earth or of the solar system by observing optical phenomena at the surface of the earth, which are in the main conditioned by the velocity and direction of the light. But all experiments performed with this end in view led to a negative result. Hence it appears that the velocity and direction of the light-rays are quite independent of the motion of the astronomic body on which the observations are carried out. Or, expressed in other words, optical phenomena depend only on the relative motions of material bodies.
This is a principle of relativity which seems quite similar to the classical principle of mechanics, and yet it has a different meaning. For it refers to velocities and directions of motional events, and in mechanics these are not independent of the motion of the system of reference.
Now there are two possible points of view. One of these
FUNDAMENTAL LAWS OF OPTICS 128
starts from the assumption that optical observations actually introduce something that is fundamentally new, namely, that light behaves differently from material bodies as regards direction and velocity. So soon as optical observations are taken as convincing evidence this point of view will be adopted if all speculations about the nature of light are left out of con- sideration. We shall see that Einstein finally pursued this path. It, however, requires utter freedom from the conventions of the traditional theory, which is attained only when the Gordian knot of constructions and hypotheses has become so intricate that the only solution left is to cut it.
But in our above discussion we were still living in the most flourishing period of the theory of the mechanical ether. This theory was compelled to regard the optical principle of relativity as a secondary, in a certain sense half accidental phenomenon, brought about by the compensating effect of causes that were acting in opposition to each other. The fact that such is possible to a certain extent is due to the circumstance that it is still open to make hypotheses about how the ether moves and how it is influenced in its motion by moving bodies. Now it is a great achievement of Fresnel's convection hypothesis that it actually accounts for the optical principle of relativity, so far as quantities of the first order are concerned. So long as the accuracy of optical measurements did not attain the great improvement necessary to measure quantities of the second order, this theory sufficed all demands of experiment with one possible exception, to which, curiously enough, very little attention was paid. For if improved accuracy in astro- nomical measurement should arrive at the result that by observing the eclipses of Jupiter's satellites according to the old method of Romer (see p. 80) an influence of the motion of the solar system on the velocity of light were to be revealed, then certainly the ether theory would be confronted with a problem that would appear insoluble. For it is clear that this effect of the first order could be argued away by no hypothesis about the convection of the ether.
So we recognize the importance of the experimental task of measuring the dependence of optical events on the earth's motion as far as quantities of the second order. Only the solution of this problem can give us a decision as to whether the optical principle of relativity holds rigorously or only approximately. In the former case Fresnel's ether theory would fail ; we should then be confronted with a new state of affairs.
Historically, this occurred only about 100 years after Fresnel's time. In the meanwhile the ether theory was de- veloped in other directions. For at the outset there was not
124 THE THEORY OF RELATIVITY
one ether but a whole series, an optical, a thermal, an electrical, a magnetic ether, and perhaps a few more. A special ether was invented, as a carrier, for every phenomenon that occurs in space. At first all these ethers had nothing to do with each other, but existed in the same space independently of each other, side by side or, rather, interwoven. This state could not, of course, last in physics. Relationships were soon found be- tween the phenomena of different branches that were at first separate, and so there emerged finally one ether as the carrier of all physical phenomena that bridge over space free of matter. In particular, light showed itself to be an electromagnetic process of vibration, of which the carrier is identical with the medium that transmits electric and magnetic forces. These discoveries first gave the ether theory strong support. At last, indeed, the ether came to be identified with Newtonian space. It was to persist in absolute rest and was to transmit not only electromagnetic effects but also indirectly to generate the Newtonian inertial and centrifugal forces.
We shall next describe the development of the theory. The process has features resembling the trial of a case in court. The ether is alleged to be the universal culprit, the pieces of evidence accumulate overwhelmingly, until at the end the undeniable proof of an alibi, namely, Michelson and Morley's experiment about the quantities of the second order, and its interpretation by Einstein puts an end to the whole business.
CHAPTER V THE FUNDAMENTAL LAWS OF ELECTRODYNAMICS
i. Electro- and Magneto-statics
THE fact that a certain kind of ore, magnetite, attracts iron, and that rubbed amber (elektron in Greek) attracts and holds light bodies was known even to the ancients. But the sciences of magnetism and electricity are products of more recent times, which had been trained by Galilei and Newton to ply Nature with rational questions and to read the answer out of experiment.
The fundamental facts of electric phenomena were estab- lished from the year 1600 onwards. We shall recapitulate them briefly. At that time friction was the exclusive means of producing electrical effects. Gray discovered (1729) that metals, when brought into contact with bodies that have been electrified by friction, themselves acquire similar properties.
He showed that the electrical effects can be passed along in the metals. This led to the classification of substances as conductors and non-conductors (insulators). It was dis- covered by du Fay (1730) that electrical action is not always attraction but may also be repulsion. He interpreted this fact by assuming two fluids (nowadays we call them positive and negative electricity), and he established that similarly charged bodies repel each other, oppositely charged bodies attract each other.
We shall here define the conception of the electric charge quantitatively at once. In doing so we shall not follow rigor- ously the very often circuitous steps of argument that led historically to the enunciation of the conceptions and laws, but we shall rather select a series of definitions and experi- ments in which the logical sequence comes out most clearly.
Let us imagine a body M that has somehow been electrified by friction. This now acts attractively or repulsively on other electrified bodies. To study this action we shall take small test bodies, say spheres, whose diameters are very small com- pared with their closest approach to the body M, at which we
125
126 THE THEORY OF RELATIVITY
still wish to investigate the force. If we bring such a test body P near the body M, whose action we wish to study, P experiences a statical force of definite magnitude and direc- tion, which may be measured by the methods of mechanics, say,' by balancing it against a weight with the help of levers or threads.
We next take two such test bodies Px and P2, bring them in turn to the same point in the vicinity of M, and measure in each case the forces Kx and K2 as regards size and direction. We shall henceforth adopt the convention that opposite forces are to be regarded as being in the same direction, but their values are to have opposite signs attached in calculations.
Experiment shows that the two forces have the same direc- tion, but their values may be different and they may have different signs.
Now let us bring the two test bodies to a different point near M and let us again measure the forces Kx' and K2' as regards value and direction. They have again the same direction, but in general they have different values and a different sign.
If we next form the ratio Kx : K2 of the forces at the first point, and then the ratio K/ : K2' at the second, it is found that both have the same value, which may be positive or nega- tive :
K2 K2'"
From this result we may conclude :
i. The direction of the force exerted by an electrified body M on a small test-body P does not depend at all on the nature and the electrification of the test-body, but only on the pro- perties of the body M.
2. The ratio of' the forces exerted on two test-bodies brought to the same point in turn is quite independent of the choice of the point, that is from the position, nature, and electrifica- tion of the body M. It depends only on the properties of the test-bodies.
We now choose a definite test-body, electrified in a definite way, as a unit body, and we ascribe to it the charge or amount of electrification +i. With the aid of this we everywhere measure the force that the body M exerts. Let it be denoted by E. Then this also determines the direction of the force K exerted on any other test-body P. The ratio K : E, however, depends only on the test-body P and is called its electric charge e. This may be positive or negative according as K and E are
LAWS OF ELECTRODYNAMICS 127
in the same or in opposite directions, in the narrower sense. Thus we have
- - e or K = eE . . . (44)
The force E on the charge 1 is also called the electric intensity of field of the body M. When once the unit charge has been fixed it depends only on the electrical nature of the body M and determines its electrical action in the surrounding space, or as we usually say, its " electric field."
