Relativity

"Relativity. \- The progress of physical science during the decade 1910-20 was specially remarkable for the definite emergence into general public discussion of the principle of Relativity, as expounded by Prof. Albert Einstein, professor. of Physics in the Kaiser Wilhelm Institut, Berlin. Its meaning and its history as part of present-day physical theory are discussed below.

## Introduction

The primary aim of the investigator in pure science is the discovery of natural laws. As a secondary and hardly less important aim, he tries to invent a mechanism which shall account for the laws already known. The secondary aim is forced upon him partly by the constitution of the human mind; our intellects, unsatisfied with a mere accumulation of facts, impel us ever to search for the causes underlying the facts: Vere scire est per causas scire. But to the working scientist the discovery of a mechanism has an additional and more practical value. When he has found a mechanism which will account for certain laws, he can proceed to examine the complete set of laws which the mechanism demands. If his mechanism corresponds with sufficient closeness to reality he may in this way be led to the discovery of new natural laws. On the other hand, the new laws deduced from the supposed mechanism may be false. If the falsity of the new laws is not at once revealed science may for a time be led into wrong paths. When more accurate experimenting or observation discloses that the laws are not true, a recasting of ideas becomes necessary, and the branch of science concerned may experience a time of revolution followed by a period of rapid growth.

An obvious illustration of these general statements is provided by the history of astronomy. The laws of the motions of the planets, as observed from the earth, were tolerably well known to the Greeks. They had also evolved an explanatory mechanism, starting from the metaphysical premise that the paths of the planets must necessarily be circles. The earth was the centre of the universe and round this revolved spheres to which the planets were attached. To explain the retrograde motion of the outer planets, these were supposed attached to secondary spheres revolving about points on the primary spheres which in turn revolved about the earth. This mechanism of cycles and epicycles held the field as an explanation of planetary motion for eighteen centuries. Finally the observations of Tycho Brahe provided a test which revealed the falsity of the whole structure. The position of Mars was found to differ from that required by the mechanism of epicycles by an amount as great as eight minutes of arc. " Out of these eight minutes," said Kepler, " we will construct a new theory that will explain the motions of all the planets." The history of the succeeding century of astronomy need not be recapitulated here (see 2.811). The earth yielded its place as the centre of the universe, and the structure of cycles and epicycles crumbled away. The laws of planetary motion were determined with a precision which for the time appeared to be final. The mechanism underlying these laws was supposed to be a " force " of gravitation. This force was supposed to act between every pair of particles in the universe, its intensity varying directly as the product of the masses of the particles and inversely as the square of the distance separating them - the famous law of Newton.

In science, history repeats itself. Recent years have provided a further instance of the general processes we have been considering. Under the Newtonian mechanism every planet would describe a perfect ellipse about the sun as focus, and these elliptic orbits would repeat themselves indefinitely except in so far as they were disturbed by the gravitational forces arising from the other planets. But, after allowing for these disturbing influences, Leverrier found that the orbit of the planet Mercury was rotating in its own plane at the rate of 43 seconds a century. Various attempts have been made to reconcile this observed. motion with the Newtonian mechanism. The gravitational forces arising from the known planets were demonstrably unable to produce the motion in question, but it was possible that Mercury's orbit was being disturbed by matter so far unknown to us. Investigations were made as to the disturbance to be expected from various hypothetical gravitating masses - a planet, or a ring of planets, between Mercury and the sun, a ring of planets outside the orbit of Mercury, a belt of matter extended in a flattened disc in a plane through the sun's centre, an oblateness, greater than that suggested by the shape of the sun's surface, in the arrangement of the internal layers of the sun's mass. In every case the mass required to produce the observed disturbance in the motion of Mercury would have also produced disturbances not observed in the motions of the other planets. The solution of the problem came only with the theory of relativity. Just as Tycho's eight minutes of arc, in the hands of Kepler and Newton, revolutionized mediaeval conceptions of the mechanism of the universe, so Leverrier's 43 seconds of arc, in the hands of Einstein, has revolutionized our 19th-century conceptions, not only of purely astronomical mechanism, but also of the nature of time and space and of the fundamental ideas of science. The history of this revolution is in effect the history of the theory of relativity. It falls naturally into two chapters, the first narrating the building of an earlier physical theory of relativity, and the second dealing with its extension to gravitation.

