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George B Jeffery Inaugural Lecture
In 1922 George Barker Jeffery was appointed Professor of Mathematics at King's College London. He gave the Inaugural Lecture on 9 October 1922 entitled Einstein's Theory of Relativity. A version of his lecture follows.
Einstein's Theory of Relativity
As I conceive the office of a professor, it is that he should stand before his students as the living representative of those great men who in the past have laboured in that branch of human knowledge which he has made his own; that by means of a reverent yet unflinching criticism he should strive to reveal the workings of these master minds, to the end that he may impart, not mere knowledge, but that more precious gift - the art of acquiring knowledge, the art of discovery. If we approach our task in this spirit, we shall find the key to the solution of much that is difficult and perplexing in our present knowledge, and the inspiration which will lead us on to further discoveries.
It seems natural, therefore, that I should seek to illustrate this theme by means of the subject which throughout my mathematical career has inspired me more than any other branch of mathematics or physics into which my work has led me, and the subject which, as far as one may venture to prophesy as to the future course of scientific thought, seems marked out for great advances in the immediate future.
Einstein's theory of relativity has proved full of difficulty to layman and expert alike. It seems to invite us to cut ourselves loose from all that has gone before, to scrap all our old ideas and to start afresh with new. It is true that the theory does profoundly modify our fundamental ideas of space, time, and motion, but a deeper study reveals the fact that it is nevertheless the natural and almost inevitable sequel to the work of the great masters of the past, and more particularly to the work of Isaac Newton himself; how natural and inevitable it will be the main purpose of this lecture to show.
The story of modern mechanics begins in the sixteenth century. Tycho Brahe, with no telescope, and the most primitive instruments in place of the equipment of the modern astronomical observatory, sustained through years of labour by a most extraordinary patience, observed night after night; the positions of the planets among the surrounding stars. Tycho was one who sowed but did not reap. Those who have any experience of observational astronomy find it difficult to imagine a duller task than Tycho's - the accumulation of volumes of figures whose meaning it was not given to him to read - the construction of a Nautical Almanac without its beautiful system and order. Nevertheless his work has provided the necessary foundation for all that has followed.
The task was taken up by Tycho's pupil and assistant John Kepler. He succeeded in clothing his master's data with the form of three simple descriptive laws. His was a great achievement. All Tycho's volumes of figures, all those strange motions of the bodies which men have most appropriately called wanderers, were summed up in three simple statements. Kepler had no theory; he made no attempt to explain the motions he studied. The Archangels who kept the celestial spheres in motion were dismissed, but no subtle scientific hypothesis was imported to perform their office. In effect he said: Viewed from the earth, the motions of the planets are very complicated. Now they wander forwards; now they retrace their steps; now they move in loops; and now in long sweeping curves. But viewed from the Sun, these motions are very simple, for then each planet describes a perfectly regular ellipse.
Meanwhile Galileo had brought the same on the problems of terrestrial dynamics. He investigated the laws which govern the motions of falling bodies. It had been taught that every body had its "proper place." The proper place of heavy bodies was low down, and the proper place of light bodies was high up. A body tended to move to its proper place; the heavier a body, the more quickly it fell, since it was presumably at a greater distance from its proper place. It is a striking commentary on medieval thought, that it seems to have occurred to nobody before the time of Galileo to test this conclusion by means of a simple experiment. Galileo made such an experiment at the leaning tower of Pisa, and found that all bodies, heavy or light, fall towards the ground in precisely the same way. By careful laboratory experiments, he ascertained the law of this fall. Any body falls towards the ground with a speed which increases in proportion to the time, so that its speed is increased by about 32 feet per second in every second of its fall. Thus Newton found ready to hand two sets of descriptive laws: Kepler's laws, which embraced the motions of the planets; and Galileo's law, which covered a very important case of the motion of terrestrial bodies. His first step was to throw the laws of Kepler into a different form. No doubt he took the hint from Galileo's law of falling bodies, and he investigated the motion of a planet, moving in accordance with Kepler's laws, from the point of view of the change of its velocity, or, as we should say, its acceleration. He found that Kepler's laws are equivalent to the statement that the acceleration of a planet is always directly towards the sun, and that this acceleration depends in no way on the planet, but only on its distance from the sun, diminishing with increasing distance in accordance with the law of the inverse Square.
Newton observed the similarity between the motions of the planets round the sun and the motion of bodies falling towards the earth. They too fall with an acceleration which in no way depends on the falling body. Is this gravitation subject to the same laws as the gravitation of the sun which keeps the planets in their orbits? Does it also diminish as the inverse square of the distance? It is difficult to answer these questions in the narrow range of height we can employ at the earth's surface, but Newton took the heavens for his laboratory. The moon, though somewhat disturbed by the sun, moves round the earth approximately in accordance with Kepler's laws, and has an acceleration towards the earth. Is this acceleration just what 32 feet per second per second would become if it diminished in accordance with the inverse square law up to the moon's distance? Newton worked the sum and found that it was so.