As for the choice of the unit charge, it would be almost impossible to fix this practically by a decree concerning the electrification of a definite test-body ; rather, one will seek to find a mechanical definition for it. This is successfully done as follows :
We can first charge two test-bodies equally strongly. The criterion of equal charges is that they experience the same force from the third body M when placed successively at the same point near it. The two bodies will then repel each other with the same force. We now say that their charge is 1 if this re- pulsion is equal to the unit of force when the distance between the two test-bodies is equal to unit length. Nothing at all is herein assumed about the dependence of the force on the distance.
Through these definitions the amount of electricity or the electric charge becomes just as much a measurable quantity as length, mass, or force.
The most important law about amounts of electricity, which was enunciated independently in 1747 by Watson and Franklin, is the law that in every electrical process equal amounts of positive and negative electricity are always formed. For example, if we rub a glass rod with a piece of silk, the glass rod becomes charged with positive electrification ; an exactly equal negative charge is then found on the silk.
This empirical fact may be interpreted by saying that the two kinds of electrification are not generated by friction but are only separated. They are represented as two fluids that are present in all bodies in equal quantities. In non-electrified neutral " bodies they are everywhere present to the same amount so that their effects outwards are counterbalanced. In electrified bodies they are separated. One part of the positive electrification, say, has flowed from one body to another, and just as much negative has flowed in the reverse direction.
But it is clearly sufficient to assume one fluid that can flow independently of matter. Then we must ascribe to the matter that is free of this fluid a definite charge, say positive,
128 THE THEORY OF RELATIVITY
and to the fluid the opposite charge, that is, negative. The electrification consists in the negative fluid flowing from one body to the other. The first will then become positive because the positive charge of the matter is no longer wholly compen- sated ; the other becomes negative because it has an excess of negative fluid.
The struggle between the supporters of these two hypo- theses, the one-fluid theory and the two-fluid theory, lasted a long time, and of course remained fruitless and purposeless until it was decided by the discovery of new facts. We shall not enter further into these discussions, but shall only state briefly that characteristic differences were finally found in the be- haviour of the two electrifications ; these differences indicated that the positive electrification is actually firmly attached to matter, but the negative can move freely. This doctrine still holds to-day. We shall revert to this point later in dealing with the theory of electrons.
Another controversy gathered round the question as to how the electrical forces of attraction and repulsion are trans- mitted through space. The first decades of electrical research were not yet carried out under the influence of the Newtonian theory of attraction. Action at a distance seemed unthinkable. Metaphysical theorems were held to be valid, such as that matter can act only at points where it is itself present, and thus diverse hypotheses were evolved to explain electrical forces, that emanations flowed from the charged bodies and exerted a pressure when they impinged on bodies, and similar assump- tions. But after Newton's theory of gravitation had begun to reap its victories the idea of a force acting directly at a dis- tance gradually became a habit of thought. For it is, indeed, nothing more than a habit of thought when an idea impresses itself so strongly on minds that it is used as the last principle of explanation. It does not then take long for metaphysical speculation, often in the garb of philosophic criticism, to evolve the proof that the correct or accepted principle of explanation is a logical necessity and that its opposite cannot be imagined. But, fortunately, progressive empirical science does not, as a rule, trouble about this, and, when new facts demand it, it often has recourse to ideas that have been condemned. The development of the doctrine of electric and magnetic forces is an example of such a cycle of theories. At the be- ginning we see a theory of contiguous action based on meta- physical grounds ; it is replaced by a theory of action at a distance on Newton's model. At the end this becomes trans- formed, owing to the discovery of new facts, into a general theory of contiguous action again. But this fluctuation is
LAWS OF ELECTRODYNAMICS 129
no sign of weakness. For it is not the pictures that are con- nected with the theories which are the essential features, but the empirical facts and their conceptual relationships. Yet if we follow these we see no fluctuation but only a continuous development full of inner logical consistency. We may justifiably omit the first theoretical attempts of pre-Newtonian times from the series because the facts were known too incom- pletely to furnish really convincing starting-points for theory. But the fact that the theory of action at a distance then arose in conformity with the model of Newtonian mechanics is founded quite naturally in the nature of electrical facts. A branch of research which had at its disposal only the experimental means of the 18th century could not do otherwise, on the ground of the observations possible at that time, than come to the de- cision that the electrical and the magnetic forces act at a dis- tance in the same way as gravitation. Nowadays, too, it is absolutely permissible, from the point of view of the highly developed theories of contiguous action of Faraday and Max- well, to represent electro- and magneto-statics by means of actions at a distance, and when properly used, they always lead to correct results.
The idea that electric forces act like gravitation at a distance was first conceived by Aepinus (1759). He even went so far as to regard gravitation and electricity as effects of the same fluid. He supposed, in the sense of the one-fluid theory, that matter devoid of electric fluid would repel other matter, but that there is always a little excess of fluid present that effects the gravitational attraction. Curiously enough he did not succeed in setting up the correct law for the dependence of electrical actions on the distance, but he was able to explain the phenomenon of influence qualitatively. This consists in a charged body acting attractively not only on other charged bodies but also on uncharged bodies, particularly on conducting bodies, for a charge of the opposite sign is induced on the side of the influenced body nearest the acting body, whereas a charge of the same sign is driven to the further side (Fig. 78) ; hence the attraction outweighs the repulsion.
The true law was presumably first found by Priestley, the discoverer of oxygen (1767). He discovered it by an ingenious indirect way which essentially carries more con- viction with it than that of direct measurement. Independently of him Cavendish (1771) derived this law by the same method. But it receives its name from the physicist who first proved it by measuring the forces directly, namely, Coulomb (1785).
The argument of Priestley and Cavendish ran somewhat as follows :
130 THE THEORY OF RELATIVITY
If an electric charge is given to a conductor then it cannot remain in equilibrium in the interior of the conducting sub- stance since particles of the same charge repel each other. Rather, they must tend to the outer surface at which they distribute themselves in a certain way so as to be in equilibrium.
Now experiment teaches very definitely that no electric field exists within a space that is enclosed on all sides by metallic walls, no matter how strongly the envelope is charged. The charges on the outer surface of the empty space must thus distribute themselves so that the force exerted at each point in the interior must vanish. Now, if the empty space has the particular form of a sphere, reasons of symmetry convince us that the charge can only be distributed uniformly over the surface. If p is the charge per unit area of surface (density of charge, then amounts of electricity pfx + pf2 are on the two por-
+ +
+ +
Fig. 78.
