THE THEORY OF RELATIVITY
MACMILLAN AND CO., LIMITED
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TORONTO
THE THEORY OF RELATIVITY
BY
L. SILBERSTEIN, PH.D.
LECTURER IN NATURAL PHILOSOPHY AT THE UNIVERSITY OF ROME
MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON
1914
COPYRIGHT
PREFACE
THIS introduction to the Theory of Relativity is based in part upon a course of lectures delivered in University College, London, 1912-13. The treatment, however, has been made much more systematical, and the subject matter has been extended very con- siderably; but, throughout, the attempt has been made to confine the reader's attention to matters of prime importance. With this aim in view, many particular problems even of great interest have not been touched upon. On the other hand, it seemed advantageous to trace the connexion of the modern theory with the theories and ideas that preceded it. And the first three chapters, therefore, are devoted to the fundamental ideas of space and time underlying classical physics, and to the electromagnetic theories of Maxwell, Hertz-Heaviside and Lorentz, from the last of which Einstein's theory of relativity was directly derived. In the exposition of the theory itself free use has been made not only of the matrix method of representation employed by Minkowski, but even more of the language of quaternions. Very little indeed of these mathematical methods is required to follow the exposition, and this little is given in Chapter V., in a form which will be at once accessible to those acquainted with the elements of the ordinary vector algebra.
It is hoped that the book will give the reader a good insight into the spirit of the theory and will enable him easily to follow the more subtle and extended developments to be found in a large number of special papers by various investigators.
vj PREFACE
I gladly take the opportunity of expressing my thanks to Mr. William Francis and Dr. T. Percy Nunn for their kindness in reading a large portion of the MS., to Prof. Alfred W. Porter, F.R.S., for reading all the proofs and for many valuable suggestions, and to the Publishers and the Printers for the care they have bestowed on my work.
L. S.
LONDON, April, 1914.
CLASSICAL RELATIVITY -
CONTENTS
CHAPTER I
PAGE
i
CHAPTER II
p.*+
MAXWELLIAX EQUATIONS FOR MOVING MEDIA AND FRESNEL'S
DRAGGING COEFFICIENT. LORENTZ'S EQUATIONS 21
CHAPTER III
THEOREM OF CORRESPONDING STATES. SECOND-ORDER DIFFICULTIES. THE CONTRACTION HYPOTHESIS. LORENTZ'S GENERALIZED THEORY 64
CHAPTER IV
EINSTEIN'S DEFINITION OF SIMULTANEITY. THE PRINCIPLES OF RELATIVITY AND OF CONSTANT LIGHT- VELOCITY. THE LORENTZ TRANSFORMATION 107 -142- 92
CHAPTER V
VARIOUS REPRESENTATIONS OF THE LORENTZ TRANSFORMATION 123
CHAPTER VI
COMPOSITION OK VELOCITIES AND THE LORENTZ GROUP 163
CHAPTER VII
PHYSICAL QUATERNIONS. DYNAMICS OF A PARTICLE 182
viii CONTENTS
CHAPTER VIII
FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 205
CHAPTER IX
ELECTROMAGNETIC STRESS, ENERGY AND MOMENTUM. EXTENSION
TO GENERAL DYNAMICS 232
CHAPTER X
MINKOWSKIAN ELECTROMAGNETIC EQUATIONS FOR PONDERABLE
MEDIA 260
INDEX - ----- 290
CHAPTER I.
CLASSICAL RELATIVITY.
BEFORE entering upon the subject proper of this volume, namely, the modern doctrine of Relativity and the history of its origin and develop- ment, it seems desirable to dwell a little on the more familiar ground of what might be called the classical relativity, and to consider
CORRIGENDA
60, line 4 from the foot, for on read ou
65, line 8, for P read P'
71, „ ^for */<* read tf/c* 119, ,, 14, for 0(z/) = i read 0(z;)=i/a 149, line 9 from the foot, for W^ W.2 read 153. » 7 „ „ for A2 read Az 189, line 3, for right-hand read left-hand 222, line 2 from the foot, for vectors read vector operators 229, line 13, interchange the words 'real' and 'imaginary ' 235, „ 20, for P'p' read (P'p') 235> „ 25,>r/n/ readl'n, 276, ,, 6, for i/c read i/c
viii CONTENTS
CHAPTER VIII
FUNDAMENTAL ELECTROMAGNETIC EQUATIONS 205
CHAPTER IX
ELECTROMAGNETIC STRESS, ENERGY AND MOMENTUM. EXTENSION
TO GENERAL DYNAMICS - 232
CHAPTER X
MINKOWSKIAN ELECTROMAGNETIC EQUATIONS FOR PONDERABLE
MEDIA 260
INDEX ------ ----- 290
CHAPTER I.
CLASSICAL RELATIVITY.
BEFORE entering upon the subject proper of this volume, namely, the modern doctrine of Relativity and the history of its origin and develop- ment, it seems desirable to dwell a little on the more familiar ground of what might be called the classical relativity, and to consider two particular points which are of fundamental importance, not only for the appreciation of the whole subject to follow, but also for an adequate understanding of almost all physico-mathematical considera- tions. What I am alluding to are the following questions: i° the choice of a framework of axes or, more generally, of a system of reference in space, and 2° the definition of physical time, or the selection of a clock or time-keeper, to be employed for the quantita- tive determination of a succession of physical events.
Both of these questions existed and were solved, at least implicitly, a long time before the invention of the modern Principle of Relativity, in fact centuries ago, in their essence as early as Copernicus founded his system.*
The question of a space-framework is obvious enough and widely known ; it will require therefore only a few simple remarks.
The most superficial observation of everyday life would suffice to show that the form and the degree of simplicity of the statement of the laws of physical phenomena, more especially of the laws of motion of what are called material bodies, depend essentially on our selec- tion of a system of reference in space. Certain frameworks of reference are peculiarly fitted for the representation of a particular
*A clear and beautiful statement of the fundamental importance of the Copernican idea is to be found in P. Painleve's article ' Mecanique ' in the collec- tive volume De la Mtthode dans les Sciences, edited by Emile Borel. (Paris, F. Alcan, 1910.)
S.R. A
£ {THEORY OF -RELATIVITY
instance of motion of a particular body or also of almost any observable motion of bodies in general, leading to a high degree of completeness, exactness and simplicity, while other frame- works (moving in an arbitrary manner relatively to those) give of the same phenomena a most complicated, intricate and confused picture.*
Suppose that somebody, ignorant of the work of Copernicus, Galileo and Newton, but otherwise gifted with the highest experi- mental abilities and mathematical skill (a quite imaginary supposition, being hardly consistent with the first one), chooses the interior of an old-fashioned coach, driven along a fairly rough road, as his laboratory and tries to investigate the laws of motion of bodies enclosed together with him in the coach — say, of a pendulum or of a spinning top — and selects that vehicle as his system of reference. Then his tangible bodies and his conceptual * material points,' starting from rest or any given velocity, would describe the most wonderful paths, in incessant shocks and jerky motions; the axis of his 'free gyroscope' would oscillate in a most complicated way, — never disclosing to him the constancy of the vector known to us as the 'angular momentum,' i.e. the rotatory analogue of Newton's first law of motion. Nor would the uniform translational motion have for him any peculiarly simple or generally noteworthy properties at all. His mechanical experience being, in a word, full of surprises, he would soon give up his task of stating any laws of motion whatever with reference to the coach. Getting out of it on to firm ground, he will readily find out that the earth is a much better system of reference. With this framework, smoothness and simplicity will take the place of hopeless irregularity. Undoubtedly, this property must have been remarked in a very early stage of man's history, and the above example will appear to the least trained student of mechanics of our present times trivial and simply ridiculous. ' Of course,' he would say, ' the motions of material bodies relatively to that coach are so very complicated, for that vehicle is itself moving in a highly complicated way.' He would hardly consider it worth while to add ' relatively to the earth.' The coach being such a small, insigni- ficant thing in comparison with the terrestrial globe, it would seem extravagant to our interlocutor, if somebody insisted rather on saying that it is the earth which moves in such a complicated way relatively
* And as to 'absolute motion,' regardless of any system of reference, it is need- less to mention that it is devoid of meaning in exactly the same way as ' absolute position.'
FRAME OF REFERENCE 3
to the coach on its particular journey. But, as a matter of fact, both of these reference-systems move relatively to one another, and the comparative insignificance of one of them would, by itself, be but a very feeble argument (as we shall see presently, from another example).
At any rate the earth, the 'firm ground,' allowance being made for occasional large shocks and for very small but incessant oscil- lations of every part of its surface,* has proved to be an excellent system of reference for almost all motions, especially those on a small scale with regard to space and time, and practically without any reservation for all pieces of machinery and technical contriv- ance. In fact, the earth as a system of reference offered at once the advantage of a high degree of simplicity of description of states of equilibrium and motion, opening a wide field for the application of Newton's mechanics, at least as regards purely terrestrial observations and experiments.! The earth is then a reference-system which is constantly used also by the most advanced modern student of mechanics.
But things become altogether different when we look up to the sky and desire to bring into our mechanical scheme also the motions of those luminous points, the celestial bodies, including, of course, our satellite, the moon, and our sun. Then the earth loses its privilege as a framework of reference. If it were only for the so-called ' fixed stars,' which form the enormous majority of those luminous points (and the moon too), we could still satisfy our vanity and continue to consider our globe as an universal mechanical system of reference, the system of reference, as it were. On our plane drawings, or in our three-dimensional models, we could then represent the earth by a fixed disc, or sphere, respectively, with a smaller sphere moving round it in a circular orbit, to imitate our moon, the whole sur- rounded by a large spherical shell of glass sown with millions of tiny stars, spinning gently and uniformly round the earth's axis, — very
* Which gave so much trouble to the late Sir G. H. Darwin and his brother in their attempts to measure directly the gravitational action of the moon, as described in Sir G. H. Darwin's attractive popular book, The Tides and Kindred Phenomena in the Solar System, London, 1898 (German edition by A. Pockels, enlarged ; Teubner, Leipzig, 1911).
t With the exception of those of the type of Foucault's pendulum experiments, performed with the special purpose ' of showing the earth's rotation. ' In more recent times the pendulum could be successfully replaced by a gyroscope, as originally suggested, and tried, by Foucault himself.
4 THE THEORY OF RELATIVITY
much like, in fact, some primitive mental pictures of the universe.* But the case becomes entirely different when we come to consider the far less numerous class of luminous points or little discs, the planets, and the comets, moving visibly among the ' fixed ' shining points in a complicated way. Then, even before touching any dynamical part of the celestial problem, we are compelled to give up our earth as a system of reference and replace it by that of the ' fixed stars,' originally so inconspicuous, or — what turns out to be equally good — by a framework of axes pointing from an initial point fixed in the sun towards any given triad of fixed stars. It is needless to tell here again the long story of that admirable and ingenious system which was founded by Ptolemy (born about 140 B.C.), which held the field during fourteen centuries, to be replaced finally and definitely by the system of Copernicus (1473-1543), which transferred to the sun the previous dignity of the earth, f The Copernican system of reference had the enormous advantage of simplicity, quite inde- pendently of any mechanical, i.e. (to put it more strictly) dynamical considerations. Its superiority to the geocentric system manifested itself already in the simplicity it gave to the paths of the solar family of bodies, the wonderfully simple shapes of the orbits of the planets. In the geocentric scheme we had the complicated system of 'excentrics and epicycles' of Ptolemy, whereas taking, in our drawing or model, the sun as fixed, the orbits of the planets became simple circles, which in the next step of approximation turned out to be slightly elliptic. Thus the Copernican system of reference had its enormous advantages before any properly mechanical point of the subject was entered upon. Historically, in fact, the mechanics of Galileo and Newton came a long time after Copernicus, so that the
*The earth as the centre of the universe, with the ' crystal spheres,' with the stars stuck to them, spinning round the earth, still formed part of the teachings of the Ionian school of philosophers founded by Thales (born about 640 B.C.). The first to suggest the rotation of the earth round its axis and its motion round the sun seems to have been Pythagoras, one of Thales' disciples, though it has been later unjustly attributed to Philolaus, one of Pythagoras' disciples (born about 450 B.C.).
t Although I do not claim to give here anything like a history of astronomy, it may be worth mentioning that the Pythagoreans already taught that the planets and comets were circling round the sun. But at any rate the Ptolemaean geo- centric system reigned universally from the second till the fifteenth century, the only serious objection against its complexity having been raised in the thirteenth century by Alphonso X. , king of Castile, the author of the astronomical ' Tables ' associated with his name (published during 1248-1252).
INERTIAL SYSTEM OF REFERENCE 5
privilege of reference-system was taken away from our earth and transferred to the sun on the ground of purely kinematical con- siderations of simplicity, a few centuries before Newton. But after- wards the Copernican or the ' fixed-stars ' system of reference appeared to be wonderfully appropriate to Newtonian mechanics, both in its original shape and in its later (chiefly formal) development by Laplace for celestial and by Lagrange for terrestrial and general problems. It soon became the final reference-system of mechanics. It is relatively to this ' fixed-stars ' system of reference that the law of inertia has proved to be valid. We will call it, therefore, following the modern habit, the inertial system, or sometimes, also, the New- tonian system of reference* It is relatively to this system that spin- ning bodies behave in the characteristically simple manner which has led many authors to speak of their property of 'absolute orientation.' Or, to put it in less obscure words, it is relatively to the inertial system that the vector called angular momentum is preserved, both in size and in direction, — this property being a consequence of the funda- mental laws of Newton's mechanics, and, at the same time, a perfect and most instructive analogue to Newton's First Law of motion.! The most immediate and tangible manifestation of this property is that the axis of a free gyroscope (practically coinciding in direction with its angular momentum) points always towards the same fixed star ; thus having the simplest relation to the inertial system, since it is invariably orientated in this system of reference. Notice that it would, therefore, be more extravagant to say that the axis of such a gyroscope moves relatively to the earth than vice versa, — though apparently, bodily, the gyroscope of human make is such an incon- spicuous tiny thing in comparison with our planet. The conservation of the angular momentum, or moment of momentum, 2wVrv,| of the whole solar system, which is best known in connexion with Laplace's ' invariable plane,' is but the same thing on a larger scale than that exhibited by our spinning tops. But this only by the
* We speak of it in the singular, instead of infinite plural, only for the sake of shortness. For, as is well known, if S, say the * fixed ' stars, be such a system, then any other system 2' having relatively to S any motion of uniform (rectilinear) translation is equally good for all purposes.
tThis point is expressly insisted upon and successfully applied to didactic purposes in Professor A. M. Worthington's Dynamics of Rotation, sixth edition, new impression 1910 ; Longmans, Green & Co., London.
J See, for example, the author's Vectorial Mechanics, Chap. III. ; Macmillan & Co., London, 1913.
6 THE THEORY OF RELATIVITY
way. What mainly concerns us here is that the ' fixed-stars ' system — or, more rigorously, any one out of the oo 3 multitude of equivalent inertial systems — has gradually turned out to be peculiarly fitted as a system of reference for the representation of the motion of material bodies.
But also with this system of reference the laws of motion have their simple, Newtonian form only for a / measured in a certain way, i.e. for a certain clock or time-keeper, e.g. approximately the earth in its diurnal rotation, or, more exactly (in connexion with what is known as the frictional retarding effect of the tides), a time-keeper slightly different from the rotating earth. This is equivalent to defining as equal intervals of time those in which a body not acted on by ' external forces,' i.e. very distant from other bodies or otherwise suspected sources of disturbance, describes equal paths.* In main- taining the motion of such and such a body in such and such circum- stances to be uniform, we do not make a statement, but rather are defining what we strictly mean by equal intervals of time. Selecting quite at random a different time-keeper, we could not, of course, expect the same simple laws to hold, with respect to the inertial system of reference. But with another space-framework of reference another time-keeper might do as well.
Thus we see that, to a certain extent, the choice of a system of reference in space has to be made in conjunction with the selection of a time-keeper. Our x, y, z, /, the whole tetrad, the space and time framework must be selected as one whole. That kind of 'union' emphasized by the late Hermann Minkowski, the joint selection of x-> y-> z> t> manifesting itself in the modern relativistic theory by the consideration of a four-dimensional 'world' (instead of time and space, separately), is not altogether such an entirely new and revolu- tionary idea as is generally believed ; for to a certain extent, and in a somewhat different sense, it is as well a requirement of Newtonian mechanics, and, more generally, of the classical kind of Physics, as of modern Relativity. What difference there really is between the two we shall see in the following chapters.
* Thus it is manifest that the science of mechanics does not describe the motion of bodies in its quantitative dependence upon ' time, flowing at a constant rate ' {Newton), but literally gives only sets of simultaneous states of motion of the various bodies, the time-keeper itself being included. What is besides contained in these sets or successions is a non-quantitative element, namely, of what is vaguely called 'before' and 'after.'
CHOICE OF TIME-KEEPER 7
Meanwhile we have touched, in passing, the fourth variable /, and this brings us to our second point, namely, the definition of physical time, the selection of 'the independent variable /' of our physico-mathematical equations, but viewed more generally, and more carefully, than above, where we have touched it only incidentally.
To explain this question, of capital importance for almost every quantitative physical research, I must ask you to direct your attention to the following considerations.
Suppose we do not limit ourselves to the investigation of motion only, but are concerned with every possible kind of physical pheno- mena, such as conduction of heat or electricity, diffusion of liquids or gases, melting of ice, evaporation of a liquid, etc., etc., and that we propose to describe the progress of these phenomena in time, to trace their history, past and future. How are we, then, to select our time- quantity /?
First of all, we cannot, of course, take it to be Newton's * absolute time,' which is defined, according to a quotation from Maxwell,* as follows :
'Absolute, true, and mathematical Time is conceived by Newton as flowing at a constant rate, unaffected by the speed or slowness of the motions of material things. It is also called Duration.'
For, supposing there is such a thing,! we do not know how to find or to construct a clock which measures this 'absolute time,' even approximately; that is to say, we have no criterion to distinguish such a clock from a 'wrong' one. And thus, certainly, we cannot use this kind of definition for physical purposes. How are we then to measure our /? Granting that the selection of a chronometer indicating our t is (at least within certain wide limits) arbitrary or free, what is the requirement on which we have to base our choice ?
Now, it seems to me that the first and most general requirement, which may also be seen to be tacitly assumed in all the investigations of both the more recent and classical natural philosophers, especially physicists and astronomers, is
that our differential equations, representing the laws of physical (and other) phenomena, should not contain the time, the variable /, explicitly,
* Matter and Motion, page 19.
t But, as a matter of fact, the phrase ' flowing at a constant rate ' is simply meaningless.
8 THE THEORY OF RELATIVITY
i.e. that for any sufficiently comprehensive physical system, of which the instantaneous state is defined, say, by /lf /2> •••Ai» tne differential equations should be of the form
-^7 =/t(A» A>---A)> i /Ax
I
J
2=1, 2, ... «.
This requirement is also intimately connected with a certain form of what Maxwell* calls 'the General Maxim of Physical Science" and what is commonly called the Principle of Causality.
To make my above statement more intelligible to a wider circle of (non-mathematical) readers, let us consider s6me very simple examples which will enable us also to see the exact meaning of instantaneous ' state' of a system and to learn to distinguish between two very important and large classes of systems: i) complete or [undisturbed^ and 2) incomplete or ''disturbed' systems.
Suppose we have a small metallic sphere,! suspended somewhere in a large dark cellar kept at constant temperature a, receiving no heat, radiant or other, from without. Suppose we heated the sphere to 100° C, which is to be >a (say, a = o°C), and from that instant left it to its own fate. We return to it after an hour, as measured, say, on one of our common clocks (i.e. rotating earth as time-keeper ), and we find it has cooled down, say, to 90°. Thus :
/ 0
/0 100°, 1 .'. A0= - 10°,
/0 + i h. 90°. J for A/= i h.
Now, if we repeated the whole experiment to-morrow or next week, we should find that during one hour the fall of temperature of our suspended sphere would again be from 100° to 90°, i.e. A#= - 10° for A/= ih. We could make similar observations for any other stage of the cooling process of our little sphere (say down from 50° instead of 100°) and for other time-intervals (say Jh. instead of i h.), arbitrarily small, J and, repeating our observations, we should find again and again the same permanency of results, — only with different values of A 6 for different intervals A / and for different starting temperatures.
* Matter and Motion, p. 20, first paragraph of Art. xix. ; see also p. 21, lines t]-\\.
•\' Small' only so as not to be obliged to consider the different temperatures of its various parts.
£ Or practically so, at least.
COMPLETE AND INCOMPLETE SYSTEMS 9
Thus, the temperature 6 of our sphere, placed in the specified conditions of its environment, varies with time (ordinary clock-time) in a certain determinate way, namely, so that starting from a given temperature 0, its change during a given time-interval A/=/2-/'1, is always one and the same, that is to say, no matter when this happens, independently of /^ /2, but depending only on
Now, such a system, i.e. the sphere in its above environment, I propose to call an undisturbed or, what for the beginning is more cautious, complete system. And, in this case 0 being the only- quantity on whose instantaneous value the whole (thermal) future history of our sphere depends, we shall say, in accordance with general use, that the instantaneous value of the temperature 0 defines the instantaneous state of our system (a being supposed given once and for ever). In the case before us we have a one-dimensional system, which may be called also a system of one degree of freedom.*
Take the limit of the mean rate of change A0/A/ for A/->o ; then the differential equation of our simple system will be of the form
S-yw, «>
which may be read : the instantaneous time-rate of change of the temperature is a function of its instantaneous value only.f We know in this case that f(Q)= —h(Q — a) approximately, when 6 — a is small, where h is a positive constant ; but the particular form of the function / is for our present purposes a matter of indifference.
Let us, on the other hand, consider a similar sphere suspended, say, in a window, exposed south, in a land in which the sun is wont to shine often. Then, for the same starting value 0 and same A/, the change A 6 will be different at different times of the day, e.g. larger from 7 till 8 a.m. than from 2 till 3 p.m., larger in winter than in summer, and so on. Now, a system such as this sphere we will call a disturbed system or a system * exposed to external agents,' or better an incomplete system, for this concept does not presuppose the know- ledge of what is meant by 'action' of one system upon another.
