NSTEIN'S THEORY OBRELATWITY MAX BORN -•- ''••=''•'••'•••'- BOSTON UNIVERSITY COLLEGE OF LIBERAL ARTS LIBRARY CHESTER C. CORBIN LIBRARY FUND Tc^vv,^ 2. ^?l^7 Einstein's theory of relativity GOllE •7*7* t^*^*^^*^**-* EINSTEIN'S THEORY OF RELATIVITY BY MAX BORN PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF GOTT1NGEN TRANSLATED BY HENRY L. BROSE, CHRIST CHURCH, OXFORD M.A. WITH I35 DIAGRAMS AND A PORTRAIT NEW YORK E. P. DUTTON AND COMPANY PUBLISHERS Printed in Great Britain I ' t~f\ 2.8 r/>(jS/cs Co 2 2. FROM THE PREFACE TO THE FIRST EDITION THIS book is an elaboration of certain lectures which were given last winter to a somewhat considerable audience. The difficulty which persons not conversant with mathematics and physics experience in understanding the theory of relativity seems to me to be due for the most part to the circumstance that they are not familiar with the fundamental conceptions and facts of physics, in particular of mechanics. During the lectures I therefore showed some quite simple qualitative experiments to serve as an introduction to such concep- tions as velocity, acceleration, mass, force, intensity of field, and so forth. In my endeavour to find a similar means, adapted to book purposes, the semi-historical method of representation here chosen occurred to me, and I hope I have succeeded in avoiding the uninspiring method of the elementary text books of physics. But it must be emphasised that the historical arrangement has been selected only as a cloak which is to bring into stronger relief the outline of the main theme, the logical relationship. Having once started this process I found myself compelled to con- tinue, and in this way my undertaking increased to the dimensions of this book. The reader is assumed to have but little mathematical knowledge. I have attempted to avoid not only the higher mathematics but even the use f of elementary functions, such as logarithms, trigonometrical functions, and so forth. Nevertheless, proportions, linear equations, and occasionally squares and square roots had to be intro- duced. I advise the reader who is troubled with the formulae to pass them by on the first reading and to seek to arrive at an understanding of the mathematical symbols vi THE THEORY OF RELATIVITY from the text itself. I have made abundant use of figures and graphical representations. Even those who are un- practised in the use of co-ordinates will learn to read the curves easily. The philosophical questions to which the theory of relativity gives rise will only be touched on in this book. Nevertheless a definite logical point of view is maintained am throughout. I believe I right in asserting that this view agrees in the main with Einstein's own opinion. Moritz Schlick takes up a similar view in his valuable work "Allgemeine Erkenntislehre " (The General Theory of Knowledge). Of the other books which I have used I should like to quote, above all, Ernst Mach's classical ''Mechanics" (which has appeared in English), and then the very lucidly written volume by E. T. Whittaker, "A History of the Theories of Aether and Electricity" (London, Longmans, & Green Co., 19 10), and the comprehensive account of the Theory of Relativity given by Hermann Weyl in his " Space, Time, Matter " (English translation published & by Messrs. Methuen Co., Ltd., 1922). Anyone who wishes to penetrate further into Einstein's doctrines must study the latter work. It is impossible to enumerate the countless books and essays from which I have drawn more or less directly. In conformity with the character of the book I have refrained from giving references. MAX BORN Frankfurt on the Main June, 1920 PREFACE TO THE THIRD EDITION APART from a number of minor alterations, this edition differs from its two predecessors in that the chapter on Einsteinian dynamics has been revised. Previously, in forming the acceleration, we did not dis- tinguish sharply between time and proper time, and we used Minkowski's covariant force-vector in place of ordin- ary force ; this of course increased the difficulty of under- standing a chapter which was, from the outset, not easy. Dr. W. Pauli, jun., called my attention to a method of deriving the relativistic formula of mass proposed by Lewis and Tolman, which fitted in admirably with the scheme of this book, as it linked up with the conception of momentum in the same way as the account of mechanics here chosen. The chapter on Einsteinian dynamics was revised in con- formity with this point of view this also entailed some ; alterations in the manner of presenting ordinary mechanics. It is hoped that these changes will simplify the reading. I should not like to lose this opportunity of thanking Dr. W. Pauli for his advice. His great work on the theory of relativity which has appeared as Article 19 in the fifth volume of the " Enzyklopadie der mathematischen Wissenschaften," which appeared recently, has been of great service to me. It is to be recommended foremost of all to those who wish to become intimately acquainted with the theory of relativity. MAX BORN GOTTINGEN 6th March, 1922 . CONTENTS CHAPTER I Geometry and Cosmology § i. The Origin of the Art of Measuring Space and Time 7 § 2. Units of Length and Time . 7 § 3. Origin and Co-ordinate System . 8 § 4. The Axioms of Geometry . 9 § 5. The Ptolemaic System 10 § 6. The Copernican System 11 § 7. The Elaboration of the Copernican Doctrine 13 CHAPTER II The Fundamental Laws of Classical Mechanics § 1. Equilibrium and the Conception of Force . 15 — § 2. The Study of Motions Rectilinear Motion 16 § 3. Motion in a Plane 23 § 4. Circular Motion 24 —.... § 5. Motion in Space § 6. Dynamics The Law of Inertia § 7. Impulses 26 27 28 .... § 8. The Law of Impulses . § 9. Mass 29 30 § 10. Force and Acceleration — §11. Example Elastic Vibrations 32 34 §12. Weight and Mass 36 §13. Analytical Mechanics 39 § 14. The Law of Energy . 4i § 15. Dynamical Units of Force and Mass 4S CHAPTER III The Newtonian World-System § 1. Absolute Space and Absolute Time . 4S § 2. Newton's Law of Attraction ..... General Gravitation ..... §4- Celest'al Mechanics 5* 53 56. § 5. The Relativity Principle of Classical Mechanics .... § 6. Limited Absolute Space .... §7- Galilei Transformations ...... §8. Inertial Forces 59' 61 62 67 §9- Centrifugal Forces and Absolute Space 69 b ix THE THEORY OF RELATIVITY CHAPTER IV .... The Fundamental Laws of Optics § i. The Ether . § 2. The Corpuscular and the Undulatory Theory § 3. The Velocity of Light § 4. Fundamental Conceptions of the Wave Theory Interference § 5. Polarisation and Transversality of Light \\ aves § 6. The Ether as an Elastic Solid . § 7. The Optics of Moving Bodies § 8. The Doppler Effect . .... § 9. The Convection of Light by Matter § 10. Aberration §11. Retrospect and Future Prospects CHAPTER V The Fundamental Laws of Electrodynamics § 1. Electrostatics and Magnetostatics § 2. Voltaic Electricity and Electrolysis § 3. Resistance and Heating due to Currents § 4. Electromagnetism § 5. Faraday's Lines of Force § 6. Magnetic Induction . § 7. Maxwell's Contact Theory § 8. Displacement Currents § 9. The Electromagnetic Theory of Light § 10. The Luminiferous Ether . § 1 1. Hertz' Theory of Moving Bodies §12. Lorentz' Theory of Electrons § 13. Electromagnetic Mass § 14. Michelson and Morley's Experiment . §15. The Contraction Hypothesis CHAPTER VI Einstein's Special Principle of Relativity § 1. The Conception of Simultaneity . § 2. Einstein's Kinematics and Lorentz' Transformations A § 3. Geometrical Representation of Einstein's Kinematics § 4. Moving Measuring-rods and Clocks § 5. Appearance and Reality § 6. The Addition of Velocities . § 7. Einstein's Dynamics . § 8. The Inertia of Energy § 9. The Optics of Moving Bodies § 10. Minkowski's Absolute World CHAPTER VII .... Einstein's General Theory of Relativity §1. The Relativity of Arbitrary Motions § 2. The Principle of Equivalence .... §3. The Failure of Euclidean Geometry ..... § 4. Geometry on Curved Surfaces 75 76 79 83 90 93 102 104 no 120 122 125 134 137 139 142 147 149 153 155 160 162 168 175 180 184 192 198 200 206 210 217 221 230 237 242 247 250 254 257 CONTENTS XI § 5. The Two-dimensional Continuum .... PAOI 2 62 § 6. Mathematics and Reality 264 § 7. The Measure-determination of the Space-time Continuum 268 § 8. The Fundamental Laws of the New Mechanics . 272 § 9. Mechanical Consequences and Confirmations 275 § 10. Optical Consequences and Confirmations ...... §11. Macrocosm and Microcosm § 12. Conclusion 2 80 286 289 [SDKS. 591 EINSTEIN'S THEORY OF RELATIVITY INTRODUCTION Das schonste Gliick des denkenden Menschen ist, das Erforschliche — erforscht zu haben und das Unerforschliche ruhig zu verehren. Goethe. THE world is not presented to the reflective mind as a finished product. The mind has to form its picture from innumerable sensations, experiences, communica- tions, memories, perceptions. Hence there are probably not two thinking people whose picture of the world coincides in every respect. When an idea in its main lines becomes the common property of large numbers of people, the movements of spirit that are called religious creeds, philosophic schools, and scientific systems arise ; they present the aspect of a chaos of opinions, of articles of faith, of convictions, that resist all efforts to disentangle them. It seems a sheer impossibility to find a thread that will guide us along a definite path through these widely ramified doctrines that branch off perchance to recombine at other points. What place are we to assign to Einstein's theory of rela- tivity, of which this book seeks to give an account ? Is it only a special part of physics or astronomy, interesting in itself but of no great importance for the development of the human spirit ? Or is it at least a symbol of a particular trend of thought characteristic of our times ? Or does it itself, We indeed, signify a "world-view" (Weltanschauung) ? shall be able to answer these questions with confidence only when we have become acquainted with the content of Einstein's doctrine. But we may be allowed to present here a point of view which, even if only roughly, classifies the totality of all world-views and ascribes to Einstein's theory a definite position within a uniform view of the world as a whole. The world is composed of the ego and the non-ego, the inner world and the outer world. The relations of these two poles 2 THE THEORY OF RELATIVITY are the object of every religion, of every philosophy. But the part that each doctrine assigns to the ego in the world is different. The importance of the ego in the world-picture seems to me a measure according to which we may order confessions of faith, philosophic systems, world-views rooted in art or science, like pearls on a string. However enticing it may be to pursue this idea through the history of thought, we must not diverge too far from our theme, and we shall apply it only to that special realm of human thought to which Einstein's theory — belongs to natural science. Natural science is situated at the end of this series, at the point where the ego, the subject, plays only an insignificant part ; every advance in the mouldings of the conceptions of physics, astronomy, and chemistry denotes a further step towards the goal of excluding the ego. This does not, of course, deal with the act of knowing, which is bound to the subject, but with the finished picture of Nature, the basis of which is the idea that the ordinary world exists independently of and uninfluenced by the process of knowing. The doors through which Nature imposes her presence on us are the senses. Their properties determine the extent of what is accessible to sensation or to intuitive perception. The further we go back in the history of the sciences, the more we find the natural picture of the world determined by the qualities of sense. Older physics was subdivided into mechanics, We acoustics, optics, and theory of heat. see the connexions with the organs of sense, the perceptions of motion, impressions of sound, light, and heat. Here the qualities of the subject are still decisive for the formation of conceptions. The deve- lopment of the exact sciences leads along a definite path from this state to a goal which, even if far from being attained, yet lies clearly exposed before us : it is that of creating a picture