Fig. 79.
tions flf f2 of the surface. The force that a small portion of surface fx of this kind exerts on a test-body P situated in the interior of the sphere and carrying the charge e is then Kx = tfp/iRi, where Rx denotes the force which is exerted between two units of charge placed at P and flf and which somehow depends on the distance rx between P and fv Now corre- sponding to each portion of surface ft there is an opposite por- tion f2, which is obtained by connecting the points of the boun- dary of fx with P and producing these lines through P until they intersect the sphere. The two portions of area fx and f2 are thus cut out of the surface of the sphere by the same double cone with its apex at P (Fig. 79), and the angles between them and the axis of the double cone are equal. The values of f1 and f2 are thus in the ratio of the squares of the distances from P :
The charge pf2 on /3 exerts the force K2 = epf2R2 on P,
LAWS OF ELECTRODYNAMICS 131
where K2 depends on r 2 in some way ; K2 is of course oppositely directed to Kv
It readily suggests itself to assume that all the forces acting on P can only then neutralize each other if the forces due to two opposite portions of area exactly counterbalance, that is, when Kj = K2. It is possible to prove this assumption, but that would take us too far here. If we take it for granted, then it follows that/^ =/2R2, or
R2 ft r/
Accordingly R^2 = R2r'f = c
where c is a quantity independent of the distance r. This
determines Rx and R2, namely,
R,-A
R. c
I
I
Hence, in general, the force R between two unit charges at a distance r apart must have the value
R = £,
r2
In conformity with our convention about the unit of electric charge we must set c = I. The force between two unit charges unit distance apart is to be equal to I. With this convention the force that two bodies carrying the charges ex and e2 and at a distance r apart exert on each other is
K = ^2 • • • • (45)
This is Coulomb's law. In its formulation we assume that the greatest diameter of the charged bodies is, of course, small compared with their distances apart. This restriction ex- presses that this law, just like the law of gravitation, is an idealized elementary law. To deduce from it the action of bodies of finite extent we must consider the electricity dis- tributed over them to be divided into small parts, then calculate the effects of all the particles of the one body on all those of the others in pairs and sum them.
Formula (45) fixes the dimensions of quantity of electricity
e2 since we have for the repulsion of two equal charges - = K,
that is, e = r^/K, hence
[•] = [WK] = [i^J] = [VMl].
132 THE THEORY OF RELATIVITY
This, at the same time, fixes the unit of charge in the C.G.S.
. , cm. ^/errn. cm.
system : it must be written ^-5
sec.
The electric intensity of field E, defined by K = eE, has the
dimensions
™=a=[E7iH#]-[#]-[ws
and its unit is — a/- — '• sec. V cm.
After Coulomb's law had been set up electrostatics became a mathematical science. Its most important problem is this : given the total quantity of electricity on conducting bodies, to calculate the distribution of charges on them under the action of their mutual influence and also the forces due to these charges. The development of this mathematical problem is interesting in that it very soon became changed from the original formulation based on the theory of action at a distance to a theory of pseudo-contiguous action, that is, in place of the summations of Coulomb forces there were obtained differential equations in which the intensity of field E or a related quantity called potential occurred as the unknown. But we cannot here further discuss these purely mathematical questions in which Laplace (1782), Poisson (1813), Green (1828), and Gauss (1840) have achieved meritorious results. We shall emphasize only one point : in this treatment of electrostatics, which is usually called the theory of potential, we are not dealing with a true theory of contiguous action in the sense which we attached to this expression above (IV, 6, p. 95). For the differential equations refer to the change in the intensity of field from place to place, but they contain no member that expresses a change in time. Hence they entail no transmission of electric force with finite velocity but, in spite of their differential form, they represent an instantaneous action at a distance.
The doctrine of magnetism developed in the same way as electrostatics. We may, therefore, express ourselves briefly. The most essential difference between these two regions of phenomena is that there are bodies that conduct electricity, whereas magnetism is always bound to matter and can only move with it.
A lozenge-shaped magnetized body, a magnet needle, has two poles, that is, points from which the magnetic force seems to start out, and again the law holds that like poles repel, unlike poles attract. If we break a magnet in halves, the two parts do not carry opposite magnetic charges, but each part
LAWS OF ELECTRODYNAMICS 133
receives a new pole at the broken surface and again represents a complete magnet with two equal but opposite poles. This holds, no matter into how many parts the magnet be broken.
From this it has been concluded that there are indeed two kinds of magnetism as in the case of electricity, that they cannot move freely, and that they are present in the smallest particles of matter, molecules, in equal quantities but separate. Thus each molecule is itself a small magnet with a North and a South pole (Fig. 80). The magnetization of a finite body consists in all the elementary magnets that were originally in complete disorder being brought into the same direction. Then the effects of the alternate North (+) and South (— ) poles counterbalance except for those at the two end faces, from which therefore all the action seems to start.
By using a very long thin magnet needle it is possible to ar- range so that in the vicinity of the one pole the force of the other becomes inappreciable. Hence in magnetism, too, we may ope- rate with test-bodies, namely, with the poles of very long thin
OOOOG
Fig. 80.
magnetic rods. These allow us to carry out all the measure- ments that we have already discussed in the case of electricity. We thus succeed in defining the amount of magnetism or the pole strength p and the magnetic intensity of field H. The magnetic force that a pole p experiences in the field H is
K = pU . . . . (46)
The unit of pole is chosen so that two unit poles at unit distances apart exert the repulsive force 1 on each other. The law according to which the force between two poles px and p2 changes with the distance was also found by Coulomb from direct measurement. Just like Newton's law of attraction, it has the form
K=Hs .... (47)
Clearly the dimensions of magnetic quantities are the same as those of the corresponding electric quantities, and their units have the same notation in the C.G.S. system.
The mathematical theory of magnetism runs fairly parallel
134 THE THEORY OF RELATIVITY
with that of electricity. The most essential difference is that the true quantities of magnetism remain attached to the molecules, and that the measurable accumulations that condition the occurrence of poles in the case of finite magnets arise only owing to the summation of molecules that point in the same direction.
2. Voltaic Electricity and Electrolysis
The discovery of so-called contact electricity by Galvani (1780) and Volta (1792) is so well known that we may pass it by here. For however interesting Galvani's experiments with frogs' legs and the resulting discussion about the origin of electric charges may be, we are here more concerned with formulating conceptions and laws clearly. Hence we shall recount only the facts.
If two different metals be dipped into a solution (Fig. 81), say, copper and zinc into dilute sulphuric acid, the metals manifest electric charges that exert exactly the same action as frictional electricity. Ac- cording to the fundamental law of electricity charges of both sign occur on the metals (poles) to the same amount. The system composed of the solution and the metals, which is also called Voltaic element or cell, thus has the power of separating the two kinds of electricity. Now, it is remarkable that this power is apparently inexhaustible, for if the poles are connected by a wire, so that their Fig. 81. charges flow around and neutralize each
other, then, as soon as the wire is again re- moved, the poles are still always charged. Thus the element continues to keep up the supply of electricity so long as the wire connexion is maintained. Hence a continuous flow of electricity must be taking place. How this is to be imagined in detail depends on whether the one-fluid or the two-fluid theory is supported. In the former case only one current is present, in the latter two opposite currents, one of each fluid, flow.
Now, the electric current manifests its existence by showing very definite effects. Above all it heats the connecting wire. Everyone knows this fact from the metallic threads in our electric glow-lamps. Thus the current continually produces heat-energy. Whence does the Voltaic element derive the power of producing electricity continually and hence thereby indirectly generating heat ? According to the law of conservation of energy, wherever one kind of energy appears during a process another kind of energy must disappear to the same extent.
H
h
\
-=IZ
' :
_m
- —
--
•
LAWS OF ELECTRODYNAMICS 135
The source of energy is the chemical process in the cell. The one metal dissolves as long as the current flows, whereas a constituent of the solution separates out on the other. Com- plicated chemical processes may take place in the solution itself. We have nothing to do with these but satisfy ourselves with the fact that the Voltaic element is a means of generating electricity in unlimited quantities and of producing considerable electric currents.