* Observe that n mechanical • degrees of freedom ' amount to 2n degrees of freedom in the sense here adopted. t See Note 1 at the end of the chapter.
io THE THEORY OF RELATIVITY
In the present case the differential equation of our system will be of the form
7/3
§-*•(*, /), (a)
/ being again measured with the ordinary (earth-)clock, and g being some function involving / in a very complicated manner.
Now, according to the above general requirement, our /-clock would be the right one, the peculiarly fitted one, for our first physical system, (i), but not for the second, (2).
By selecting a different time-keeper we might possibly convert some (not all) ' disturbed ' into ' undisturbed ' or complete systems ; but then we should spoil the completeness of (i). Let us see, first of all, what other clocks we can take instead of our original one without spoiling the simple property of (i). Instead of /, take
then (i) will be transformed into
Thus, if the property of completeness is to be preserved, <£(/) must be a constant, and consequently T a linear function of /, say
amounting only to a different initial point of time-reckoning and to the choice of a different time unit.
Now (2), the equation of our second sphere, is not of the form d6jdt=^(t).f(B\ but rather of the form
7/3
£-/[f- •(*)]+ <?<4;
consequently, if we wished also to sacrifice the completeness of (i), we certainly cannot transform (2) into an undisturbed or complete system, by any T=<j>(t). Hence the moral: certain incomplete systems cannot be made complete by merely selecting a new clock instead of the old one, and such systems I propose to call essentially incomplete systems.
But suppose we had a system obeying a law of the form
SYSTEMS MADE COMPLETE n
i.e. a sphere as in (i), but having a coefficient h (coefficient of what Fourier called external conduction, divided by specific thermal capacity), which due to some visible changes of the sphere's surface, such as oxidation, is variable^ instead of being constant. Then we could represent it as a complete system by taking instead of the /-clock another clock indicating the time
T=\h(f)dt, say = ^(/); Jo
but, F(f) not being a linear function of the old time, this innovation would at once spoil the completeness of (i).
At this stage we would find ourselves in face of an alternative : which of the two systems, (i) or (3), is to be saved, which is to be sacrificed ? And, correspondingly : which of the two clocks, the /-clock or the T^clock is to be selected as time-keeper? If we could detect no differences between the spheres (i), (3) — besides that of their respective thermal histories — the choice would be difficult, or rather arbitrary, quite a matter of taste or caprice. But, say, the latter sphere, (3), gets oxidized, shrinks or expands, and what not, and the former, (i), remains sensibly unaffected by the process of repeated cooling and heating. Therefore, following the maxim or principle of causality, we would conserve our /-clock, best fitted for (i), and would try to convert (3) into a complete system in a different way, namely, by taking account explicitly of the oxidation of the sphere's surface, of its dilatation, and so on, i.e. by introducting besides 6 other quantities, say, the amount m of free oxygen present in the enclosure and the radius r of the sphere, and by defining the state of the system by the instantaneous values of 0, m r.
In this way, retaining our old clock, we should have converted the originally disturbed system of one degree of freedom into a complete system of three or more degrees of freedom. As a rule, we do not reject our traditional time-keeper at once. Encountering an incom- plete or disturbed system, every physicist will, first of all, try to throw the ' disturbances ' on some ' external agent ' rather than on his clock. He will look round him for external agents, almost instinctively following the voice -of the maxim of causality, whispering to him, as Maxwell puts it (Matter and Motion, p. 2 1) : ' The difference between one event and another does not depend on the mere difference of the times.' And finding nothing particularly suspect in the nearest
12 THE THEORY OF RELATIVITY
neighbourhood, he will look farther round, or deeper into, the system in question.
Similarly, if we amplified the system of our second example (the sphere cooling before an open window), taking in the sun varying in position, the atmosphere, and possibly a host of other things, we would obtain a larger, more comprehensive system which, though more complicated than the original one, would satisfy us as being undisturbed, with our old /-clock.
So it is in many other cases. Thus, we can say :
Adding to a given fragment of nature (system), which in terms of a certain /-clock behaves like a disturbed or incomplete system (/D/2> •••At)> *•*• obeys the equations
,...A, 0, (4)
* = I, 2, ... »,
fresh fragments of nature (with the corresponding parameters Pn+\ i - • - Pn+m)i we often obtain a new, larger,* system which, still with the same /, is undisturbed or complete :
i= i, 2, ... n + m.
In short, we complete the system Sn to Sn+m. The /, implied here, is practically the time indicated by that clock which proved peculiarly fitted for the description of our previous stock of experi- ence. Thus, for example, Fourier's theory of conduction of heat was preceded by the triumphs of classical mechanics ; and if asked what the / in his fundamental equation
meant, he would, doubtless, answer that it is to be measured by the rotating earth as time-keeper, though he hardly ever stopped in his researches to consider this matter explicitly.
Thus, generally, we do not reform our traditional clocks, but we make our systems complete as in (5), by amplifying them. But
* Not necessarily larger in volume ; for often we introduce new parameters by going deeper into the original system itself, sometimes as deep as the molecular, atomic or even sub-atomic structure, say, of a piece of matter ; or being originally concerned with the thermic history only, we supplement the temperature by the pressure, volume, electric potential, and so on.
AMPLIFIED SYSTEMS 13
sometimes, when we think that we have made our system Sn+m sufficiently comprehensive, that we have exhausted all reasonably suspected material as possible 'external agents,' and when Sn+m nevertheless continues to behave as an incomplete system, i.e. when still
then, to make it finally complete, we decide ourselves to change our /, our traditional clock, — especially if the change required is a slight one. This procedure, of course, is possible only when the F^ in (6) are all of the form
Otherwise, we feel obliged to help the matter by introducing yet fresh parameters /n+m+1, A+m+2> etc., and not finding real (perceivable) supplementary material round us, we introduce fictitious supplements, which sometimes turn out to be real afterwards, thus leading to new discoveries.
From this it is also manifest that the Principle of Causality has the true character of a maxim ; though of inestimable value both in science and in everyday life, it is not a law of nature, but rather a maxim of the naturalist.
We have classical examples of both the procedures sketched above, viz. of reforming our clocks and of supplementing or amplifying a system with the view of securing its completeness. In the first place, to get rid of one of the inequalities in the motion of the moon round the earth, astronomers have had recourse to the supposition that there is a gradual slackening in the speed of the earth's rotation. Of course, they did it in connexion with the tides and with immediate regard to the fundamental principles of mechanics, implying also the law of gravitation. But at any rate, in doing so, and in declaring that the earth as a clock is losing at the rate of 8-3, or (according to another estimate) of 22 seconds per century, they gave up the earth as their time-keeper and substituted for the sidereal time / a certain function T=$(t\ slightly differing from /*, as their new Akinetic time,' as Prof. Love calls it.* Secondly, as is widely known, the perturbations of the planet Uranus have led Adams and Le Verrier
*A. E. H. Love, Theoretical Mechanics \ second edition, Cambridge, 1906, page 358. In connexion with our subject, the whole of Chapter XL of Prof. Love's book may be warmly recommended to the reader.
14 THE THEORY OF RELATIVITY
(working independently) to complete the system by a celestial body, at first fictitious, but then, thanks to admirable calculations based on
the -g-law, actually discovered and called Neptune. Notice that
both kinds of procedure have essentially the character of successive approximations.
Any future researches of mechanical, thermal, electromagnetic and other phenomena, either new or old ones but treated with increasing accuracy, if leading to 'disturbed' systems, obstinately withstanding the supplementing procedure (i.e. that consisting in the introduction of fresh parameters pn+\> etc.), may oblige us to reform also the newer, slightly corrected earth-clock, to give up the * kinetic time ' of modern astronomy for a better one, more exactly fitted for the representation of a larger field of phenomena, and so on by successive approximation. It may well happen that we shall have to give up the kinetic time for the sake of the ' electromagnetic time,' — if I may so call the variable / entering in Maxwell's differential equations of the electromagnetic field.* For suppose for a moment that some future experimental investigations of high precision were to prove that the variable / in
3E 9M
^- = c . curl M, -^- r = - c . curl E
ot ot
is not proportional to the kinetic time ; then the electricians would hardly give up these admirably simple and comprehensive equations ; they would rather sacrifice the kinetic time. Thus, in the struggle for completeness of our physical universe, we shall have always to balance the mathematical theory of one of its fragments, or sides, against that of another. A great help in this struggle is to us the circumstance that, though, rigorously, all parts of what is called the universe interact with one another, yet we are not obliged to treat at once the whole universe, but can isolate from it relatively simple
* Thus we read in Painleve's article (loc. cit. page 91): 'La duree d'une ondulation lumineuse correspondant a une radiation determined (ou quelque duree deduite d'un phenomene electrique constant} sera vraisemblablement la prochaine unite de temps. ' This idea seems to be suggested first by Maxwell ; the cor- responding wave-length would at the same time be the standard of length, when the platinum ' metre ttalon ' will be given up. Thus it may happen that the 'kinetic length' (i.e. that based on our notion of a 'rigid' body) will be sacrificed for the benefit of an optical or ' electromagnetic length ' in the same way as the 'kinetic time' may be replaced by an 'electromagnetic time.'
NEWTON'S EQUATIONS 15
parts or fragments, which behave sensibly as complete systems, or are easily converted into such.
Herewith I hope to have explained to you, at least in its fundamental points, the question of selection of a time-keeper.
Thus, we know, essentially, how to measure our /, at least in or round a given place (taken relatively to a certain space-framework). We do not yet know what is the precise meaning of simultaneous events occurring in places distant from one another. But the notion of simultaneity, especially for systems moving relatively to one another, belongs to the modern Theory of Relativity, and is, in fact, a characteristic point in Einstein's reasoning. Therefore it will best be postponed until we come to treat of the principal subject of this volume.
We could now pass immediately to the history of the electro- magnetic origin of the modern principle of relativity, extending from Maxwell to Lorentz. But since we already have come to touch, more than once, Newtonian or classical mechanics, let us dwell here another moment upon this subject.
Let us call 2 one of the ' inertial ' systems of reference, say the system of 'fixed' stars, and let xit yit zt be the rectangular co- ordinates of the z'-th particle* of a material system, relatively to 2, at the instant /. Then the Newtonian equations of motion are
d~X{ „ 1 0 »
mi~dfl= *' *' ' '
or
dxi _ dyi _ dzi _ ~di~U" ~di~V" ~di = W"
dt l dt dt
where mit the masses, are constant scalars belonging to the individual particles, / is the 'kinetic time' and Xit etc., are functions of the instantaneous state of the material system, i.e. of the instantaneous configuration and (in the most general case) of the instantaneous velocities of the particles relatively to one another, which for .certain systems may, but for a sufficiently comprehensive system do not, contain explicitly the time /. If the material system is subject to
constraints, say
m = o, Y = o» etc.,
* The material ' particle ' may also play the part of a planet or of the sun, as in celestial mechanics.
1 6 THE THEORY OF RELATIVITY
then X{, etc., contain, besides the components of what are called the impressed forces, also terms like
which depend only upon the relative positions and relative velocities of the parts of the system (i.e. of the mass-particles) to one another or to the surfaces or lines on which they are constrained to remain, or to the points of support or suspension entering in such constraints. Thus the bob of a pendulum is constrained to remain at a constant distance relatively to the point of suspension, the friction of a body moving on a rough surface depends on its velocity relative to that surface, and so on. Consequently, if instead of 2 any other system of reference 2'(#', y'9 z) is taken, having relatively to 2 a purely translational, uniform, rectilinear motion, Xi, YI, Zi are not changed. And the same thing is true of the left-hand sides of the equations of motion. For, if #/, etc., be the coordinates of the z-th particle relatively to 2' at the instant t, and if we take, for simplicity, the axes of x ', y, z parallel to and concurrent with those of x, y, z respectively, then
. .
t =t,
where (u, v, w) is the constant velocity of 1! relatively to 2, and where the fourth equation is added to emphasize that the old time t is retained in the transformation. Consequently,
, dxl dXi Ui = ~jfr = -jj--u = Ui-u, etc.
(and for any pair of particles ul - «/ = ut - ujt etc.), and dul _dui dvl _dVi dwl dwi
W = ~dt' W = ~di' ~W = ~di*
which proves the statement.
Thus, the equations of motion (8), or, in vector form,
remain unchanged by the transformation (9), or, written vectorially, by the transformation
NEWTONIAN TRANSFORMATION 17
where v, the resultant of the above u, v, w, is the vector-velocity of 2' relatively to 2. As regards the time, we could write also t' = at + b (a, b being constants), but this would amount only to a change of units and shifting of the beginning of time-reckoning.
In view of the above property, the linear transformation (9) or (90), v being any constant vector, is called the Newtonian (and by some authors the Galileian) transformation. Thus we can say, shortly :
The equations of classical mechanics are invariant with respect to the Newtonian transformation. J if ^^ &^
Notice that v being quite arbitrary, both as regards its size (or tensor) and direction, we have in (90) a manifold of oo3 transforma- tions, and all of these form a group of transformations. For, if
r^ri-Vj/; /' = /, and
r/' = r/-v./; /" = /', then
r/' = ri-v/; /" = /', where
v = V! + v2. (10)
We shall refer sometimes to (9) or (90) as the Newtonian group.
Notice the simple additive property (10), to be compared later on with a less simple property of the corresponding group in modern Relativity.
Thus, there is no unique frame of reference for classical mechanics ; if the Newtonian equations of motion are strictly valid relatively to the framework 2 of the ' fixed ' stars, they are equally valid relatively to any other out of the oo3 frameworks 2', connected with 2 by (9), say relatively to the solar-system frame, which has relatively to 2 a uniform velocity of something like 25 kilometres per second, towards the constellation of Hercules.* Therefore, by purely in- ternal mechanical experiment and observation, i.e. not looking out- side to external systems, we could never distinguish the solar frame 2' from 2, that is to say, 2', like 2, does not show any anisotropy with regard to mechanical phenomena. The same remark applies, • with sufficient approximation, to the earth's annual motion : it is not ascertainable by purely terrestrial mechanical experiments.
Physicists hoped to detect this motion which they called also * the motion relative to the aether,' by the means of purely terrestrial
* Quoted after Painleve, loc. cit. page 117. S.R. B
1 8 THE THEORY OF RELATIVITY
optical or electromagnetic experiments, — we shall see later how unsuccessfully.
In other words, seeing that there is no unique 'kinetic' space- framework, they tried to find a unique ' optical ' or ' electromagnetic ' reference-system, the 'aether,' or rather to show that this wonderful medium, already invented for other purposes, was such a unique frame of reference. But the results of all experiments of this kind have been obstinately negative.
It is chiefly this which has led to the construction of the new theory of relativity.
NOTES TO CHAPTER L
Note 1 (to page 9). To show, generally, the connexion between the integral form of the properties of a complete system, as stated in the above illustrations, and its differential form, of which eq. (i) is an example, let us consider such a system of n degrees of freedom. Let its state at any instant / be determined by
Then, /0 = o being any other, say, past instant,
A(0=^[A(o),-A(o); 4 *»i,2,,,.«,
where Pi is a symbol of an operation or a function, implying besides the 'initial' state p(o) the time-interval t=t-tQ elapsed, but independent of the choice of the initial instant. This is the finite or integral way of expressing that the system is complete. Now let t=a be any particular instant and t=c another instant of time, such that
c—a + l). Then
4
so that the transformations corresponding to the passage of the system from any of its states to its successive states form a group of transforma- tions, t being the (only) 'parameter' of the group. Thus we can imitate Lie's general proof of his Theorem 3 (Sophus Lie, Theorie der Transfor- mationsgruppen, Leipzig, 1888 ; Vol. I.) for this simplest case of one
COMPLETE SYSTEM 19
parameter. Considering ^(o), ...pn(o\ a, c as independent variables, differentiate pi(c) with respect to a ; then
t(tf) da ^n(a} da 'db 'da
%w
but 3£/dtf= - i ; therefore
) PM*
da
/=!, 2, ...«.
Now pi(c\ ...pn(c) are mutually independent ; otherwise less than n quantities p would suffice for the determination of the state of the system, contrary to the supposition. Therefore the functional determinant
does not vanish identically, and the above system of n equations can be solved with respect to dp^(a)\da^ etc., leading to
*->«[AC*i -AW; *1 '=', v~*
But these equations must be valid for all values of the mutually in- dependent magnitudes b and a. Giving therefore to b any constant value, and writing t instead of #, we obtain for any /,
and this is the differential form alluded to, flt /2, ... fn being functions of the instantaneous state only.
It is instructive to consider the instantaneous state of a system as a point in the w-dimensional space, or domain of states 5n, (/i, pz ---pn\ and to trace in this ' space ' the lines of states, i.e. the linear continua of states assumed successively by different copies (exemplars) of the system, starting from given initial states. The differential equations of these lines of states, or, as Lie calls them, the 'paths (Bahncurven} of the corresponding infinitesimal transformation,' are
_= =
A A fn '
A complete system may then be characterized by saying that the lines of states are fixed in the corresponding space 5re, like the lines of flow of an incompressible fluid in steady motion. A copy of the system, or rather its representative point, placed on one of these lines remains on
20 THE THEORY OF RELATIVITY
it, moving along it in a determined sense. (For particulars of physical application of these concepts, see the author's paper in Ostwald's Annalen d. Naturphilosophie, Vol. II. pp. 201-254.)
Note 2 (to page 12). Systems obeying partial differential equations, as for instance that of Fourier,
_
adduced in the text, may be considered as systems of infinite degrees of freedom. The instantaneous state of such a system implies an infinite number of data^-, or say p=p(x,y, z\ given as a function of _r, j, z for every point of a portion of space coextensive with the system, as for example the instantaneous temperature for every point of a cooling body of finite dimensions, in which case the system will have co3 degrees of freedom. Instead of one we may have also two or more functions of x, y, z, defining the instantaneous state, as for example two vectors, amounting to six scalars, for an electromagnetic system (field), the differential equations being in this case those of Maxwell,
c>E .__ 3M lt,
-~-- = £.curlM, -~TT- = - c . curl E.
Here, as in the above example, the right-hand sides do not contain the time explicitly, but depend only on the space-distribution of magnitudes referring to the instantaneous state. If such be the differential equations and if also the limit or surface-conditions do not contain the variable / explicitly, the system of infinite degrees of freedom will be a complete or undisturbed one, in the sense of the word adopted throughout the chapter. Thus a heat-conducting sphere, of finite radius R, obeying in its interior Fourier's equation and whose surface is thermally isolated or radiates heat into free space, will be a complete system ; for its boundary con- ditions, viz.
30
Wr=° or
~j- = const, x (9- const.)
respectively, do not contain the time explicitly. But a sphere (like the earth), whose surface is kept at a generally variable temperature by means of external sources (like the sun), will be an incomplete system, unless we amplify it by taking in the ' sources ' themselves.
CHAPTER II.
MAXWELLIAN EQUATIONS FOR MOVING MEDIA AND FRESNEL'S DRAGGING COEFFICIENT. LORENTZ'S EQUATIONS.
THE modern principle of relativity arose on the ground of Lorentz's electrodynamics and optics of moving bodies. Einstein's work, in fact, consisted mainly in deducing logically, on the basis of plausible and sufficiently general considerations, certain formulae of space and time transformation, which in Lorentz's theory had partly a purely mathematical meaning and partly the character of an hypothesis invented ad hoc (' local time ' and the contraction hypothesis, respectively). In a word, Einstein has given a plausible support to, and a different interpretation of, what appeared already in the theory of the great Dutch physicist. In its turn, the theory of Lorentz, based on the macroscopic treatment of a crowd of electrons (though later supported and made vital by physical evidence of an entirely different kind), was constructed by its author chiefly with the purpose of accounting for optical phenomena in moving bodies, which may be best grouped summarily under the head of Fresnel's * dragging coeffi- cient ' and with which the equations of Maxwell and of Hertz- Heaviside have proved to be in complete disagreement.
Now, it seems to me that the best, most natural and most efficient way of propagating new ideas (if indeed there is such a thing arising in the collective mind of humanity) is to show their intimate connexion with older ones, and the more so when the new ideas have the reputation, widespread but partly unjustified in our case, of being of a very revolutionary character. It will be advisable, therefore, before entering upon our proper subject, to turn back to Lorentz and Maxwell. In doing so, I must warn the reader at the outset that the new Relativity, though grown on electromagnetic soil, does not — in spite of a current opinion — require us at all to adopt an electro-
22 THE THEORY OF RELATIVITY
magnetic view of all natural phenomena. Nor does it force upon us a purely mechanistic view, which till recently held the field, before the pan-electric tendencies arose. Modern Relativity is broader than this : it subordinates mechanical, electromagnetic and other images to a much wider Principle which is colourless, as it were.
Thus, the reason of returning here to Maxwell is, in the first place, of an historical (and partly didactic) character. But we have yet another reason for dwelling in the present chapter upon the great inheritance left to Science by Clerk Maxwell. It is widely known that but a few things of the old system of physics have remained untouched by the modern principle of relativity, though the changes required are generally but very slight. In fact, almost nothing of the old structure has been spared by the new theory of relativity ; but Maxwell's fundamental equations, namely those known as his equations for * stationary •' media, have been spared. More than this : not only have they been preserved entirely in their original form, without the slightest modification of any order of magnitude whatever, but they form one and the best secured of the actual possessions of the new theory, the largest and brightest patch of colour, as it were, on the vast and as yet mostly colourless canvas contained within the frame of the new Principle. Moreover, a peculiar union or combination of the electric and magnetic vectors which appear in Maxwell's equations of the electromagnetic field became the standard and prototype (not as regards physical meaning, but mathematical transformational properties) of a very important class of entities admitted by the new theory (the so-called 'world- six-vectors ' or * physical bi vectors ').
So much to justify the insertion of the following topics of the present chapter.