of nature which, confined within no limits of possible perception or intuition, represents a pure structure of conception, con- ceived for the purpose of depicting the sum of all experiences uniformly and without inconsistencies. Nowadays mechanical force is an abstraction which has only its name in common with the subjective feeling of force. Mechanical mass is no longer an attribute of tangible bodies but is also possessed by empty spaces filled only by ether radiation. The realm of audible tones has become a small province in the world of inaudible vibrations, distinguishable physically from these solely by the accidental property of the human ear which makes it react only to a definite interval of frequency numbers. Modern optics is a special chapter out of the theory of electricity and magnetism, and it treats of the INTRODUCTION 8 electro-magnetic vibrations of all wave-lengths, passing from the shortest 7-rays of radioactive substances (having a wavelength of one hundred millionth of a millimetre) over the Rontgen rays, the ultraviolet, visible light, the infra-red, to the longest wireless (Hertzian) waves (which have a wave-length of many kilometres). In the flood of invisible light that is accessible to the mental eye of the physicist, the material eye is almost blind, so small is the interval of vibrations which it converts into sensations. The theory of heat, too, is but, a special part of mechanics and electro-dynamics. Its fundamental conceptions of absolute temperature, of energy, and of entropy belong to the most subtle logical configurations of exact science, and, again, only their name still carries a memory of the subjec- tive impression of heat or cold. Inaudible tones, invisible light, imperceptible heat, these constitute the world of physics, cold and dead for him who wishes to experience living Nature, to grasp its relationships as a harmony, to marvel at her greatness in reverential awe. Goethe abhorred this motionless world. His bitter polemic against Newton, whom he regarded as the personification of a hostile view of Nature, proves that it was not merely a question of an isolated struggle between two investigators about individual questions of the theory of colour. Goethe is the representative of a world-view which is situated somewhere near the opposite end of the scale suggested above (constructed according to the relative importance of the ego), that is, the end opposite to that occupied by the world-picture of the exact sciences. The essence of poetry is inspiration, intuition, the visionary comprehension of the world of sense in symbolic forms. But the source of poetic power is experience, whether it be the clearly conscious perception of a sense-stimulus, or the powerfully represented idea of a relationship or connexion. What is logically formal and rational plays no part in the world- picture of such a type of gifted or indeed heaven-blessed spirit. The world as the sum of abstractions that are connected only indirectly with experience is a province that is foreign to it. Only what is directly presented to the ego, only what can be felt or at least represented as a possible experience is real to it and has significance for it. Thus to later readers, who survey the development of exact methods during the centurv after Goethe's time and who measure the power and significance of Goethe's works on the history of natural science by their fruits, these works appear as documents of a visionary mind, as the expression of a marvellous sense of one-ness with (Ein- fuhlung) the natural relationships, but his physical assertions will seem to such a reader as misunderstandings and fruit]- SS 4 THE THEORY OF RELATIVITY rebellions against a greater power, whose victory was assured even at that time. Now in what does this power consist, what is its aim and device ? It both takes and renounces. The exact sciences presume to aim at making objective statements, but they surrender their absolute validity. This formula is to bring out the following contrast. All direct experiences lead to statements which must be allowed a certain degree of absolute validity. If I see a red flower, if I experience pleasure or pain, I experience events which it is meaningless to doubt. They are indubitably valid, but only for me. They are absolute, but they are subjective. All seekers after human knowledge aim at taking us out of the narrow circle of the ego, out of the still narrower circle of the ego that is bound to a moment of time, and at establishing common ground with other thinking creatures. It first establishes a link with the ego as it is at another moment, and then with other human beings or gods. All religions, philosophies, and sciences have been evolved for the purpose of expanding the ego to the wider community that " we " represent. But We the ways of doing this are different. are again confronted by the chaos of contradictory doctrines and opinions. Yet we no longer feel consternation, but order them according to the importance that is given to the subject in the mode of com- prehension aimed at. This brings us back to our initial prin- ciple, for the completed process of comprehension is the world-picture. Here again the opposite poles appear. ' , The minds of one group do not wish to deny or to sacrifice the absolute, and they therefore remain clinging to the ego. They create a world-picture that can be produced by no sys- tematic process, but by the unfathomable action of religious, artistic, or poetic means of expression in other souls. Here faith, pious ardour, love of brotherly communion, but often also fanaticism, intolerance, intellectual suppression hold sway. — — The minds of the opposite group sacrifice the absolute. They discover often with feelings of terror the fact that inner experiences cannot be communicated. They no longer fight for what cannot be attained, and they resign themselves. But they wish to reach agreement at least in the sphere of the attainable. They therefore seek to discover what is common in their ego and in that of the other egos ; and the best that was there found was not the experiences of the soul itself, not sensations, ideas, or feelings, but abstract conceptions of the — simplest kind numbers, logical forms ; in short, the means of expression of the exact sciences. Here we are no longer con- INTRODUCTION cerncd with what is absolute. The height of a cathedral does not, in the special sphere of the scientist, inspire reverence, but is measured in metres and centimetres. The course of life is no longer experienced as the running out of the sands of time, but is counted in years and days. Relative measures take tin- place of absolute impressions. And we get a world, narrow, one-sided, with sharp edges, bare of all sensual attraction, of all colours and tones. But in one respect it is superior to other world-pictures : the fact that it establishes a bridge from mind to mind cannot be doubted. It is possible to agree as to whether iron has a specific gravity greater than wood, whether water freezes more readily than mercury, whether Sirius is a planet or a star. There may be dissensions, it may sometimes seem as if a new doctrine upsets all the old " facts," yet he who has not shrunk from the effort of penetrating into the interior of this world will feel that the regions known with certainty are growing, and this feeling relieves the pain which arises from solitude of the spirit, and the bridge to kindred spirits becomes built. We have endeavoured in this way to express the nature of scientific research, and now we can assign Einstein's theory of relativity to its category. In the first place, it is a pure product of the striving after the liberation of the ego, after the release from sensation and We perception. spoke of the inaudible tones, of the invisible We light, of physics. find similar conditions in related sciences, in chemistry, which asserts the existence of certain (radioactive) substances, of which no one has ever perceived the smallest — trace with any sense directly or in astronomy, to which we refer below. These " extensions of the world," as we might call them, essentially concern sense-qualities. But everything takes place in the space and the time which was presented to mechanics by its founder, Newton. Now, Einstein's discovery is that this space and this time are still entirely embedded in the ego, and that the world-picture of natural science becomes more beautiful and grander if these fundamental conceptions are also subjected to relativization. Whereas, before, space was closely associated with the subjective, absolute sensation of extension, and time with that of the course of life, they are now purely conceptual schemes, just as far removed from direct perception as entities, as the whole region of wave-lengths of present-day optics is inaccessible to the sensation of light except for a very small interval. But just as in the latter case, the space and time of perception allow themselves to be ordered without giving rise to difficulties, into the system of physical conceptions. Thus an objectivation is attained, which has 6 THE THEORY OF RELATIVITY manifested its power by predicting natural phenomena in We a truly wonderful way. shall have to speak of this in detail in the sequel. Thus the achievement of Einstein's theory is the relativization and objectivation of the conceptions of space and time. At the present day it is the final picture of the world as presented by science. CHAPTER I GEOMETRY AND COSMOLOGY i. The Origin of the Art of Measuring Space and Time THE physical problem presented by space and time is nothing more than the familiar task of fixing numerically a place and a point of time for every phy event, thus enabling us to single it out, as it were, from the chaos of the co-existence and succession of things. The first problem of Man was to find his way about on the earth. Hence the art of measuring the earth (geodesy) became the source of the doctrine of space, which derived its name " geometry " from the Greek word for earth. From the v outset, however, the measure of time arose from the regular change of night and day, of the phases of the moon and of the seasons. These phenomena forced themselves on Man's attention and first moved him to direct his gaze to the stars, which were the source of the doctrine of the anivc cosmology. Astronomic science applied the teachings of g metry that had been tested on the earth to the heavenly regions, allowing distances and orbits to be defined. For this purpose it gave the inhabitants of the earth the celestial (astronomic) measure of time which taught Man to distinguish between Past, Present, and Future, and to assign to each thing its place in the realm of Time. 2. Units of Length and Time The foundation of every space- and time-measurement is laid by fixing the unit. A datum of length, " so and so many metres." denotes the ratio of the length to be measured to the " length of a metre. A time-datum of " so and so many seconds denotes the ratio of the time to be measured to the duration of a second. Thus we are always dealing with ratio-nunV: relative data concerning the units. The latter themselves are to a high degree arbitrary, and are chosen for reasons of their being capable of easy reproduction, of being easily transportable, durable, and so forth. . 