But we shall now have have to consider the reverse process, in which the electric current produces a chemical decomposition. For example, if we allow the current between two undecompos- able wire leads (electrodes), say of platinum, to flow through slightly acidified water, the latter resolves into its components, hydrogen and oxygen, the hydrogen coming off at the negative electrode (cathode), the oxygen at the positive electrode (anode). The quantitative laws of this process of " elec- trolysis," discovered by Nicholson and Carlisle (1800), were found by Faraday (1832). The far-reaching consequences of Faraday's researches for the knowledge of the structure of matter are well known ; it is not the consequences them- selves that lead us to discuss these researches but the fact that Faraday's laws furnished the means of measuring electric currents accurately, and hence allowed the system of electro- magnetic conceptions to be still further elaborated.
This experiment in electrolytic dissociation can be carried out not only with a Voltaic current, but just as well with a current discharge, which occurs when oppositely charged metallic bodies are connected by a wire. Care must indeed be taken that the quantities of electricity that take place in the discharge are sufficiently great. We have apparatus for storing electricity, so-called condensers, whose action depends on the induction principle, and which give such powerful discharges that measurable amounts are decomposed in the electrolytic cell. The amount of the charge that flows through the cell may be measured by the above discussed methods of electrostatics. Now, Faraday discovered the law that twice the charge produces twice the dissociation, three times the charge three times the dissociation, in short, that the amount m of dissociated substance (or of one of the products of dis- sociation) is proportional to the quantity e of electricity that has passed through the cell :
Cm = e.
The constant C also depends on the nature of the substances and of the chemical process.
A second law of Faraday regulates this dependence. It
136 THE THEORY OF RELATIVITY
is known that chemical elements combine together in perfectly definite proportions to form compounds. The quantity of an element that combines with i grm. of the lightest element, hydrogen, is called its equivalent weight. For example, in water (H20) 8 grms. of oxygen (O) are combined with i grm. of hydrogen (H), hence oxygen has the equivalent weight 8. Now Faraday's law states that the same quantity of electricity that separates out i grm. of hydrogen is able to separate out an equivalent weight of every other element, thus, for example, 8 grms. of oxygen.
Hence the constant C need only be known for hydrogen, and then we get it for every other substance by dividing this value by the equivalent weight of the substance. For we have for I grm. of hydrogen
Cf . I = e
and for any other substance with the equivalent weight fi
C/jl = e.
By dividing these equations we get
C i . r Q
~- = -. i.e. C = — . L9 p /x
Thus C0 = e is the exact quantity of electricity that separates out i grm. of hydrogen. Its numerical value has been de- termined by exact measurements and amounts in the C.G.S. system to
C0 = 2 90 . io14 units of charge per gramme . . (48)
Now we may combine Faraday's two laws into the one formula :
e = — m .... (49)
Thus electrolytic dissociation furnishes us with a very convenient measurement of the quantity of electricity e that has passed through the cell during a discharge. We need only determine the mass m of a product of decomposition that has the equivalent weight fx and then we get the desired quantity of electricity out of equation (49). In this it is of course a matter of indifference whether this electricity is obtained from the discharge of charged conductors (condensers) or whether it comes from a Voltaic cell. In the latter case the electricity flows continuously with constant strength ; the quantity that passes per unit of time through any cross-section of the con- ducting circuit, and hence also through the decomposing cell, is called the intensity of current or current strength. This may
LAWS OF ELECTRODYNAMICS 187
be measured simply by allowing the Voltaic current to flow for a unit of time (i sec.) through the electrolytic cell and then determining the mass m of a product of dissociation. Then equation (49) again gives us the charge e, which is equal to the current strength. If the current flows not for 1 sec. but for / sees., then the quantity of electricity e and the mass m of each product of decomposition separated out is / times as great. Hence the intensity of current J is
J-|-7?-7 • • • • (50)
Its dimensions are
[j] - [t] = [t^k] = %jm\
and its unit is
cm. x/grm. cm. sec.2
3. Resistance and Heat of Current
We must next occupy ourselves a little with the process of conduction or flowing itself. It has been customary to compare the electric current with the flowing of water in a conducting tube and to apply the conceptions there valid to the electrical process. If water is to flow in a tube there must be some driving force. If the water be allowed to flow down from a higher vessel through an inclined tube to a lower vessel, gravitation is the driving force (Fig. 82). This is so much the greater, the higher the upper surface of the water is above the
lower. But the velocity of the current of water or its current strength depends not only on the amount of the impulse given by gravitation but also on the resistance that the water ex- periences in the conducting tube. If this is long and narrow the amount of water sent through per unit of time is less than in the case of a short wide tube. The current strength J is thus proportional to the difference of level V that drives the water and inversely proportional to the resistance W. We set
Fig. 82.
J= *orJW = V
(5i)
138 THE THEORY OF RELATIVITY
in which the unit of resistance chosen is that which allows the current of strength i to flow when the difference of level is I.
G. S. Ohm (1826) applied precisely the same ideas to the electric current. The difference of level that effects the flow corresponds to the electric force. For a definite piece of wire of length I we must set V = El, where E is the field strength, which is regarded constant along the wire. For if the same electric field acts over a greater length of wire, it furnishes a stronger impulse to the flowing electricity. The force V is also called the electromotive force (difference of potential or level). It is moreover identical with the conception of electric potential which we mentioned above (p. 132).
Since the current-strength J and the electric intensity of field E, hence also the potential difference or electromotive force V = El, are measurable quantities, the proportionality between J and V expressed in Ohm's law may be tested experi- mentally.
The resistance W depends on the material and the form of the conducting wire ; the longer and thinner it is, the greater is W. If / is the length of the wire and q the size of the cross- section, then W is directly proportional to /, and inversely proportional to q. We set
aW = - or W = L . . . (52)
q aq
where the factor of proportionality a depends further only on the material of the wire V and is called the conductivity. If we substitute W from (52) and V = el in (51), we get
JW = J - = V = El.
qa
By cancelling / we get
1- = E or J = ctE.
But ^ denotes the current strength per unit cross-section. q This is called the current density and is denoted by i. We thus
have
i = oE . . . (53)
In this form Ohm's law is left with only one constant peculiar to the conducting material, namely, the conductivity a, but in no other way depending on the form of the conducting body (wire).
LAWS OF ELECTRODYNAMICS 189
In the case of insulators a = O. But ideal insulators do not exist. Very small traces of conductivity are always present except in a complete vacuum. There is an unbroken sequence known leading from bad conductors (such as porce- lain, amber) over the so-called semi-conductors (such as water and other electrolytes), to the metals, which have enormously high conductivity.
We have already pointed out above that the current heats the conducting wire. The quantitative law of this pheno- menon was found by Joule (1841). It is clearly a special case of the law of conservation of energy, in which electric energy becomes transformed into heat. Joule's law states that the heat developed per unit of time by the current J in traversing the potential difference V is
Q = JV . . . . (54)
where Q is to be measured not in calories but in mechanical units of work. We shall make no further use of this formula, and state it here merely for the sake of completeness.