Thus the inward nature of gravitation was laid bare. There is a gravitation of the sun, in virtue of which any planet, comet, or meteorite which may happen to find itself in a given position experiences an acceleration which depends only upon that position. There is a gravitation of the earth, in virtue of which the moon, or any unsupported body near the earth, experiences an acceleration which again depends only upon the position of the accelerated body.
Thus far Newton was on very safe ground, for he was merely expressing the results of observation in a concise and compact form. He then proceeded to frame a theory which should account for the observed facts. Here we can trace the influence of Galileo very clearly. From his experiments on the motion of a body down an inclined plane, Galileo inferred that a body moving on a horizontal plane would continue to move with a constant velocity in a straight line. Earlier thinkers had felt the necessity of ascribing some cause to the motion of bodies; if a body moves some agency must be at work to maintain its motion. The experiments of Galileo, and Newton's interpretation of Kepler's laws, conspired to promote the view that it was the change of motion, the acceleration of a body, for which a cause must be found, rather than the motion itself. Newton adopted the view that when the motion of a body changes it does so because the body is acted upon by a force, and that this force is measured by the product of the mass of the body and its acceleration. Gravitation is explained by the action of forces arising from, and directed towards, attracting bodies. This in the barest outline is the Newtonian system of mechanics as commonly understood. Before we proceed to criticise it, it may be well for a moment to pause to consider the achievement which stands to its credit. The motions of the planets are not, in fact, quite so simple as the laws of Kepler would indicate. Someone has said that if Kepler had possessed a modern telescope he would never have discovered his laws. Nevertheless, with a few small outstanding differences, the deviations from Kepler's laws are all explained when we take into account the gravitation of the planets upon each other. The history of dynamical astronomy has been very largely the verification, to an ever-increasing degree of refinement, of Newton's law of universal gravitation. Cavendish observed the workings of this same law in the attraction between quite small bodies in the laboratory. The laws of motion, originally deduced from the motions of the planets, are verified day by day in every engineering workshop.
It seems to me that the supposed conflict between Newton and Einstein rests very largely upon a failure to apprehend a distinction upon which Newton was always insisting, the distinction between what he called mathematical principles and philosophical principles. Mathematical principles were to Newton, not ultimate causes, but merely concise descriptions of the phenomena of Nature, which could be verified by observation and experiment. He distinguishes them very clearly from philosophical principles, whose function it is to explain and interpret phenomena. This distinction, maintained in actual scientific work, is one of the great debts which we owe to Newton. It defines at once the purpose and the limitation of Science. When Science shall have accomplished its purpose and described the whole material universe in the simplest way, it must leave us face to face with the philosophical problem of the mystery and meaning of the things which it has described. But Newton, like many of us, had within him something of the philosopher. He might jeer at the metaphysicians, but at times he could not help speculating, and rightly speculating, as to the meaning of those great descriptive laws which he found running throughout the whole fabric of Nature. He was, however, always careful to distinguish these speculations from the formulation of the mathematical principles which he regarded as the main part of his work, and we find them for the most part in the scholia in the Principia, and in the queries in the Optics. These speculations have been the subject of controversy ever since, and it is towards them that the criticism of Relativity is, for the most part, directed.
In a scholium which follows the definitions in the Principia, Newton sets forth his views on time, space, and motion. He distinguishes between absolute time and relative time which is measured by some motion. He says:-
The natural days, which, commonly, for the purpose of the measurement of time, are held as equal, are in reality unequal. Astronomers correct this inequality, in order that they may measure by a truer time the celestial motions. It may be that there is no equable motion, by which time can accurately be measured. All motions can be accelerated or retarded. But the flow of absolute time cannot be changed. Duration, or the persistent existence of things, is always the same, whether motions be swift or slow or null.In the same way he distinguishes between absolute space and relative space, and between absolute motion and relative motion. He says:-
We use in common affairs, instead of absolute places and motions, relative ones; and this without any inconvenience. But in physical disquisitions, we should abstract from the senses. For it may be that there is no body really at rest, to which the places and motions of others can be referred.Thus we need go no further than Newton himself, to find a clear statement of the problem to which the theory of relativity has attempted to supply an answer. Our experience is entirely of relative motions. We are at rest relatively to our immediate surroundings; we are moving at a rate of 100,000 miles an hour relatively to the sun; we are moving relatively to Sirius at such and such a speed; but how we are moving in an absolute sense, without reference to any other body, is a question which experimental science has often tried, but always failed, to answer. The statement that we are moving at a rate of 100,000 miles an hour is devoid of all physical meaning whatsoever, unless we state what we conceive to be at rest. This something, which for a particular purpose we assume to be at rest, we call our "frame of reference."
Now, if we consider Newton's work in its proper setting, there is no doubt at all as to what his frame of reference was. It was implicit in Tycho's data, and Tycho observed the motions of the planets relatively to the fixed stars. Newton's frame of reference was one in which the distant fixed stars are at rest. It seems likely that Newton, who boasted that he did not frame hypotheses, adopted the hypothesis of absolute space because in the fixed stars he found ready to hand a frame of reference which transcended the domestic motions of the solar system the chief objects of his study. Was not Newton's absolute space after all just the physical space mapped out by the fixed stars, rather than the metaphysical concept we have usually taken it to be?