8 THE THEORY OF RELATIVITY In physics the measure of length is the centimetre (cm.), the hundredth part of a metre rod that is preserved in Paris. This was originally intended to bear a simple ratio to the circumference of the earth, namely, to be the ten-millionth part of a quadrant, but more recent measurements have disclosed that this is not accurately true. The unit of time in physics is the second (sec), which bears the well-known relation to the time of rotation of the earth on its axis. 3. Origin and Co-ordinate System But if we wish not only to determine lengths and periods of time, but also to designate places and points of time, further conventions must be made. In the case of time, which we regard as a one-dimensional con- figuration, it is sufficient to specify an origin (or zero-point) Historians reckon dates by counting the years from the birth of Christ. Astronomers choose other origins or initial points, according to the objects of their researches ; these they call epochs. If the unit and the origin are fixed, every event may be singled out by assigning a number-datum to it. In geometry in the narrower sense, the determination of position on the earth, two data must be given to fix a point. To say " My house is in Baker Street," is not sufficient to fix it. The number of the house must also be given. In many American towns the streets themselves are numbered. The address No. 25, 13th Street, thus consists of two number-data. It is exactly what mathematicians call a " co-ordinate determination." The earth's surface is covered with a network of intersecting lines, which are numbered, or whose position is determined by a number, distance, or angle (made with respect to a fixed initial or zero- line). Geographers generally use geographic longitude (east of Greenwich) and latitude (north or south) (Fig. 1). These determinations at the same time fix the zero-lines from which the co-ordinates are to be counted, namely, for geographical longitude the meridian of Greenwich, and for the latitude the GEOMETRY AND COSMOLOGY 9 equator. In investigations of plane geometry we generally use rectilinear [Cartesian) co-ordinates (Fig. 2), x, y, which signify the distances from two mutually perpendicular coordinate axes ; or, occasionally, we also use oblique co-ordinates (Fig. 3), polar co-ordinates (Fig. 4), and others. When the Fig. 2. Fig. 3. co-ordinate system has been specified, we can seek out each point or place if two numbers are given. In precisely the same way we require three co-ordinates to fix points in space. It is simplest to choose mutually per- pendicular rectilinear co-ordinates again ; we denote them by x,y,z (Fig. 5). Fig. 4. Fig. 5. 4. The Axioms of Geometry Ancient geometry, regarded as a science, was less concerned with the question of determining positions on the earth's surface, than with determining the size and form of areas, We figures in space, and the laws governing these questions. see traces of the origin of this geometry in the art of surveying and of architecture. That is also the reason why it managed without the conception of co-ordinates. First and foremost, 10 THE THEORY OF RELATIVITY geometric theorems assert properties of things that are called points, straight lines, planes. In the classic canon of Greek geometry, the work of Euclid (300 B.C.), these things are not defined further but are only denominated or described. Thus we here recognize an appeal to intuition. You must already know what is a straight line if you wish to take up the study of geometry. Picture the edge of a house, or the stretched cable of your surveying instruments, form an abstraction from what is material and you will get your straight line. Next, laws are set up that are to hold between these configurations of abstraction, and it is to the credit of the Greeks to have made the great discovery that we need assume only a small number of these theorems to make all others come out of them correctly with logical inevitableness. These theorems, which are used as the foundation, are the axioms. Their correctness cannot be proved. They do not arise from logic but from other sources of knowledge. What these sources are has formed the subject of the theories of all the philosophies of the succeeding centuries. Scientific geometry itself, up to the end of the 18th century, accepted these axioms as given, and built up its purely deductive system of theorems on them. We shall not be able to avoid discussing in detail the question of the meaning of the elementary configurations called point, straight line, and so forth, and the grounds of our knowledge of the geometric axioms. For the present, however, we shall adopt the standpoint that we are clear about these things, and we shall thus operate with the geometric conceptions in the way we learned (or should have learned) at school, and in the way numberless generations of people have done, without scruples. The intuitional truth of numerous geometric theo- rems, and the utility of the whole system in giving us bearings in our ordinary real world is to suffice for the present as our justification for using them. 5. The Ptolemaic System To the eye the heavens appear as a more or less flat dome to which the stars are attached. But in the course of a day the whole dome turns about an axis whose position in the heavens is denoted by the pole-star. So long as this visual appearance was regarded as reality an application of geometry from the earth to astronomic space was superfluous, and was, as a matter of fact, not carried out. For lengths and distances measurable with earthly units were not present. To denote the positions of the stars only the apparent angle that the line of vision from the observer to the star formed with the GEOMETRY AND COSMOLOGY 11 horizon and with another appropriately chosen plane had to be known. At this stage of knowledge the earth's surface wa considered at rest and was the eternal basis of the universe. The words " above " and " below " had an absolute meaning and when poetic fancy or philosophic speculation undertook to estimate the height of the heavens or the depth of Tartarus, the meaning of these terms required no word of elucidation. At this stage scientific concepts were still being drawn from the abundance of subjective data. The world-system called after Ptolemy (150 a.d.) is the scientific formulation of this mental attitude. It was already aware of a number of detailed facts concerning the motion of the sun, the moon, and the planets, and it had a considerable theoretical grasp of them, but it retained the notion that the earth is at rest and that the stars are revolving about it at immeasurable distances. Its orbits were determined as circles and epi-cycles according to the laws of earthly geometry, yet astronomic space was not actually through this subjected to geometry. For the orbits were fastened like rings to the crystal shells, which, arranged in strata, signified the heavens. 6. The Copernican System It is known that Greek thinkers had already discovered the spherical shape of the earth and ventured to take the first steps from the geometric world-systems of Ptolemy to higher abstractions. But only long after Greek civilization and culture had died, did the peoples of other countries accept the spherical shape of the earth as a physical reality. This is the first truly great departure from the evidence of our eyes, and at the same time the first truly great step towards relativization. Again centuries have passed since that first turningpoint, and what was at that time an unprecedented discovery has now become a platitude for school-children. This makes it difficult to convey an impression of what it signified to thinkers to see the conceptions " above " and " below " lose their absolute meaning, and to recognize the right of the inhabitants of the antipodes to call " above " in their regions what we call " below " in ours. But after the earth had once been circumnavigated all dissentient voices became silent. For this reason, too, the discovery of the sphericity of the earth offered no reason for strife between the objective and the subjective view of the world, between scientific research and the church. This strife broke out only after Copernicus (1543) displaced the earth from its central position in the universe and created the helio- centric world-system. 12 THE THEORY OF RELATIVITY In itself the process of relativization was hardly advanced by this, but the importance of the discovery for the develop- ment of the human spirit consisted in the fact that the earth, mankind, the individual ego, became dethroned. The earth became a satellite of the sun and carried around in space the peoples swarming on it. Similar planets of equal importance accompany it in describing orbits about the sun. Man is no longer important in astronomy, except for himself. But still more, none of these amazing facts arise from ordinary observation (such as is the case with a circumnavigation of the globe), but from observations which were, for the time in question, very delicate and subtle, from different calculations of planetary orbits. The evidence was at any rate such as was neither accessible to all men nor of importance for everyday life. Ocular evidence, intuitive perceptions, sacred and pagan tradition alike speak against the new doctrine. In place of the visible disc of the sun it puts a ball of fire, gigantic beyond imagination; in place of the friendly lights of the heavens, similar balls of fire at inconceivable distances, or spheres like the earth, that reflect light from other sources; and all visible measures are to be regarded as deception, whereas immeasurable distances and incredible velocities are to represent the true state of affairs. Yet this new doctrine was destined to be victorious. For it drew its power from the burning wish of all thinking minds to comprehend all things of the material — world, be they ever so unimportant for human existence, as a co-ordinate unity to make them a permanent possession of the intellect and communicable to others. In this process, which constitutes the essence of scientific research, the human spirit neither hesitates nor fears to doubt the most striking facts of visual perception, and to declare them to be illusions, but prefers to resort to the most extreme abstractions rather than exclude from the scientific description of Nature one established fact, be it ever so insignificant. That, too, is why the church, at that time the carrier of the subjective worldview then dominant, had to persecute the followers of the Copernican doctrine, and that is why Galilei had to be brought before the inquisitorial tribunal as a heretic. It was not so much the contradictions to traditional dogmas as the changed attitude towards spiritual events that called this struggle into being. If the experience of the soul, the direct perception of things, was no longer to have significance in Nature, then religious experience might also one day be subjected to doubt. However far even the boldest thinkers of those times were removed from feelings of religious scepticism, the church scented the enemy. GEOMETRY AND COSMOLOGY 18 The great relativizing achievement of Copernicus was the root of all the innumerable similar but lesser relativizations of growing natural science until the time when Einstein's discovery ranged itself as a worthy result alongside that of its great predecessor. But now we must sketch in a few words the cosmos as mapped out by Copernicus. We have first to remark that the conceptions and laws of earthly geometry can be directly applied to astronomic space. In place of the cycles of the Ptolemaic world, which were supposed to occur on surfaces, we now have real orbits in space, the planes of which may have different positions. The centre of the world-system is the sun. The planets describe their circles about it, and one of them is the earth, which rotates about its own axis, and the moon in its turn revolves in its orbit about the earth. But beyond, at enormous dis- tances, the fixed stars are suns like our own, at rest in space. Copernicus' constructive achievement consists in the fact that with this assumption the heavens must exhibit all these pheno- mena which the traditional world-system was able to explain only by means of complicated and artificial hypotheses. The alternation of day and night, the seasons, the phenomena of the moon's phases, the winding planetary orbits, all these things become at one stroke clear, intelligible, and accessible to simple calculations. 