4. Electromagnetism
Up to that time electricity and magnetism had been re- garded as two regions of phenomena that were similar in some respects, but quite separate and self-dependent. A bridge was eagerly sought between the two regions, but for a long time without success. At last Oersted (1820) discovered that magnet needles are deflected by Voltaic currents. In the very same year Biot and Savart discovered the quantitative law of this phenomenon, which Laplace formulated as an action at a dis- tance. It is very important for us for the reason that in it there occurred a constant, peculiar to electromagnetism and of the nature of a velocity, which showed itself later to be identical with the velocity of light.
Biot and Savart established that the current flowing in a straight wire neither attracts nor repels a magnetic pole, but strives to drive it round in a circle around the wire (Fig. 83), so that the positive pole moves in the sense of a right- handed screw turned from below (contrary to the hands of a watch) about the (positive) direction of the current. The gravitative law can be brought into the simplest form by sup- posing the conducting wire to be divided into a number of short pieces of length I and writing down the effect of these current-elements, from which the effect of the whole current is obtained by summation. We shall do no more than state the law of a current element for the special case in which
140 THE THEORY OF RELATIVITY
the magnetic pole lies in the plane that passes through the middle part of the element and is perpendicular to its direction (Fig. 84). Then the force that acts on the magnet pole of unit strength, i.e. the magnetic intensity of field H in this plane, is perpendicular to the line connecting the pole with the mid- point of the current-element, and is directly proportional to the current intensity J and to its length /, and inversely pro- portional to the square of the distance r :
cH
I1 r2
(55)
Outwardly this formula has again similarity with Newton's law of attraction or with Coulomb's law of electrostatics and magnetostatics, but the electromagnetic force has nevertheless a totally different character. For it does not act in the direc- tion of the connecting line but perpendicular to it. The three
y
Fig. 83.
Fig. 84.
directions J, r, H are perpendicular to each other in pairs. From this' we see that electrodynamic effects are intimately connected with the structure of Euclidean space ; in a certain sense they furnish us with a natural rectilinear co-ordinate system.
The factor of proportionality c introduced in formula (55) is completely determined since the distance r, the current- strength J, and the magnetic field H are measurable quantities. It clearly denotes the strength of that current which when it flows through a piece of conductor of length 1 produces the magnetic field 1 at unit distance. It is customary and often convenient to choose in place of the unit of current that we have introduced (namely, the quantity of static electricity that flows through the cross-section per unit of time) and that is called the electrostatic unit, this current of strength c in
LAWS OF ELECTRODYNAMICS 141
electrostatic measure as the unit of current ; it is then called the electromagnetic unit of current. Its use brings with it the advantage that the equation (55) assumes the simple form
H = 1? or J = •— -, whereby the measurement of the strength
r'2 I
of a current is reduced to that of two lengths and of a magnetic field. Most practical instruments for measuring currents depend on the deflection of magnets by currents or the con- verse, and hence give the current strength in electromagnetic measure. To express this in terms of the electrostatic measure of current first introduced the constant c must be known ; for this, however, only one measurement is necessary.
Before we speak of the experimental determination of the quantity c, we shall get an insight into its nature by means of a simple dimensional consideration. According to (55)
it is defined by c = ^—. Hr2
Now the following dimensional formulae hold :
j=[t} »-[!-■]
hence the dimensions of c become
But we know that the electric charge e and the magnetic strength of pole p have the same dimensions because Coulomb's law for electric and magnetic force is exactly the same. Hence we get
w - &
that is, c has the dimensions of a velocity.
The first exact measurement of c was carried out by Weber and Kohlrausch (1856). These experiments belong to the most memorable achievements of physical precision measure- ments, not only on account of their difficulty but also on account of the far-reaching consequences of the result. For the value obtained for c was 3.1010 cm. /sec, which is exactly identical with the velocity of light.
This equality could not be accidental. Numerous thinkers, above all Weber himself, the mathematicians Gauss and Rie- mann, and the physicists Neumann, Kirchhoff, Clausius felt the close relationship that the number c = 3.1010 cm./sec. established between two great realms of science, and they sought
142 THE THEORY OF RELATIVITY
to discover the bridge that necessarily led from electromagnetism to optics. Riemann came very near to solving the problem, but this was actually accomplished by Maxwell, after Faraday's wonderful and ingenious method of experimenting had brought to light new facts and new views. We shall next pursue this development.
5. Faraday's Lines of Force
Faraday came from no learned academy ; his mind was not burdened with traditional ideas and theories. His sensational rise from a book-binder's apprentice to the world-famous physicist of the Royal Society is well known. The world of his ideas, which arose directly and exclusively from the abun- dance of his experimental experiences, was just as free from the conventional scheme as his life. We discussed above his researches on electrolytic dissociation. His method of trying all conceivable changes in the conditions of experiment led him (1837) to insert a non-conductor like petroleum and turpentine between the two metal plates (electrodes) of the electrolytic cell in place of a conducting fluid (acid or solution of a salt). These non-conductors did not dissociate, but they were not without influence on the electrical process. For it is found that when the two metal plates are charged by a definite Voltaic battery with a definite potential difference, they take up totally different charges according to the substance that happens to be between them (Fig. 85). The non-conducting substance thus influences the power of taking up electricity or the capacity of the system of conductors composed of the two plates, which is called a condenser.
The discovery impressed Faraday so much that from that time onwards he gave up the usual idea that electrostatics was based on the direct action of electric charges at a distance, and developed a peculiar new interpretation of electric and magnetic phenomena, which is to be called a theory of con- tiguous action. What he learned from the experiment above described was the fact that the charges on the two metal plates do not simply act on each other through the intervening space, but that this intervening space plays an essential part in the action. From this he concluded that the action of this medium is propagated from point to point, and is thus an action by contact or a contiguous action.
We are familiar with the contiguous action of elastic forces in deformed rigid bodies. Faraday, who always kept to em- pirical facts, did indeed compare the electric contiguous action in non-conductors with elastic tensions, but he took good care
LAWS OF ELECTRODYNAMICS 14.3
not to apply the laws of the latter to electrical phenomena. He used the graphical picture of " lines of force " that run in the direction of the electric intensity of field from the positive charges through the insulator to the negative charges. In the case of a plate-condenser the lines of force are straight lines perpendicular to the planes of the plate (Fig. 86). Faraday regards the lines of force as the true substratum of electric events ; for him the}' are actually material configurations that move about, deform themselves, and hereby bring about electrical effects. For Faraday the charges play a quite subordinate part, as the places at which the lines of force start out or end. He was strengthened in this view by those experi- ments which prove that in conductors the total electric charge resides on the surface whilst the interior remains quite free. To give a drastic proof of this he built a large cage fitted out all round with metal, into which he entered with sensitive
Fig. 85.
Fig. 86.
electrical measuring instruments. He then had the cage very strongly charged, and found that in the interior not the slightest influence of the charges was to be detected. Above (V, 1, p. 130) we used just this fact to derive Coulomb's law of action at a distance. But Faraday concluded from it that the charge was not the primary element of electrical events, and that it could not be imagined as a fluid which had the power of exerting forces at a distance. Rather, the primary element is the state of tension of the electric field in the non-conductors, which is represented by the picture of the lines of force. The conductors are in a certain sense holes in the electric field, and the charges in them are only fictitious conceptions, in- vented to explain the pressures and tensions arising through the strains in the field as actions at a distance. Among the non-conductors or dielectric substances there is also the vac it ion, the ether, which we here again encounter in a new form.