Maxwell's fundamental laws of the electromagnetic field in a * fixed ' or * stationary ' non-conducting dielectric medium * may be expressed in integral form as follows :
I. Electric displacement-current through any surface <r bounded by the circuit s = c x line integral of magnetic force M round s.
II. Magnetic current through <r = -rxline integral of electric force E round s,
* Practically, fixed with respect to the earth, or, if not, then with respect to a definite system of reference S, to be ascertained on further examination.
MAXWELL'S EQUATIONS 23
i.e. in mathematical symbols :
ii.
where (£, Jft denote the dielectric displacement or polarization and the magnetic induction respectively, c a scalar constant, the velocity of light in vacuum, n a unit vector normal to o-, the sense of the integration round s, of which ^s is a vectorial element, being clock- wise for a spectator looking along n (see Fig. i). Here, as throughout
FIG. i.
the volume, ((£n), etc., generally (AB), in round parentheses, denote the scalar product of a pair of vectors :
(AB) = ^cos(A, B),
A, B being the sizes or absolute values of the vectors A, B.* Thus, the surface element do- being considered as an ordinary scalar, the
surface integral I ((£n)dcr stands for the total number of Faraday tubes
(unit tubes) crossing o-, and the surface integral in n. has a similar meaning with respect to the tubes of magnetic induction.
* If it were only for purely vectorial algebra and analysis, we could write, after Heaviside, for the scalar product simply AB. But since we shall have to recur in the sequel to Hamilton's quaternionic calculus, we reserve AB for the full quaternionic product, and write therefore (AB) for the scalar product, i.e. for the negative scalar part of the Hamiltonian product, and VAB for the vector product, thus
AB = S. AB + V. AB
= -(AB) + VAB.
24 THE THEORY OF RELATIVITY
Remembering the definition of ' curl ' by means of the line integral, we may write I. and n. at once in differential form,
Sf-,c.,M,
or, in Cartesian expansion,
(la)
pf p
Every point or surface element of <r being fixed, relatively to the system of coordinates x, y, z, round d's have been written on the left hand to express partial differentiations with respect to /, i.e. local time-rates of change of the corresponding vectors.
(la) or (i) is the Hertz-Heaviside form of Maxwell's differential equations, although, if I am not mistaken, Maxwell himself on one occasion employed this form. At any rate, the Hertz-Heaviside equations for a stationary medium differ only formally from the equations of Maxwell as given in his monumental ' Treatise ' and in several papers ; the auxiliary potentials being easily eliminated.
As regards the relations obtaining between (£, JE and E, M respectively, it will be enough to remember here that the first pair of vectors are linear functions of the second, say,
(g = ^E and Jft = /*M, (2)
where A", /A are in the general case, of crystalline bodies, symmetrical or self-conjugate linear vector operators, which in the simplest case of an isotropic medium degenerate into ordinary scalar coefficients, the dielectric ' constant ' or the permittivity, and the magnetic permeability or the inductivity, — to adopt Heaviside's nomenclature.* Notice that, using the relations (2), K and //. being supposed given, we have in (i) two vectorial equations of the first order for two vectors, so that if the 'initial' state, say E0, M0, and eventually the limit-conditions, be given, the whole history of the field, past
* As yet we have no need to touch upon the subject of conducting media.
MAXWELL'S EQUATIONS 25
and future, is uniquely determined, — though in most cases the mathematician may have the greatest difficulties in finding it out. The electromagnetic field, as far as it obeys these equations, is at any rate a complete system in the sense of the word previously explained. It will be noticed later that the funda- mental equations of the electron theory do not possess this simple property.
From i., ii. we see immediately that the total current, electric or magnetic, through all possible surfaces a- bounded by one and the same circuit (s), has the same value. Taking therefore a pair of such surfaces crl, <r2, which together form a surface (o-), enclosing completely a certain portion r of the medium, and inverting one of the normals of the component surfaces (Fig. 2), so that the
n=n.
FIG. 2.
normal n is directed everywhere outwards (or everywhere inwards) with respect to the enclosed space, we see that, for any closed surface (<r),
I (£n)^cr, (Jttn)dr = const, in time, J(o-) J(<r)
the second constant being everywhere equal to zero, by experience. In other words, the total electric charge enclosed by (<r) does not vary in time, its magnetic analogue being constantly non-existent. The same property being valid for any volume T, and remembering that ' div ' or divergence is defined as the surface integral of a vector per unit of enclosed volume, we may write also, in differential form,
div (& = p = const, div JE = const. = o ;
26 THE THEORY OF RELATIVITY
p is the volume density of (true) electricity. The second property is commonly expressed by saying that the tubes of magnetic induc- tion are always closed, or that Jft has a purely solenoidal distribution. The invariability of both divergences may be seen with equal ease from (i), remembering that the operations div and 3/9/ are com- mutative, while div curl = o, identically.
Thus, the full system of Maxwell's equations for a stationary dielectric, which we will put here together for future reference, is
(3)
-^.curlM
-r.curlE; divjtt = o
the equation
/o-div(g
being here considered as the definition of the density p of electric charge. Notice, in passing, that the ' electric charges ' have been driven to the background by the Maxwellian theory (especially as propagated by Hertz, Heaviside and Emil Cohn), as rather secondary derivate entities, but to return later with increased vigour and to reacquire their dominant position, viz. as fundamental elements of the electron theory.
We shall not stop here to consider the general Maxwellian ex- pressions of energy, ponderomotive force and of the corresponding stress.
In vacua, and practically also in air under ordinary conditions,
so that Maxwell's equations (3) become
3E
-~- = c . curl M ot
^r- = - c . curl E ot
div M = o, to which in the present case may be added also
divE = o (4^
expressing the absence of electric charge. Notice in passing that these equations are not invariant with respect to the Newtonian
(4)
MAXWELL'S EQUATIONS 27
transformation. The transformation which does preserve their form is of a different kind, as will be seen later.
The independent variable t appearing in Maxwell's equations (4) for empty space may be taken, provisionally at least, as far as experience goes, to be the ordinary or the kinetic time. And as regards the (or a) space-framework, with respect to which they are intended to be rigorously valid, let us call it once and for ever the system S, whatever it may be. If the reader wants to fix his ideas he may think of S as the ' fixed-stars ' system ; but as yet we cannot and need not discuss this point thoroughly, being forced by the very nature of the question to postpone it to a later chapter. At first sight it might seem that (4) are wholly independent of a space-frame of reference ; for the curls and div's can be, and primarily are, defined in terms of line integrals and surface integrals respectively, and thus depend only upon the distributional peculiarities of the respective vector fields. But this means only that the equations in question are independent of the choice of axes (#, jr, z) within S, the only condi- tion being that they must be immovable relatively to 6" ; in other words, curl E, curl M are vectors as good as E, M themselves * and div E, div M are true scalars like a volume, for instance. Notice, however, that, on the left hand of the equations, 3/3 / is to be the local time rate of change of E or M, i.e. the variation in a point P kept fixed. Now, this would be altogether meaningless if it is not explained with respect to what frame the point P is to be fixed. It would not help us very much if somebody told us that P is to be a fixed point of the field or of a Faraday tube ; for we have no means of identifying such a point. The truth of what has just been said may be seen even more im- mediately from the integral form of Maxwell's equations, i. and n., where for the present case <£, JE are to be identified with E, M ; for the circuit (s) is to be kept 'fixed,' i.e. fixed with respect to something.! Therefore we necessarily want a frame of reference, and call it S.
* The distinction of what are called axial and polar vectors does not concern us here.
fin the more general case of a ponderable medium, say in a piece of glass, the circuit (s) is, of course, to be fixed in the glass ; but this would not be enough : the whole piece of glass, as will be explained presently, must not move in an arbitrary manner relatively to some external frame or other, if the laws I., n. are to be valid, whether the observer does or does not share its motion.
28 THE THEORY OF RELATIVITY
To see the property of the scalar constant c, eliminate, in the usual way, E or M, employing their solenoidal properties ; then
?&-*•+• (5)
where <£ means E or M, or any one of their Cartesian components £lt ... M3; hence, in the case of plane waves, for example,
and
/ being an arbitrary function of the linear argument. Thus c, in round figures 3.io10 cm. sec.'1, is the velocity of propagation in empty space, relatively to S, of transversal electromagnetic waves or disturbances, their transversality being an immediate consequence of the solenoidal conditions, which, in the present case, reduce to 'd£l/'dx = o, fdMll'dx = o. Henceforth c will be referred to shortly as 'the velocity of light,' and sometimes as the 'critical' velocity.
What is properly called a wave is a non-stationary surface of discon- tinuity of E, M themselves or of their derivatives, which is individually recognizable as such and can be watched when moving about. It is the velocity of motion of such a wave, normal to itself, which is properly called the velocity of propagation, as distinguished from the phase-velocity of a continuous train of disturbances. Now it may be easily shown that c is precisely the value of this true velocity of pro- pagation for any form of the wave, plane or not, the property belonging to every surface element of the wave, considered separately. (See Note i at the end of the chapter.)
Notice that this property is quite independent of the direction of the wave normal, i.e. of its orientation with respect to any axes drawn in S. In other words :
Maxwell's equations imply isotropic as well as uniform * propaga- tion in empty space relatively to S, i.e. to that system in which they are valid. There are no privileged places or directions for the electromagnetic disturbances.
Thus a continuous train of spherical waves, with centre O, will remain spherical for ever, which may be seen also from (5). For a
* By ' uniform ' we mean homogeneous or constant in space and invariable in time, c being constant with respect to both.
ISOTROPIC PROPAGATION 29
particular integral of that equation, adaptable to any initial state <t>0 = -f(r)t is <£ = -f(r ± ct\ r being the scalar distance measured
from O. Again — which is more satisfactory — if cr be at any instant a spherical surface of transversal discontinuity or a proper electro- magnetic wave, then, expanding (or shrinking) with time, it will remain spherical for ever, with centre O coinciding always with that of the original cr, fixed once and for ever with respect to the frame S, — quite independently of whether and how the material source was moving at the instant when it originated that wave. Thus a * point-source ' (and notice that a physical source of any shape or finite dimensions may be regarded as such, provided we go away from it far enough) producing a solitary disturbance, say a flash of light, at the instant /0, will originate a wave which always will be spherical of radius
having its centre where the source was at the instant /0, no matter whither it went afterwards or whence it came, or how swiftly it flashed through that place.
We shall have to return to this argument, of capital importance, more than once ; but meanwhile we must leave it.
As has been already remarked, Maxwell's equations for '•stationary ' dielectrics, i.e. I. and n. with their supplements as given together with their differential form under (3), have not only survived the general massacre, but have very substantially enriched the new theory. In fact, both the most particular and simple equations (4) for the vacuum and the more general ones, (3), for ponderable media have been incorporated into the possessions of modern Relativity, the former in a quite easy way by Einstein (1905), and the latter in a less easy and very ingenious way by Minkowski (1907). On the other hand, it is needless to tell here again about the wide field of experience covered by these equations and about their numerous and successful applications in proper Electro- magnetism, to say nothing about the electromagnetic theory of light which soon after its creation proved to be much superior to the elastic theory.
Serious difficulties arose only in connexion with the electro- dynamics, and more especially with the optics of moving media, a long time before the dates just quoted.
30 THE THEORY OF RELATIVITY
There are two different sets of what are commonly called Max- wellian equations for moving media : i° a system of equations which may be gathered together from different chapters of Maxwell's ' Treatise,' and which we shall call shortly the equations of Maxwell, though it can be reasonably doubted whether Maxwell himself would consent to attribute to them general validity, especially with the inclusion of optics; and 2° a system of equations which Hertz obtained by a certain, apparently the most obvious, extension of the meaning of the form i., 11., and which Heaviside, inde- pendently, constructed by introducing into Maxwell's equations a supplementary term dictated by reasons of electro-magnetic sym- metry ; these are widely known as the Hertz-Heaviside equations for moving bodies.
We shall use for i° and 2° the abbreviations (Mx), (HH). Neither has been able to stand the test of experience. Though contrary to the historical order, it will be more instructive to con- sider first the latter and then the former system of equations.
Let us return to the semi-integral form of electromagnetic laws i. and ii., given, in words and symbols, on pp. 22-23. These are valid for a ponderable dielectric medium or body, stationary with respect to our frame S, and for any surface a- which, together with its bounding circuit s, is fixed in the body. Thus the surface o-, through which the 'current' is to be taken, is itself fixed in S. Now, what Hertz did in order to obtain the required extension, was simply to suppose that i. and 11. are still valid for a body, rigid or deformable, moving with respect to S in any arbitrary manner, provided that the currents on the left-hand side of these equations are taken through a surface composed always of the same particles of the body, or— to put it shortly — through an individual o-, together with its s. This gives for the current per unit area of <r, instead of the local time-rate of change 9®/3/, if v be the velocity of a particle relatively to S,
^ + vdiv@ + curlV<£v, (6)
and a similar expression for the magnetic current,* while the right- hand sides of i., ii., containing only the instantaneous values of line integrals, remain obviously unaffected by the Hertzian require- ment. The distribution of JE being supposed solenoidal, as
*See Note 2.
HERTZ-HEAVISIDE EQUATIONS
previously, the second term in the above expression is absent in the magnetic current. Thus, transferring the curl-terms of the currents to the right-hand sides, we obtain the required equations
(HH)
Heaviside calls V(£v/<: the ' motional magnetic force ' and Vvjft/r the 'motional electric force,' considering them as a kind of impressed forces.
In what we have called Maxwell's equations, the former of these ' motional forces ' and the convection current pv are absent ; other- wise they are as (HH) ; thus
= c . curl M
(MX)
The connexions between <£, Jft and E, M are as in (3), except that K, fj. may undergo continuous variations due to the strain of the material medium. Also, div,.|ft = o, as in (3). Notice, in passing, that the first of (HH) gives
or
dp dt
+ p div v = o,
where <//9/*#=3/>/d/-f(vV)p is the variation at an individual point of the body. Now, div v being the cubic dilatation, per unit time and per unit volume, the last equation may at once be written
where dr is an individual volume-element of the material medium, i.e. an element composed always of the same particles. Thus the charge pdr of any such element remains invariable, being attached to it once and for ever. The charge, being preserved in quantity, moves
32 THE THEORY OF RELATIVITY
with the body. In this respect it behaves like the mass, according to classical mechanics. As regards the equations (Mx), they must be considered as referring to the particular case of an uncharged body ; Maxwell happened not to consider explicitly charges in motion ; otherwise he wo^ld doubtless have brought in the term pv.
Now, both of these systems of equations, (Mx) as well as (HH), are in full disagreement with experience, especially with optical experience, terrestrial and astronomical, i.e. with experiments on the propagation of electromagnetic waves (light) in bodies moving relatively to the observer, and also in bodies moving with the observer and with his apparatus relatively to the source, say relatively to a star. *
The equations in question have also been manifestly contradicted by electromagnetic experiments properly so called, viz. those of H. A. Wilson and of Roentgen and Eichenwald ; * but it will be enough to consider here only the difficulties met with on optical ground, the other deviations being of essentially the same character, while the optical examples, quite conclusive by themselves, seem to be very instructive.
Let me explain to you fully what this disaccordance consists in.
To take the simplest case possible, let the material medium or
tf
FIG 3.
body move as a whole with uniform translational velocity v with respect to S, and let plane waves of light be propagated in it along the positive direction of v (Fig. 3). If the unit-vector i be the wave normal, concurrent with the propagation, then v = vi. Let fo' be the scalar velocity of propagation of the waves, when the material medium
* H. A. Wilson, Phil. Trans., A. Vol. CCIV. p. 121 ; 1910.— W. C. Roentgen, Berl. Sitzber., 1885; Wiedem. Ann., Vol. XXXV. 1888, and Vol. XL. 1890.— A. Eichenwald, Ann. der Physik, Vol. XI. 1903.
THE DRAGGING COEFFICIENT 33
is stationary in S, and b their velocity of propagation, . as judged from the ^-standpoint, when the medium is moving with its actual velocity. What is the relation between b and b', v? If we were concerned with waves of sound, instead of light waves, then b would be simply the sum of b' and of the whole v; the waves would be entirely dragged by the medium, say air or water, with its full velocity. But the case before us is different. Write, generally,
or
then K, whatever its value, will be what is called the dragging coefficient, indicating the fraction (if it happens not to be the whole) of the medium's velocity conferred upon the waves. What is, then, the dragging coefficient in the case of electromagnetic, and especially of luminous waves?
According to (HH) it is, obviously, equal to unify. To see this we have no need to integrate these differential equations,* but simply to remember Hertz's interpretation of the laws i., n., which furnished him with these equations (p. 30). For according to that interpreta- tion, and extension, of i., n., the electromagnetic disturbances behave relatively to the material medium (generally in each of its elements, and in the present case, of rigid translation, throughout the whole medium) just as if it were stationary. Hence, on the ground of classical kinematics of course, the velocity of the medium is simply added to that of the waves, precisely as in the case of sound. Thus, K = i, according to (HH).
Let us now see what is the value of the dragging coefficient according to (Mx). Take the simplest case of an isotropic medium ; then
where, by the way, /* = i for light waves. Measuring x along i in the system S, take E, M, and therefore also (£, JH, proportional to a function of the argument ^-b^, so that b will be the velocity of
* Though the reader, to satisfy himself, may do so. Proceeding similarly as in the case of (Mx), worked out in Note 3 at the end of this chapter, he will soon find that b = ti' + v.
S.R. C
34 THE THEORY OF RELATIVITY
propagation relatively to S, as above, and by a simple calculation (Note s)
or
b = *'(i+*«W* + ^, (7)
where /3 = v/c and where 72 = <r/V is the index of refraction of the medium. Now, in all actual experiments, by means of which the dragging of light can be determined, /? is a small fraction, viz. io~4 in the case of Airy's astronomical, and much smaller in that of Fizeau's terrestrial experiment, both to be considered later. There- fore terms of the order of /24 can certainly be rejected, so that
and
but here even the /3-term may be safely omitted, so that finally
Thus, we have for the dragging coefficient according to (HH) and (Mx), respectively,
K=I, (HH)
K=K. (MX)
Now, both of these are radically wrong, the true one, i.e. that showing excellent agreement with experiment, being Fresnel's widely known dragging coefficient (coefficient d? entrainement)
•irtu j
K - i - — , (Frsnl)
/ /• ^"
• • -
where n is the index of refraction. It is, for more than one reason, worth our while to dwell here upon the interesting history of Fresnel's coefficient.
The phenomenon of stellar aberration, discovered by Bradley in 1728, found its immediate explanation when the assumption was made that the light-waves do not share in the earth's orbital motion,
*This result was obtained by J. J. Thomson. See Heaviside's Electromagnetic Theory, Vol. Ill, §471 et seq., where some interesting remarks regarding this and allied subjects may be found.
ABERRATION 35
and, consequently, in the motion of the tube of the telescope (if filled with air or empty). In fact, making this assumption, the aberrational formula
and, for 0 = Tr/2, - $ °
- = sin </> = tan </>, (8a)
is easily obtained by using the widely known analogy of a ship in motion pierced by a shot fired from a gun on the shore.
Formula (8) gave, from Bradley's observations (<f> = 2o"-44) and from the known velocity v of the earth's motion (30 kilom. per second), a value for c, the velocity of propagation of light, which agreed very
Earth
FIG. 4.
closely with that obtained by Romer in 1676 from observations of the eclipse of Jupiter's satellites. Thus (8) was verified. To state the bare facts, it would have been enough to say simply that the tube of the telescope, or the air contained in it, does not carry with it the light coming from the star, whatever it may consist in (corpuscles or waves). But to make the statement more tangible, it has been said that the 'corpuscles' or the 'aether,' respectively, do not share in the telescope's motion. Whereas aberration was explained by its discoverer in terms of the corpuscular theory (each corpuscle of light corresponding then most immediately to the shot in the above analogy), it was Young who first showed (1804) how it may be explained on the wave-theory of light and on the hypothesis that the aether ' pervades the substance of all material bodies with little or no resistance, as freely perhaps as the wind passes through a grove of
36 THE THEORY OF RELATIVITY
trees.' * This picturesque analogy fitted altogether the case of air, which behaves very nearly like a vacuum, but not glass or water, for which the ' grove of trees ' had to be replaced by a rather dense thicket. But at any rate the above words of Young hit very near the truth.
To put it shortly, in the case of air" the dragging is «//, or nearly so, K === o.
But the case is different for optically denser media, having, for light of a given frequency, an index of refraction ;/, sensibly different from unity. For if K were nil also for such media, we should have to replace c in (8) by the smaller velocity of propagation cjn^ (so) tharTthe angle of aberration ^vould be) different for optically different media, whereas it has been proved experimentally to be just the same as in the case of air. More generally, Arago concluded from his experi- ments on the light of stars that the earth's motion has no sensible influence on the refraction (and reflection) of the rays emitted by these light-sources, i.e. that the rays coming from a star behave, say, in the case of a prism or a slab of glass, precisely as they would if the star were situated at the point in which it appears to us in conse- quence of ordinary Bradleyan (air-telescope) aberration, and the earth were at rest relatively to the star. Arago himself tried to explain this result of his experiments on the corpuscular theory, and on the supplementary hypothesis that the sources of light impress upon the corpuscles an infinity of different velocities, and that out of these none but those endowed with a certain velocity (±-oi %) have the power of exciting our organ of sight. But this strange hypothesis entangled him in a maze of difficulties, and the whole theory, not free from other difficulties, does not seem to have satisfied its author. At any rate, Arago proposed to Fresnel to investigate whether the above result of his observations could not be more easily reconciled with the wave theory of light.
It was in answer to this invitation that Fresnel wrote in 1818 his celebrated letter to Arago * on the influence of the earth's motion upon certain optical phenomena,'! in which he gives a beautiful
* Phil. Trans., 1804, p. 12, — as quoted by Whittaker in A History of the Theories of Aether and Electricity -, p. 115 ; London, 1910.
t ' Lettre d'Augustin Fresnel a Francois Arago, sur 1'influence du mouvement terrestre dans quelques phenomenes d'optique,' Annales de chim. et de phys.y Vol. IX. p. 57, cahier de septembre, 1818; reprinted in Fresnel's CEuvres com- pletes, Vol. II., Paris, 1868; No. XLIX. pp. 627-636.