## The Physical Theory of Relativity

The earliest successful attempt to formulate the laws governing the general motion of matter is found in Newton's laws. The first law states that- " Every body perseveres in its state of rest or of uniform motion in a right line unless it is compelled to change that state by forces impressed thereon."  In this law no distinction is made between rest and uniform motion in a straight line, and the same is true of the remaining laws. Hence follows the remarkable property to which Newton draws explicit attention in his fifth corollary to the laws of motion: " The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion."  As a concrete application of this principle, Newton instances " the experiment of a ship, where all motions happen after the same manner whether the ship is at rest or is carried uniformly forward in a right line." Just as a passenger on a ship in a still sea could not determine, from the behaviour of bodies inside the ship, whether the ship was at rest or moving uniformly forward, so we cannot determine from the behaviour of bodies on our earth whether the earth is at rest or not. We believe the earth to be moving round the sun with a speed of about 30 km. a second, so that there can be no question of the earth being permanently at rest, but we are unable to determine whether it is at rest at any specified point of its orbit, or, in the probable event of its not being at rest, what its absolute velocity may be. There is no more reason for thinking the sun, than the earth, to be at rest. Newton wrote as follows: " is possible that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest, but impossible to know, from the positions of bodies to one another in our regions, whether any of these do keep the same position to that remote body. It follows that absolute rest cannot be determined from the position of bodies in our regions." The above quotations are all from the first book of the Principia Mathematica. Previous to them all Newton writes: " I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies." The two centuries which elapsed after the publication of the Principia witnessed a steady growth of the belief in the reality of such an all-pervading medium. It was called the aether, and by the end of these two centuries (1887) it was almost universally believed that light and all electromagnetic phenomena were evidence of actions taking place in this aether. Light from the most distant stars was supposed to be transmitted to us in the form of wave motions in the aether, and we could see the stars only because the sea of aether between us and these stars was unbroken. It had been proved that if this sea of aether existed it must be at rest, for the alternative hypothesis that the aether was dragged about by ponderable bodies in their motions had been shown to be incompatible with the observed phenomenon of astronomical aberration and other facts of nature (see 1.292). On this view it was no longer necessary to go to Newton's " remote regions of the fixed stars, or perhaps far beyond them," to find absolute rest. A standard of absolute rest was provided by the aether which filled our laboratories and pervaded all bodies. Owing to our motion it would appear to be rushing past us, although without encountering any hindrance - " like the wind through a grove of trees," to borrow the simile of Thomas Young. The determination of the absolute velocity of the earth was reduced to the problem of measuring the velocity of an aether current flowing past us and through us.

In this same year (1887) the first experimental determination of this velocity was attempted by the Chicago physicist A. A. Michelson. The velocity of light was known to be, in round numbers, 300,000 km. a second, a velocity which was believed to represent the rate of progress of wave motion through the aether. If the earth were moving through the aether with a velocity of i,000 km. a second, the velocity of light relative to a terrestrial observer ought to be only 299,000 km. a second when the light was sent in exactly the direction of the earth's motion through the aether, but would be 301,000 km. a second if the light was sent in the opposite direction. In more general terms, if the earth were moving through the aether, the velocity of light, as measured by a terrestrial observer, would depend on the direction of the light, and the extent of this dependence would give a measure of the earth's velocity. The velocity of light along a single straight course does not permit of direct experimental determination, but the same property of dependence on direction ought to be true, although to a less extent, of the average to-and-fro velocity of a beam of light sent along any path and then reflected back along the same path.

It was through this property that Michelson attempted to measure the earth's velocity through the aether.

The apparatus was simple in principle. A circular table ABCD was arranged so as to be capable of slow rotation about its centre O. Light sent along CO was divided up at 0 into two beams which were made to travel along perpendicular radii OA, OB. The arms OA, OB were made as equal as possible and mirrors were placed at A and B to reflect the beams of light back to O. An extremely sensitive optical method made it possible to detect even a very slight difference in the times of the total paths of the two beams from 0 back to O. There would in any case be a difference owing to the necessarily imperfect equalization of the lengths of the arms OA, OB, but if the earth is moving through the aether in some direction OP, and if the table is made to rotate slowly about 0, then this difference ought itself to vary on account of the earth's motion through the aether. Michelson, and afterwards Michelson and Morley in collaboration, attempted to estimate the amount of this variation.

No variation whatsoever could be detected, although their final apparatus was so sensitive that the variation produced by a velocity through the aether of even i km. a second ought to have shown itself quite clearly.

Thus to the question " What is our velocity through the aether ? " Nature appeared to give the answer " None." was never suggested that this answer should be accepted as final; it would have brought us back to a geocentric universe. Clearly either the question had been wrongly framed or the answer wrongly interpreted. It was pointed out in 1893 by Fitzgerald, and again, independently, in 1895, by Lorentz, that the null result of the Michelson-Morley experiment could be explained if it could be supposed that motion through the aether altered the linear dimensions of bodies.

To be explicit, it was found that the experiment would invariably and of necessity give a null result if it was supposed that every body moving through the aether with a velocity u was contracted in the direction of its motion in the ratio V 1 - 2 2, c being :the velocity of light. The supposition that such a contraction occurred was not only permissible - it was almost demanded by electrical theory. For Lorentz had already shown that if ,matter were a purely electrical structure, the constituent parts would of necessity readjust their relative positions when set in motion through the aether and the final position of equilibrium would be one showing precisely the contraction just mentioned.

On this view, there was no prima-facie necessity to abandon the attempt to measure the earth's velocity through the aether. The answer to the problem had merely been pushed one stage farther back, and it now became necessary only to measure the shrinkage of matter produced by motion. It was obvious from the first that no direct material measurement could disclose the amount of this shrinkage, since any measuring rod would shrink in exactly the same ratio as the length to be measured; but optical and electrical methods appeared to be available. Experiments to this end were devised and performed by Rayleigh, Brace, Trouton and Noble, Trouton and Rankine and others. In every case a null result was obtained. It appeared then that if the earth moved through the aether this motion was concealed by a universal shrinkage of matter, and this shrinkage was in turn concealed by some other agency or agencies whose wit, so far, appeared to be greater than that of man.