This strange view of Faraday's at first found no favour among
144 THE THEORY OF RELATIVITY
the physicists and mathematicians of his own time. The view of action at a distance was maintained, and this was possible even when the "dielectric" action of non-conductors discovered by Faraday was taken into account. Coulomb's law only needed to be altered a little ; to every non-conductor there is assigned a peculiar constant €, its " dielectric constant," which is defined by the fact that the force acting between two charges elt e2 embedded in the non-conductor is smaller in the ratio i : e than that acting in vacuo :
K = le^ .... (56) e r2
For a vacuum e = 1, for every other body € > 1
With this addition the phenomena of electrostatics could actually all be explained even when the dielectric properties of non-conductors were taken into account. We have already mentioned above that electrostatics had formally long ago passed over into a theory of pseudo-contiguous action, the so- called theory of potential. This likewise easily succeeded in assimilating the dielectric constant e. Nowadays we know that this really already signified that the mathematical formulation of Faraday's conception of lines of force had been obtained. But as this method of potential was regarded only as a mathe- matical artifice, the antithesis between the classical theory ot action at a distance and Faraday's idea of contiguous action still remained.
Faraday developed precisely similar views about magnetism. He discovered that the forces between two magnet poles like- wise depend on the medium that happens to lie between them, and this again led him to the view that the magnetic forces, just like the electric forces, are produced by a peculiar state of tension in the intervening media. The lines of force served him to represent these tensions. They can, as it were, be made actually visible by scattering iron filings over a sheet of paper and holding the latter closely over a magnet (Fig. 8j).
The theory of action at a distance leads to the formal introduction of a constant characteristic of the substance, the magnetic penetrability or permeability /x, and gives Coulomb's law in the altered form :
K = IM? .... (57) jjl r2
Physicists have not, however, remained satisfied with this formal addition, but have devised a molecular mechanism that makes the magnetic and dielectric power of polarization in-
LAWS OF ELECTRODYNAMICS 145
telligible. We have already seen above that the properties of magnets lead ns to regard their molecules themselves small elementary magnets that are made to point in parallel directions by the process of magnetization. It is assumed that they retain this parallelism of themselves, say, through fric- tional resistances. Now it may be assumed that in the case of most bodies that do not occur as permanent magnets this friction is wanting. The parallel position is then indeed pro- duced by an external magnetic field, but will at once disappear if the field is removed. Such a substance will then be a magnet only so long as an external field is present. But it need not even be assumed that the molecules are permanent magnets that assume a parallel position. If each molecule contains the two magnetic fluids, then they will separate under the action of the field and the molecule will become a magnet of
..
r-^;>
Fig. 87.
itself. But this induced magnetism must have exactly the action that the formal theory describes by introducing the per- meability. Between the two magnet-poles (N, S) in such a medium there are formed chains of molecular magnets whose opposite poles everywhere compensate each other in the interior, but end with opposite poles at N and S, and hence weaken the actions of N and S (Fig. 88). (The converse effect, namely, strengthening, also occurs, but we shall not enter into its inter- pretation.)
Exactly the same as has just been illustrated for magnetism may also be imagined for electricity. A dielectric, in this view, is composed of molecules that are either electric dipoles of themselves and assume a parallel position in an external field or that become dipoles through the separation of the positive and negative electricity under the action of the field. Between two plates of a condenser (Fig. 89) chains of molecules
10
146 THE THEORY OF RELATIVITY
again form whose charges compensate each other in the interior but not on the plates. Through this a part of the charge on the plates is itself neutralized, and a new charge has to be imparted to the plates to charge them up to a definite tension or potential. This explains how the polarizable dielectric increases the receptivity or capacity of the condenser.
According to this idea of the theory of action at a distance the effect of the dielectric is an indirect one. The field in the vacuum is only an abstraction. It signifies the geometrical distribution of the force that is exerted on an electric test- body carrying a unit charge. But the field in the dielectric is in a state of real physical change, the molecular displace- ment of the two kinds of electricity.
Faraday's theory of contiguous action knows no such differ- ence between the field in the ether and in insulating matter.
<& ^ ^
qOGGGB
N
C30Q
Fig. 88.
Fig. 89.
Both are dielectrics. For the ether the dielectric constant e=i, for other insulators e differs from 1. If the graphical picture of electric displacement is correct for matter, it must also hold for the ether. This idea plays a great part in the theory of Maxwell, which is essentially nothing else than the translation of Faraday's idea of lines of force into the exact language of mathematics. Maxwell assumes that in the ether, too, the genesis of an electric or a magnetic field is accompanied by " displacements " of the fluids. It is not necessary for this purpose to imagine the ether to have an atomic structure, yet Maxwell's idea comes out most clearly if we imagine ether molecules which become dipoles just like the material mole- cules in the field. The field is not, however, the cause of the polarization, but the displacement is the essence of the state of tension which we call electric field. The chains of ether molecules are the lines of force and the charges at the surface
LAWS OF ELECTRODYNAMICS 147
of the conductors are nothing more than the end-charges of these chains. If there are material molecules present besides the ether particles, the polarization becomes strengthened and the charges at the ends become greater.
Are Faraday's and Maxwell's ideas or those of the theory of action at a distance right ?
So long as we confine ourselves to electrostatic and mag- netostatic phenomena, both are fully equivalent. For the mathematical expression of Faraday's idea is what we have called a theory of pseudo-contiguous action, because it does, indeed, operate with differential equations but recognizes no finite velocity of propagation of tensions. Faraday and Maxwell, however, themselves disclosed those events which, analogously to the inertial effects of mechanics, effect the delay in the transference of an electromagnetic state from point to point and hence bring about the finite velocity of propagation. These events are the magnetic induction and the displace- ment current.
6. Magnetic Induction
After Oersted had discovered that an absolute current produces a magnetic field and Biot and Savart had formulated this fact as an action at a distance, Ampere discovered (1820) that two Voltaic currents exert forces on each other, and he in turn succeeded in expressing the law underlying this pheno- menon in the language of the theory of action at a distance. This discovery had far-reaching consequences, for it made it possible to reduce magnetism to a form of electricity. Ac- cording to Ampere small closed currents are supposed to flow in the molecules of magnetized bodies. He showed that such currents behaved exactly like elementary magnets. This idea has stood the test of examination ; from his time onwards magnetic fluids became superfluous. Only electricity was left, which when at rest produced the electrostatic field, and when flowing the magnetic field besides. Ampere's discovery may also be expressed thus : According to Oersted a wire in which the current Jx is flowing produces a magnetic field in its neigh- bourhood . A second wire in which the current J 2 is flowing then experiences force effects in this magnetic field. Thus this field clearly tends to deflect or accelerate flowing electricity.
Hence the following question suggests itself. Can the mag- netic field also set electricity that is at rest into motion ? Can it produce or " induce " a current in the second wire which is initially without a current ?
Faraday found the answer to this question (1831). He
148 THE THEORY OF RELATIVITY
discovered that a static magnetic field has not the power of producing a current, but that one arises as soon as the magnetic field is changed. For example, when he suddenly approached a magnet to a closed conducting wire, a current flowed in the wire so long as the motion lasted ; or when he produced the magnetic field by means of a primary current a short impulse of current occurred in the secondary wire whenever the first current was started or stopped.