FRESNEL'S DRAGGING COEFFICIENT 37
solution of the problem, and which has since become one of the most solid supports of modern inquiry into the optics of moving media. Here appears for the first time his 'coefficient d'entrainement,' already mentioned above. Fresnel based the theory of aberration, and associated matters, on the following hypothesis, which turned out to be a very happy guess indeed :
Fresnel supposed that the excess, and only the excess, of the aether contained in any ponderable body over that in an equal volume of free space is carried along with the full velocity, v, of the body ; while the rest of the aether within the space occupied by the body, like the whole of the free aether outside, is stationary, — with respect to the fixed stars, of course.
This amounts * to supposing that the velocity of propagation of the light-waves is augmented only by the velocity of the ' centre of gravity ' (centre of mass) of the whole mass of the aether contained in the body. This velocity will, generally, be but a fraction of v. Call it KV ; then K will be what has above been called the dragging coefficient. Let pQ be the density of the aether outside the body, and p its density within the body; then, by Fresnel's hypothesis,
or
* = I ~ PO/P'
Now, e being the coefficient of elasticity of the aether within the body, and eQ that of the free aether, the body's refractive index n is given by
*-A./f.
PojP
But Fresnel's aether has throughout the same elasticity, within ponderable bodies and interplanetary space, so that e = e0 and
Thus we obtain Fresnel's celebrated formula for the dragging coefficient :
(Frsnl)
Notice that considering the excess of the aether, i.e. p- pQ per unit volume, as a permanent part of material bodies, it can be said simply that the aether proper is not moved at all, that it is entirely
* See the letter in question, p. 631 of reprint in Vol. II. of (Euvres completes.
38 THE THEORY OF RELATIVITY
uninfluenced by the moving bodies. Fresnel's theory is therefore usually alluded to as the theory of a fixed aether. Implicitly, this aether of Fresnel is supposed to be fixed relatively to the stars, or at least to those stars which have been concerned in the aberrational observations.
For a vacuum, or air, n = i and K = o. Thus, first of all, Fresnel's theory is in perfect agreement with Bradley's observations. For other media n>i and O<K<I, or the dragging is partial^ and increases with the optical density of the medium.
By means of his dragging coefficient Fresnel treated fully the problem of refraction in a prism, showing that it must be sensibly * uninfluenced by the earth's motion, in agreement with Arago's obser- vations. This problem, in fact, was the chief object of the letter quoted.
To close his admirable letter, Fresnel gives an application of his theory to an experiment, suggested previously, in 1766, by Boscovich,f consisting in the observation of the phenomenon of aberration with a telescope filled with water, — commonly called 'Airy's experiment.' Fresnel infers from his formula for K, by simple and most elegant reasoning, that if observations were made with such a telescope, the aberration would be unaffected by the presence of the water. This result was verified, for the first time, by Sir G. B. Airy in 1871, in the observatory of Greenwich. His observations on 7 Draconis, during 1871-1872, proved indeed that the presence of water, in place of air, has jip sensible, i.e. no first-order (vjc) influence on the aberration.
* i.e. as far as theyfrj1/ power of v\c goes.
t R. J. Boscovich (or Boskovic), born in Ragusa 1711, died in Milan 1787. The principle of the water-telescope was first explained by Boscovich in a letter to Beccaria in 1766, and then fully developed in the second volume of his optical and astronomical papers, Opera pertinentia ad opticam et astronomiam ; Basani, 1785? Vol. Ill, opusculum III. pp. 248-314. An interesting account of the work (and life) of Boscovich is given by G. V. Schiaparelli in a manuscript, SulF attivith del Boskovic quale astronomo in Milano, edited recently by Dr. V. Varicak (Agram, South Slavic Acad. of Sc., 190; 1912). In connexion with the subject of our Chap. I., the reader may also be warmly recommended to consult another paper of Boscovich, edited by Dr. Varicak (ibidem, 190 ; 1912) : De motu absolute, an possit a relative distingui, originally a supplement of Boscovich to Philosophiae recentioris a Benedicto Stay versibus traditae, Libri X, ; Vol. I. p. 350 ; Rome, J755' This paper, which is missing even in Duhem's bibliography of the subject (Le moitvement absohi et le mouvement relatif, 1909), contains many remarkably clear and radical ideas regarding the relativity of space, time and motion.
For both of these pamphlets I am indebted personally to Dr. Varicak.
THE BOSCOVICH-AIRY EXPERIMENT 39
Though Fresnel's own reasoning, reprinted at the end of the present chapter (Note 4), exhausts the subject entirely, let us yet dwell upon it a moment.
If the aether behaved in optically denser bodies as in air, i.e. if there were no dragging at all, we should have, by the ship and shot analogy, instead of (8),
v _sm<j> cfn~ sin 6»'
f/n being the velocity of propagation of light in water, or in any other medium filling the tube of the telescope. Then Airy's experiment would have given a positive result. But he obtained precisely the same <£ as for air. This negative result suggested to him (at least as it is usually represented in text-books) the supposition that the k water carries with it the aether ' with only a certain part of its velocity, namely such that, in the above formula, we have to write v instead of v, where
v = v/n, so that
sin</>_ v _v
sin0 ~f/n~f1
as for air. In reality the process of compensation is not so simple as this; but in Airy's experiment the compensation — sensibly complete — is produced in a slightly different way. Considering a slab of water moving perpendicularly to its axis, and neglecting second-order terms (i.e. vi\cL= io~8), you will easily obtain*
sm4>_(v-v)c , .vn* ( }
-—=v-«>— '
where, v-v being the relative velocity of the aether and telescope, K = vjv has been written for the dragging coefficient, as yet supposed to be unknown. Hence, to account for Airy's negative result, i.e. to make (9) identical with (8), we have to write (i — K)«2= i, or
as in Fresnel's formula.
* See, if necessary, for instance N. R. Campbell's Modern Electrical Theory, Cambridge, 1907 ; pp. 293-294 (but interchange the dashes at P> C, 0, Q in his Figure 28, which are placed the wrong way ; correct also some dashes on p. 294 and read at the bottom of the page ' presence ' instead of ' pressure.' As regards Fizeau's experiment, amend the shocking anachronism on p. 295 : ' Fizeau tried ' — 1851 — 'to test the correctness of Airy's hypothesis' — 1871).
THE THEORY OF RELATIVITY
Thus, Airy's negative result is perfectly accounted for by Fresnel's dragging coefficient, terms of the order of io~s being, of course, beyond the possibility of observation.
But Fresnel's formula found also, twenty years earlier, an im- mediate verification in Fizeau's optical interference-experiment with flowing water.* The arrangement of the apparatus which was used by Fizeau is seen at a glance from Fig. 5. Light from a narrow slit, S, after reflection from a plane parallel plate of glass, A A, is rendered parallel by a lens L and separated into two pencils by apertures in a screen EE placed in front of the tubes T19 T2 containing running water. The two pencils, after having traversed (towards the left hand) the respective columns of water, are focussed, by the lens £, upon a plane mirror Z, which interchanges their paths : the upper pencil returns towards L by the tube T2, the lower by T^. On
FIG. 5.
emerging finally from the water, both pencils are brought, by Z, to a focus behind the plate AA, at S' (and partly also at S). Here a system of interference fringes is produced which can be observed and measured in the usual way. Thus, each pencil traverses both tubes, T^ and T2, i.e. the same thickness of flowing water, say /. Moreover, the (originally) upper pencil is travelling always with, the other against the current. If, therefore, v be the velocity of the water and K the dragging coefficient, the difference in light-time for the two pencils will be given by
cln — KV c\n + KV) '
where n is the refractive index of water. Passing from stationary to flowing water, Fizeau observed a measurable displacement of the interference fringes, namely with #=700 cm./sec. ; and by reversing
*H. Fizeau, Comptes rendus, Vol. XXXIII., 1851; Annales de Chimie, Vol. LVIL, 1859.
FIZEAU'S EXPERIMENT 41
the direction of the current of water the displacement of the fringes could be doubled. From the observed displacement it is easy to find the difference of times A, and by equating it to the above expression of A to find the dragging coefficient K in terms of /, ;/, ?-, which can be measured. The result of Fizeau's experiment was that K is a fraction, sensibly less than unity. How much less, could not be ascertained with sufficient precision. Fizeau's experiment was therefore repeated in a form modified in several important points by Michelson and Morley* (1886), who found, for water (moving with the velocity of 800 cm. per second) at 18° C, and for sodium light,
K = 0-434 ±0-02, (MM)
i.e. 'with a possible error of ±0-02.'
Now, n being, in the case in question, equal to 1-3335, Fresnel's formula gives
K= i -4, = 0-438, (Frsnl)
a value agreeing very closely with Michelson and Morley's experi- mental result.
Thus, Fresnel's formula, deduced from what in our days may be deemed an assumption of naive simplicity, proved to be in admirable conformity with experiment, like everything predicted by Fresnel in optics^ His dragging coefficient has acquired a special importance in recent times, and every modern theory is proud to furnish his *, which has become, in fact, one of the first requirements demanded from every theory of electrodynamics and optics of moving bodies which is being proposed. 'Agreeing with Fresnel' has become almost a synonym of 'agreeing with experience.'
Now Maxwell's and Hertz-Heaviside's equations for moving media, (Mx) and (HH), giving, as we have just seen, *=4 and *c= i, or half and full drag, respectively, for any medium, be it as dense as water or glass or as rare as air, proved thereby to be in full disagreement with Fresnel, i.e. with experiment.
The first successful attempts to smooth out this discordance of (Mx) and (HH) from experiment, which — as has been mentioned — manifested itself also in the case of electromagnetic experiments properly so called, were made by H. A. Lorentz in 1892. The
* Michelson and Morley, American Journ. of 'Science, Vol. XXXI. p. 377 ; 1886. See also A. A. Michelson's popular book, Light Waves and their Uses ; Chicago
1907 ; p- 155-
42 THE THEORY OF RELATIVITY
theory proposed in a paper published in that year,* and which led with sufficient approximation to Fresnel's dragging coefficient, was then simplified and extended in 1895, m a paper f which has since become classical.
Stokes' moving aether (1845) leading to serious difficulties,! Lorentz decided in favour of Fresnel's immovable, stationary aether, as the all-pervading electromagnetic medium.
Thus, Lorentz's theory, presently known widely as the Electron Theory, is, first of all, based on the assumption of a stationary, isotropic and homogeneous aether. In calling it shortly ' stationary ' (ruhend\ Lorentz states expressly that to speak of the aether's 4 absolute rest ' would be pure nonsense, and that what he means is only that the several parts of the aether do not move relatively to one another (Essay , p. 4). In other words, Lorentz's aether is not deformed, it is subjected to no strain, and does not, consequently, execute any mechanical oscillations. And this being the case, it has, of course, no kind of elasticity, nor inertia or density. It is thus far less corporeal than Fresnel's aether. One fails to see what properties, in fact, it still has left to it, besides that of being a colourless seat (we cannot even say substratum) of the electromagnetic vectors E, M. And although Lorentz himself continues to tell us, in 1909,! that he 1 cannot but regard the ether as endowed with a certain degree of substantiality,' yet, for the use he ever made of the aether, he might as well have called it an empty theatre of E, M, and their perform- ances, or a purely geometrical system of reference, stationary with regard to the (or at least to some) ' fixed ' stars. This aether, having been deprived of many of its precious properties, was at any rate already so nearly non-substantial, that the first blow it had to sustain from modern research knocked it out of existence altogether, — as will be seen later. Still, substantial or not, for the theory of Lorentz we are now considering, it is something, namely its unique system of reference. So long, therefore, as it was thought that there is such an
* H. A. Lorentz, La thtorie flectroinagnttique de Maxwell et son application aux corps motivants ; Leiden, E. J. Brill, 1892 (also in Arch. ngerL, Vol. XXV.).
t H. A. Lorentz, Versiich einer Theorie der elect rise hen ^lnd optischen Erschein- zingen in bewegten Korpern; Leiden, E. J. Brill, 1895. This paper will be shortly referred to as 'Essay* (= Versuch],
% See Note 5 at the end of this chapter.
§ Lorentz, The Theory oj Electrons •, etc. , Lectures delivered in Columbia University, 1906; Leipzig, Teubner, 1909; p. 230.
LORENTZ'S EQUATIONS 43
unique system, Lorentz's all-pervading medium could continue its scanty existence.
For this free aether, i.e. where it is not contaminated by the presence of ponderable matter, Lorentz assumes the exact validity of Maxwell's equations, (4), i.e.
— = c . curl M ; -^-7 = —c. curl E ; div M = o,
Of Ct
with /) = divE = o. (As to terminology, Lorentz calls the above E the dielectric displacement ', and M the magnetic force.}
Then, to account for the optical and, more generally, electro- magnetic phenomena in moving ponderable matter, he has recourse to electro-atomism, an hypothesis already employed (1882-1888) by Giese, Schuster, Arrhenius, Elster and Geitel, and others, and later also by Helmholtz (1893) in his famous electromagnetic theory of dispersion, and in various writings of Sir Joseph Larmor. According to Lorentz, matter by itself has no influence whatever on the electromagnetic phenomena : in this respect it behaves like the free aether. Only when and as far as matter is the seat of 'ions,' in Lorentz's, or electrons in modern terminology,* it modifies the electromagnetic field and its variations. In other words, Maxwell's equations, (4), are assumed to be strictly valid not only in the free aether, but also in all those portions of ponderable molecules in which there is no charge, i.e. wherever p = o. And as to the question whether ponderable matter consists entirely of electrical particles (charges) or not, Lorentz leaves it an open question. If I may venture an opinion, it was very wise of him not to have had M. Abraham's ambition to construct a purely electromagnetic " Weltbild,' as the Germans call it. (This remark will be under- stood better later on, when we shall see that, as far as we know, even the mass of the free electrons, such as the kathode ray- or /3-particles, may not be of purely electromagnetic origin.) The part played in Lorentz's theory by matter itself consists only in keeping the electrons, or at least some of them, at or round certain places, say, restraining them from too wide excursions. Maxwell's equations, as written above for the free aether, are modified only where
div E = p =f o,
*' Electron ' is due to Johnstone Stoney (1891). The distinction made now between ' ions ' and ' electrons ' does not concern us here ; besides, it is generally known from a host of popular writings.
44 THE THEORY OF RELATIVITY
i.e. where there is, at the time being, some electric charge or electricity, and where, moreover, the electricity is moving.* The * modification is the slightest imaginable,' to put it in Lorentz's own words (Electron Theory, p. 12). If p be the velocity of electricity at a point, relatively to the aether, i.e. relatively to that system of reference, S, in which the free-aether equations (4) are valid, then the left-hand member of the first of these equations, or the displace- ment current, is supplemented by the convection current, per unit area, i.e. by pp, while the second and third equations remain unchanged.
Thus, Lorentz's differential equations, assumed to be valid exactly or microscopically^ throughout the whole space, are
^T + PP = c . curl M, where p = div E
I /T \
3M I
-—=-<:. curl E ; div M = o.
These have been since generally called the fundamental equations of the electron theory. They contain, of course, the equations for the free aether as a particular case, namely for p = o.
An important supplement to the above system of equations con- sists in the formula for the ponderomotive force ' acting on the electrons and producing or modifying their motion,' which, guided by obvious analogies, Lorentz assumes to be, per unit volume,
(ii.) or, per unit charge,
(10)
This ' force ' is supposed to be exerted by the aether on electrons or matter containing electrons. Vice versa, as Lorentz states it expressly, matter, whether containing electrons or not, exerts no action at all on the aether, — since the aether has already been supposed to undergo no deformations, etc. Of course, Lorentz's aether is massless as well. Lorentz tells us, with emphasis, not to
*This, of course, implies the possibility of our following an individual portion or element of charge in its motion, — a subtle point (due to circuital indeterminate- ness, etc. ), which, however, need not detain us here.
fTo be contrasted afterwards with his macroscopic (or average) equations.
LORENTZ'S EQUATIONS 45
bring in even the notion of a ' force on the aether.' It is true — he adds — that this is against Newton's third law (action = reaction), ' but, as far as I see, nothing compels us to elevate that proposition to a fundamental law of unlimited validity' (Essay, p. 28).
But there is no need to keep in mind all these, and similar, re- marks and verbal explanations, — especially as the absence of force on the free aether is seen from (11.) at a glance, by putting /> = o.
It is perfectly sufficient to state that the basis of Lorentz's theory is entirely contained in the above (microscopically valid) equations (i.), (n),* all other things being obtained from these equations by more or less pure deduction, without new hypo- theses.!
Notice, in passing, that (i.) is not a complete system in the sense of the word explained in Chap. I. For to trace the electromagnetic history, not only E0, M0 for / = o and for the whole space, but also p and p for all values of / must be given. In (i.) we have, essentially, two vector equations of the first order for three vectors E, M, p, and the formula (n.) does not complete the system, since, on further research, it does not lead to an equation of the form Sp/3/ = 12 (E, M, p), + but in the most favourable case to an integral equation extending over a certain interval of time, generally finite, but sometimes indefinitely prolonged. But this 'incompleteness' is no disadvantage in (i.), (ii.), especially for the purpose of macroscopic treatment, in which consisted Lorentz's main object of constructing these equations.
The equations assembled in (i.), which, together with the formula for the ponderomotive force, have been received into the domain of modern Relativity, as will be seen later, can be easily condensed into a single quaternionic equation. First of all, put
B = M-*E (u)
(where t = \/ - i ), and call it the electromagnetic bivector. Also write, for convenience,
l=ict. (12)
* These are also the equations of Larmor, who started from the conception of a quasi-rigid aether and deduced the equations in question from the principle of least action. (Aether and Matter •, Cambridge, 1900.)
tTill he comes to Michelson and Morley's famous interference experiment.
£f2 being some space-operator and E, M, p the instantaneous values of the three vectors or vector-fields.
46 THE THEORY OF RELATIVITY
Then, the first and third, and the second and fourth of (i.) coalesce respectively into the bivectorial equations
and
div B = - ip ;
or, in Hamilton's symbols,
SVB = - (VB) = - div B = ip.
Add up, and remember that the full quaternionic * product ' of the Hamiltonian V and of the bivector B is
then
Next, introduce the operator
which will turn out to be of fundamental importance for our subse- quent relativistic considerations, and the quaternion
('4)
which we may call the current-quaternion. Then the last equation becomes
£>B=C. (i.a)
Thus, the four vectorial equations in (i.) coalesce into a single quaternionic equation (i. a), whose form will be most convenient for relativistic electromagnetism. It is scarcely necessary to say that what we have done here has nothing to do with Relativity. We are not as yet so far. (i. a) is simply a formal condensation of the fundamental electronic equations (i.).
What we are mainly concerned with in the present chapter is the macroscopic or average result of these equations and of the force formula (n.). But before passing to consider Lorentz's macroscopic equations, it will be good to dwell here a little upon the exact or
ELECTROMAGNETIC ENERGY 47
microscopic formulae (i.), (11.), and some of their immediate and most important consequences.
First, as regards the conservation of energy, multiply the first of (i.) by E and the third by M, both times scalarly. Then, remembering that, by (n.), /o(Ep) = (Pp), and, by vector algebra,
(E curl M) - (M curl E) = - div VEM, the result will be
where u = $(E? + M*) (16)
and g=rVEM. (17)
Now, (Pp) is the activity of the ponderomotive force or the work done ' by the ether on the electrons ' per unit time, and unit volume. Thus, by (15), the principle of conservation of energy will be satisfied for every portion of space, however small, if u is inter- preted as the density, and at the same time Ji as tne flux> °f electro- magnetic energy. The possibility of adding to J3 any vector of purely solenoidal distribution need not detain us here. Ji ls widely known as the Poynting vector, in commemoration of the fact that this vector and the corresponding conception of the flow of energy were first formulated by Poynting (1884). Thus we see that the density and the flux of electromagnetic energy, given by (16) and (17), are in Lorentz's theory precisely as in Maxwell's and Hertz-Heaviside's theory.
Next, as regards the pojideromotive force P, in comparison with that of Maxwell as expressed by his electromagnetic stress, use the first and third of the fundamental equations (i.) ; then (n.) will become
P = pE - VE curl E - VM curl M--V — M--VE ~,
C Ot C Ct
or, introducing the Poynting vector,
P = pE - VE curl E - VM curl M - -^ ^. ( 1 8)
C~ (3£
This is the expression of Lorentz's force, equivalent, in virtue of (i.), to the original expression (n.). Now, MaxweWs pondero- motiTe force, per unit volume, is given by
PMX = />E - VE curl E - VM curl M. (19)
48 THE THEORY OF RELATIVITY
This is the resultant of Maxwell's well-known electromagnetic stress
f» = UR - E(En) - M(Mn), (20)
i.e. PMx= -idivf1-jdivf2-kdivf3, (21)
fn being the pressure* per unit area, on a surface element whose unit normal is n, and f1? f2, f3 meaning the same things as fn for n = i, j, k respectively. We do not stop here to show the equi- valence of (19) and (21), for we shall have an opportunity to do so later. What concerns us here is the comparison of Lorentz's with Maxwell's ponderomotive force. From (18) and (19) we see that the former is
P P * 3$ (22}
P~PM,-?^.