7. The Elaboration of the Copernican Doctrine The circular orbits of Copernicus soon no longer sufficed to account for the observations. The real orbits were evidently considerably more complicated. Now, an important point for the new view of the world was whether artificial constructions, such as the epicycles of the Ptolemaic system or an improvement in the calculations of the orbits could be successfully carried out without introducing complications. It was the immortal achievement of Kepler (1618) to discover the simple and striking laws of the planetary orbits, and hence to save the Copernican system at a critical period. The orbits are not, indeed, circles about the sun, but curves closely related to circles, namely, ellipses, in one focus of which the sun is situated. Just as this law determines the form of the orbits in a very simple manner, so the other two laws of Kepler determine the sizes of the orbits and the velocities with which they are traversed. Kepler's contemporary, Galilei (1610), directed a telescope, which had just then been invented, at the heavens and 14 THE THEORY OF RELATIVITY discovered the moons of Jupiter. In them he recognized a microscopic model of the planetary system and saw Coper- nicus' ideas as optical realities. But it is Galilei's greater merit to have developed the principles of mechanics, the application of which to planetary orbits by Newton (1867) brought about the completion of the Copernican world-system. Copernicus' circles and Kepler's ellipses are what modern science calls a kinematic or phoronomic description of the orbits, namely, a mathematical formulation of the motions which does not contain the causes and relationships that bring about these same motions. The causal expression of the laws of motion is the content of dynamics or kinetics, founded by Galilei. Newton has applied this doctrine to the motions of the heavenly bodies, and by interpreting Kepler's laws in a very ingenious way he introduced the causal conception of mechanical force into astronomy. Newton's law of gravitation proved its superiority over the older theories by accounting for all the deviations from Kepler's laws, the so-called perturbations of orbits, which refinements in the methods of observation had in the meantime brought to light. This dynamical view of the phenomena of motion in astro- nomic space, however, at the same time demanded a more precise formulation of the assumptions concerning space and time. These axioms occur in Newton's work for the first time as explicit definitions. It is therefore justifiable to regard the theorems that held up to the advent of Einstein's theory as expressions of Newton's doctrine of space and time. To under- stand them it is absolutely necessary to have a clear survey of the fundamental laws of mechanics, and that, indeed, from a point of view which places the question of relativity in the foreground, a standpoint that is usually neglected in the We elementary text-books. shall therefore next have to discuss the simplest facts, definitions, and laws of mechanics. CHAPTER II THE FUNDAMENTAL LAWS OF CLASSICAL MECHANICS i. Equilibrium and the Conception of Force HISTORICALLY, mechanics took its start from the doctrine of equilibrium or statics ; logically, too, the development from this point is the most natural one. The fundamental conception of statics is force. It is derived from the subjective feeling of exertion experienced when we perform work with our bodies. Of two men he is the stronger who can lift the heavier stone or stretch the stiffer bow. This measure of force, with which Ulysses established his right among the suitors, and which, indeed, plays a great part in the stories of ancient heroes, already contains the germ of the objectivation of the subjective feeling of exertion. The next step was the choice of a unit of force and the measurement of all forces in terms of their ratios to the unit of force, that is, the relativization of the conception of force. Weight, being the most evident manifestation of force, and making all things tend downwards, offered the unit of force in a convenient form, namely, a piece of metal which was chosen as the unit of weight through some decree of the state or of the church. Nowadays it is an international congress that fixes the units. The unit of weight in technical matters is the weight of a definite piece of platinum in Paris. This unit, called the gramme (grm.) will be used in the sequel till otherwise stated. The instrument used to compare the weights of different bodies is the balance. Two bodies have the same weight, or are equally heavy, when, on being placed in the two scales of the balance, they do not disturb its equilibrium. If we place two bodies found to be equally heavy in this manner in one pan of the balance, but, in the other, a body such that the equilibrium is again not disturbed, then this new body has twice the weight of either of the other two. Continuing in this way we get, starting from the unit of weight, a set of weights with the help of which the weight of every body may be conveniently deter- mined. 15 16 THE THEORY OF RELATIVITY It is not our task here to show how these means enabled man to find and interpret the simple laws of the statics of We rigid bodies, such as the laws of levers. here introduce only just those conceptions that are indispensable for an understanding of the theory of relativity. Besides the forces that occur in man's body or in that of his domestic pets he encounters others, above all in the events that we nowadays call elastic. The force necessary to stretch a cross-bow or any other bow belongs to this category. Now, these can easily be compared with weights. If, for example, we wish to measure the force that is necessary to stretch a spiral spring a certain distance (Fig. 6), then we find by trial what weight must be suspended from it to effect equilibrium for just this extension. Then the force of the spring is equal to that of the weight, except that the former exerts a pull upwards but the latter X~ ? downwards. The principle that * action and reaction are equal and opposite in the condition of equilib- rium has tacitly been applied. If such a state of equilibrium be disturbed by weakening or re- moving one of the forces, motion occurs. The raised weight falls when it is released by the hand sup- porting it and thus furnishing the reacting force. The arrow shoots forth when the archer releases the FlG 6 string of the stretched bow. Force tends to produce motion. This is the starting-point of dynamics, which seeks to discover the laws of this process. — 2. The Study of Motions Rectilinear Motion It is first necessary to subject the conception of motion itself to analysis. The exact mathematical description of the motion of a point consists in specifying at what place relative to the previously selected co-ordinate system the point is situated from moment to moment. Mathematicians use We formulae to express this. shall as much as possible avoid this method of representing laws and relationships, which is not familiar to everyone, and shall instead make use of a graphi- cal method of representation. Let us illustrate this for the simplest case, the motion of a point in a straight line. Let the unit of length be the centimetre, as usual in physics, and let the ; LAWS OF CLASSICAL MECHANICS 17 = moving point be at the distance x i cm. from the zero point or origin at the moment at which we start our considerations = and which we call the moment t o. In the course of i sec. suppose the point to have moved a distance of \ cm. to the = right, so that for t i the distance from the origin amounts to i-5 cms. In the next second let it move by the same amount = to x 2 cms., and so forth. The following small table gives the distances x corresponding to the times t. 8... t. \ O T 2 ^ A ^ 6 7 x i i-5 2 2-5 3 3*5 4 4-55... \ We see the same relationship pictured in the successive lines of Fig. 7, in which the moving point is indicated as a small circle on the scale of distances. Now, instead of drawing a number of small diagrams, one above the other, we may also t'6 US t'-<+ t*3 t=2 t-1 t'O >>A 2 3 Fig. 7. draw a single figure in which the x's and the t's occur as coordinates (Fig. 8). In addition, this has the advantage of allowing the place of the point to be depicted not only at the beginning of each full second but also at all intermediate times, We need only connect the positions marked in Fig. 7 by a continuous curve. In our case this is obviously a straight line. For the point advances equal distances in equal times the co-ordinates x, t thus change in the same ratio (or proportionally), and it is evident that the graph of this law is a straight line. Such a motion is called uniform. The name velocity v of the motion designates the ratio of the path traversed to the time required in doing so, or in symbols : v= ) 18 THE THEORY OF RELATIVITTYV In our example the point traverses \ cm. of path in each second. The velocity remains the same throughout and amounts to J cm. per sec. The unit of velocity is already fixed by this definition ; it is the velocity which the point would have if it traversed i cm. per sec. It is said to be a derived unit, and, without introducing a new value, we call it cm. per sec. or cm. /sec. To express that the measurement of velocities may be referred back to measurements of lengths and times in accordance with formula (i) we also say that velocity has the dimensions length = divided by time, written thus : [v] [_yj or [L.T"" 1 ]. In the same way we assign definite dimensions to every quantity that allows itself to be built up of the fundamental quantities, >x length /, time t, and weight G. When the latter are known the unit of the quantity may at once be expressed by means of those of length, time, and weight, say, cm., sec. and grm. In the case of great velocities the path % traversed in the time t is great, thus the graph line has only a small inclination to the x- axis : the smaller the velocity, the steeper the graph. A point that is at rest has zero velocity and is represented in our diagram by a straight line parallel to the i!-axis, for the points of this straight line have the same value of % for all times t (Fig. 9 a) . If a point is firstly at rest and then at a certain moment suddenly acquires a velocity and moves on with this velocity, we get as the graph a straight line one part of which is bent, the other being vertical (Fig. 9 b). Similarly broken lines LAWS OF CLASSICAL MECHANICS 19 represent the cases when a point that is initially moving uni- formly for a while to the right or to the left suddenly changes its velocity (Figs. 9 c and 9 d). If the velocity before the sudden change is v x (say, 3 cms. — — = per sec), and afterwards v t (say, 5 cms. per sec), then the increase of velocity is v 2 v x (that is, 5 3 2 cms. per sec, =i added in each sec). If v 2 is less than v x (say, v 1 cm. per — — = — sec), then v z v t is negative (namely, 1 3 2 cms. per sec), and this clearly denotes that the moving point is suddenly retarded. If a point experiences a series of sudden changes of velocity then the graph of its motion is a succession of straight lines joined together (polygon) as in Fig. 10. If the changes of velocity occur more and more frequently t kt FlQ. 10. ->x fc- Fig. 11. and are sufficiently small, the polygon will no longer be distinguishable from a curved line. It then represents a motion whose velocity is continually changing, that is, one which is non-uniform, accelerated or retarded (Fig. 11). An exact measure of the velocity and its change, accelera- tion, can be obtained in this case only with the aid of the methods of infinitesimal geometry. It suffices for us to imagine the continuous curve replaced by a polygon whose straight sides represent uniform motions with definite velocities. The bends of the polygon, that is, the sudden changes of velocity, may be supposed to succeed each other at equal intervals of time, = say, t 1 - sees. If, in addition these changes are equally great, the motion is said to be " uniformly accelerated." Let each such change 20 THE THEORY OF RELATIVITY of velocity have the value w, then if there are n per sec. the total change of velocity per sec. is (2) LAWS OF CLASSICAL MECHANICS 21 Then the velocity after the first interval of time is : ,, second ,, ,, „ third ,, ,, = v x w, = = v 2 w v x -f- 2w = + = v3 v2 w 31V, and so forth. The point advances = after the first interval of time to : x, t v- n second ,, ,, = = + x 2 x x -f v 2 n (v x n v 2 ) -, third „ „ x9 =x t +vr-=(v n 1 +v t +v9)-n» and so forth. After the nth. interval of time, that is, at the end of the time /, the point will have arrived at x= + + (v ± v2 . . . vn )~. n But V V -\1 -\- 2 . . . = + + w + Vn IW 2Z£> Z • • • Htf' = + + + (1 2 3 . . . n)w. The sum of the numbers from 1 to n can be calculated quite simply by adding the first and the last ; the second and the second to last ; and so forth ; in each case we get for the sum + of the two numbers n 1, and altogether we have — of such + + = sums or pairs. Thus we get 1 2 n . . . - (n -\- 1). If, further, we replace w by b -, we get 11 + + = + - = - + v x v2 vn . . . nt . x bt - (» 1) 2 w bt (n , , x 1), 2 thus — = + = + — ^ (n 1) - (1 -). 