At this time the word " conspiracy " found its way into the technical language of science. There was supposed to be a conspiracy on the part of the various agencies of nature to prevent man from measuring his velocity of motion in space. If this motion produced a direct effect x on any phenomenon, the other agencies of nature seemed to be in league to produce a countervailing effect - x. A long train of experiments had not revealed, as was intended, our velocity through the aether; they had merely created a conviction that it was beyond the power of man to measure this velocity. The conspiracy, if such there was, appeared to have been perfectly organized.

A perfectly organized conspiracy of this kind differs only in name from a law of nature. To the inventor who tries to devise a perpetual-motion machine it may well appear that the forces of nature have joined in a conspiracy to prevent his machine from working, but wider knowledge shows that he is in conflict not with a conspiracy, but with a law of nature - the conservation of energy. In 1905 Einstein, crystallizing an idea which must FIG. 2.  have been vaguely present in many minds, propounded the hypothesis that the apparent conspiracy might be in effect a law of nature. He suggested, tentatively, that there might be a true law to the effect that " it is of necessity impossible to determine absolute motion by any experiment whatever." This hypothetical law may again be put in the equivalent form: " The phenomena of nature will be the same to two observers who move with any uniform velocity whatever relative to one another." This may be called the hypothesis of relativity.

The hypothesis in itself was not of a sensational character. Indeed, from the quotations which have already been given from Newton's works, it appears probable that Newton himself would have accepted the hypothesis without hesitation: he might even have regarded it as superfluous. The true significance of the hypothesis can only be understood by a reference to the scientific history of the two centuries which had elapsed since Newton. The Newtonian view that absolute rest was to be found only " in the remote regions of the fixed stars, or perhaps far beyond them," had given place to a belief that absolute rest was to be found all around us in an aether which permeated all bodies. What was striking about the hypothesis was its implication - either that we could not measure the velocity relative to ourselves of a medium which surrounded us on all sides, or else that no such medium existed.

The hypothesis demanded detailed and exhaustive examination. It was for the mathematician to test whether the hypothesis was in opposition to known and established laws of physics, and to this task Einstein, Lorentz and others set themselves. If a single firmly established law proved to be in opposition to the hypothesis, then of course the hypothesis would require to be abandoned. It was unlikely that such an event would occur among the well-established laws, for if it did, the phenomena governed by that law would enable direct measurement to be made of the earth's velocity through the aether, a measurement which had so far eluded all attempts of experimenters. It was among the more obscure and less well-established laws, if anywhere, that discrepancies were to be looked for.

It is impossible here to give a complete account of the many tests to which the relativity hypothesis has been subjected. The result of all can be summed up in one concise and quite general statement: - Wherever the hypothesis of relativity has appeared to be in conflict with known or suspected natural laws, further experiment, where possible, has, without a single exception, shown the laws to be erroneous, and has moreover shown the alternative laws suggested by the hypothesis of relativity to be accurate. It is only in somewhat exceptional cases that the hypothesis of relativity of itself suffices to determine fully the form of a natural law; these cases constitute the most striking triumphs of the theory. As instances may be mentioned the determination of the law connecting the mass of an electron with its velocity; of the law expressing the velocity of light through a transparent medium in motion (Fizeau's water-tube experiment); and of the formulae for the magnetic forces on moving dielectric media (experiments of Eichenwald and H. A. Wilson).1 Befo e passing on from the general statement which has been made, particular mention must be made of one special case.

A natural law which was at an early stage seen to be in conflict with the hypothesis of relativity was Newton's famous law of gravitation - namely, that every particle of matter attracts every other particle with a force proportional to the product of the two masses, and to the inverse square of their distance apart. Either, then, Newton's great law had to be abandoned, or else the hypothesis of relativity had to be discarded, in which case it would immediately become possible, in theory at least, to determine the earth's velocity through space by gravitational mans. It is the choice between these two alternatives that has led to the most surprising developments of the theory of relativity; and to these we shall return later.

1 For references to the original papers dealing with these and other tests of the hypothesis of relativity see Cunningham, The Principle of Relativity, or J. H. Jeans, Mathematical Theory of Electricity and Magnetism (4th ed.).

## Space and Time

The hypothesis of relativity, as has already been explained, postulates that the phenomena of nature will be the same to any two observers who move relative to one another with any uniform velocity whatever. The hypothesis has been so amply tested as regards all optical and electromagnetic phenomena that no doubt is felt, or can rationally be felt, as to its truth with respect to these phenomena. The hypothesis can be examined and developed in two opposite directions. We may, on the one hand, proceed from the general hypothesis to the detailed laws implied in it; this has already been done, with completely satisfactory results as regards confirmation of the hypothesis. Or we may regard the hypothesis of relativity as being itself a detailed law and attempt to generalize upward to something still wider. It is this possibility which must for the moment claim our attention.

In 1905 Einstein examined in full the consequences of the hypothesis that one simple optical phenomenon - namely, the transmission of a ray of light in free space - was, in accordance with the hypothesis of relativity, independent of the velocity of the observer. If an aether existed, and provided a fixed framework of reference, then light set free at any instant would obviously travel with a velocity which would appear to an observer at rest in this a,ether to be the same in all directions, and the wave front at any instant would be a sphere having the observer as centre. On the hypothesis of relativity the phenomenon of light transmission must remain unaffected by the motion of the observer, so that the light must appear to a moving observer also, to travel with a uniform velocity in all directions, and thus to the moving observer also the wave front must appear to be a sphere of which he will be the centre. It is, however, quite obvious that the same spherical wave front cannot appear to each of two observers who have moved some distance apart to be centred round himself, unless the use either of the common conceptions of science or of the ordinary words of language is greatly changed. In fig. 2 it is not possible in ordinary language that both 0 and P should at the same instant be at the centre of the sphere A B C. The change to which Einstein was forced is one which has an intimate bearing upon our fundamental conceptions of the nature of space and time; this change it will be necessary to explain in some detail.