Maxwell's force on the free aether, i.e. for p = o, is, by (19) and the system (i.), which in this case coincides with Maxwell's equations,
PMx = -VEM+-VEM,
i 3$ i.e. PMX = - ^, for p = o. (I90)
Thus, in a variable field, Maxwell's ponderomotive force on the free aether is, generally, different from zero. The supposed existence of such a force, which has been treated on various occasions by Heaviside, suggested to Helmholtz the argument of his last paper, namely an investigation of the possible motions of the free aether, f On the other hand, Lorentz's force on the free aether is always nil, according to his fundamental formula (n.) ; as has been already remarked, he forbids us even to talk about a force on the aether, since its elements are supposed once and for ever to be immovable. According to (22) the Maxwellian force on the aether is just com- pensated by Lorentz's supplementary term - -$$fdt. In using the
Maxwellian stress fn in his theory, Lorentz cbnsiders it, of course, as a system of 'merely fictitious tensions' (cf. Essay, p. 29). In
* Pressure proper being counted positive, and tension proper negative.
t H. v. Helmholtz, Folgerimgen aus maxwell's Theorie tiber die Bewegungen des reinen Aethers ; Berl. Sitzber., July 5, 1893 ; Wied. Ann., Vol. LIII. p. 135, 1894.
PONDEROMOTIVE FORCE 49
Maxwell's theory the ponderomotive actions observed in electric and magnetic fields were physically accounted for by the tensions and pressures of the aether. But Lorentz, in order to be consistent, avoids considering the ' aether tensions ' as something physical, since these would mean forces exerted by the different parts of the aether on one another. Thus, the Maxwellian stress is to him but a con- venient instrument for calculation.
Returning to the general case, p^o, Lorentz's ponderomotive force (n.) may be written, by (22) and (21),
P= -idivf-jdivf^kdivfg-. (23)
It thus consists of two parts, the first of which is deducible from the Maxwellian stress, while the second, foreign to Maxwell's theory, is given by the negative time-rate of local change of the vector Ji/^2. It is this second term which always compensates the Maxwellian action on the pure aether.
Finally, to obtain Lorentz's resultant force
n= (Wr
on the whole system of electrons (T being any volume containing all the electrons), use the expression (23), and observe that
f div iid-r = f
d<r, i= i, 2, 3,
where n is the outward unit normal of the surface <r enclosing the region r. Also remember that
i(f1n)+j(f2n) + k(f3n) = fn3 (24)
since the Maxwellian stress is irrotational or self-conjugate. Then the result will be
$</r, (25)
<r being supposed fixed in the aether, i.e. relatively to the framework S in which the fundamental equations are to be valid. Formula (25) states simply the same thing for the whole system, contained in T, which is expressed by (23) for each of its elements. Of course, in passing from (23) to (25), the continuity of the vector fn (or at least of its components normal to surfaces of discontinuity, if there be any) S.R. D
50 THE THEORY OF RELATIVITY
has been tacitly assumed throughout r* The last formula, again, may be written :
which needs no further explanation. Now, as the mathematicians say, let cr expand to infinity, or at least so that, £, M decreasing in the usual way as i/r2, the surface integral may vanish. Then
nMx = o,
while
where the vector G is defined by
r T
(27)
and is called the electromagnetic momentum.
Thus Maxwell's resultant force is strictly nil, satisfying Newton's third law (actio est par reactioni\ while Lorentz's resultant force is generally different from zero, against the third law, — a result which has been already stated in a slightly different form. Thus Maxwell's theory, admitting an action on the pure aether, did, while Lorentz's theory, denying it, does not satisfy Newton's third law. But, as was observed by Lorentz himself, there is nothing to compel us to universalize that law of Newtonian mechanics. At first, Poincare tried to use this as an argument against Lorentz's theory ; f but he soon gave it up. This was to be only one of a whole series of sacrifices, and not the greatest one, made by modern physicists.
Similarly, the resultant moment of the ponderomotive forces,
(28)
where r is the vector drawn to any point of the field from a point O fixed in the aether, or fixed relatively to S, may be easily put into the form
*The treatment of possible exceptions to this assumption, as electromagnetic surfaces of discontinuity or -waves properly so called [which exceptions seem to be overlooked by the leading electronists, who claim for (25) general validity], need not detain us here.
fH. Poincare, Arch. Nterland.^ Vol. V. ; 1900.
ELECTROMAGNETIC MOMENTUM 51
Thus, for the whole space, and with the usual assumption as to the behaviour of £, M at infinity,
and
where
H =
is called the electromagnetic moment of mome?itum. Its analogy to the ordinary, mechanical, moment of momentum
is obvious. So is also the analogy of the above G with the ordinary momentum
2wv
and the corresponding interpretation of (26) and (29). Both G- and H are so constructed as if the aether contained (electromagnetic) momentum in each of its elements amounting to
(30)
per unit volume.
So much as regards the chief consequences of the fundamental formulae (i.) and (H.).
Now for Lorentz's macroscopic equations. These are obtained from (i.), (n.) by averaging over ' physically infinitesimal ' regions of space. Lorentz calls a length / 'physically infinitesimal' (in dis- tinction from what is called 'mathematically infinitesimal') if the values of any observable magnitude obtaining in two points distant / from one another are sensibly equal to, i.e. indiscernible from, one another. Molecular, and, a fortiori, electronic, dimensions and mutual distances of molecules constituting a ponderable medium, are assumed to be small fractions of /. Let ^ be any magnitude, scalar or vectorial. Round a point P draw a sphere of physically infinitesimal radius ; let T be the volume of this sphere. Then
is called the ' mean value of ^ at P,1 and is denoted by ^. If ^ be any of the magnitudes involved in the fundamental (microscopic)
52 THE THEORY OF RELATIVITY
equations, as for instance p or M, then ^ is what is macroscopically observable.
We cannot reproduce here the details of the process of averaging based upon the above fundamental notion,* but shall simply write down the resulting macroscopic equations, limiting ourselves to the case of a 'perfectly transparent (i.e. non-conducting), non-magnetic ponderable medium, and leaving out of account dispersion. We must, however, explain first the meaning of the symbols involved in these equations.
Assuming that the molecules of the ponderable medium or body contain electrons,! to which belong certain positions of equilibrium within the individual molecules, Lorentz supposes their displacements from these positions, q, and their velocities relative to the cor- responding molecule,
to be infinitesimal. In other words, he neglects the squares and products of q, q, or any of their components in presence of their first powers. Notice that the only part played by the molecules of ponderable matter consists here in restraining the electrons, i.e. in keeping them near certain positions. For, as has already been remarked, one of Lorentz's fundamental assumptions is, that matter by itself, apart from electricity, behaves like the free aether, its presence having no influence whatever upon the electromagnetic field.
Let e be the charge of an electron which has experienced the displacement q, as explained above. Then Lorentz brings in the notion of electrical moment, not unfamiliar to older theories, defining this vector to be, per unit volume, the average of eq,, i.e.
^q.
Taking the sum of this and of the average of our above E, Lorentz introduces the macroscopic vector
<£ = E + ^q (31)
*See Sections II. and IV. of Lorentz's Essay, or his article in Encykl. d. math. Wiss., Vol. V.2, pp. 2OO et seq. ; Leipzig, 1904.
fViz. 'polarization-electrons,' and leaving out of account circling or ' mag- netization- ' and free or 'conduction-electrons.'
LORENTZ MACROSCOPIC EQUATIONS 53
which he calls the dielectric polarization* Thus, in the free aether <£ reduces to E, and generally (£ is what Maxwell called the dielectric displacement.
Next, the macroscopic magnetic force is denned to be the average of our above M, i.e. M, instead of which, however, we shall write shortly M.
Finally, the macroscopic electric force is introduced, being denned as the average of E', i.e. of the ponderomotive force per unit charge, as given by the formula (10). Instead of E' we shall, again, write more conveniently E'. Thus Lorentz's macroscopic electric force will be
(32)
Notice that here p means the resultant velocity of an electron, i.e. the vector sum of its velocity relatively to the molecule in question and of the velocity of the ponderable body as a whole, say v, 'relatively to the aether,' so that p = q + v.
With these meanings of the symbols, Lorentz's macroscopic equations for a transparent, non-magnetic, ponderable body, moving with constant^ velocity v * through the stagnant aether,' i.e. relatively to the framework S, are as follows (Essay, p. 76) :
(33)
-75— = c . curl M' ; div (£ = o
3M
-^— = - c . curl E' ; div M = o
of
M' = M - * VvE'
' /*± *
c
Here the system of coordinates involved in div and curl, is rigidly attached to the ponderable body, thus sharing in its motion through the aether. But the time / is the same as in the fundamental equations (i.); obviously, therefore, 3/3/ is the time rate of change (for constant values of those coordinates, />.) at a fixed point of fhe body, not of the aether or of S.
*The above <£ is Lorentz's ^.
t Constant in space and time, that is to say for a body having a uniform purely translational, rectilinear motion.
54 THE THEORY OF RELATIVITY
The second of (33) is an obvious expression of the (assumed) absence of macroscopic charge, i.e. of /5 = o. In the more general case of a sensibly charged body we should have div (& = p, where J> is the observable density. As to K, appearing in the last of (33), it is a linear vector operator in crystalline, and a simple scalar coefficient in isotropic bodies, known as the ' dielectric constant ' or permittivity, and depending in a complicated way on the distributional properties of the electrons. The numerical value of K in an isotropic, and its principal values, K^ K^ Kz in a crystalline body, are not constant, of course, but vary with the period T of the incident light- or, generally, electromagnetic oscillations. However, to avoid unneces- sary complication, we may think here of the simple case of homo- geneous light, of a particular kind (colour). Then K, or K^ A"2, K^, are constants, whose numerical values are to be considered as deduced from the observable refractive properties of the body with regard to light of that particular kind. In case of isotropy we have to write K=riL, if n be the corresponding index of refraction.*
Notice that (33) contains, besides the solenoidal conditions for <£ and M, four vector equations for as many vectors,
<B, M, E', M',
the velocity of motion v being given. And since the differential equations are of the first order with regard to /, the electromagnetic history of the whole medium is determined by its initial state, say, by <£0, M0 given for /=o.
It must be kept in mind that, to obtain the system of equa- tions (33) from the fundamental ones, Lorentz has consciously neglected not only various small terms concerning the minute influence of electrons, but also all terms of second order in /3, or, to put it shortly, all fit-terms, where
This is especially true of the fifth of (33), which has been obtained from the more exact formula M' = M - VvE/<r by writing E' instead of E, and thus [cf. (32)] omitting VvVpM/<r2, which is a /52-term.
* As to dispersion, which need not detain us here, it can be accounted for in the well-known way by attributing to the body (or to its molecules) one or more internal, 'natural periods,' and, to introduce these, plenty of opportunities are offered by the hypothesis of the electronic structure of molecules and atoms.
LORENTZ MACROSCOPIC EQUATIONS 55
Let us now consider some of the most important consequences flowing from the above system of macroscopic equations.
First of all, as the reader may easily find by himself,* they give the right value for the dragging coefficient, viz. sensibly Fresnel's
coefficient, K= i --^. This, in fact, is a consequence of (33), when
/32-terms are neglected and when dispersion is not taken into account. For a dispersive medium that value of the index of refraction is to be taken which corresponds to the ' relative ' period of oscillation, T', — a concept to be explained further on. This gives a slight correction term, — n~ ~lTdn /3 T (Essay, p. 101), where n is the refractive index of the medium corresponding to the 'absolute' period T, i.e. the period of the oscillations emitted by the source, say, in Fizeau's experiment. Thus, Lorentz's formula is
For water, at i8°C, and for sodium light, this becomes
" = 0-451, (Lor)
whereas Fresnel's value, and that obtained experimentally by Michelson and Morley, have been 0-438 and 0-434 ± 0-02 respec- tively. Thus Lorentz's dragging coefficient agrees with the experi- mental value (MM) quite as well as Fresnel's, especially if the 'possible error of ±0-02' be taken into account. In a word, Lorentz's equations give the right value of the dragging coefficient. And, from what has been said previously, it can be argued that these equations will also give correct results for all first order phenomena. Next, putting v = o, we see at once that (33) become
cX£ 3/
^=-<-.cur!E; divM = o ( (33o)
that is to say, identical with Maxwell's equations, for a station- ary (non-magnetic) medium, (i), p. 24. Taking account of
* Proceeding, mutatis mutandis, similarly as in Note 3, concerning (HH). Another, more simple, method of obtaining the dragging coefficient is to apply Lorentz's 'theorem of corresponding states,' to be considered later.
56 THE THEORY OF RELATIVITY
magnetization-electrons, we would have, in the second and third equation, JE instead of M, where Jft = /*M, //. being the permeability.
This is a very satisfactory result, for, as already mentioned, Maxwell's equations for stationary media, agreeing fully with experiment, have been able to stand even the severe criticism of the modern relativists, who have adopted them without the slightest modification whatever.
' Stationary ' means, of course, in Lorentz's theory, fixed relatively to the aether.
In order to exhibit the properties of his equations, (33), in the general case of any constant v, i.e. for a material medium having any uniform motion of rectilinear translation relative to the aether, Lorentz transforms these equations by introducing instead of the time t a new variable of very remarkable properties. This, the so- called ' local time,' which was to become one of the most immediate forerunners of Einstein's relativistic theory, deserves a rather more extended treatment. It will occupy our attention in the next chapter.
NOTES TO CHAPTER II.
Note 1 (to page 28). Let cr be a surface of electromagnetic discon- tinuity of first order, for example ; that is to say, the vectors E, M being themselves continuous across cr, let their space- and time-derivatives of first order be different in absolute value and direction on the two sides of the surface. Call one of its sides I, and the other 2 ; draw the normal unit vector n from I towards 2, and denote by [a] the jump of any magni- tude a, i.e. the difference 012-04. Then the so-called identical conditions, to be fulfilled in any case, are
[div E] = (ne) ; [curl E] = Vne ; (a)
and the kinematical condition of compatibility, valid under the supposition that the surface is neither being split into two or more nor dissolved, is
6 being the same vector as in (a\ characterizing the electrical discon- tinuity, and b (an independent scalar) the velocity of propagation of cr, counted positively along n. Both e and i) remain so far indeterminate, in numerical value and direction. Similarly, for the magnetic discontinuity,
[divM] = (nm), [curl M] = Vnm, (^,)
WAVE OF DISCONTINUITY 57
m being a new vector and to the same scalar as above, since the electric and magnetic discontinuities are supposed not to part from one another. (For the deduction of the above conditions see J. Hadamard's Lemons sur la propagation des ondes et les equations de Vhydrodynamiquc, Paris, 1903, or, in vectorized form, my book on Vectorial Mechanics, London, Mac- millan £ Co., 1913 ; also Annalen der Physik, Vol. XXVI., 1908, p. 751 and Vol. XXIX., 1909, p. 523.)
If 6, m are normal to <r, we have a longitudinal, and if tangential, a transversal discontinuity.
So far everything has been independent of any electromagnetic con- nections. Now use Maxwell's equations (4), with (4^ ; since they are valid on both sides of cr, we have also
, etc., and, using (a\ (b) with their magnetic analogues,
(mn)=o ; (en)=o.
Notice that if b does not vanish, i.e. if there is propagation at all, the second pair of equations becomes superfluous, since it then follows identically from the first pair. Now, eliminating m from the first pair of (c), we have
|«j
-2 e = VnVen = e - n(en),
n being a unit vector. But (en) = o ; hence
and similarly
-m=m
Thus, if e, m do not vanish, i.e. if there is at all a discontinuity,
that is to say, each element da- of the wave is propagated normally to itself with the velocity c. Q.E.D.
Notice that the sign of to, left undetermined in (d\ due to the quadratic result of elimination, may be defined uniquely by means of the original pair of equations (c\ which are linear in to. In fact, multiply the first scalarly by e (or the second by m), then
to = s (eVmn) = .j-(nVem),
where s is a positive scalar, namely c/e2. Thus, if n, e, m is a right- handed system, like the usual i, j, k then to is positive, i.e. the sense of
58 THE THEORY OF RELATIVITY
propagation is that of n, and if n, e, m is left-handed, then the propaga- tion is along — n. Thus, the sense of propagation coincides always with that of the vector
Vem.
If e points upwards and m to the right, then the wave is propagated forwards. Notice the similarity with the sense of the flux of energy, or the Poynting vector, in relation to E, M,
Finally, notice, in passing, that by the first pair of (c\
similarly to the known characteristic, E'2 = M\ of the usual 'pure3 waves. The above results may easily be extended to waves of discontinuity of any order.
Note 2 (to page 30). Take as a surface element the parallelogram constructed on two coinitial line elements a, b, composed always of the same particles, so that, n being its positive normal,
Write, generally, R for (£ or JE. Then the induction through da- will be given by the volume of the parallelepiped R, a, b, i.e.
(RnX<r=(RVab).
The current through dcr, say (pn)rt<T, being the rate of change of this induction, is
(pn) da- = (Rn) da- + (RVab) + (RVab), (a)
where the dots stand for individual variation. Thus
and \Vectorial Mechanics, Chap. V., formula (75)]
b = (bV)v.
Now, i, j, k being the usual right-handed system of mutually normal unit vectors, take a rectangular rfb-, say
and, consequently,
n = i, da-=dy.dz. Then
CURRENT IN MOVING MEDIA 59
so that the sum of the last two terms in (a) will be
or, per unit area,
hence, substituting (£) in the first term of (a) and remembering that (pn)=(pi)=A,
with similar expressions for P^p^ \t d& be taken normal to j or k respec- tively. Thus the resultant current will be
p = current (R) = ^ + (vV)R-(RV)v + Rdiv v, or
p = current (R) = -^ 4- V di v R + curl VR v, (c)
which is the required formula.
In the simplest case, considered on p. 33, in which the material medium moves as a whole with purely translational velocity v=-z/i, we have to take only the first term of (a\ so that in this case
Note 3 (to page 34). Take E, etc., proportional to an exponential function of the argument
*-C*-*0,
where g is an imaginary constant, as usual. Then
and, consequently, curl = W =g Vi. Introducing this in the equations (Mx), remembering that v = vi and omitting the common factor g^ we obtain at once
60 THE THEORY OF RELATIVITY
but divJR=o gives in the present case (JEi) = o. Thus
and, the medium being isotropic,
Eliminate E, remembering that (Mi) = o; then the result will be
where to' would be the velocity of propagation, if the medium were stationary in 5. Thus
and, the sense of propagation being that of V,
which is the required formula.
Note 4 (to page 39). To spare me trouble and to give the reader a sample of Fresnel's charming manner of exposition, I quote here simply the closing passages of his letter to Arago (loc. cit. pp. 633-636), in which he treats in a masterly manner the water-telescope experiment, both on the corpuscular and on the undulatory theory of light :
' Je terminerai cette lettre par une application de la meme theorie a 1'experience proposee par Boscovich, consistant a observer le phenomene de 1'aberration avec des lunettes remplies d'eau, ou d'un autre fluide beaucoup plus refringent que 1'air, pour s'assurer si la direction dans laquelle on aperc,oit une etoile peut varier en raison du changement que le liquide apporte dans la marche de la lumiere. Je remarquerai d'abord qu'il est inutile de compliquer de 1'aberration le resultat que Ton cherche, et qu'on peut aussi bien le determiner en visant un objet terrestre qu'une etoile. Void, ce me semble, la maniere la plus simple et la plus commode de faire 1'experience.'
' Ay ant fixe a la lunette meme, otaf plutot au microscope FBDE [figure 2 of Fresnel's letter], le point de mire M, situe dans le prolongement de son axe optique CA, on dirigerait ce systeme perpendiculairement a Pecliptique, et, apres avoir fait 1'observation dans un sens, on le retour-
BOSCOVICH'S EXPERIMENT
61
nerait bout pour bout, et 1'on ferait 1'observation en sens contraire. Si le mouvement terrestre depla^ait 1'image du point M par rapport au fil de 1'oculaire, on la verrait de cette maniere tantot a droite et tantot k gauche du fil.'
' Dans le systeme d'emission, il est clair, comme Wilson 1'a deja re- marque, que le mouvement terrestre ne doit rien changer aux apparences du phenomene. En eflfet, il resulte de ce mouvement que le rayon partant de M doit prendre, pour passer par le centre de 1'objectif, une direction MA' telle que 1'espace AA' soit parcouru par le globe dans le meme intervalle de temps que la lumiere emploie a parcourir MA', ou MA (k cause de la petitesse de la vitesse de la terre relativement a celle
CC'G g
de la lumiere). Representant par v la vitesse de la lumiere dans 1'air, et par / celle de la terre [i.e. our c and v respectively], on a done :
MA : A A' \\v\t ou
AA' t MA v
c'est le sinus d'incidence. i/ etant la vitesse de la lumiere dans le milieu plus dense que contient la lunette \y' is our cjri\> le sinus de Tangle de
refraction C'A'G sera egal a -^ ; on aura done C'G=A'C'—, ; d'ou 1'on tire la proportion
C'G : A'C : : t : -J.
Par consequent le fil C de 1'oculaire place dans 1'axe optique de la lunette arrivera en G en meme temps que le rayon lumineux qui a passe par le centre de 1'objectif.'
So far the corpuscular or emission theory. Again :
'La theorie des ondulations conduit au meme- resultat. Je suppose, pour plus de simplicite, que le microscope est dans le vide, ^/et d' etant les vitesses de la lumiere dans le vide et dans le milieu que contient la
62 THE THEORY OF RELATIVITY
lunette, on trouve pour le sinus de Tangle d'incidence AM A', - , et pour
, »/ ci
celui de Tangle de refraction C'AG, — . Ainsi, independamment du ddplacement des ondes dans le sens du mouvement terrestre,
Mais la vitesse avec laquelle ces ondes sont entrainees par la partie mobile du milieu dans lequel elles se propagent est egale a
\i.e. in our notation i> ( i — jj J ; done leur deplacement total Gg, pen- dant le temps qu'elles emploient a traverser la lunette, est egal k A'C d*-
d' ainsi
On a done la proportion C'g : A'C' :: t : d' ; par consequent Timage du point M arrivera en ^en meme temps que le fil du micrometre. Ainsi les apparences du phenomene doivent toujours rester les memes quel que soit le sens dans lequel on tourne cet instrument. Quoique cette experience n'ait point encore ete faite, je ne doute pas qu'elle ne confirmat cette con- sequence, que Ton deduit egalement du systeme de Temission et de celui des ondulations.'