2 «2 n Here we may choose n to be as great as we please. Then becomes arbitrarily small and we get x=- bt 2. 2 This signifies that in equal times the paths traversed are proportional to the squares of the times. If, for example, 22 THE THEORY OF RELATIVITY = the acceleration b 10 metres per sec, then the point traverses = = 5 metres in the first sec, 5 . 2 2 5 . 4 20 metres in the second = sec, 5 . 2 3 45 in the third sec, and so forth. This relationship is represented by a curved line, called a parabola, in the xt plane (Fig. 13). If we compare the figure with Fig. 12 we see how the polygon approximately represents the continuously = curved parabola. In both figures the acceleration b 10 has been chosen, and this determines the appearance of the curves, whereas the units of length and time are unessential. We may also apply the conception of acceleration to non- uniformly accelerated motions, by using instead of 1 sec. a time of observation which is so small that, during it, the motion may be regarded as uniformly accelerated. The acceleration itself then becomes continuously variable. All these definitions become rigorous and at the same time convenient to handle if the process of sub-division into small l 6 5 V 3 Z 1 LAWS OF CLASSICAL MECHANICS 23 3. Motion in a Plane If we wish to study the motion of a point in a plane, our method of representation at once allows itself to be extended to this case. We take in the plane an ^-co-ordinate system and erect a rf-axis perpendicular to it (Fig. 14). Then a straight line in the #y/-space corresponds to a rectilinear and uniform motion Fig. 14. Fig. 15. in the #y-plane. For if we project the points of the straight = line that correspond to the points of time t o, I, 2, 3, . . . on to the %y-plane, we see that the positional displacement takes place along a straight line and at equal intervals. Every non-rectilinear but uniform motion is said to be accelerated even if, for example, a curved path is traversed with constant velocity. For in this case the direction of the velocity changes although its numerical value remains constant. An ac- celerated motion is represented in the #v/-plane (Fig. 15) by an arbitrary curve. The projection of this curve into the ^y-plane is the orbit in the plane (or plane- orbit). The velocity and the ac- -*~JT celeration are again calculated by Fig. 16. supposing the curve replaced by a polygon closely wrapped round the curve. At each corner of this polygon not only the amount but also the direction of A the velocity alters. more exact analysis of the conception of acceleration would take us too far. It is sufficient to mention that it is best to project the graph of the moving point on to the co-ordinate axes x, y, and to follow out the rectilinear motion of these two points, or what is the same, the change 24 THE THEORY OF RELATIVITY in time of the co-ordinates x, y. The conceptions denned for rectilinear motions as given above may now be applied to these We projected motions. thus get two components of velocity vx , vy , and two components of acceleration bx , by , that together fix the velocity or the acceleration of the moving point at a given instant. In the case of a plane motion (and also in one that occurs in space) velocity and acceleration are thus directed magnitudes (vectors). They have a definite direction and a definite magni- tude. The latter can be calculated from the components. For example, we get the direction and magnitude of the velocity from the diagonal of the rectangle with the sides vx and vy (Fig. 16). Thus, by Pythagoras' theorem, its magnitude is v = VV + V • • • (3) An exactly corresponding result holds for the acceleration. 4. Circular Motion There is only one case which we wish to consider in greater detail, namely, the motion of a point in a circular orbit with Fig. 17. constant speed (Fig. 17). According to what was said above, it is an accelerated motion, since the direction of the velocity constantly alters. If the motion were unaccelerated the moving point would move forward from A in a straight line with the uniform velocity v. But in reality the point is to remain on the circle, and hence it must have a supplementary velocity or acceleration that is directed to the central point M. This is called the centripetal acceleration. It causes the velocity at a neighbouring point B, which is reached after a short interval t, to have a direction different from that at the point A. From a point c we next draw the velocities at A and B in a . LAWS OF CLASSICAL MECHANICS 25 separate diagram (Fig. 17), paying due regard to their magnitude and direction. Their magnitude will be the same, namely v, since the circle is to be traversed with constant speed, but their direction is different. If we con- nect the end-points D and E of the two velocity lines, then the connecting line is clearly the supplementary velocity w, which transforms the first velocity state We into the second. thus get an isosceles triangle CED, having the base w and the sides v, and we at once see that the angle a at the vertex is equal to the angle sub- tended by the arc AB, which the Fig. 18. point traverses, at the centre of A the circle. For the velocities at and B are perpendicular MA to the radii and MB, and hence include the same angle. MAB Consequently the two isosceles triangles and CDE are similar, and we get the proportion DE AB CD MA = = MA Now DE w, CD v, and further, is equal to the AB radius r of the circle, and is equal to the arc s except for a small error that can be made as small as we please by choosing the time-interval t sufficiently small. Hence we have w— = s - or w = s—v v r r = — = We now divide by t and notice that S - IS) v, 0. t t the acceleration b= Hence • (4) that is, the centripetal acceleration is equal to the square of the velocity in the circle divided by the radius. This theorem, as we shall see, is the basis of one of the first and most important empirical proofs of Newton's theory of gravitation. Perhaps it is not superfluous to have a clear idea of what this uniform circular motion looks like in the graphical representation in the *y£-space. This is obviously produced by allowing the moving point to move upwards regularly 26 THE THEORY OF RELATIVITY We parallel to the *-axis during the circular motion. thus get a helix (screw line), which now represents the orbit and the course of the motion in time completely. In Fig. 1 8 it is drawn on the surface of a cylinder that has its base on the #y-plane. 5. Motion in Space Our graphical method of representation fails for motions in space, for in this case we have three space co-ordinates x, y, z, and time has to be added as a fourth co-ordinate. But unfortunately our visual powers are confined to three-dimensional space. The symbolic language of mathematics must now lend us a helping hand. For the methods of analytical geometry allow us to treat the properties and relationships of spatial configurations as pure matters of calculation without requiring us to use our visual power or to sketch figures. Indeed, this process is much more powerful than geometric construction. Above all, it is not bound to the dimensional number three but is immediately applicable to spaces of four or more dimensions. In the language of mathematics the conception of a space of more than three dimensions is not at all mystical but is simply an abbreviated expression of the fact that we are dealing with things that allow themselves to be fully determined by more than three number data. Thus the position of a point at a given moment of time can be fixed only by specifying four number data, the three space-co-ordinates x, y, z and the time t. After we have learned to deal with the xyt-spa.ce as a means of depicting plane motion it will not be difficult also to regard the motions in three- dimensional space in the light of curves in the xyzt-spa.ce. This view of kinematics as geometry in a four-dimensional xyzt-spa,ce has the advantage of allowing us to apply the well-known laws of geometry to the study of motions. But it has a still deeper significance that will become clearly apparent in Einstein's theory. It will be shown that the conceptions space and time, which are contents of experience of quite different kinds, cannot be sharply differentiated at all as objects of physical measurement. If physics is to retain its maxim of recognizing as real only what is physically observ- able it must combine the conceptions space and time to a higher unity, namely, the four-dimensional xyzt-spa.ee. Minkowski called this the " world " (1908), by which he wished to express that the element of all order of real things is not place nor point of time but the " event " or the " world-point/' that is, a place at a definite time. He called the graphical picture of a moving point " world-line," an expression that we shall LAWS OF CLASSICAL MECHANICS 27 continue to use in the sequel. Rectilinear uniform motion thus corresponds to a straight world-line, accelerated motion to one that is curved. — 6. Dynamics The Law of Inertia After these preliminaries we revert to the question with which we started, namely, as to how forces generate motions. The simplest case is that in which no forces are present at A all. body at rest will then certainly not be set into motion. The ancients had already made this discovery, but, above this, they also believed the converse to be true, namely, that wherever there is motion there must be forces that maintain them. This view at once leads to difficulties if we reflect on why a stone or a spear that has been thrown continues to move when it has been released from the hand. It is clearly the latter that has set it into motion, but its influence is at an end so soon as the motion has actually begun. Ancient thinkers were much troubled in trying to discover what forces actually maintain the motion of the thrown stone. Galilei was the first to find the right point of view. He observed that it is a prejudiced idea to as- sume that wherever there is motion there must always be force. Rather it must be asked what quantitative property of motion has a regular relationship with force, whether it be the place of the moving body, its velocity, its acceleration, or some composite quantity dependent on all of these. No amount of reflection will allow us to evolve an answer to these questions We by philosophy. must address ourselves directly to nature. The question which she gives is, firstly, that force has an influence in effecting changes of velocity. No force is necessary to maintain a motion in which the magnitude and the direction of the velocity remain unaltered. And conversely, where there are no forces, the magnitude and direction of the velocity remain unaltered ; thus a body which is at rest remains at rest, and one that is moving uniformly and rectilinearly continues to move uniformly and rectilinearly. This law of inertia (or of persistence) is by no means so obvious as its simple expression might lead us to surmise. For in our experience we do not know of bodies that are really withdrawn from all influences from without, and if we use our imaginations to picture how they travel on in their solitary rectilinear paths with constant velocity throughout astronomic space, we are at once confronted with the problem of the absolutely straight path in space absolutely at rest, with which we shall have to deal in detail later on. For the present, then, we shall interpret the law of inertia in the restricted sense in which Galilei meant it. 28 THE THEORY OF RELATIVITY Let us picture to ourselves a smooth exactly horizontal table on which a smooth sphere is resting. This is kept pressed against the table by its own weight, but we ascertain that it requires no appreciable force to move the sphere quite slowly on the table. Evidently there is no force acting in a horizontal direction on the sphere, otherwise it would not itself remain at rest at any point on the table. But if we now give the sphere a velocity it will continue to move in a straight line and will lose only very little of its speed. This retardation was called a secondary effect by Galilei, and it is to be ascribed to the friction of the table and the air, even if the frictional forces cannot be proved to be present by the statical methods with which we started. It is just this depth of vision, which correctly differentiates what is essential in an occurrence from disturbing subsidiary effects, that characterizes the great investigator. The law of inertia is at any rate confirmed for motion on the table. It has been established that in the absence of forces the velocity remains constant in direction and magnitude. Consequently the forces will be associated with the change of velocity, the acceleration. In what way they are associated can again be decided only by experiment. 