Suppose that two observatories, say Greenwich and Paris, wish to synchronize their clocks, with a view to, let us say, an exact determination of their longitude difference. Paris will send out a wireless signal at exact midnight as shown by the Paris clock, and Greenwich will note the time shown by the Greenwich clock at the instant of receipt of the signal. Greenwich will not, however, adjust their clock so as to show exact midnight when the signal is received; a correction of about oor second must be made to allow for the time occupied by the signal in traversing the distance from Paris to Greenwich. To turn to mathematical symbols, if t o is the time at which a signal is sent out from one station, the time of receipt at a second station is taken to be t o + `-, where x is their distance apart, and c is the velocity of light. This represents the ordinary practice of astronomers, but it is clear that if the earth is travelling through a fixed aether with a velocity 2t in the direction of the line joining the two observatories, the velocity of transmission of the signal relative to the two observatories will not be c but c?-u, and the time of receipt at the second station will be to+ x Thus it cd-U  appears that it is impossible to synchronize two clocks unless we know the value of u, and that the ordinary practice of astronomers will not, as they expect, synchronize their clocks, but set them at an interval apart equal to I I x - c c+u  which may, to an approximation, be put equal to u 2 
    
    
    c According to the hypothesis of relativity, it is impossible ever to determine the value of u, and so is impossible ever truly to synchronize two clocks. Moreover, according to this hypothesis, the phenomena of nature go on just the same whatever the value of u, so that the want of synchrony cannot in any way show itself - in fact, if it did, it would immediately become possible to measure the effect and so arrange for true synchrony.

As the earth moves in its orbit, the value of u changes, so that its value in the spring, for instance, will be different from its value in the autumn. One pair of astronomers may attempt to synchronize a pair of clocks in the spring, but their synchronization will appear faulty to a second pair who repeat the determination in the autumn. There will, so to speak, be one synchrony for the spring and another for the autumn, and neither pair of astronomers will be able to claim that their results are more accurate than those of their colleagues. More generally we may say that different conceptions of synchrony will correspond to different velocities of translation.

These elementary considerations bring us to the heart of the problem which we illustrated diagrammatically in fig. 2. The observer at 0 in the diagram will have one conception of simultaneity, while the second observer who moves from 0 to P will, on account of his different velocity, have a different conception of simultaneity. The instants at which the wave front of the light signal from 0 reaches the points A, B, C in the diagram will be deemed to be simultaneous by the observer who remains at 0, but the observer who moves fron 0 to P will quite unconsciously have different ideas as to simultaneity. At instants which he regards as simultaneous the wave front will have some form other than that of the sphere A B C surrounding O. If the hypothesis of relativity is to be true in its application to the transmission of light signals, this wave front must be a sphere having P as its centre.

Einstein examined mathematically the conditions that this should be possible. Unfortunately a precise statement of his conclusions can only be given in mathematical language.

The observer who is supposed to remain at 0 in fig. 2 may be supposed to make exact observations and to record these observations in mathematical terms. To fix the positions of points in space he will map out a " frame of reference " consisting of three orthogonal axes, and use Cartesian coordinates x, y, z, to specify the projections along these axes of the radius from the origin to any given point. He will also use a time coordinate which may be supposed to specify the time which has lapsed since a given instant, as measured by a clock in his possession. Any observations he may make on the transmission of light signals can be recorded in the form of equations between the four coOrdinates x, y, z, t. For instance, the circumstance that light travels from the origin with the same velocity c in all directions will be expressed by the equation (of the wave front): - x 2 + 5, 2 + z 2 - c 2 t 2 o... (r) The second observer who moves from 0 to P will also construct a frame of reference, and we can simplify the problem by supposing that his axes are parallel to those already selected by the first observer. His coOrdinates, to distinguish them from those used by the first observer, may be denoted by the accented letters x', y', z', t'. If his observations also are to show light always to travel with the same velocity c in all directions, the equation of the wave front, as observed by him, must be: x ,2 + y ,2 +z '22 t '= o. 2 (2) A 19th-century mathematician would have insisted that x, y, z, t must be connected with x', y', z', t by the simple relations: y z' =z t' =t  but it is obvious that if these relations hold, then equation (r) cannot transform into equation (2). Einstein finds that equation (i) will transform into equation (2) provided the coOrdinates x, y, z, t of the first observer are connected with the coOrdinates x', y', z', t of the second observer by the equations:- x'=$ (x - ut) l  z' = y z. .. (B)) t'i (t,ux c where u: ?. ? stands for (I - c2 To form some idea of the physical meaning of these equations, it will be advantageous to consider the simple case in which the first observer is at rest in the aether while the second moves through the aether with velocity u. The points of difference between equations and (A) then admit of simple explanation. The factor 1 3 in the first of equations (B) is simply, according to the suggestion of Fitzgerald and Lorentz already mentioned, the factor according to which all lengths parallel to the axis of x must be adjusted on account of motion through the aether with velocity u. The moving observer must correct his lengths by this factor, and he must correct his times by the same factor in order that the velocity of propagation of light along the axis of x may still have the same velocity c; this explains this presence of the multiplier)3 in the last of equations (B). The one remaining difference between the two sets of equations, namely the replacement of in (A) by 1- - 2 in (B), represents exactly the G want of synchrony which, as we have already seen, is to be expected in the observations of two observers whose velocity differs by a velocity u.  Although the equations admit of simple illustration by considering the case in which one observer is at rest in a supposed aether, it will be understood that the equations are more general than the illustration. They are in no way concerned with the possibility of an observer being at rest in an aether, or indeed with the existence of an aether at all. Their general interpretation is this: If one observer 0, having any motion whatever, finds, as a matter of observation, that light for him travels uniformly in all directions with a constant velocity c, then a second observer P, moving relative to 0 with a constant velocity u along the axis of x, will find, as a matter of observation, that light, for him also, travels uniformly in all directions with the same constant velocity c, provided he uses, for his observations, coOrdinates which are connected with the coOrdinates of 0 by equations (B).