Note 5 (to page 42). Stokes' theory of aberration (' On the Aberration of Light,' Phil. Mag., Vol. XXVII., 1845, P- 9> reprinted in Math, and Phys. Papers, Vol. I. p. 134) was based on the assumption that the aether surrounding the earth is dragged by this planet in its annual motion, in such a way that the velocity of the aether relative to the earth is nil near its surface, and, increasing gradually, becomes equal and opposite to the earth's orbital velocity at very considerable distances from our planet. It is obvious that this hypothesis led at once to a rigorous independence of purely terrestrial optical phenomena from the earth's annual motion. But in order to explain correctly astronomical aberration, Stokes had to assume that the aether's motion, between the earth and the ' fixed ' stars, is purely irrotational, which assumption could not be reconciled with the absence of sliding over the earth's surface, so long as the aether was regarded as incompressible. It is true that this difficulty, as has been shown by Planck, can be overcome by giving up the incompressibility, namely by supposing the aether to be condensed around the earth and the celestial bodies, as if it were subjected to
STOKES' ABERRATION THEORY 63
gravitation and behaved more or less like a gas. But the condensation around the earth, required to reduce the sliding to, say, one half per cent, of the earth's orbital velocity, would be something like el\ i.e. corre- sponding to a density of the aether near the earth about 60,000 times as great as its density in celestial space. Now, it is certainly difficult to admit that the velocity of light is not to any sensible extent altered by this enormous condensation of the aether around the earth.
Particulars concerning the discussion of this most interesting subject will be found in Lorentz's book on Theory of Electrons (Chap. V.), and in his original paper on * Stokes' Theory of Aberration in the Supposition of a Variable Density of the Aether,' Amsterdam Proceedings^ 1898-1899, p. 443, reprinted in Abhandlungen ub. theor. Physik, Vol. I. p. 454.
CHAPTER III.
THEOREM OF CORRESPONDING STATES. SECOND ORDER DIFFICULTIES. THE CONTRACTION HYPOTHESIS. LORENTZ'S GENERALIZED THEORY.
LET us return to Lorentz's macroscopic equations, for a material medium moving relatively to the aether with uniform velocity v,
-~- = c . curl M' ;
a/
div (B = o = - c . curl E' ; div M = o
= M--VvE'
c
(L)
In the simplest case of a medium fixed in the aether, i.e. for v = o, these, as already noticed, become identical with Maxwell's equations for a stationary dielectric,
a*
BM
c . curl M ; div (£ = o - c . curl E ; div M = o
(L0)
In order to exhibit the properties of the more general equations (L), Lorentz introduces instead of the ' universal time,' as he calls /, a new variable /', which will now be explained.
Let O' be a point fixed in the material body, chosen arbitrarily but once and for ever as the origin of coordinates, x\ y, z', measured
LOCAL TIME 65
along axes rigidly attached to the body. From O' draw to any individual point of the body P'(x, y', z') the vector r', so that the three Cartesian coordinates are condensed in
Let us call the framework of reference rigidly attached to the body the system S'. For comparison and to impress better upon your mind the meaning of r', take also an initial point O fixed in the aether, i.e. relatively to the system S, and draw from O to P the vector r, or in semi-Cartesian expansion, using the same unit vectors as above,*
If O' is taken to coincide with O at the instant simply
r' = r - v/.
= o, we have
FIG. 6.
Remember that the equations (L) hold for t and x\ y z (not x, y, z) as independent variables, or, more shortly, for
r', /.
This fixes the meaning of curl, div and 3/3/, as already mentioned in Chap. II. As regards the curls and divergences, they are, of course, the same in x'9 y', z as in .v, y, z.
*This is always possible, since the material body or medium moves relatively to S in a purely translational manner.
S.R. E
66 THE THEORY OF RELATIVITY
Now, r' being the above vector characterising any given point P' of the moving body or medium, the new variable /' is defined by
/' = /-^(r'v), (i)
and is called the local time at P'. Since the scalar product in the second term vanishes for r'J_v, the local time coincides with the ' universal ' one at all points lying on the plane passing through O' and perpendicular to the direction of motion. But at all other places the new and the old time differ from one another, the local time being behind the ' universal ' time in the anterior portion of the body, and the reverse being the case in its posterior portion (Fig. 6). In Cartesians, if v^iz^+j^ + ke^, the local time is
or if i be taken along the direction of motion, /' = / - x'vjc1.
Notice that Lorentz's local time, as just defined, has nothing physical about it. It is merely an auxiliary mathematical quantity to be used instead of the 'universal' time / in order to simplify the form of equations (L). It is constructed expressly for this purpose, and serves it excellently.
In fact, taking instead of r', / (or #', y , z\ t)
r', /'
as the new independent variables, and denoting the divergence and curl in terms of the new variables by
div' and curl', we obtain, for example, by (i) and by the third of equations (L),
divM = div'M + -vcurlE' c
= div'M--divVvE',
since curlv = o, by hypothesis. But for VvE', as for any vector normal to v, we have, obviously, div = div'. Hence, by the fifth of (L),
div M = div' (M - - VvE') = div' M'.
CORRESPONDING STATES 67
Thus, the fourth of equations (L), divM = o, becomes, in the new variables, div'M' = o. Similarly, the second of (L), div(£ = o, is transformed into div' (£' = o, where (£' is a new vector defined by the formula
<£' = <£+ 1 VvM. (2)
Using this new vector and the vector M', denned by the fifth equation, the remaining equations (L) may be transformed, with equal ease, to the new variables.
The result is surprisingly simple. The system of Lorentz's equations (L) for a moving medium takes with the new variables r', t'(x',y, z, /) the form
-^-T- = c . curl' M' ; ut
r = - c . curl' E' : div' M' = o
-~- of
(L')
that is to say, precisely the same form as for a stationary medium, (L0), the only difference being that the electromagnetic vectors E, (£, M are replaced by their dashed correspondents, as are also the independent variables r, /.
This remarkable discovery, made by Lorentz, has played a most important role not only in his own theory, but also in the subsequent evolution of ideas concerning electromagnetism and optics. Un- doubtedly, it may, to a great extent, be regarded as the germ of modern relativistic tendencies. It will therefore be worth our while to treat this subject at some length, and not only as an historical episode.
The above result may be put into the form of what has been called by Lorentz the Theorem of corresponding states :
If we have for a stationary medium or system of bodies any solution (of Maxwell's equations L0), in which
E, <£, M
are certain functions of
x, y, s, /,
68 THE THEORY OF RELATIVITY
we will obtain a solution for the same system of bodies moving with uniform translation-velocity v, taking for
E', <£', M'
exactly the same functions of the variables
x', y, z and t' = t-\ (vr').
In other words, and somewhat more shortly :
For each state in which E, (£, M depend in a certain way on xi y\ z-> t m tne stationary system, there is a corresponding state in the moving system characterised by E', (£', M' which depend in the same way on x , y', 2', t'.
It will be useful to put here together the scattered definitions of the dashed vectors. These are, by (32), Chap. II.,* by (2) and by the fifth of equations (L),
-VvM
c
* VvM
- VvE'.
c
(3)
As to the coordinate systems, notice that they are in both cases rigidly attached to the material medium or to the system of bodies in question, x, y, z being fixed together with it in the aether, and x't y', z sharing its motion through the aether.
The above theorem of corresponding states has, of course, like the equations (L) themselves, the character of a first approximation only, terms of the order of j3'2 = v2/c2 having been neglected.
The broad and easy applicability of this beautiful theorem of Lorentz is obvious. It will be enough to quote here a few illus- trative examples.
* Remembering that M itself is of the first order, so that
i VpM == i VvM = ^ WM, i.e. in the adopted short notation, -VvM.
OPTICS OF MOVING SYSTEMS 69
If, in the stationary medium or system S of bodies, E, (£, M are periodical functions of /, with period T, then, in the moving system S', the vectors E', (£', M' are periodical functions of the local time /', and consequently, at a point P' fixed in S', also of /, with the same relative period T. What Lorentz calls the relative period is the period of changes going on at a fixed point of the system S' moving relatively to the aether, i.e. for a constant r', whereas the period of changes taking place at a point fixed in the aether, i.e. for a constant r, is called the absolute period. Similarly, relative rays are distinguished from absolute rays, and so on. Thus, to luminous vibrations in 5 of a given absolute period correspond luminous vibra- tions in S' of the same relative period.
If, in certain regions of the stationary system, J5 = o, etc., then also E = o, etc., in the corresponding regions of the moving system. Thus, to darkness corresponds darkness. Also, limitations of beams in S and S' correspond to one another. Luminous rays in S\ of relative period T, are refracted and reflected according to the same laws as rays of (absolute) period T in S. The same is true of the distribution of dark and bright interference fringes, and consequently also of the concentration of light in a focus, by mirrors or lenses, this being a limiting case of diffraction.
But, although the lateral limitations of beams for corresponding states are the same, corresponding wave normals in S, S' have generally dijferent directions, this being again an immediate conse- quence of the theorem of corresponding states. In fact, if we have in S, say, plane waves whose normal is given by the unit vector n and whose velocity of propagation is b, i.e. if E, (£, M are proportional to a function of the argument
(rn) - b/,
then, in the moving system, E', etc., will be the same functions of the argument
(r'n)-b/' = (r'n) + (r'v)-b/. (4)
Consequently, the direction of the wave normal in the moving system will be given by that of the vector
(5)
70 THE THEORY OF RELATIVITY
Thus, unless n || v, the directions of the wave normals in S and S' are different. To state the same thing in Cartesians, the direction- cosines of the wave normal in the moving system will be given by the proportions
In particular, for a vacuum or, very approximately, for air, in which case to = t,
N' = n + ^v, (50)
or, in clumsy Cartesians,
These formulae may, after a slight transformation, be applied at once to the case of astronomical aberration, the relative period being here that reduced according to Doppler's law. Thus Lorentz obtains immediately the right results for air- and water-telescope aberration. (Cf. Essay , p. 89.)
To obtain the dragging coefficient it is enough to write the argument (4)
Since here n' is a unit vector, the velocity of propagation in S' is
or, neglecting the term containing ft2 = (vjcf, developing the square root and neglecting again the second and higher powers of (vn)/V,
(6)
In particular, if the propagation is in the direction of motion or against it, as in Fizeau's experiment,
»• =•,*(*)%. '
TERRESTRIAL OPTICS 71
Thus, the velocity of propagation relative to the aether will be
»{-©>
and the value of the dragging coefficient
Here v = cjb is the refractive index of the medium, say water, corre- sponding to the relative period which is connected with the period T of the emitted light by the formula
second order terms being neglected. Thus, if n be the refractive index for the period T,
whence Lorentz's formula for the dragging coefficient,
i i 3«
* = I -- 9 -- -I ^7^ J
n- n 3T
closely agreeing with experiment, as already mentioned in Chapter II. For purely terrestrial experiments, in which not only the observer but also every part of his apparatus and the source of light are attached to the earth, the theorem of corresponding states leads to the following result :
The earth's motion has no first order influence whatever on any of such experiments.
The possibility of a second order influence remains, of course, in this stage of the research, an open question. For, as will be re- membered, before arriving at the macroscopic equations (L), from which the theorem of corresponding states has been seen to follow, /^-terms have been throughout neglected. In other words, that beautiful theorem, developed and illustrated by a series of most important examples in the fifth section of Lorentz's classical JEssay, is but a first order approximation.
72 THE THEORY OF RELATIVITY
So far everything is quite satisfactory. But now, in the sixth, and last, section of Lorentz's Essay the difficulties begin. * In this section Lorentz investigates three problems, of which two concern the rotation of the plane of polarization and Fizeau's polarization experiments. But without dwelling on these, we shall pass straight on to the third one, namely to the famous inter- ference experiment of Michelson and Morley. This second order or /^-experiment, originally suggested by Maxwell,! was performed by Michelson in 1881, and six years later repeated on a larger scale and writh a higher degree of exactness by Michelson and Morley. J A beam of luminous rays coming from the source s, after having been made parallel in the usual way, is divided by the semi-transparent
B
FIG. 7.
plane mirror (half-silvered plate) ab, which is inclined at an angle of 45° to sOA, into a transmitted beam OA, and a reflected one OB, After having been reflected by the mirrors placed at A and B (at right angles to OA, OB, which directions are perpendicular to each other), the two beams of light return to the central mirror ; here a part of the first beam is reflected along OC and a part of the second
* As is explicitly stated in the title : ' Abschnitt VI. — Versuche, deren Ergeb- nisse sich nicht ohne Weiteres erklaren lassen.'
t See Note at the end of chapter.
\ A. A. Michelson, ' The relative motion of the earth and the luminiferous ether,' Amer. Journ. of Science, 3rd Ser. Vol. XXII., 1881. A. A. Michelson and E. W. Morley, Sill. Journ., 2nd Ser. Vol. XXXI. , 1886; Amer. Journ. of Science, 3rd Ser. Vol. XXXIV., 1887; Phil. Mag., 5th Ser. Vol. XXIV., 1887. What is given above is but the usual rough scheme ; details of the actual arrange- ment will be found in the original papers quoted and, to a certain extent, also in Michelson's popular book on Light Waves and their Uses, where a diagram of the actual apparatus is given (Fig. 108).
THE MICHELSOX EXPERIMENT 73
beam is transmitted towards C, thus producing with one another a system of bright and dark interference fringes, which can be observed through a telescope placed on the line OC. To resume it shortly, the paths, taken relatively to the earth, of the two interfering beams of light are :
sOAAOC and sOBBOC.
Let OA (Fig. 7) be in the direction of the motion of the earth, and consequently also of the apparatus, source and all, with respect to the aether of Fresnel and Lorentz, and let v be the velocity of this motion, i.e. the resultant of the earth's orbital velocity, at the time being, and of the velocity of the solar system with respect to the 'fixed stars' or to those 'fixed' stars relatively to which the aether is supposed to be at rest. (Cf. Note 2.) On this assumption , let us calculate the times taken by the two beams in travelling along I their paths. Since the parts sO and OC are common to both, we have only to consider the intervals of time, say T^ and T2, taken to traverse
OAAO and OB BO
respectively, where the letters denote, of course, points attached to the apparatus.
Now, as has been already said in Chapter II., in connexion with Maxwell's equations for the 'free aether,' the velocity of light with respect to the aether is always equal c=$. io10 cm. sec."1, quite independently of the motion of its source. This is no novel idea at all ; Fresnel himself considers it apparently as an obvious matter, when he says (in an early part of his letter, already mentioned) without any further explanations : ' car la vitesse avec laquelle se propagent les ondes est independante du mouvement du corps dont elles emanent.' Thus, according to both the classical and the more recent adherents of the aether, the velocity of light relative to the aether does not depend on the source's motion : and on the wave-theory there is no reason why it should. Newton's corpuscular theory, revived in a more elaborate form in the writings of the late Dr. Ritz, need not detain us here.
Thus, the mirror A, receding from the waves on the part OA of their journey, and the mirror O moving toward them on their return from A to O, we have
—+— = -r,
c-v c+r) c1 -v1
74
THE THEORY OF RELATIVITY
where the index i is to remind us that OA is ' longitudinal,' i.e. along the direction of motion. Putting vjc^P and
&>••/
(7)
we may write shortly, without yet making any use of the smallness of /3s,
Ti = *--fOAi. (8)
To obtain T2, the time for the second beam, we could say simply, after the manner of some authors, that the relative velocity of light, being the vector sum of the velocity c parallel to OB and of the velocity v of the aether with respect to the apparatus, perpendicular to OB and directed backwards, is equal (c2 - vrf, so that
7;= 2
or
T2 = -cyOJ3t,
(9)
0 0' 0'
FIG. S.
where the index t is to remind us that OB is ' transversal ' or per- pendicular to the direction of motion. But since this may not seem very satisfactory, we can support it by the following, equally frequent, reasoning which is but formally different from the above short statement. Contemplate for a moment Fig. 8, the paper on which it is drawn being now supposed to be stationary in the aether, and the apparatus moving past it from left to right. Let the centre of the inclined mirror be at O at the instant t = o, when the light leaves it, and at O" at the instant /= T2, when the light returns to it ; let B' be the position of B when the beam reaches it, and let O' be the simultaneous position of O, If it be granted
THE MICHELSON EXPERIMENT 75
that the three distinct points of the aether, O, O\ O", are the consecutive positions of exactly the same point of the inclined mirror, that is to say, that the ray in question returns to exactly, or sensibly, the same point of the mirror from which it started, then OB'O" will be an isosceles* triangle, so that OB' = \cT^ and
This gives T* = 2OBl(c--vr)~^, which is identical with (9).
By (8) and (9) we get for the time-difference of the two beams, by which the phenomenon of their interference is determined,
T^-T^^OA.-OB,}. (10)
Let us now turn round the whole apparatus through 90°, so that OA becomes transversal, and OB longitudinal. Then we shall have, using dashes to distinguish this case from the above one,
so that the time-difference of the two beams will become
(10')
If therefore the fixed-aether theory is true, such a rotation of the apparatus should produce a shift in the position of the inter- ference fringes, corresponding to the change of the time-difference of the two beams, A = (io) - (10'), i.e.
The indices , and t, distinguishing between longitudinal and trans- versal orientation, have been introduced here (contrary to the his- torical order) only for the sake of subsequent discussions. To Michelson and Morley there was no question of distinguishing be- tween the lengths of a segment in different orientations. To put
* That the above assumption is satisfied with a sufficient degree of accuracy may be seen from Note 3 at the end of the chapter, where the corresponding Huygens construction is worked out.
76 THE THEORY OF RELATIVITY
ourselves into agreement with their manner of treatment we have, therefore, to write simply
To secure these equalities Michelson and Morley mounted the mirrors* and, in fact, the whole of the apparatus, on a heavy slab of stone mounted on a disc of wood which floated in a tank of mercury, so as to be able ' to rotate the apparatus without intro- ducing strains.' In a word, they made the configuration of O, A, etc., 'rigid,' that is to say as rigid as a stone is. On this understanding, formula (n) may be written
A = -7(y-i).(O4 + a#). (12)
As to the mutual relation of OA, OB, they were made 'nearly equal,' to suit the well-known requirements for producing neat interference fringes, in each of the two orientations of the appa- ratus. Moreover, since these lengths or distances enter in the formula only by their sum, their equality or non-equality is of no essential importance. We may therefore, without any more ado, write OA = OB = L or else call the sum of these lengths 2Z. Then, as regards the factor depending on the velocity of motion, we have,
<7)'
or, up to quantities of the second order, i.e. neglecting /34-terms, etc.,
Thus, the second-order effect to be expected on the stationary- aether theory would be determined by the change of the time- difference of the two beams
A=^Z. (120)
If T be the period of the light and A = r7^the wave-length, the corresponding shift s = &/Tof the interference bands, measured as a fractional part of the distance of two neighbouring bands, would be given by
* = /82X' <I3)
* In the actual experiment not three but sixteen in number.
THE MICHELSOX EXPERIMENT 77
The length 2Z, which in Michelson's original apparatus was too small, was in Michelson and Morley's experiment (1887) increased to about 22 metres, by multiple reflection from suitably placed mirrors. And since, for sodium light, A = 5«89.io~5 cm., the ratio 2Z/A. had nearly the value -37 . io8. As regards /32, we should have, taking for v simply the earth's orbital velocity, i.e. 30 kilom. per second, f&= io~s. It is true that, at least in some of the experiments, the rays of light, being horizontal, made a considerable angle with the earth's orbit, but on the other hand the motion of the whole solar system exerted a favourable influence, so as to double the value of J3'2 (as was already mentioned). So that to put [P equal io s is certainly not to overestimate its value considerably. Thus the shift should be on the stationary-aether theory, in round figures,
s = 0-4 of a fringe width.
In no case, however, did the actual displacement of the fringes exceed -02, and probably it was less than -or, i.e. less than Joih of the expected value. The experiment was repeated in 1905 by Morley and Miller* with considerably increased accuracy, and their result was that, if there is any fringe-shift of the kind expected, it is something like ^ = -0076 instead of 1-5, i.e. not greater than one two-hundredth of the computed value, t
Thus, not nearly the expected second-order effect of the earth's motion relatively to the aether was observed. It seems, therefore, reasonable to say at least that, as far as we know, the above A is nil.
In order to explain this negative result and to save, at the same time, the stationary-aether theory, Lorentz has had recourse to a peculiar hypothesis, constructed ad hoc^ which occurred to him independently of Fitzgerald, who was the first to suggest it. \ It is
*E. W. Morley and D. C. Miller, Phil. Mag., Vol. VIII. p. 753, 1904; Phil. Mag., Vol.' IX. p. 680, 1905.
fAs to various objections raised against the correctness of the interference experiment by Sutherland, Liiroth and Kohl, and their refutation by Lodge, Lorentz, Debye and Laue, see the ' Literaturiibersicht ' in J. Laub's report * Uel>er die experimentellen Grundlagen des Relativitatsprinzips,' Jahrbuch der Radioaktivitat und Elektronik, Vol. VII. p. 405, 1910.
£Cf. Lorentz's Essay, p. 122 (1895), where reference is made to a p£per of his, dated 1892-93. As regards Fitzgerald, we read in The Ether of Space by Sir Oliver Lodge (London, 1909, p. 65), referring to that hypothesis: 'It
78 THE THEORY OF RELATIVITY
now widely known under the name of the contraction hypothesis, and it consists in assuming that, in Lorentz's words, 'the dimensions of a solid body undergo slight changes, of the order /3'2, when it moves through the ether,' namely a longitudinal contraction amounting to ^P'2 per unit length or, more generally, both a transversal' and a longitudinal lengthening, e and S, per unit length, such that e-S = l/3'2. This would amount for the whole earth to about 6-5 centimetres only.
To see at once that the negative result of the Michelson experi- ment is thus accounted for and to grasp as clearly as possible the nature of the hypothesis, let us return to the more general formula (n) for A, from which (12) or (i2«) followed by identifying OAj with OAt, and similarly OB\ with OBt. Now, to simplify matters, assume OBi = OAl and OBt = OAt (which, as we saw, is of no essential importance), but on the other hand distinguish between OAl and OAt. Then formula (n), valid by the fixed-aether theory, will become
OAt}- (14)
and since A = o, by experience, we have to write, in order to respect both that theory and experience,
or, up to quantities of the second order,
which is the Fitzgerald-Lorentz hypothesis.