7. Impulses We have presented the acceleration of a non-uniform motion as a limiting case of sudden changes of velocity of brief uniform motions. Hence we shall first have to enquire how a single sudden change of velocity is produced by the application of a force. For this a force must act for only a short time ; it is then what we call an impulse or a blow. The result of such a blow depends not only on the magnitude of the force but also We on the duration of the action, even if this is very short. therefore define the intensity of a blow or impulse as follows : K n impulses J, each of which consists of the force acting = during the time t — sees., will, if they follow each other withn out appreciable pauses, have exactly the same effect as if the K force were to continue to act throughout the whole second. Thus we should have J = J ,J = K, t or, J = ?K n = *' above, that expresses a displacement along the Fig. 38- #-axis. It is clear that the difference of the ^-co- — ordinates of two points P and Q, namely, x 2 x lf does not change. As a matter of fact (Fig. 39), x\ == (x. = — a) x2 % If the two co-ordinate systems S and S' are inclined to each other, then the distance s of any point P from the origin is an invariant (Fig. 40). It has the same expression in both systems, for, by Pythagoras' theorem, we have = + = + / s2 x2 y2 x' 2 2 . . . (28) In the more general case, in which the co-ordinate system is simultaneously displaced and turned, the distance P, Q of two points becomes an invariant. The invariants are par- ticularly important because they represent the geometrical relations in themselves without reference to the accidental choice of the co-ordinate system. They will play a considerable part in the sequel. If we now return after this geometrical digression to our THE NEWTONIAN WORLD-SYSTEM 65 starting-point, we have to answer the question as to what are the transformation laws that allow us to pass from one inertial system to another. We defined the inertial system as a co-ordinate system in which the law of inertia holds. Only the state of motion is important in this connexion, namely, the absence of accelerations with respect to the absolute space, whereas the nature and position of the co-ordinate system is unessential. If we choose it to be rectangular, as happens most often, its position still We remains free. may take a displaced or a rotated system, only it must have the same state of motion. In the foregoing we have always spoken of system of reference wherever we were concerned with the state of motion and not with the +y fy' Q 6 *1 L VA/ *2 Fig. 39. Fig. 40. *-J? nature and position of the co-ordinate system, and we shall use the expression systematically from now onwards. If an inertial system S' is moving rectilinearly with respect to S with the velocity v, we may choose rectangular co-ordinates in both systems of reference such that the direction of motion becomes the x- and the #'-axis, respectively. Further, we may = assume that at the time t the origin of both systems co- incides. Then, in the time t the origin of the S'-system will = have been displaced by the amount a vt in the ^-direction : thus at this moment the two systems are exactly in the position that was treated above purely geometrically. Hence the equations (27) hold, in which a is now to be set equal to vt. Consequently we get the transformation equations x = x — vt y =y = z' z . . (29) in which we have added the unchanged co-ordinate. This 66 THE THEORY OF RELATIVITY law is called a Galilei transformation in honour of the founder of mechanics. We may also enunciate the principle of relativity as follows : The laws of mechanics are invariant with respect to Galilei transformations. This is due to the fact that accelerations are invariant, as we have already seen above by considering the change of velocity of a moving body with respect to two inertial systems. We showed earlier that the theory of motions or kinematics may be regarded as a geometry in four-dimensional xyzt-sp&ce, the " world " of Minkowski. In this connexion it is not without interest to consider what the inertial systems and the Galilei transformations signify in this four-dimensional geometry. This is by no means difficult, for the y- and the z-co-ordinate do not enter into the trans- formation at all. It is thus sufficient to operate in the ^-plane. We represent our iner- tial system S by a rectangu- lar ^-co-ordinate system A (Fig. 41). second inertial system S' then corresponds to another co-ordinate sys- tem x't', and the question is : what does the second look like and how is it situated relatively to the Fig. 41. first ? First of all, the time-measure of the second system S' is exactly the same as that of the first, namely, the — = one absolute time t t' ; thus the #-axis, on which t lies, = coincides with the #'-axis, t' 0. Consequently the system S' can only be an oblique co-ordinate system. The /'-axis is the = world-line of the point %' 0, that is, of the origin of the system S'. The ^-co-ordinate of this point which moves with the velocity v relatively to the system S is equal to vt in this system at the time t. For any world-point P whatsoever the figure then at once gives the formula of the Galilei transforma- = — tion x' x vt. Corresponding to any other inertial system there is another oblique ^-co-ordinate system with the same #-axis, but a differently inclined /-axis. The rectangular system from which we started has no favoured position among all these oblique systems. The unit of time is cut off from all the /-axes of the various co-ordinate systems by the same parallel to the #-axis. THE NEWTONIAN WORLD-SYSTEM 67 This is in a certain sense the " calibration curve " of the xt- plane with respect to the time. We compress the result into the sentence : In the xt-plane the choice of the direction of the t-axis is quite arbitrary ; in every xt-co-ordinate system having the same x-axis the fundamental laws of mechanics hold. From the geometric point of view this manifold of equivalent co-ordinate systems is extremely singular and unusual. The fixed position or the invariance of the %-axis is particularly remarkable. When we operate in geometry with oblique co-ordinates there is usually no reason for keeping the position of one axis fixed. But this is required by Newton's fundamental law of absolute time. All events which occur simultaneously, that is for the same value of t, are represented by a parallel to the #-axis. Since, according to Newton, time flows " absolutely and without reference to any object whatsoever," simultaneous events must correspond to the same world-point in all allowable co-ordinate systems. We shall see that this unsymmetrical behaviour of the world-co-ordinates x and t, here only mentioned as an error of style, is actually non-existent. Einstein has eliminated it through his relativization of the conception of time. 8. Inertial Forces After having recognized that the individual points in Newton's absolute space have at any rate no physical reality, we enquire what remains of this conception at all. Well, it asserts itself quite clearly and emphatically, for the resistance of all bodies to accelerations must be interpreted in Newton's sense as the action of absolute space. The locomotive that sets the train in motion must overcome the inertial resistance. The shell that demolishes a wall draws its destructive power from inertia. Inertial actions arise wherever accelerations occur, and these are nothing more than changes of velocity in absolute space ; we may use the latter expression, for a change of velocity has the same value in all inertial systems. Systems of reference that are themselves accelerated with respect to inertial systems are thus not equivalent to the latter, We or equivalent among themselves. can, of course, also refer the laws of mechanics to them, but they then assume a new and more complicated form. Even the path of a body left to itself is no longer uniform and rectilinear in an accelerated system (see III, i, p. 48). This may also be expressed by saying that in an accelerated system apparent forces, inertial forces, act besides the true forces. A body on which no true 68 THE THEORY OF RELATIVITY forces act is yet subject to these inertial forces, and its motion is therefore in general neither uniform nor rectilinear. For example, a vehicle when being set into motion or stopped is such an accelerated system. Railway journeys have made everyone familiar with the jerk due to the train starting or stopping, and this is nothing other than the inertial force of which we have spoken. We shall consider the phenomena individually for a system S moving rectilinearly, whose acceleration is to be equal to k. If we now measure the acceleration b of a body with respect to this moving system S, then the acceleration with respect to absolute space is obviously greater to the extent k. Hence the fundamental dynamical law with respect to space is = + m(b k) K. If we write this in the form mb = K — mk, we may say that in the accelerated system S a law of motion of Newtonian form, namely, mb = K' again holds, except that now we must write for the force K' the sum K' = K - mk K — where is the true, and mk the apparent or inertial force. K = Now, if there is no true force acting, that is, if o, .... then the total force becomes equal to the force of inertia K' = — mk (30) We Thus this force acts on a body left to itself. may recog- We nize its action from the following considerations. know that the gravitation on the earth, the force of gravity, is determined = by the formula G mg, where g is the constant acceleration = — due to gravity. The force of inertia K' mk thus acts exactly like weight or gravity ; the minus sign denotes that the force of acceleration is in a direction opposite to the system of reference S used as a basis. The value of the apparent gravitational acceleration k is equal to the acceleration of the system of reference S. Thus the motion of a body left to itself in the system S is simply a motion such as that due to falling or being thrown. This relationship between the inertial forces in accelerated systems and the force of gravity still appears quite fortuitous here. It actually remained unobserved for two hundred years. THE NEWTONIAN WORLD-SYSTEM 69 But even at this stage we must state that it forms the basis of Einstein's general theory of relativity. 