This is the meaning that was attached to the equations by Einstein in 1905, but the equations had been familiar to mathematicians before this date. They had in fact been discovered by Lorentz in 1895 as expressing the condition that all electromagnetic phenomena, including of course the propagation of light, should be the same for an observer moving through the aether with velocity u as for an observer at rest in the aether. For this reason the transformation of coOrdinates specified by these equations is universally spoken of as a " Lorentz transformation." What Einstein introduced in r905 was not a new system of equations but a new interpretation of old equations. The two observers who used the coordinates x, y, z, t and x', y', z', t' had been regarded by Lorentz as being one at rest in an aether and one in motion with a velocity u; for Einstein they were observers moving with any velocities whatever subject to their relative velocity being u. Lorentz had regarded t as the true time and t as an artificial time. If the observer could be persuaded to measure time in this artificial way, setting his clocks wrong to begin with and then making them gain or lose permanently, the effect of his supposed artificiality would just counterbalance the effects of his motion through the aether. With Einstein came the conception that both times, t and t', had precisely equal rights to be regarded as the true time. The measure t is precisely that which would be adopted naturally by any set of observers, or race of men, who disregarded their steady motion through space; their adoption of it would be above criticism if, as Einstein suggested, their motion through space had no influence on material phenomena, and it represents, as we have seen, the usual practice of astronomers in comparing time at different places. From this point of view, . (A) neither measure of time is more accurate or more logical than the other. There are as many ways of measuring time as there are observers, and all are right.

The investigator who is trying to discover laws of nature will, in general, require to measure either directly or indirectly both time and space. If, to take a simple case, he is studying the motion of a single particle, he will measure out the position of the particle at definite instants as determined by his clock. He may specify the position of the particle at any instant by three measurements in space - for instance, he may say that two seconds after his particle started it was 6 ft. to the E. of the point from which it started, 9 ft. to the N. and 12 ft. vertically upward. The mathematician would express this by taking axes x, y, z to the E., to the N. and vertically upwards, and saying that at time 1=2 the particle had coordinates x = 6, y=9, z = 12. Or he might, putting his time coordinate t on the same footing as the space coordinates x, y, z, simply say that x = 6, y= 9, z = 12, t= 2 represented one position of the particle.

A complete set of readings of this type, each consisting of values of four coordinates, would give the complete history of the motion of the particle.

Such sets of simultaneous measurements form the common material of investigations in both pure and applied science. For instance, the engineer may measure the extension of a sample of steel corresponding to different loads; the electrician may measure the amount of light given by an electric filament corresponding to different amounts of current passed through it. In each of these cases there are only two quantities to be measured simultaneously, and an investigator can conveniently represent the result of the whole series of his measurements in graphical form; a single reading is represented by a point whose distances from two fixed perpendicular lines represent the quantities measured, and the curve obtained by joining these single points will give all the information contained in the whole set of readings.

We have seen that, in studying the motion of a particle in space, four sets of quantities must be measured, so that the results obtained cannot be plotted graphically on a piece of paper. Their proper representation demands a four-dimensional space, in which x, y, z and t are taken as coordinates. The practical importance of such graphical representation is nil, since it is impossible to construct a four-dimensional graph, but its theoretical importance to the theory of relativity is immense. For if the hypothesis of relativity is true, then the four-dimensional graphs of any natural event constructed by all observers, no matter what their relative motions, will be identical. The influence of their motion will be shown only in that the axes of x, y, z and t will be different for different observers, and the relations between these sets of axes will be those given by the foregoing equations (B).

The importance of this conception can hardly be overestimated, and it may be well to consider it further with the help of an illustrative example. Imagine a number of aeroplanes flying over England, and, in order to eliminate one of the three directions in space - the vertical - let us limit them to fly always at the same height, say i,000 ft. above sea-level. Imagine a number of similar plates of glass prepared, each marked faintly with an outline map of England and with lines of latitude and longitude. Suppose that at 12 h. o m. G.M.T. a plate is taken and the position of each aeroplane marked by a thick black dot. At z 2 h. r m. let a second plate be taken and similarly marked, and let this be done every minute for an hour. The 60 plates so marked will constitute a record of the motion of each aeroplane within this hour. If, now, we place the plates in order, one above the other, on a horizontal table, the mass of glass so formed will present a graphical representation, in three dimensions, of the motions of all the aeroplanes. In this graph the two horizontal coordinates represent motions in any two rectangular directions over England, say E. and N., while the third coordinate - the vertical - represents time. The individual black dots which represent the positions of any one aeroplane will form a dotted curve, and this curve gives a graphical representation of the motion of the particular aeroplane. Our rectangle of glass contains the history, for one hour, of all the aeroplanes in graphical form.