Notice that it would be a perfectly idle thing to quarrel whether OAt is shortened, while OAt remains unchanged, by the earth's motion through the aether, or whether OAt alone is lengthened, or, finally, whether both are changed in suitable proportions. The only thing we are required by the aether theory and by experiment to do is to consider the ratio of the lengths of one and the same ' material '
was first suggested by tfie late Professor G. F. Fitzgerald, of Trinity College, Dublin, while sitting in my study at Liverpool and discussing the matter with me. The suggestion bore the impress of truth from the first.' Happy are those who are gifted with that immediate feeling for 'truth.'
THE CONTRACTION HYPOTHESIS 79
segment OA, or shortly Z, in those two orientations as being equal to i - J/3-, or, more rigorously,
Z,:Z,Wi-/?2. (15)
This implies that for /? = o, i.e. if the earth stopped moving through the aether, or nearly so, we should have Ll = Lt, say, both equal to Z0. But it cannot inform us as to the ratio which either length bears to Z0, when the earth is moving through that medium ; more- over, such considerations are, thus far, physically meaningless.
At any rate, Lorentz soon decided in favour of a purely longitudinal contraction, which amounts to writing
Z, = Z0 and Z, = y° = Z0s/f^ (15*)
In doing so he based himself on certain results obtained from the fundamental (microscopic) equations in an early part of his classical Essay, to be mentioned presently. That this, in fact, was his choice we see explicitly from the shape attributed by him to moving electrons. While Abraham's electron is and remains always a sphere, being rigid in the classical sense of the word, Lorentz's electron is a sphere of radius ^?, say, when at rest, and becomes- flattened longitudinally, when in uniform motion, to a rotational ellipsoid of semiaxes
•. -^, ./?, ./?.
y
Such an electron, of, homogeneous surface- or volume-charge, is now generally known as the Lorentz electron. The history of its rivalry with the rigid one, and of its rather victorious issue from the contest, need not detain us here. It is, besides, sufficiently well known.
Lorentz's attitude towards the contraction hypothesis may be seen best from his own words, written in 1909 (Electron Theory, p. 196) :
'The hypothesis certainly looks rather startling at first sight, but we can scarcely escape from it, so long as we persist in regarding the ether as immovable. We may, I think, even go so far as to say that, on this assumption, Michelson's experiment proves the changes of dimension in question, and that the conclusion is no less legitimate than the inferences concerning the dilatation by heat or the changes of the refractive index that have been drawn in many other cases from the observed positions of interference bands.'
80 THE THEORY OF RELATIVITY
The obvious criticism of the above comparison may be left to the reader.
As regards the justification of the contraction hypothesis which to an unprepared mind certainly does 'look rather startling,' Lorentz observes in his original Essay of 1895 (P- I24-) tnat we are ^e(^ precisely to the change of dimensions defined by (15^), if, dis- regarding the molecular motion, we assume that the attractive and repulsive forces acting on any molecule of a solid body which ' is left to itself are in mutual equilibrium, and if we apply to these molecular forces the same law which, by the fundamental equations, holds for electrostatic actions. It is true, as Lorentz himself con- fesses, that ' there is, of course, no reason ' for making the second of these assumptions. But those who entertain the hope of constructing an electromagnetic theory of matter will easily adhere to it. To obtain the law in question return to the fundamental electronic equations (i.), Chap. II., and introduce the so-called vector potential A and the scalar potential <£, satisfying the differential equations
(i6)
) c }
and subject to the condition
(17) Then all of the equations (i.) will be satisfied by
(18)
= curlA,
so that every electromagnetic problem is reduced to finding the potentials according to (16) and (17). Suppose, now, that a material body moves as a whole, relatively to the aether or to the system S, with uniform translational velocity v, and that all the electrons it carries are at rest with respect to the body. Then the above p will have throughout the constant value v, so that, by (16),
A = v</>. (19)
THE CONTRACTION HYPOTHESIS 81
Thus everything is made to depend on </> alone. Take the x-axis in »S along the direction of motion, so that v = £i, A = !/?<£, and suppose that the electromagnetic field is invariable with respect to the material body. This assumption will be satisfied if 4> is supposed to depend only on the coordinates attached to the body,
Thus we shall have
3 333
and the equation for <£ will become
i 32<£ B2^ B2^ ?^ + S?+3F--*
while the condition (17) will be satisfied identically. Here
?-'= (i -/**),
as above. Again, by (18),
whence the ponderomotive force per unit charge, or Lorentz's electric force, E + /8V1M, (10), Chap. II., which we shall now denote by Jf (since the dashed E would be misleading),
(.0
where V«iB/d£+j3/di|+kB/9f*i3/aar+... is the Hamiltonian (here acting as the slope), taken with respect to the aether or, which in our case is the same thing, with respect to the material body. Thus, the electric force is derived from a scalar potential </>/y2, precisely as in ordinary electrostatics. By the way, <j>/y2 is called the convection potential. Notice that it is Jf, the electric force, and not the 'dielectric displacement' E, that has a scalar potential.
S.R. F
82 THE THEORY OF RELATIVITY
Now, supposing always fi'2 < i and consequently y real, write
x = y£, y =i~i, z' = tt (22)
and denote the corresponding Hamiltonian, i3/9#' + etc., by V. Then (20) will become
V2<j>= -p. (23)
To adopt for the moment Lorentz's notation, call the moving material body or system of bodies the system Slt and compare it with a system S? which is fixed in the aether and which is obtained from Sl by stretching all its constituent bodies, together with the electrons, longitudinally in the ratio y:i, so that to any point £, ry, f of S1 corresponds the point ,T', y, z of S^ and so that corresponding volume-elements, dr and dT=ydr, contain equal charges. Then, p and p being the densities of electric charge at
corresponding points,
, i
and, by (23),
If then <£' be the scalar, electrostatic, potential in S.2, so that
V'2<£'= -p', we shall have
~7 ' and consequently, instead of (21), using (22),
y .v y
But the electric force in the stationary system S2 is
Therefore, using the indices \ and t to denote the longitudinal and the transversal components of the electric forces,
<$i = £i'; £t = - (#/ = A'N/T -/32, (24)
THE CONTRACTION HYPOTHESIS 83
and since charges of corresponding elements are equal, exactly the same relations will hold between the ponderomotive forces acting on each electron in the moving system St and on the corresponding electron in the stationary system S2.
This is the ' law ' alluded to. Now, suppose that it is obeyed by the molecular forces keeping together the parts of a moving solid which, disregarding its interior molecular and electronic motions, is to be taken for the system Sl. Then, if the molecular forces balance each other in the corresponding stationary body S2, they will do so in the moving body Sl. But, by (22), S1 is the body ,& contracted longitudinally with preservation of its transversal dimensions, exactly as in (150), and the motion would produce this flattening 'by itself.' Whence Lorentz's justification of the contraction hypothesis.
Thus, the longitudinal contraction, though at first manifestly invented ad hoc, to account for the negative result of the Michelson experiment, found a kind of legitimate support by being brought into connexion with the fundamental assumptions of the electron theory. But the cure of the disease has not been radical. In fact, the idea naturally suggested itself, that the Lorentz-Fitzgerald contraction, like an ordinary strain, might give rise to double refraction, of the order /3'2, in solids or liquids, a property which should be directionally connected with the earth's motion round the sun. But here again the result of experiments has been sensibly negative. Lord Rayleigh's* experiments (1902) with liquids (water and carbon disulphide) as well as those with solids, with glass plates piled together, have given no trace of an effect of the expected rkind. At least, if there was any effect on turning round the apparatus, it was less than jj^th of that sought for. Rayleigh's experiment was then repeated (1904) by Brace t with considerably increased accuracy, and the result has again been negative : the relative retardation of the rays due to the supposed double refraction should be of the order io~8, whereas, if existent at all, it was certainly less than 5 . io~n, in the case of glass, and even less than 7 . io~13, in the case of water.
To account for these obstinately negative results, and with a view to settle the matter once and for ever, Lorentz undertook what he
*Lord Rayleigh, Phil. Mag,, Vol. IV. p. 678, 1902.
tD. B. Brace, Phil. Mag., Vol. VII. p. 317, 1904; Boltzmann- Festschrift, p. 576, 1907-
84 THE THEORY OF RELATIVITY
thought a radical discussion of the whole subject, that is to say, of the electromagnetic phenomena in a uniformly moving system, not as hitherto for small values of v, but for any velocity of transla- tion smaller than that of light, i.e. for any /3< i. Lorentz's ideas, laid down in a paper published in 1904,* are fully developed in his Columbia University Lectures, already quoted (p. 196 et seq.). His aim was now to reduce, 'at least as far as possible,' the electro- magnetic equations for a moving system to the form of those that hold for a system at rest — always, of course, relatively to the aether — without neglecting either /3'2- or, in fact, terms of any order whatever. It will be remembered that even in his first approximation, i.e. when neglecting /32-terms, Lorentz employed the ' local time ' /' = / - (vr)/<:2, or, measuring x along the line of motion,
f-'-ji*.* («)
Then the necessity of accounting for the negative result of Michelson's interference experiment brought him to the contraction hypothesis, according to which the longitudinal dimensions of the moving system are reduced in the ratio i : y"1, where y = ( i - /32)~% while the transversal ones remain unchanged. This contraction corre- sponds to /= const, and consequently may easily be shown to be equivalent to transforming x, y, z, the coordinates of a point with respect to axes fixed in the aether, or the 'absolute' coordinates, into
x' = y(x-vt), y'=y, z' = z. (b)
It is true that the transformation (a) was as yet purely formal, and that the contraction, or (/£), was introduced by Lorentz first ad hoc, but afterwards to be justified. But at anyrate, having already (a) and (£), Lorentz has been naturally led to investigate in a general way the consequences of introducing, instead of x, y, z, t,
* H. A. Lorentz, ' Electromagnetic phenomena in a system moving with any velocity smaller than that of light,' Proc. Amsterdam Acad., Vol. VI. p. 809; 1904.
t Here, according to the original definition of ' local time,' p. 66, we should have rigorously (instead of the coordinate x, measured in the fixed framework)
x - vt, so that /' = ( I + /32) t - -g*. But, since at that stage /32-terms were neglected,
we could write simply x instead of x-vt. The symbols x', etc., in what follows are not to be confounded with the x', etc. , of page 66.
LORENTZ GENERALIZED THEORY 85
new independent variables, called by him the effective coordinates and the effective time,
//==, . . , , . (25)
where y is as above and A. is a numerical coefficient of which Lorentz, provisionally, assumes only that it is a function of v alone, whose value equals i for v = o and differs from i by an amount of the order fi'2 for small values of the ratio f$ = vjc* Introducing the new variables (25) into the fundamental electronic equations, (i.), Chap. II., and defining new vectors E', M',
!
and also, instead of the relative velocity p - v of an electric particle, the vector
i.e. with the above choice of axes, simply
P' = 7{i?(A -»)+JA + kA}» (27)
and, instead of the density p,
P=y*.~*P, (28)
Lorentz obtained again the equations (i.) with dashes,
BE'/d/' + p'p = c . curl' M', etc., but with the difference that divE = /> was replaced by
, (29)
* Columbia University Lectures, p. 196. The above v, 7, X stand for Lorentz's w, k, I respectively. A transformation equivalent to (25) was previously applied
by Voigt, as early as 1887, to equations of the form -^ ^-^-V2=o; ' Ueber das
Doppler'sche Princip, Gottinger Nachrichten, 1887, p. 41. Lorentz himself states (loc. cit., p. 198 ; 1909) that Voigt' s paper had escaped his notice all these years, and adds: *The idea of the transformation' (25) 'might therefore have been borrowed from Voigt, and the proof that it does not alter the form of the equations for \hejree ether is contained in his paper.'
86 THE THEORY OF RELATIVITY
not by div'E' = /o'. Thus, the fundamental equations for the free aether (p = p = o) turned out to be rigorously invariant with respect to the transformation (25), which, especially for A.= i, has since been universally called the Lorentz transformation. The same invariance holds also in the general case, that is to say, in the presence of electric charges, but for the slight deviation given by (29).
Using this result, Lorentz generalized his Theorem of corresponding states for any velocity v smaller than c, and succeeded in showing that the theorem thus extended not only accounts for the con- traction required by the result of the Michelson experiment, but that it explains, among other things, why Lord Rayleigh and Brace failed to detect a double refraction due to the earth's orbital motion. A discussion of the formulae for the longitudinal and transversal masses of an electron, which need not detain us here,* led Lorentz to attribute to the coefficient X (his /) the value i, whereby the transformation formulae (25) and (26) were reduced to
= y(x- vt\ / =y, z = z,
(3°) "-**
and
\
, M3' = y(M3- /3E2).)
With this specialization, Lorentz's modified theory, which in its essence was built up in 1904, satisfied the requirements of self- consistency and accounted for the negative results of all, second as well as first order, terrestrial experiments intended to show our planet's motion through the aether. In other words, by modifying and gradually extending his original theory, Lorentz obtained the desired physical equivalence of the ' moving ' system S', with its effective coordinates and time x\ y', z', /', and of a corresponding * stationary ' system with its absolute coordinates and time x, y, z, /.
But still one of the two systems S, S', namely S, was privileged, being regarded by Lorentz as 'fixed in the aether.' Their equival- ence, as indicated persistently by such numerous experiments, was not placed as the basis of the theory, but followed as the result of long, laborious, and rather artificial constructions, intended to com-
*See Columbia University Lectures, pp. 211-212.
LORENTZ GENERALIZED THEORY 87
pensate gradually the pretended play of the ' aether.' For, to repeat, Lorentz continued to assume this hypothetical medium of his classical Essay in his extended theory, dated 1904, and adheres to it even now, if we may judge from the last sentences of his American Lectures (p. 230). Not only is the aether for Lorentz a unique framework of reference, but he * cannot but regard it as endowed with a certain degree of substantiality.' According to this standpoint, then, there certainly is such a thing as the aether, though every physical effect of the motion of ordinary, ponderable matter through it, being compensated by more or less intricate processes, remains undis- coverable for ever.
As regards the above transformation of Lorentz, we may further notice here that Poincare made, in 1906, an extensive use of its more general form (25) \Rend. del Circolo mat. di Palermo >, Vol. XXI. p. 129] for the treatment of the dynamics of the electron and also of universal gravitation. Some of Poincare's results con- tinue even now to be of considerable interest.
In the meantime, 1905, Einstein published his paper on 'the electrodynamics of moving bodies,'* which has since become classical, in which, aiming at a perfect reciprocity or equivalence of the above pair of systems, S, S', and denying any claims for primacy to either, he has investigated the whole problem from the bottom. Asking himself questions of such a fundamental nature, as what is to be understood by 'simultaneous' events in a pair of distant places, and dismissing altogether the idea of an aether, and in fact of any unique framework of reference, he has succeeded in giving a plausible support to, and at the same time a striking interpretation of, Lorentz's transformation formulae and the results of Lorentz's extended theory. Einstein's fundamental ideas on physical time and space, opening the way to modern Relativity, will occupy our attention in the next chapter.
*A. Einstein, AnnaL der Physik, Vol. XVII. p. 891 ; 1905.
88 THE THEORY OF RELATIVITY
NOTES TO CHAPTER III.
Note 1 (to page 72). It seems desirable to quote here after Lorentz (Abhandlungen iiber theor. Physik, Vol. I. p. 386, footnote) a passage from Maxwell's letter ' On a possible mode of detecting a motion of the solar system through the luminiferous ether,' published after his death in Proc. Roy. Soc., Vol. XXX. (1879-1880), p. 108 :
' In the terrestrial methods of determining the velocity of light, the light comes back along the same path again, so that the velocity of the earth with respect to the ether would alter the time of the double passage by a quantity depending on the square of the ratio of the earth's velocity to that of light, and this is quite too small to be observed.'
Note 2 (to page 73). Usually, at least in all text-books, it is simply said: 'Suppose that the aether remains at rest, and let ?y = the velocity of the apparatus, i.e. of the earth in its orbit.' For this to be correct, the aether would have to be at rest with respect to our sun. But when astrono- mical aberration is in question, we are told that the aether is stationary with respect to the 'fixed stars,' say, with respect to the constellation ot Hercules, which, I hope, is 'fixed' enough. Now, as has incidentally been mentioned (p. 17), the sun or the whole solar system has a uniform velocity of something like 25 kilometres per second towards that con- stellation, which, being nearly equal in absolute value to the earth's orbital velocity (30 klm. per sec.), certainly cannot be neglected. Thus, the velocity (y) of Michelson's interferometer with respect to the aether would oscillate to and fro, in half-year intervals, between considerably distinct maximum- and minimum-values. According to Lorentz (' De Pinfluence du mouvement de la terre sur les phenomenes lumineux,' 1887, reprinted in Abhandlungen^ Vol. I. ; see p. 388) the resultant of the earth's orbital and the solar system's velocity had at the time when Michelson was performing his experiment both a direction and an absolute value ' very favorable ' to the effect sought for, even so much as to double the displace- ment of the fringes expected. I am not aware whether or no the defenders and the adversaries of the aether have discussed this circumstance with sufficient care. But at any rate it seemed worth noticing here. Of course, it is for the adherents of the aether (and not those of empty space) to tell us explicitly with respect to what celestial bodies, the sun, or Hercules or other groups of stars, the aether is to be stationary, if it be granted that the parts of that medium do not move relatively to each other. For these stars certainly move relatively to one another.
I cannot help remarking here that it is repugnant to me to think of an omnipresent rigid aether being once and for ever at rest relatively rather to one star than to another. For, this medium, unlike Stokes's aether, being non-deformable and not acted on by any forces whatever, none of the celestial bodies, be it ever so conspicuous in bulk or mass, can claim for itself this primacy of holding fast the aether. The bare idea
REFLECTION FROM MOVING MIRROR
89
of action exerted upon the aether by material bodies being dismissed at the outset, there is nothing which could confer this distinctive privilege upon any one of them. But, then, I am quite aware that what 'is re- pugnant to think of may not necessarily be wrong altogether. There are other reasons to be urged against the aether.
Note 3 (to page 75). Let a plane wave <r (Fig. 9) proceed towards the inclined mirror (half-silvered plate) Oa in the direction of its motion, i.e. from left to right. Let sO, sma represent the incident wave normals, limiting a part of the beam of breadth Om = b, and let CLY'be the normal to the mirror, so that B=sOX is the angle of incidence. Let the wave reach the centre O of the mirror at the instant /=o. Let Ol and a± be the positions of the points O and a of the mirror (both taken in the plane
of the figure) at a later instant /=T, when the wave of disturbance reaches «!, so that
aal = OOl = vr.
Draw round O a circle with the radius
then the tangent to this circle, drawn from alt will represent the re- flected wave, and ON will be the reflected wave normal. To obtain the angle of reflection, & = XON, consider the triangles ONa^ and Omalt having the side Oa^ in common and right angles at m and at N. Since, moreover, their sides CWand avm are equal to one another, alN=Om = b, so that the breadth of the beam remains unchanged by reflection, as for a stationary mirror, and
where f=<«C?a1. But ^=7r/2-€ + f. Thus, the angle of reflection & and the angle of incidence 6 are connected by the relation
(A)
90 THE THEORY OF RELATIVITY
where the angle { is determined by the given properties of the parallelo- gram OaalOl. Writing
we have at once
Oa^= (2/r)2 + /2 + 2VTl . sin 0 and
whence
cos20=i /2 2* jnfl sinaf (vrf vr S1]
But -t/T = 7//sin Q\(c-v\ or ljvr= /J--^--/}* so that the required formula for C is ' Sm
(A) and (B) contain the rigorous solution of the problem, based, of course, on the assumption of a stationary aether.
In Michelson and Morley's experiment, as treated above (Fig. 8), 20=90°, so that (B) becomes
To connect Fig. 9 with Fig. 8, notice that, according to (A), the angle BOB' should be equal to 2^. The approximate treatment given in connexion with Fig. 8 (p. 74) amounts to writing
*\*(BOff} = v\c=$. (c)
Now, developing (BX) and remembering that ft is a small fraction, we have, up to quantities of the second order,
or, neglecting the third and higher powers of the small angle f,
But the term J/32 appearing in this formula for the angle would give in the final formula for T2 only terms of the order of )83 and /34. Thus, aiming at results which are correct only up to quantities of the second order, we may write the last formula
mi (a{)- A
in agreement with (c). Our Huygens-construction shows then that the treatment adopted on page 74 is sufficiently correct for the purpose in question.
That treatment, which is given in all text-books (including also such valuable modern works as Laue's Relativitcitsprinzip, 1913) without any further remark, would be rigorously correct if O were, say, a point
REFLECTION FROM MOVING MIRROR 91
source of (spherical) waves spreading out in all directions, and not, as it actually is, one of the points of a mirror at which reflection of plane waves is taking place.
A different way of treating rigorously the above question will be found in Lorentz's paper entitled ' De Pinfluence du mouvement de la terre sur les phenomenes lumineux,' Arch, neerL, Vol. XXI. (1887), pp. 169-172 (reprinted in Abhandlungen iiber theor, Physik, Vol. I. pp. 389-392) and partly also in his Columbia University Lectures, p. 194.
The discussion of our general formulae (A), (B) connecting the angle of reflection with that of incidence, for large values of /?, may be left to the reader as a curious exercise.
CHAPTER IV.
EINSTEIN'S DEFINITION OF SIMULTANEITY. THE PRIN- CIPLES OF RELATIVITY AND OF CONSTANT LIGHT- VELOCITY. THE LORENTZ TRANSFORMATION.
WE are now sufficiently prepared to grasp the meaning of Einstein's ideas* and to appreciate their relation to the work of his prede- cessors, especially of Lorentz.