9. Centrifugal Forces and Absolute Space In Newton's view the occurrence of inertial forces in acceler- ated systems proves the existence of absolute space or, rather, the favoured position of inertial systems. Inertial forces present themselves particularly clearly in rotating S3'stems of reference in the form of centrifugal forces. It was from them that Newton drew his main support for his doctrine of absolute space. Let us give the substance of his own words : " The effective causes which distinguish absolute and relative motion from each other are centrifugal forces, the forces tending to send bodies away from the axis of rotation. In the case of a motion that is only relatively circular these forces do not exist, []77 , __j^ but they are smaller or greater in proportion to the amount of the (absolute) motion." " Let us, for example, hang a vessel by a very long thread and turn it about its axis until the thread becomes very stiff through the torsion (Fig. 42). Then let us fill it with water and wait till both vessel and contents are com- pletely at rest. If it is now made to rotate in the opposite direction by a force applied suddenly, and if this lasts for some time whilst the thread unwinds itself, the surface of the water will first be plane, just as before the vessel began to move, and then when the force gradually begins to act on the water, the vessel will make the water participate appreciably in the motion. It (the water) gradually moves away from the middle and mounts up the walls of the vessel, assuming a hollow shape (I have carried out this experiment personally)." " At the beginning when the relative motion of the water in the vessel ( with respect to the walls) was greatest, it displayed no tendency to move away from the axis. The water did not seek to approach the periphery by climbing up the walls, but remained plane, and thus the true circular motion had not yet begun. Later, however, as the relative motion of the water decreased, its ascent up the walls expressed the tendency to 70 THE THEORY OF RELATIVITY move away from the axis, and this tendency showed the con- tinually increasing true circular motion of the water, until this finally reached a maximum, when the water itself was resting relatively to the vessel." " Moreover, it is very difficult to recognize the true motions of individual bodies and to distinguish them from the apparent motions, because the parts of that immovable space in which the bodies are truly moving cannot be perceived bjf the senses." " Yet the position is not quite hopeless. For the necessary auxiliary means are given partly by the apparent motions, which are the differences of the real ones, and partly by the forces on which the true motions are founded as working causes. If, for example, two spheres are connected at a given distance apart by means of a thread and thus turned about the usual centre of gravity (Fig. 43), we recognize in the tension of the thread the tendency of the spheres to move away from the axis of the motion, and from this we can get the magnitude of the circular motion ... In this way we could find both the magnitude and the direction of this circular motion in every infinitely great space, even if there were nothing external and perceptible in it, with which the spheres could be com- pared." These words express most clearly the meaning of absolute space. We have only a few words of explanation to add to them. Concerning, firstly, the quantitative conditions in the case of the centrifugal forces we can at once get a survey of these if we call to mind the magnitude and the direction of the accelera- tion in the case of circular motions. It was directed towards the centre and, according to formula (4), p. 25, it had the = — value b v2 , where r denotes the circular radius, and v the r velocity. Now, if we have a rotating system of reference S that ro- tates once in T sees., then the velocity of a point at the distance r from the axis (see formula (18), p. 52) is 277?' hence the acceleration relative to the axis, which we denoted by k (see p. 68) is k =_ A^r Now, if a body has the acceleration b relatively to S, its + absolute acceleration is b k. Just as above in the case of THE NEWTONIAN WORLD-SYSTEM 71 rectilinear accelerated motion there then results an apparent force of the absolute value '" ' a A 77 1 (3i: which is directed away from the axis. It is the centrifugal force. It is well known that the centrifugal force also plays a part in proving that the earth rotates (Fig. 44). It drives the masses away from the axis of rotation and through this causes, firstly, the flattening of the earth at the poles, and, secondly, We the decrease of gravity from the pole towards the equator. became acquainted with the latter phenomenon above, when we were dealing with the choice of the unit of force (II, 15, p. 45) without going into its cause. According to Newton it is a proof of the earth's rotation. The centrifugal force, acting outwards, acts against gravity and reduces the weight. The oo Fig. 43. decrease of the acceleration g due to gravity has the value 4*L? at the equator, where a is the earth's radius. If we here insert for a the value given above (III, 3, (23), p. 55), = = a 6*37 . 10 cms., and for the time of rotation T 1 day = = 24 . 60 . 60 sees. 86,400 sees., we get for the difference of the gravitational acceleration at the pole and at the equator the value 3*37 cm./sec. 2, which is relatively small compared with 981 ; this value has to be increased slightly, owing to the flattening of the earth. According to Newton's doctrine of absolute space these phenomena are positively to be regarded not as due to motion relative to other masses, such as the fixed stars, but as due to absolute rotation in empty space. If the earth were at rest, and if, instead, the whole stellar system were to rotate in the opposite sense once around the earth's axis in 24 hours, 72 THE THEORY OF RELATIVITY then, according to Newton, the centrifugal forces would not occur. The earth would not be flattened and the gravitational force would be just as great at the equator as at the pole. The motion of the heavens, as viewed from the earth, would be exactly the same in both cases. And yet there is to be a definite difference between them ascertainable physically. The position is brought out perhaps still more clearly in Foucault's pendulum experiment (1850). According to the laws of Newtonian dynamics a pendulum swinging in a plane must permanently maintain its plane of vibration in absolute space if all deflecting forces are excluded. If the pendulum is suspended at the North Pole, the earth rotates, as it were, below it (Fig. 45). Thus the observer on the earth sees a rotation of the plane of oscillation in the reverse sense. If the earth were at rest but the stellar system in rotation, then, ^x Fig. 45. Fig. 46. according to Newton, the position of the plane of oscillation should not alter with respect to the earth. The fact that it does so again appears to prove the absolute rotation of the earth. — We shall consider a further example the motion of the moon about the earth (Fig. 46). According to Newton the moon would fall on to the earth if it had not an absolute rotation about the latter. Let us imagine a co-ordinate system, with its origin at the centre of the earth, and the #y-plane as that of the moon's orbit, the #-axis always passing through the moon. If this system were to be absolutely at rest, then the moon would be acted on only by the gravitational force towards the centre of the earth, which, by formula (26) on p. 56, has the value K = *M?. THE NEWTONIAN WORLD-SYSTEM 78 Thus it would fall to the earth along the *-axis. The fact that it does not do so apparently proves the absolute rotation of the co-ordinate system xy. For this rotation produces a centrifugal force that keeps equilibrium with the force K, and we get y r2 This formula is, of course, nothing other than Kepler's m third law. For if we cancel the mass of the moon on both = sides and express v by the period of revolution T, v -=-, we get or, by (25) on p. 56, 47rV = JM : : T2 r2 fr3 2 " &M £T* _" rU An exactly corresponding result holds, of course, for the rotation of the planets about the sun. These and many other examples show that Newton's doctrine of absolute space rests on very concrete facts. If we run through the sequence of arguments again, we see the follow- ing : The example of the rotating glass of water shows that the relative rotation of the water with respect to the glass is not responsible for the occurrence of centrifugal forces. It might be that greater masses in the neighbourhood, say the whole earth, are the cause. The flattening of the earth, the decrease of gravity at the equator, Foucault's pendulum experiment show that the cause is to be sought outside the earth. But the orbits of all moons and planets likewise exist only through the centrifugal force that maintains equilibrium with gravitation. Finally, we notice the same phenomena in the case of the farthermost double stars, the light from which takes thousands of years to reach us. Thus it seems as if the occurrence of centrifugal forces is universal and cannot be due to inter-actions. Hence nothing remains for us but to assume absolute space as their cause. Such modes of conclusion have been generally current and regarded as valid since the time of Newton. Only few thinkers have opposed them. We must name among these few above all Ernst Mach. In his critical account of mechanics he has analysed the Newtonian conceptions and tested their logical bases. He starts out from the view that mechanical experience can never teach us anything about absolute space. Relative 74 THE THEORY OF RELATIVITY positions and relative motions alone may be ascertained and are hence alone physically real. Hence Newton's proofs of the existence of absolute space must be illusory. As a matter of fact, everything depends on whether it is admitted that if the whole stellar system were to rotate about the earth no flattening, no decrease of gravity at the equator, and so forth, would occur. Mach asserts rightly that such statements go far beyond possible experience. He reproaches Newton very energetically with having become untrue to his principle of allowing only facts to be considered valid. Mach himself has sought to free mechanics from this grievous blemish. He was of the opinion that the inertial forces would have to be regarded as actions of the whole mass of the universe, and sketched the outlines of an altered system of dynamics in which only relative quantities occurred. Yet his attempt could not succeed. In the first place the importance of the relation between inertia and gravitation that expresses itself in the proportion- ality of weight to mass escaped him. In the second place he was unacquainted with the relativity theory of optical and electro-magnetic phenomena which eliminated the prejudice A in favour of absolute time. knowledge of both these facts was necessary to build up the new mechanics, and the dis- covery of both was the achievement of Einstein. CHAPTER IV THE FUNDAMENTAL LAWS OF OPTICS i. The Ether MECHANICS is both historically and logically the foun- dation of physics, but it is nevertheless only a part of it, and, indeed, a small part. Hitherto to solve the problem of space and time we have made use only of mechanical observations and theories. We must now enquire what the other branches of physical research teach us about it. It is, above all, the realms of optics, of electricity, and of magnetism that are connected with the problem of space ; this is due to the circumstance that light and the electric and magnetic forces traverse empty space. Vessels out of which the air has been pumped are completely transparent for light no matter how high the vacuum. Electric and magnetic forces, too, act across such a vacuum. The light of the sun and the stars reaches us after its passage through empty space. The relationships between the sun-spots and the polar light on the earth and magnetic storms show inde- pendently of all theory that electromagnetic actions take place through astronomic space. The fact that certain physical events propagate themselves through astronomic space led long ago to the hypothesis that space is not empty but is filled with an extremely fine imponderable substance, the ether, which is the carrier or medium of these phenomena. So far as this conception of the ether is still used nowadays it is taken to mean nothing more than empty space associated with certain physical states or " fields." If we were to adopt this abstract conception from the very out- set, the majority of the problems that are historically connected with the ether would remain unintelligible. The earlier ether was indeed regarded as a real substance, not only endowed with physical states, but also capable of executing motions. We shall now describe the development, firstly, of the prin- ciples of optics, and, secondly, of those of electrodynamics. This will for the present make us digress a little from the problem 75 76 THE THEORY OF RELATIVITY of space and time, but will then help us to take it up again fortified with new facts and laws. 2. The Corpuscular and the Undulatory Theory * I say then that pictures of things and thin shapes are emitted from things off their surfaces . . . Therefore in like manner idols must be able to scour in a moment of time through space unspeakable . . . But