To represent the motion of particles in the whole world of space a four-dimensional graph is required. The four-dimensional space in which it is constructed may, following the usual terminology, be spoken of as a four-dimensional continuum. The history of any particle in the universe - just as that of anyaeroplane flying over England - will be represented by a continuous line in the continuum, and this is called the " world line " of the particle. If the hypothesis of relativity is true the same continuum and the same world lines will represent the history of the particles of the universe equally well for all. observers, the influence of their motions being shown only through their choosing different axes in the continuum for their axes of space and time. Thus the continuum must be thought of as something real and objective, but the choice of axes is subjective and will vary with the observer, the relation between different choices being expressed mathematically by our equations (B), the equations of the Lorentz transformation. An inspection of these equations shows that the sets of axes chosen by different observers have different orientations in the continuum, so that what one observer describes as a pure space interval will appear to another to be a mixture of time and space.

The instant of time and point in space at which any event occurs can be fixed by a single point in the continuum, so that the interval between two events will be represented by a finite line. The events and the interval between them are absolute, but the interval will be split up into time and space in different ways by different observers. The interval between any two events, such as the great fire of London and the outburst on the star Nova Persei, may be measured by one set of observers as so many years and so many millions of miles, but another set of observers may divide the interval quite differently. For instance a terrestrial astronomer may reckon that the outburst on Nova Persei occurred a century before the great fire of London, but an astronomer on the Nova may reckon with equal accuracy that the great fire occurred a century before the outburst on the Nova. A third astronomer may insist that the events were simultaneous. All will be equally right, although none will be right in an absolute sense. At this stage we may notice one respect in which our pile of glass plates failed to represent the true continuum. The mass of glass was stratified into different plates which represent different times for one particular observer. To obtain a section which would represent what an observer in motion relative to this first observer could regard as simultaneous positions of the aeroplanes, we should have to cut the mass of glass on the slant. The continuum is more closely represented by our plates of glass if they are annealed into a solid mass from which all trace of the original stratification is made to disappear. All observers, no matter what their motion, are then equally free to cut a section to represent their individual ideas of simultaneity.

Thus space and time fade into subjective conceptions, just as subjective as right hand or left hand, front and behind, are in ordinary life. The continuum alone is objective and may be thought of as containing an objective record of the motion of every particle of the universe. The curve in which this record is embodied is spoken of as the world line of the particle in question. To use the words of Minkowski: " Space in itself and time in itself sink to mere shadows, and only a kind of union of the two retains an independent existence." Gravitation and Relativity. - Since all the phenomena of light and of electromagnetism are believed, on almost incontrovertible evidence, to be in accordance with the hypothesis of relativity, it is necessarily impossible to determine absolute velocity by optical or gravitational means. On the other hand, as we have already mentioned, the Newtonian law of gravitation is readily seen to be inconsistent with the hypothesis of relativity. Three alternatives arc open: (i.) The Newtonian law may be true, in which case it must be possible to determine absolute velocity by gravitational means.

(ii.) The Newtonian law may be untrue in its original form, but may become true when amended so as to conform to the relativity hypothesis.

(iii.) Neither of the foregoing possibilities may be true.

Alternative (i.) was explored by Sir Oliver Lodge, who, assuming the exact truth of the Newtonian law of gravitation, deduced that the observed motion of the perihelion of Mercury could be accounted for if the sun were moving through space with a velocity of about 70 km. a second in a certain direction. This investigation had to be abandoned when it was shown by Eddington that a similar discussion of the motions of the other planets would lead to vastly different values for the sun's velocity. Alternative (ii.) was explored by Einstein and others, but was found to lead to a motion of the perihelion of Mercury equal only to one-sixth part of that actuallyobserved.

Alternative (iii.) remained with its innumerable possibilities. Einstein commenced his attack on the problem by eliminating all possibilities which did not conform to two general principles. The first of these was the principle of relativity. Inasmuch as .all physical phenomena except gravitation were believed to conform to this principle, it was natural to try, as a working hypothesis, the effect of assuming gravitation also to conform. The second principle was the so-called principle of equivalence, and this demands a word of explanation.

To our children we explain that an apple falls to the ground because a force of gravitation inherent in the earth's mass impels the apple towards the centre of the earth. Most schoolboys know that this is not quite the whole story; the path of the apple is more accurately determined by supposing the apple to be acted on simultaneously by two forces - a gravitational force of attraction towards the earth's centre and the centrifugal force arising from the earth's rotation. It is only because the earth's rotation is comparatively slow that the conception of an attraction towards the earth's centre gives a tolerably plausible account of the fall of the apple. If the earth rotated at i 7 times its present rate objects would not fall, even approximately, towards the earth's centre; they would fall always parallel to the earth's axis, and the inhabitants of the northern hemisphere might explain this as arising from a force of repulsion inherent in the pole star. If the earth rotated many times faster even than this, bodies would fall always perpendicularly away from the earth's axis, and this might be interpreted as arising from a gravitational repulsion residing in the earth's axis.