In Chapter I. we have seen how it is possible to define the time as a physically measurable quantity fulfilling certain reasonable and fairly general requirements. Practically, it was the variable / measured by the rotating earth as time-keeper or what, with a slight correction connected with tidal friction, has been called the 'kinetic time.' It has certainly not escaped the reader's notice that the requirements on which that choice was based had nothing absolute or necessary about them, being merely recommended by their simplicity and convenience. But this circumstance need not detain us here any further. Suppose we have secured a clock indicating, with a sufficient degree of precision, the kinetic time /. Suppose we keep that clock at a certain place «, relatively to a given space-framework of reference, say in a certain physical laboratory or astronomical observatory. Thus far we have tacitly assumed that the time /, measured by such a chronometer, is universal, if I may say so, i.e. that it is valid for all points of space, for all parts of any system, be it near to our clock or very far from it, be it at rest or moving with respect to it. It is very likely that nobody has ever
* As laid down in his paper of 1905, already quoted, and then (1907) developed by him more fully in a paper, ' Ueber das Relativitatsprinzip und die aus demselben gezogenen Folgerungen,' Jahrbitch der Radioaktivitiit und Elektronik, Vol. IV. p. 411. In what follows we shall refer principally to the former of these papers by quoting simply the original numbers of its pages.
DEFINITION OF SIMULTANEITY 93
asserted explicitly this universality and uniqueness of time, but everybody has certainly given to it his tacit consent, and would willingly endorse it if asked to do so. As far as we know, the first to question this pretended universality of time was Einstein.
Our clock, placed at a, indicates the time /, i.e. marks different time-instants and measures the intervals between them, to begin with, only at the place a, or nearly so. It is, to give it a short name, the time ta. Suppose that some well-marked instant is chosen as the initial instant, /a = o. Then, if any event is happening at a or near «, we give to it that date or, as it wrere, label it with that number ta which is simultaneously shown by the index of our clock. We are exempted from defining what ' simultaneous ' (as well as ' earlier ' or ' later ') means when applied to a pair of events occurring at the same place or near that place, as the passage of the index through a given division of the dial of our clock and the production of an electric spark closely to it* But we do not know, beforehand, what we are to understand by saying that of two events occurring at places «, b distant from one another the first occurs earlier or later than the second, or that both are simultaneous. The meaning of these words has to be defined. If the labelling of all possible kinds of events, occurring at distant points, fixed or moving relatively to one another, is to be of any use at all, wTe must establish the rules according to which we are going to label them with the /-numbers. And first of all we have to decide which of these events have to receive the same labels, i.e. we have to define simultaneity at distant points.
This notion is to be defined in terms of simultaneity at the same place, which alone is assumed to be known to us, and of some other things or processes which are actually realizable. In other words, distant simultaneity has to be reduced to local simultaneity by some physical process. Abstractly speaking, the choice of such a process is arbitrary, in very wide limits at least; but practically the choice will be reduced to siich processes as are of possibly universal occurrence, and wrhich are independent of the capricious peculiarities of different sorts of matter. Einstein has chosen for this purpose the propagation of light in vacuo. Gravitation being, chiefly due
* We need not stop here to consider such apparatus as Siemens' ' spark- chronometer,' in which the visible marks corresponding to pairs of events are brought very close to one another, and which enable the modern physicist to fix with a high degree of precision their time-relations.
94 THE THEORY OF RELATIVITY
to its alleged instantaneous action, out of question, this has been, in fact, the only possible choice. Moreover, it was not unprecedented in the history of physics and astronomy, and it suggested itself most obviously because the recent difficulties met with lay in the optical and, more generally, electromagnetic departments of physics.
To an unbiassed mind the question may present itself : Why label everything in the world with Anumbers at all? Such a question is not altogether unreasonable, and it may deserve some careful attention. But once we decide to attach a time-label to every event, we are forced to reduce in some kind of way distant simultaneity to local simultaneity (for pairs of points at rest or moving relatively to one another), and not to delude ourselves with thinking that we know what 'universal simultaneity' means, or that it is, in fact, a self-consistent notion. To have initiated a critical analysis of the concept of simultaneity at all is certainly a great merit of Einstein's.
But let us leave aside these generalities and pass to the definition in question. We shall have to consider in the first place the simpler case of distant points a, />, etc., in relative rest, and then the somewhat intricate case of distant points belonging to systems which are uniformly moving with respect to one another.
Let a, b) etc., be points or 'places' fixed relatively to one another and with respect to a certain space-framework or system S, say, the system of the fixed stars.* Suppose we succeeded in manufacturing at the place a a number of equal clocks, each measuring the same, say the * kinetic,' time / and set equally or synchronously, and that retaining one of them at a we sent the others to b, etc., together with an equal number of observers who are to remain at those distant places with their clocks for ever. Then, to begin with, we should have as many ' times ' as there are places in consideration, 4) 4) etc., valid, respectively, for the places a, b, etc., and for their nearest neighbourhoods. For, though all of these clocks were manufactured equally at a, we do not know whether they continue to be * equal ' or permanently synchronous, when one of them is
*In his paper (p. 892) Einstein begins with taking, for the purpose of his definition of simultaneity, that 'system of coordinates in which Newton's mechanical equations are valid.' But it seems advisable not to appeal at the outset, and in connexion with such a fundamental definition, to Newtonian mechanics, especially as it requires, according to the relativistic view itself, some essential, though numerically slight, modifications. On the other hand, the physical specification of what has been called above the system S will appear presently without recourse to any theory of mechanics.
SYNCHRONOUS CLOCKS 95
still kept at a, while the others are sent far away, to the places b, etc. More than this, we do not know what their being synchronous or not, when far apart, means. We have yet to fix how we are going to test it. To invoke the preservation of rate of clocks of 'good make ' in spite of their being carried to distant places, on the title of the high precision of their mechanisms, would not help us out of the difficulty. For, supposing we also decided to assert such infallible and rigorous permanence, at different places within S, of the mechanical laws, necessarily involved, still we should have to verify whether the accessorial conditions of validity of those laws (and practically there would be a host of such conditions) are fulfilled at and round each place in question. To avoid this verifi- cation, which soon would prove to be a difficult task, we must have some means of testing in a direct manner the synchronism of our distant clocks and, more generally, of correlating with one another the times /«, 4, etc., without being obliged to enter upon the properties and structure of the corresponding clock mechanisms.*
Now, the kind of test adopted by Einstein, and constituting at the same time the essence of his definition of distant simultaneity, is as follows.
Let an observer stationed at a send a flash of light at the instant /„ (as indicated by the «-clock) towards b, where it arrives at the instant 4 (according to the ^-clock). Let another observer send it back from b without any delay, or let the flash be automatically reflected at b, towards a, where it returns at the instant faf. Then the ^-clock is said, by definition, to be synchronous with the tf-clock, if
fa -4 = 4 -/«. (i)
This amounts to requiring, per definitionem, that 'the time* employed by light to pass from a to b should be equal to 'the time' employed to return from b to a. Instead of (i) we may write, equivalently,
Thus, the instant of arrival at b is expressed by the arithmetic mean of the a-times of departure and return of the light-signal. Such
*\Ve may notice in this connexion that Einstein's specification (p. 893): 1 eine Uhr [at ti\ von genau derselben Beschaffenheit wie die in A [a] befindliche ' is unnecessary and, to a certain extent, misleading.
96 THE THEORY OF RELATIVITY
being the connexion of the a-time and of the Mime, the clock placed at b is said to be synchronous with that placed at a.
This definition of synchronism is supposed to be self-consistent, for any number of clocks placed at different points of the system S, say, besides a and b, at c, d, £, etc. To secure this consistency, Einstein makes, explicitly, the following two assumptions :
1. If the clock at b is synchronous with that at a, then also the clock at a is synchronous with that at b. In other words : clock- synchronism is reciprocal, for any pair of places taken in S.
2. If two clocks, placed at a and b, are synchronous with a third clock, placed at c, they are also synchronous with one another. Or, more shortly, clock-synchronism is transitive throughout the system S.
This is the way that Einstein himself puts the matter. But it may easily be shown that the first of his assumptions will be fulfilled if we require that 'the time' employed by the light-signal to pass from a to b is always the same. In fact, let us denote the tf-time, taken generally, by a instead of 4, and similarly, let us write b instead of the general variable 4, and let us use the suffixes d> a> r to denote the instants of departure, arrival and return. Then, if the ^-clock is synchronous with the «-clock, we have, by definition, or
for the ' return ' at a may be equally well considered as an arrival at that place. Now, if at the instant aa the flash be sent again towards b, where it arrives at the instant br, we have, by our above requirement,
ba-ad = br-aa,
and, by the last equation,
aa -bo,= t>r-<*a-
But here ba is identical with the instant of departure /%', and, consequently,
i.e. the clock placed at a is synchronous with that placed at b. Q.E.D.
A similar treatment of assumption 2. may be left to the reader, who will find sufficient hints in Fig. 10. This assumption will be easily seen to imply that if a pair of flashes be sent out simultaneously
PROPERTIES OF THE SYSTEM <S' 97
from a, one via b^ c and the other via t, b, they will both return simul- taneously at a. More generally, the time elapsing between the instant of departure and that of return of the light-signal sent round abca will be equal to the time elapsing between departure and return of the signal sent round acba, and similarly for every other closed path in S, both times being measured by the clock placed at a. This form of the property attributed to the system S is worthy of being especially insisted upon, as it implies only operations to be performed at one and the same spot. To state this property of the system 6", the observer has not to move from his place.
FIG. 10.
Such then are the physical properties of this system of reference.
It is strange that Einstein, after having made explicitly the above assumptions i. and 2., considers it necessary to add (p. 894) that ' according to experience ' the quantity
ab
2
ar-ad
or, in the notation of formula (i), the quantity
ab
is to be taken as 'an universal constant (the velocity of light in empty space).' At any rate, if the last assumption is made, for any pair of points a, b in 51, once and for ever, then the above state- ments i. and 2. are certainly superfluous. But considerations of this order need not detain us here any more. S.R. G
98 THE THEORY OF RELATIVITY
The properties ascribed to the system S* may be briefly sum- marised by saying that
Isotropy and homogeneity with respect to light propagation are postulated throughout S, once and for ever.
In this way the various times, /a, 4, etc., originally foreign to one another, are all connected so as to constitute one time only, valid for the whole system, which we may denote simply by /, calling it shortly the S-time.
There is, thus far, nothing essentially new in Einstein's procedure. It was more or less unconsciously applied since people began to measure the velocity of light, and even sound, nay, since they began to exchange with one another letters or messages of any kind. The novelty does not come in until the next stage, when the time-labelling is extended to different systems moving (uniformly) with respect to one another.
Let ,5" be as above, and let us consider other systems of reference, S', S", and so on, each having with respect to S a motion of uniform (rectilinear) translation. Having settled the matter for the system S, i.e. having established the S-time, /, let us similarly establish an 6"-time, /', an S"-time, /", etc., and let us see how the times /', /", etc., are connected with the time / valid for S. It can reasonably be expected that these processes of (time-) labelling of events happening at different places, being undertaken from different standpoints, S, S', S", etc., will generally not coincide with one another, e.g. that events obtaining identical /-labels may receive different /'-labels, and so on. Such, in fact, will be the case ; the labels of different sorts, dashed and non-dashed, though none is privileged in any way, will have to be carefully distinguished from one another. In a word, it will appear that, with the above definition of simultaneity, no universal, no unique time-labelling is possible.
It will be enough to consider explicitly, besides S, one other system only, say, S'. Supposing that a consistent time-labelling of events occurring at different places of S' or an S'-time, is possible, like the above S-time, the question is, how is this time /' to be connected with the time /? We shall see that the connexion sought for will involve also the coordinates defining the position of points
* Which Einstein himself, in order to have a convenient name, provisionally, calls 'the stationary system.'
EINSTEIN'S PRINCIPLES 99
within S, and within S', In a word, the time-connexion of both systems will turn out to be entangled with their space relations.
Here we shall have, to appeal to what Einstein calls the principles of 'relativity' and of 'constancy of light-velocity,' and wrhich he enunciates in the following way :
I. The Principle of Relativity. The laws of physical phenomena* are the same, whether these phenomena are referred to the system S or to any other system (of coordinates) S' moving uniformly with respect to it.
II. The Principle of Constant Light- Velocity. Every light-disturbance is propagated, in vacito, relatively to the system S with a determinate velocity c, no matter whether it is emitted from a source (body) stationary in or moving ivith respect to S. The 'velocity' c is the light-path divided by the corresponding time-interval, c = abjt; t being the .S-time as denned above.
Here, to begin with and to fix the ideas, the system S is taken. But applying the principle I., we can say at once that the same constancy of light propagation is valid also with respect to the system S'. The constancy of the velocity of light, i.e. its independ- ence of the motion of the source, as emphasised in Chap. II., has already been appealed to by Fresnel. But there is this essential difference that Fresnel claimed this property of light propagation only for a certain, unique system of reference, namely the aether or a system fixed in the aether, while Einstein, by accepting I. and II., postulates it for any one out of an infinity (°o3) of systems moving uniformly with respect to one another. With regard to this property the systems S', S", etc., are perfectly equivalent to the system S or become so in force of Principle I., — and this is the reason why the mere notion of an ' aether ' breaks down. None of the systems in question is privileged. To make it as plain as possible, let P be a point fixed in the system S, and let a point-source, moving relatively to S in a quite arbitrary manner, emit an instantaneous flash just when it is passing through P. Then the observers rigidly attached to the system S will find that the disturbance is propagated from P in all directions with the same velocity c, i.e. that the ensuing thin
* Literally : ' The laws according to which the states of physical systems are changing,' etc. (Einstein, p. 895).
TOO THE THEORY OF RELATIVITY
pulse or wave of discontinuity is a spherical surface, of centre P and of radius
if / is reckoned from the instant of emission. Again, if P' is a point fixed in S\ and if the arbitrarily moving source emits a flash just when it is passing through P\ then the wave, as it appears to observers rigidly attached to S\ will be a sphere whose centre is permanently situated at P' and whose radius at any instant of the »S'-time is
r'~«af,
if /' is reckoned from the instant of emission. Such is, in virtue of I., the meaning of the principle of ' constancy ' of light-propagation in empty space. Of especial interest is the particular case, in which our source is fixed at a point P' of the system S', and therefore moving uniformly with respect to S. In this case the centre of the spherical wave will, to the ,S'-observers, be permanently situated at the material particle playing, the part of source, whereas for the ^-observers the centre of the spherical wave, fixed in S, will detach itself from the material source, the source moving away from it with uniform velocity together with the whole system S'. This case will be made use of presently.
Let us now return to the first of the above principles, and let us remember how the time /, valid for the whole system »S, has been defined. Since S has been endowed with physical properties required for a consistent method of time-labelling of events occurring at its various points, the same properties will, in virtue of I., hold also for S'. Again, local clocks satisfying the requirements of convenience, e.g. the causality-maxim, being possible in S, such time-keepers are, by I., possible also for various stations taken in S'. We can therefore consider first a time /</, measured by a clock placed at a point a in S ', then distant clocks placed at //, etc., leaving the task of testing their synchronism to observers attached to the system 6", and repeating in fact literally all that has been said before with regard to the system S. In this way we should obtain out of the originally local times a unique time t' applicable to the whole system S '. Let us call the time thus constructed the s'-time.
The question now is, how are the .S'-time and the »S-time con- nected with one another (and, possibly, with other things, viz. lengths or distances as measured by the .5"- and ^'-observers) ?
TIMES AND LENGTHS 'COMPARED'
101
The answer to this fundamental question may be obtained, with the help of the two above principles, in a variety of ways. But for certain reasons the following way, though not the shortest, seems to me the most instructive to begin with.* It is, moreover, intimately connected with what has been said in the last chapter with regard to the Michelson experiment.
Let us imagine an ^"-observer having at his disposal a point-source of light at a place P' fixed in the system S'. Let A and B' be a pair of distant points also fixed in S', and such that the straight line P'A is in the direction of motion of S' relatively to S, and that P' B' is perpendicular to that direction (Fig. n). As before, we shall call P'A' longitudinal, and P' B' transversal. Let /' be the
tB'
xs; o'
/ 0 q \
1 I.. , ' \ '
' length ' of the first of these segments or the ' distance ' from P' to A, according to the estimation of the .^'-inhabitants, and similarly s the length of the second segment. Suppose that our observer sends an instantaneous light-flash from Pf towards A and receives it back at P' after the lapse & of the /'-time. Then, having assured himself by any means that an assistant stationed at A sends him back his signals without any delay, our observer will write
Under similar conditions, if he sends a flash towards! B' and
* Einstein's method of reasoning, as given in his original paper (§ 3, see also Notes at the end of this Chap. ) may be mathematically interesting, but does not seem to be the fittest when a clear discussion of the physical aspect of the question is aimed at.
f To avoid unnecessary difficulties as to hitting the receiving station, now /?' and now A't it will be best to imagine that our observer sends each time a full spherical wave of discontinuity or a very thin spherical pulse. This will be found especially convenient when we come next to consider the same processes from the ^-standpoint.
102 THE 'TftfcORY OF RELATIVITY
receives it back after the interval r' of the /'-time, he will put down the equation
25
There is, in fact, by the above principles, no difference between longitudinal and transversal light signalling between stations fixed in S', as observed by the inhabitants of this same system.
Let us now see how each of the above two processes will be described by an observer attached to the system S. Call the lengths or distances P'A', P B' , as estimated by the ,S-observer, / and s respectively. Each of these is obtained by ascertaining, with the help of an appropriate number of synchronous /-clocks, which are the points of the ,5-system, through which P' and A ', or P' and B' pass simultaneously, and by measuring the mutual distances of these points by means of an ^"-standard rod. Similarly, /' and s are to be considered as the distances P A' and P B' measured by standard rods which the ^"-observers are carrying with themselves. Notice that, by Principle I., /' and s', thus measured, will be the same whether the system S', together with its observers, clocks and measuring rods, is at rest with respect to S or whether it moves uniformly with respect to that system, as it actually does. But /, j are not necessarily equal to /', s'. For although they are ' distances of the same pairs of material points,' the source and the receiving stations, they are not obtained by the same processes. Having thus explained the meaning of /, s, let us consider, from the ^-standpoint, first the longitudinal and then the transversal signal- ling. The flash sent out by the luminous source will, according to Principle II., appear to the ^-observers in both cases as a spherical wave expanding with the velocity c and having its centre at that point P§, fixed in *S, through which the source has passed when emitting the flash. Now, if v be the velocity of S' relative to S, the receiving station A' moves away from PQ with the uniform velocity v. If, therefore, 01 be the £-time required for the wave to expand from />0 to A',
and
In the same way, if 0.2 be the ,5-time employed by the light
TIMES AND LENGTHS COMPARED 103
to return from the receiving station* to the sending station />',
Thus, the .S-time 6 elapsing between the first appearance and the reappearance of a light flash at A', being the sum of 0j and 02, will be given by
o.*.,^
c- — v
A similar reasoning applied to the case of transversal signalling, in which case the sphericity of the wave will be found particularly convenient, will give us for the 5-time elapsing between the appear- ance of the first and second flash at A' the value
+*<*¥*'(*. j
KN '\f--2iyv--' r=24
&i£j%,.+^ *
Compare the last two formulae with the above ones for & and T', and denote the ratio s/s' by a. Then the result will be
*& ty-, ^j-* (3) J
where a is a number which for v = o becomes equal i, but is other- ^ J wise an unknown function of the data of the problem.
Now, each of the two processes, i.e. the longitudinal and the transversal signalling, may (by disregarding the receiving stations) ^K^ be considered as phenomena consisting in a double appearance of a flash at one and the same station, at the same individually dis- cernible point A\ fixed in S'. Thus far we have, purposely, kept these two processes separate. But now we can advantageously combine them with one another. If the receiving stations were chosen so that / = /', then we should have, by the first pair of formulae,
ff = r', say =T',
and if the two processes were started simultaneously, from the 6"-point of view, they would also have ended simultaneously for the ^"-inhabitants. In other words, we would have, in S', a pair of
* This station A' (and similarly, in the case of transversal signalling, the station B') may be imagined to become an instantaneous point-source emitting a spherical wave at the moment when it is reached by the original wave.
104 THE THEORY OF RELATIVITY
simultaneous events followed by another pair of simultaneous events, all of these occurring at the same place A'. Let us now require (what, as far as I know, is tacitly assumed by most authors) that
III. Events locally* simultaneous for an S' -observer should also be simultaneous for the S-observers.
This amounts to supposing that there is a oiie-to-o?ie correspondence between the /-labels and the /'-labels to be applied to events occurring at any given place, i.e. for fixed values of the coordinates x',y', z in S '. (The analogous one-to-one correspondence between #', /, z and x, y, z for /' = const, is tacitly assumed as a matter of course.) On the other hand, two events occurring at distinct places, being simultaneous in 6", are generally ^^/-simultaneous from the ^-standpoint.
Now, in virtue of the requirement III., call it postulate or desideratum, or whatever you prefer, the above two simultaneous processes or phenomena occurring at A will also begin and end simultaneously for the ^-observers, so that
0 = r, say =T, and
Consequently, by the equations (3),
These are the required connexions between durations and lengths, measured in 6" and in S'. They are based on the above assumptions I., II., III., the last of which is certainly the most obvious. The common coefficient a is, thus far, indeterminate. If we are to endow (empty) space with homogeneity, as well as with isotropy,f and if it be granted that the relations between the S- and ^'-measure- ments do not vary in time, the unknown coefficient a can depend only upon v — c^. The only thing we thus far know about this
* i.e. occurring at one and the same place.
fBoth properties having been already attributed to it physically, i.e. as regards propagation of light, by II.
LONGITUDINAL CONTRACTION 105
function is that it reduces to unity for /? = o, when S' is 'at rest relatively to S, when, in fact, both systems cease to be discernible from one another. Thus
a = a(/3), a(o)=l.
Notice that for v = o we have also 7=1, so that in this case T, /, s become, by (4), identical with T', /',