From this it is clear that the induced electric force depends on the velocity of alteration of the magnetic field in time. Faraday succeeded in formulating the quantitative law of this phenomenon with the help of his lines of force. We shall give it such a form that its analogy with Biot and Savart's law comes out clearly. We imagine a bundle of parallel lines of magnetic force that constitute a magnetic field H. We suppose a circular conducting wire placed around this sheath (Fig. 90). If the intensity of field H changes in the small
interval of time t by the amount h we call - its velocity of
t
change or the change in the number of lines of force. If we
represent the lines of force as chains of magnetic dipoles (which,
however, according to Ampere, is not allowed), then in the change
of H a displacement of the magnetic quantities will occur in
every ether molecule, or a " magnetic displacement current " will
occur whose current strength per unit of area or current density
is given by j = -. If the field H is not in the ether but in a substance of permeability /x, the density of the magnetic
displacement current is j = /lc— . Thus the magnetic current
t
J = qj = qp— passes through the cross-section q, that is, t
through the surface of the circle formed by the conducting
wire.
Now, according to Faraday, this magnetic current produces all around it an electric field E, which encircles the magnetic current exactly as the magnetic field H encircles the electric current in Oersted's experiment, only in the reverse direction. It is this electric field E that drives the induced current around in the conducting wire ; it is also present even if there is no conducting wire in which the current can form.
We see that the magnetic induction of Faraday is a perfect parallel to the electromagnetic discovery of Oersted. The quantitative law, too, is the same. According to Biot and Savart the magnetic field H produced by a current-element
LAWS OF ELECTRODYNAMICS 149
of length / and of strength J (Cf. Fig. 84, p. 140) in the middle plane perpendicular to the element is perpendicular to the connecting line r and to the current direction, and has the
value H = JL [Formula (55), p. 140]. cr*
Here exactly the same holds when electric and magnetic
quantities are exchanged and when the sense of rotation is
reversed (Fig. 91). The induced electric intensity of field
in the central plane is given by E = J-_«
In it the same constant c, the ratio of the electromagnetic to the electrostatic unit of current occurs, which was found by Weber and Kohlrausch to be equal to the velocity of light. It can easily be seen from considerations of energy that this must be so.
A great number of the physical and technical applications
H
AAA
CJ>
Fig. 90.
Fig. 91.
of electricity and magnetism depend on the law of induction. The transformer, the induction coil, the dynamo, and innumer- able other apparatus and machines are appliances for inducing electric currents by means of changing magnetic fields. But however interesting these things may be, they do not lie on our road of investigation, the final goal of which is to examine the relationship of the ether with the space problem. Hence we turn our attention at once to the representation of Max- well's theory, whose object was to combine all known electro- magnetic phenomena to one uniform theory of contiguous action.
7. Maxwell's Theory of Action by Contact
We have already stated above that soon after Coulomb's law had been set up electrostatics and magnetostatics were brought into the form of a theory of pseudo-contiguous action. Maxwell now undertook to fuse this theory with Faraday's
150 THE THEORY OF RELATIVITY
ideas, and to elaborate it so that it also included the newly discovered phenomena of dielectric and magnetic polarization, of electromagnetism, and magnetic induction.
Maxwell took as the starting-point of his theory the idea already mentioned above that an electric field E is always ac- companied by an electric displacement cE not only in matter, for which e is greater than i, but also in the ether, where 6 = 1. We explained above how the displacement can be visualised as the separation and flowing of electric fluids in the molecules.
The first fact that Maxwell established was that in the light of this idea of displacement Coulomb's law was essentially nothing more than an inference from the law of indestructi- bility of electricity.
Let us imagine a metallic sphere embedded in a medium whose dielectric constant is e (Fig. 92). In this sphere we construct a concentric sphere of radius 1 and another of radius
r. Now, let the metal sphere be charged with an amount of electricity -f e. Then, according to Maxwell, a displacement of the positive electricity outwards must occur in every molecule in order that the amount of electricity con- tained in any arbitrary volume re- main constant. And the amount of electricity transported across the surface of a sphere of radius 1 is to be measured by cE ac- cording to Maxwell. The same amount of electricity will pass through every concentric sphere since otherwise an accumulation of charges would occur in the dielectric. And since the surfaces of two spheres are in the ratio of the squares of the radii, an amount of electricity r2eE passes through the sphere of radius r.
Now this must also be exactly equal to the charge e of the metal sphere at which the displacement comes to an end ; thus we have r2eE = e, or
E == ±,
Fig. 92.
But this is Coulomb's law in the generalized form (56), (p. 144) ; E is the force exerted by the charge e on unit charge at the distance r.
If we are dealing, not with spheres, but with arbitrary charged bodies, Maxwell's fundamental idea still remains the same : the field is determined by the condition that the
LAWS OF ELECTRODYNAMICS 151
displacement cE of the electricity outwards in the delectric or the " divergence " of cE (div. cE) across any arbitrarily small closed surface just compensates the charges that occur in the interior of the surface. By denoting the charge per unit of volume or the density of charge of the electricity by p, we write symbolically
div.eE=p .... (58)
This is to serve us only as a mnemonic for the law formu- lated above. But Maxwell showed that it is possible to derive a definite differential expression for the conception of divergence. Hence to mathematicians formula (58) signifies a differential equation, a law of contiguous action.
Exactly the same considerations apply to magnetism, but with one important difference : according to Ampere no real magnets exist, no magnetic quantities, but only electromagnets. The magnetic field is always to be produced by electric currents, whether they be conduction currents in wires or molecular currents in the molecules. From this it follows that the magnetic lines of force never end, that is, they either merge into themselves again or stretch to infinity. This is so in the case of an electromagnet, a coil through which a current is flowing (Fig. 93) ; the magnetic lines of force run rectilinearly through the interior of the coil, partly joining outside and partly going off to infinity. If we consider the coil enclosed between two planes A and B, then just as much " mag- netic displacement " /xH will enter through A as goes out through B. As, by the way, the displacement picture is unsuitable in this case, we usually say magnetic induction instead of displacement. Hence just as many lines of force will go out through any closed surface as enter into it, or the total divergence of magnetism through an arbitrary closed surface is nil :
div./xH^O .... (59)
This is Maxwell's formula of contiguous action for magnet- ism.
We now come to Biot and Savart's law of electromagnetism. To convert this into a law of contiguous action we suppose the electric current not to be flowing in a thin wire, but to be
distributed uniformly with the density i = J over a circular
cross section q, and we then ask what is the magnetic intensity of field H at the edge of the cross section (Fig. 94). But by Biot and Savart's law this is everywhere in the direction of the tangent to the circle and, according to formula (55), (p. 140),
152 THE THEORY OF RELATIVITY
it has the value H = J-, where r is the radius of the circle, and
cr2 I the length of the current element. Now the cross section, being circular, is ttt2, hence we may write formula (55) thus:
_ = J_ = i = i, and this holds for every cross section, irl ttt2 q
however small, and for every length, however short. On the left, then, there is a certain differential quantity of the magnetic field, and the law states that this quantity is proportional to the current-density. We cannot here carry out the mathe- matical investigation as to how this differential quantity is formed. It has to take into account not only the intensity but also the direction of the magnetic field, and since this encircles or curls round the direction of the current, the dif-
Fig. 93.
o
Fig. 94.
ferential operation is called "curl" of the field H (written, curl H). Accordingly we write symbolically
c curl H = i . . . . (60)
and again regard this formula only as a mnemonic for the relationships between the intensity and direction of the mag- netic field H and the intensity of current i. To the mathe- matician, however, it is a differential equation of the same kind as the law (58).