These illustrations will show that it is easy to confuse acceleration arising from the earth's rotation with gravitational attraction. We may go further and say that it is impossible to distinguish between the effects of gravitational attraction and the effects of acceleration of any kind whatever. Every aeroplanist knows this to his sorrow; it is inherently impossible to devise any instrument which shall show the direction of the vertical in an aeroplane, since an acceleration of the aeroplane produces on any instrument whatever, effects which are indistinguishable from those of gravity. From such considerations Einstein was led to his principle of equivalence, which may be enunciated as follows: " A gravitational field of force at any point of space is in every way equivalent to an artificial field of force resulting from acceleration, so that no experiment can possibly distinguish between them." Guided by these two principles - relativity and equivalence - Einstein was led to the view that all gravitational " fields of force " must be illusions. The apparent " force " arises solely from acceleration and there is no other kind of gravitational force at all. In this statement, as in the statement of the principle of equivalence above, the word acceleration is used in its widest sense. Acceleration results not only from change in the amount of a velocity, but from a change in its direction also. For instance a motor-cyclist riding in a circle at a uniform speed of 60 miles an hour will be the subject of an acceleration towards the centre of the circle. He knows that the apparent force so produced is just as real in its effects as gravitation, and to save himself from falling as a result of its influence he must incline the direction of his machine to the vertical.

It is clear that the acceleration or curvature of path which figures as gravitation cannot be an acceleration or curvature in ordinary three-dimensional space. Before the apple starts to fall from the tree there is neither acceleration nor curvature, and yet the apple is undoubtedly acted on by gravitation. Moreover, this three-dimensional space is, as we have seen, different for different observers - it is a subjective and not an objective conception, and the gravitation resulting from such a curvature could not conform to the relativity condition. Einstein was accordingly led to suppose that gravitation arose from curvature in the four-dimensional space, or continuum, in which time formed the fourth dimension. This continuum, as has been seen, is objective and if the path of the particle can also be made objective, the resulting gravitation will conform to the relativity principle. The path of the particle in the continuum is, however, simply its " world line," which we have already had under discussion. This world line is determined by natural laws, and if these are to be objective the specification of the world line must also be objective. There is, however, only one specification of world lines in the continuum which is objective in the sense that the same specification will give the same world lines to observers moving with different velocities. It is that every world line must be so drawn as to represent the shortest path between any two points on it. Mathematically, lines which satisfy this condition are known as geodesics. Thus Einstein was led to suppose that world lines must be geodesics in the four-dimensional continuum.

Consider for a moment a page of this volume as presenting a two dimensional analogy of the continuum. The shortest distance between any two points is of course the straight line joining them, so that the geodesics are simply straight lines. These possess no curvature of path and if they formed a true analogy to the geodesics in the continuum there could clearly be no explanation of gravitation of the type we have been contemplating. There is, however, another type of two-dimensional surface. It is represented by the surface of a solid body such as a sphere - say the earth. On the earth's surface the geodesics are the great circles; every mariner or aeronaut who desires to sail the shortest course between two points sails along a great circle. To take a definite instance, the shortest course from Panama to Ceylon is not along the parallel of lat. (about q° N.) which joins them - the aeronaut wishing to fly the shortest course between the two countries will fly N.E. from Panama, he will pass over England and finally reach Ceylon from the north-west. The reader may rapidly verify this by stretching a thread tightly over the surface of an ordinary geographical globe. Let him now trace out the course on an ordinary Mercator chart, and it will be found to appear very curved indeed - the course of the aeronaut will look surprisingly like that of a comet describing an orbit under the attraction of a sun situated somewhere near the middle of the Sahara.

The reader who performs these simple experiments will understand how Einstein was led to suppose that gravitation could be explained by a curvature inherent in the continuum. The world lines of particles are geodesics but the space itself, so to speak, provides the curvature. The curvature of path is thrust upon the particle by the nature of the continuum, but we, who until recently have been unaware even of the existence of the continuum, have been tempted to ascribe it to the action of a special agency which we have invented ad hoc and called " gravitation." According to Einstein, it is no more accurate to say that the earth attracts the moon than to say that the pockets of an uneven billiard table repel the balls.

This train of thought may seem artificial. If so, the reason is that we have not been able to explore the other possibilities which have branched off our main line of thought. In point of fact, Einstein found himself practically limited to the conclusion we have stated. Not only so, but the actual type and degree of curvature in the continuum prove to be uniquely fixed in terms of the masses of the gravitating bodies. Thus Einstein, knowing the mass of the sun, found himself in a position to predict absolutely what the motion of the perihelion of Mercury ought to be. It was found to be 42.9" a century, a figure which agreed with observation to well within the limits of error of these observations. The motions of the other planets, as predicted by the theory of relativity, have also been found to agree with those observed to within the errors of observation. This latter test, however, is not a very stringent one, since the departures from the motion predicted by the Newtonian law are too small to admit of very precise measurement.