because we can see with the eyes alone, the consequence is that, to whatever point we turn our sight, there all the several things meet and strike it with their shape and colour . . . That is what we read in the poem of Titus Lucretius Carus on the Nature of Things (Book 4), that poetic guide to Epicurean philosophy, which was written in the last century before the birth of Christ. The lines quoted contain a sort of corpuscular theory of light which is elaborated by the imaginative power of the poet but at the same time developed in a true scientific spirit. Yet we can no more call this doctrine a scientific doctrine than we can other ancient speculations about light. There is no sign of an attempt to determine the phenomena quantitatively, the first characteristic of objective effort. Moreover it is particularly difficult to dissociate the subjective sensation of light from the physical phenomenon and to render it measurable. The science of optics maybe dated from the time of Descartes. His Dioptrics (1638) contains the fundamental laws of the propagation of light, the laws of reflection and refraction. The former was already known to the ancients, and the latter had been found experimentally shortly before by Snell (about 1618). Descartes evolved the idea of the ether as the carrier of light, and this was the precursor of the undulatory theory. It was already hinted at by Robert Hooke (1667), and was clearly formulated by Christian Huygens (1678). Their great contemporary, Newton, who was somewhat younger, is regarded as the author of the opposing doctrine, the corpuscular theory. Before entering on the struggle between these theories we shall explain the nature of each in rough outline. The corpuscular theory asserts that luminescent bodies send out fine particles that move in accordance with the laws of mechanics and that produce the sensation of light when they strike the eye. The undulatory theory sets up an analogy between the propagation of light and the motion of waves on the surface of water or sound-waves in air. For this purpose it has to assume the existence of a medium that permeates all trans- * From Munro's prose translation, published by Deighton, Bell & Co. FUNDAMENTAL LAWS OF OPTICS 77 parent bodies and that can execute vibrations ; this is the luminiferous ether. In this process of vibration the individual particles of this substance move only with a pendulum-like motion about their positions of equilibrium. That which moves on as the light-wave is the state of motion of the particles and not the particles themselves. Fig. 47 illustrates the process for a series of points that can vibrate up and down. Each of the diagrams drawn vertically below one another corresponds to = a moment of time, say, t 3 o, 1, 2, . . . Each individual point executes a vibration vertically. The points all taken together present the aspect of a wave that advances towards the right from moment to moment. Now there is a significant objection to the undulatory theory. p^ UO pX VP PX ft >\ t--1 ^P P >K >X t--2 ^P t--3 t=b «i P *V VP Fig. 47. It is known that waves run around obstacles. It is easy to see this on every surface of water, and sound waves also "go around corners." On the other hand, a ray of light travels in a straight line. If we interpose a sharp-edged opaque body in its path we get a shadow with a definite outline. This fact moved Newton to discard the undulatory theory. He did not himself decide in favour of a definite hypothesis but merely established that light is something that moves away from the luminescent body " like ejected particles." But his successors interpreted his opinion as being in favour of the emission theory, and the authority of his name gained the acceptance of this theory for a whole century. Yet, at that time Grimaldi had already discovered (the result was published posthumously in 1665) that light can also " bend round corners." 78 THE THEORY OF RELATIVITY At the edges of sharp shadows a weak illumination in successive striae are seen ; this phenomenon is called the diffraction of light. It was this discovery in particular that made Huygens a zealous pioneer of the undulatory theory. He regarded as the first and most important argument in favour of it the fact that two rays of light cross each other without interfering with each other, just like two trains of water-waves, whereas bundles of emitted particles would necessarily collide or at least disturb each other. Huygens succeeded in explaining the reflection and the refraction of light on the basis of the undulatory theory. He made use of the principle, now called after his name, according to which every point on which the light impinges is to be regarded as the source of a new spherical wave of light. This resulted in a fundamental difference between the emission and the undulatory theory, a difference that later led to the final experi- mental decision in favour of the latter. It is known that a ray of light which passes through the air and strikes the plane bounding surface of a denser body such as glass or water is bent or refracted so that it is more steeply inclined to the bounding surface (Fig. 48) The emission theory accounts for this by assuming that the corpuscles of light experience an Fig. 48. attraction from the denser medium at the moment they enter into it. Thus they are accelerated by an impulse perpendicular to the bounding surface and hence deflected towards the normal. It follows from this that they must move more rapidly in the denser than in the less dense medium. Huygen's construction on the wave theory depends on just the opposite assumption (Fig. 49). When the light wave strikes the bounding surface it excites elementary waves at every point. If these become transmitted more slowly in the second, denser, medium, then the plane that touches all these spherical waves and that represents the refracted wave according to Huygens, is deflected in the right sense. Huygens also interpreted the double refraction of Iceland spar, discovered by Erasmus Bartholinus in 1669, on the basis of the wave-theory, by assuming that light can propagate itself in the crystal with two different velocities in such a way that the one elementary wave is a sphere, the other a spheroid. FUNDAMENTAL LAWS OF OPTICS 70 He discovered the remarkable phenomenon that the two rays of light that emerge out of such a piece of fluor spar behave quite differently from other light towards a second piece of fluor spar. If the second crystal is turned about a ray that comes out of the first, then two rays arise out of it which are of varying intensity according to the position of the crystal, and it is possible to make one or other of these rays vanish Fig. 49. entirely (Fig. 50). Newton remarked (1717) that it is to be concluded from this that a ray of light corresponds in symmetry not to a prism with a circular but rather to one with a square cross-section. He interpreted this as evidence against the undulatory theory, for at that time, analogously with soundwaves, only waves of compression and rarefaction were thought of, in which the particles swing " longitudinally " in the direction Fig. 50. of propagation of the wave (Fig. 51), and it is clear that these must have rotatory symmetry about the direction of pro- pagation. 3. The Velocity of Light The first determinations of the most important property of light, that which will form the nucleus of our following 80 THE THEORY OF RELATIVITY reflections, namely, the velocity of light, were made independently of the controversy between the two hypotheses about the nature of light. The fact that it was enormously great was clear from all observations about the propagation of light. Galilei had endeavoured (1607) to measure it with the aid of lantern signals but without success, for light traverses earthly distances in extremely short fractions of time. Hence the measurement succeeded only when the enormous distances between the heavenly bodies in astronomic space were used. Olaf Romer observed (1676) that the regular eclipses of Jupiter's satellites occur earlier or later according as the earth • • > \ ''•'•»• p • • .«' • * • 4 i > • '• '» *. ' 6 7 8 9 K 10 71 12 13 15 Fig. 51. is nearer to or farther away from Jupiter (Fig. 52). He inter- preted this phenomenon as being caused by the difference of time used by the light to traverse the paths of different lengths, We and he calculated the velocity of light on this basis. shall in future call this velocity c. Its exact value, to which Romer approximated very closely, is = = c 300,000 km./sec. 3 . io 10 cms. per sec. . (32) James Bradley discovered (1727) another effect of the FUNDAMENTAL LAWS OF OPTICS 81 finite velocity of light, namely, that all fixed stars appear to execute a common annual motion that is evidently a count i r- part to the rotation of the earth around the sun. It is very easy to understand how this effect comes about from the point We of view of the emission theory. shall give this inter- pretation here, but we must remark that it is just this pheno- menon that raises certain difficulties for the wave-theory, about which we shall yet have much to say. We know (see III, 7, p. 64) that a motion which is rectilinear and uniform in our system of reference S is so also in another system S', if the latter executes a motion of translation with respect to S. But the magnitude and the direction of the velocity is different in the two systems. It follows from this that a stream of light corpuscles which, coming from a fixed Fig. 52. Fig. 53. star, strike the earth, appear to come from another direction. We shall consider this deflection or aberration for the particular case when the light impinges perpendicularly to the motion of the earth (Fig. 53). Let a telescope, on the objective of which a light corpuscle strikes, be in the position 1. Now, whilst the light traverses the length / of the telescope, the earth, and with it the telescope, moves into the position 2 by an amount d. Thus the ray strikes the centre of the eye-piece only when it comes, not from the direction of the telescopic axis, but from a direction lying somewhat behind the earth's motion. Hence the direction in which the telescope aims does not point to the true position of the star, but to a point of the heavens that is displaced forward. The angle of deflection is determined by the ratio d : /, and is evidently independent of the length / of the telescope. For if the latter be 6 82 THE THEORY OF RELATIVITY increased, so also is the time that the light requires to traverse it, and hence also the displacement d of the earth is increased in the same ratio. The two paths I and d, traversed in equal times by the light and the earth, must be in the ratio of the corresponding velocities : d_v This ratio, also called the aberration constant, will in future be denoted by £ : .... = v fl (33) c It has a very small numerical value, for the velocity of the = earth in its orbit about the sun amounts to about v 3° km. /sec, whereas the velocity of light, as already mentioned, amounts to 300,000 km./sec Hence j3 is of the order 1 : 10,000. The apparent positions of all the fixed stars are thus always a little displaced in the direction of the earth's motion at that Fig. 54. moment, and hence describe a small elliptical figure during the annual revolution of the earth around the sun. By measuring this ellipse the ratio jS may be found, and since the velocity v of the earth in its orbit is known from astronomic data, the velocity of light c may be determined from it. The result is in good agreement with Romer's measurement. We shall next anticipate the historical course of events and shall give a note on the earthly measurements of the velocity of light. All that was essential for this was a technical device that allowed the extremely short times required by light to traverse earthly distances of a few kilometres or even only a few metres, to be measured with certainty. Fizeau (1849) and Foucault (1865) used two different methods to carry out these measurements, and confirmed the numerical value of c found by the astronomic method. The details of the process need not be discussed here, particularly as they are to be found in every elementary textbook of physics. We call attention to only one point : in both processes the ray of