Now exactly the same holds for magnetic induction, but we shall write the opposite sign to indicate the opposite sense of rotation :
c curl E = — j . . . (61)
The four symbolic formulae (58) to (61) show wonderful sym- metry. Formal agreement of this kind is by no means a matter
LAWS OF ELECTRODYNAMICS 153
of indifference. It exhibits the underlying simplicity of pheno- mena in nature, which remains hidden from direct perception owing to the limitations of our senses, and reveals itself only to our analytical faculty.
8. The Displacement Current
But this symmetry is not perfect ; for i denotes the density of the electric current of conduction, that is, a transportation or convection of electric currents along finite distances, whereas ; is the time change of the magnetic field, and can be interpreted as a displacement current only on the basis of the very artificial hypothesis of ether dipoles.
Now Maxwell remarked (1864) that what sufficed for the magnetic field should hold no less for the electric field. The idea of dipoles compels us also to assume a dielectric displace- ment current, which flows in non-conductors when the electric field E varies. If e is the change of E in the time t, then the density of the dielectric displacement current must be set equal
This Maxwellian theory, which seems almost trivial in our description, is of the greatest importance, for it became the key to the elec- tromagnetic theory of light. We shall make its meaning clear by considering a concrete example. Suppose the poles of a galvanic cell to be connected with the plates of a condenser by means of two wires, and let there be a key in one of the two connexions (Fig. 95). If the latter is depressed, a short current flows which charges up the two plates of the condenser ; between them an electric field E is thereby pro- duced. Before Maxwell's time, this phenomenon was regarded as an " open circuit." Maxwell, however, asserts that during the growth of the field E a displacement current flows between the condenser plates, and the current becomes added to the conduction current so that the circuit becomes complete. So soon as the condenser plates are charged, both currents, the conduction and the displacement current, cease.
Now, the essential point is that Maxwell affirms that, just
Fig. 95.
154 THE THEORY OF RELATIVITY
like the conduction current, the displacement current also pro- duces a magnetic field according to Biot and Savart's law. That this is actually so has not only been proved by the success of Maxwell's theory in predicting numerous phenomena, but was also confirmed directly later by experiment.
In a semi-conductor a conduction and a displacement current will be present simultaneously. For the former, Ohm's law, » = <rE, holds; (53) (p. 138), for the latter,
Maxwell's law, i = — . If both are present simultaneously we t
thus have i = e- + aE. There is no conduction current
for magnetism, we always have j = fx- in that case. If we
t
insert this in our symbolic equations (58) to (61), we get :
(a) div. eE = p (c) c curl H — e- = <tE
h &)
(b) div. /xH = O (d) c curl E + ^ = O
These are Maxwell's laws, which have remained the founda- tion of all electromagnetic and optical theories up to our own time. To the mathematician they are perfectly definite differential equations. To us they are mnemonics which state :
(a) Wherever an electric charge occurs, an electric field
arises of such a kind that in every volume the charge is exactly compensated by the displacement.
(b) Through every closed surface just as much magnetic
displacement passes outwards as comes inwards.
(c) Every electric current, be it a conduction or a displace-
ment current is surrounded by a magnetic field.
(d) A magnetic displacement current is surrounded by an
electric field in the reverse sense.
Maxwell's " field equations," as they are called, constitute a true theory of contiguous action or action by contact, for, as we shall presently see, they give a finite velocity of propagation for electromagnetic forces.
At the time when they were set up, however, faith in direct action at a distance, according to the model of Newtonian attraction, was still so deeply rooted that a considerable interval elapsed before they were accepted. For the theory of action at a distance had also succeeded in mastering the phenomena of induction by means of formulae. This was done by assuming that moving charges exert in addition to the Coulomb attraction also the special actions at a distance
LAWS OF ELECTRODYNAMICS 155
that depend on the amount and direction of the velocity. The first hypotheses of this kind were due to Neumann (1845). A particularly famous law is that which was set up by Wilhelm Weber (1846) ; similar formulae were given by Riemann (1858), and Clausius (1877). All these theories have in common the idea that all electrical and magnetic actions are to be explained by means of forces between elementary electrical charges or, as we say nowadays, " electrons." They were thus precursors of the present-day theory of electrons, with an essential factor omitted, however, namely, the finite velocity of propagation of the forces. These theories of electrodynamics, based on action at a distance, gave a complete explanation of the motive forces and induction currents that occur in the case of closed conduction currents. But in the case of " open " circuits, that is, condenser charges and discharges, they were doomed to failure, for here the displacement currents come into play, of which the theories of action at a distance know nothing. It is to Helmholtz that we are indebted for appro- priate experimental devices, allowing us to decide between the theories of action at a distance and action by contact. He succeeded in carrying the experiment out with a certain measure of success, and he himself became one of the most zealous pioneers of Maxwell's theory. But it was his pupil, Hertz, who secured the victory for Maxwell's theory by dis- covering electromagnetic waves.
9. The Electromagnetic Theory of Light
We have already mentioned above (V, 4, p. 141) the im- pression which the coincidence, established by Weber and Kohlrausch, of the electromagnetic constant c with the velocity of light made upon the physicists of the day. And there were still further indications that there is an intimate relation be- tween light and electromagnetic phenomena. This was shown most strikingly by Faraday's discovery (1834) tnat a polarized ray of light which passes by a magnetized body is influenced by it. When the beam is parallel to the magnetic lines of force its plane of polarization becomes turned. Faraday himself concluded from this that the luminiferous ether and the carrier of electromagnetic lines of force must be identical. Although his mathematical powers were not sufficient to allow him to convert his ideas into quantitative laws and formulae, his conceptions were very abstract and were in no wise confined within the narrow limits of the trivial view which accepted as known what was familiar. Faraday's ether was no elastic medium. It derived its properties, not by analog}', from the
156 THE THEORY OF RELATIVITY
apparently known material world, but from exact experiments and from the consequent relationships that were really known. Maxwell continued Faraday's work. His talents were akin to those of Faraday, but they were supplemented by a complete mastery of the mathematical means available at the time.
We shall now make clear to ourselves that the propagation of electromagnetic forces with finite velocity arises out of Maxwell's field laws (62). In doing so we shall confine ourselves to events that occur in vacuo or in the ether. The latter has no conductivity, that is, a = 0, and no true charges, that is, p = 0 ; and its dielectric constant and permeability are equal to 1, that is, € = 1, ft s= 1. The first two field equations (62) then assert that
div. E = 0 div. H = 0 . . . (63)
or that all lines of force are either closed or run off to infinity. Even if only to obtain a rough picture of the processes we shall imagine to ourselves individual closed lines of force.
The other two field equations are then
- = c curl H - = — c curl E . . (64)
t t
We now assume that, somewhere in a limited space, there is an electric field E which alters by the amount e in the small interval
of time t ; then - is its rate of change. According to the first t
equation, a magnetic field immediately coils itself around this
electric field, and its rate of change is also proportional to
-. The magnetic field, too, will alter in time, say, by h, during t
a successive small interval, t. Again, in accordance with the
second equation, its rate of change - immediately induces an
t
interwoven electric field. In the following interval of time
the latter again induces an encircling