Einstein's theory requires us to suppose that the world line of a ray of light also shall be a geodesic in the continuum. In a gravitational field the curvature of the continuum will impose a twist on the path of a ray of light. Einstein found in particular that a ray of light which comes from a distant star and passes near the edge of the sun on its journey ought to be bent, in its passage past the sun, by an angle which should be 175" if the ray just grazes the sun, and would be less in proportion to the inverse distance from the centre of the sun for other rays. The observatories of Greenwich and Cambridge dispatched expeditions to test this prediction at the eclipse of 1919. It was found that the stars which appeared near to the sun at the instant of eclipse showed an appreciable displacement, as compared with their normal positions, of the type required by Einstein's theory. Exact measurement confirmed that the displacement varied approximately as the inverse distance from the sun, and that the displacement at the limb was sensibly equal to Einstein's predicted value of 1.7 5". The Cambridge observers, hampered by cloudy weather, obtained for this quantity the value 161" 0.30". The Greenwich observers obtained a value of I98" 0.12", but it has since been pointed out by Prof. H. N. Russell that their photographs indicate a horizontal and vertical scale difference of the order of i part in 12,000, almost certainly due to a distortion of the coelostat mirror under the sun's rays, and if the measures are corrected for this the result is brought much closer to the theoretical prediction.

The theory makes one further prediction which admits of experimental test. The atoms of any element, say calcium, may be supposed to be formed according to a definite specification, the terms of which depend neither on the velocity of a particular observer nor on his position relative to the gravitational fields of the universe. It can be deduced that the light received from a calcium atom situated in the intense gravitational field near the sun's surface ought to be of slower period, and therefore of redder colour, than the similar light emitted by terrestrial atoms. To be more precise, the Fraunhofer lines in the solar spectrum ought to show a displacement to the red; this displacement ought to be homologous, and should be of amount 0.008 A units at the cyanogen band X 3883 at which observations have been chiefly made. Attempts to test this prediction led to strangely discordant results. All observers agreed in finding some effect of the kind predicted, but its amount was always less than the predicted amount, varying from almost nil (St. John, 1917) to nearly the full amount to be expected (Evershed, 1918; Grebe and Bachem, 1919). In 1921 the position with regard to this test still remained one of great uncertainty and confusion.

It will have been seen that the restricted physical theory of relativity introduced a revolution into the foundations of scientific thought by destroying the objectivity of time and space. The gravitational theory has effected a hardly less important revolution by destroying our belief in the reality of gravitation as a " force." The physicist has, however, to deal with other " forces " besides those of gravitation, and the question inevitably arises as to whether these too must be regarded as illusions, arising only from our faulty interpretation of the special metrical properties of the continuum. Prof. H. Weyl has pointed out that the continuum imagined by Einstein, and found to be adequate to explain gravitational phenomena, is not, in respect of its metrical properties, the most general type of continuum imaginable. A further generalization is possible and the new curvatures introduced must of necessity introduce new apparent forces other than gravitational. Weyl's investigation shows that these new forces would have exactly the properties of the electric and magnetic forces with which we are familiar. Indeed, the predicted forces coincide so completely with known electromagnetic forces that no experimental test of Weyl's theory is possible. Had there been the slightest divergence between the forces predicted by Weyl and those predicted by ordinary electromagnetic theory, experiment could have been asked to decide between the two, but no such divergence exists. It may, however, be said that Weyl's theory makes it highly probable that all forces reduce to nothing more than our subjective interpretations of special properties of the continuum in which we live our lives.

Finally a thought may be given to the position, under the new conceptions introduced by the theory of relativity, of the electromagnetic aether. At one stage in the history of science there wa s a tendency to fill space with aethers, to the extent almost of one aether for every set of phenomena requiring explanation. That stage passed, and by the end of the 19th century only one aether received serious consideration, the so-called electromagnetic aether of Faraday and Maxwell. This aether gave a plausible mechanical explanation of electrostatic phenomena, although it was more than doubtful whether it could account for the electromagnetic phenomena from which it took its name, and it was comparatively certain that it could not account for gravitation. It gave, however, a satisfactory explanation of the propagation of waves of light - they were simply waves in the aether and travelled with an absolute velocity c determined once and for all by the structure of the nether. On this view it was quite certain that an observer moving through the nether with a velocity u would measure the velocity of light travelling in the same direction as himself as c - u. Relativity teaches that this velocity is always precisely c, and this in itself disposes of the nether of Faraday and Maxwell. Whether any new aether will be devised to replace it remains to be seen, but none appears to be necessary. Any aether which can be imagined would appear to depend upon an objective separation of time and space. Relativity does not deny that such an objective separation. may, in the last resort, really exist, but it shows that no material phenomena are concerned with such a separation. By a very slight turn of thought, the primary postulate of relativity may be expressed in the form that the material world goes on as though no aether existed.

To the relativist the essential background to the picture of the universe is not the varying agitation of a sea of aether in a three-dimensional space but a tangle of world lines in a fourdimensional space. Moreover, it is only the intersection of the world lines that are important. An intersection at a point in the continuum represents an event, while the part of a world line which is free from intersections represents the mere uneventful existence of a particle or a pulse of light. And so, since our whole knowledge of the universe is made up of events, it comes about that the tangle of world lines may be distorted and bent to any degree we please; so long as the order of the intersections is not altered, it will still represent the same universe. And so the last function of the aether, that of providing a scale of absolute measurements in space, becomes a superfluity. To the physicist who urges that space measurements without an underlying aether become meaningless, the relativist can reply that timemeasurements without an underlying " time-aether " are equally meaningless. A " time-aether " has never been regarded as a necessity, and the relativist feels that the " space-aether " has no greater claim to retention. (J. H. Jr.)