In the light of modern knowledge this frame of reference presents great difficulties. We can now, in many cases, measure the velocities of these stars relative to each other and to our sun. We find that they are not fixed, or at least, they are not all fixed, for they move relatively to each other with widely different velocities. The reason why, night after night, they seem to occupy the same positions in their constellations is the same as that which makes an express train seem to move so slowly when viewed from a long distance across country. It is not that their motions are slow - in many cases they are almost inconceivably great - but that the stars themselves are at such immense distances from us. Still more modern knowledge forbids us to attempt to surmount this difficulty by supposing that the motions of the stars are random, like the motions of the atoms of a gas, so that we could average them out, in order to arrive at our fixed frame of reference. If our stellar system has indeed grown out of a giant nebula, there may be an ordered system in the motions of the stars.
By the time that the discordant motions of the stars had been well established, a new hope had arisen. The undulatory theory of light seemed to call for some medium to transmit the light vibrations, and the idea of an ether pervading all space was developed. Clerk Maxwell showed the intimate relation between light and electromagnetism. Later on, the electron theory promised to explain the whole of physics in terms of electricity. Matter was simply an aggregation of electric charge, and electricity was a state or singularity of the ether. The ether had become fundamental in physics. Here it seemed that the solution of all our difficulties might lie. A body moves when it moves relatively to the ether; our frame of reference is to be fixed, not with respect to the so-called fixed stars, but with respect to the ether.
The result did not work out happily. If mechanics adopted the ether in order to simplify the problem of motion, never was foster-parent blessed with a more unruly child. If we observe a star, the ether is undisturbed by the earth's motion through it; if we fill our telescope with water, the water communicates part of its motion to the ether; if we make an interference experiment in the laboratory, we can only conclude that the earth carries the ether with it in its motion. Quite apart from the logical difficulty as to how the ether, the standard of absolute rest, can itself move at all, it moves or it does not move in a delicate accommodation to the particular experiment which we may happen to have in hand at the moment. In spite of the labours of some of the greatest English mathematicians of the latter half of the nineteenth century, the situation grew steadily worse.
In the meantime experimental physicists had concentrated on the problem of the determination of our motion relative to the ether. Many different experiments were proposed and carried out with all the skill and ingenuity of a great generation of experimenters. The result was always the same. No experiment succeeded in revealing our motion through the ether. The story is not unlike that of an earlier chapter in the history of science, which tells how for centuries men tried to construct a perpetual motion machine. They failed, and out of their failure modern physics has erected a great principle. They searched in vain, until they were led to deny the very possibility of the thing they sought. That denial has become the Second Law of Thermodynamics, one of the most powerful principles of modern physics. Relativity is the outcome of the application of the same method to our present difficulty. As the result of repeated failure, it asserts that no physical experiment can ever reveal our motion through the ether.
This was the culmination of a long sustained effort to bring the absolute space of Newton within reach of physical experiment, or perhaps we should say, rather, to restore to absolute space the physical reality which it lost on the discovery of the motions of the fixed stars. It is the starting point of the theory of relativity, that no method has yet been discovered by which this can be accomplished. If this position is accepted it constitutes a fatal criticism of Newton's laws of motion, at any rate in the form in which he stated them, for the very terms of those laws motion, change of motion have no meaning apart from some pre-determined standard of rest or frame of reference. It is obvious that the time had arrived when some fundamental reconstruction of the theory could no longer be delayed. Einstein did not bring forth his theory merely as an elaboration and refinement of physical law in order to bring theory into accord with a few isolated and newly-discovered facts; he brought it forth to meet the situation created by a complete theoretical breakdown of the older system.
If we seek a way out of the difficulty, the first suggestion which presents itself is that, since our experience is confined to relative motions, it ought to be possible to express the laws of motion in terms of relative motions alone, without any reference to absolute motions.
This in effect is what Einstein has done, though he approached the problem from a rather different point of view. If we take any frame of reference, we can obtain laws which will describe the course of natural phenomena. Since we have to recognise that the choice of a frame of reference is arbitrary, we shall expect these descriptive laws to be different if we choose another frame of reference. In other words, we shall expect to find that our descriptive laws are relative to the particular frame of reference which we have chosen. For example, if we take a frame of reference fixed with respect to the earth, we shall obtain the Ptolemaic system of astronomy with its epicycles, etc., whereas if we take a frame of reference fixed with respect to the sun, we shall obtain the very different descriptive laws of Kepler.
The question to which Einstein addresses himself is, whether the descriptive laws of physics can be framed in such a way that if they are true for one frame of reference they will also be true for any frame of reference whatever. This is essentially a mathematical question. If it is answered in the affirmative, the experimental question will arise as to whether these general laws are in fact true for one frame of reference. By the aid of the calculus of tensors, Einstein was able to give an answer to the mathematical question, and it appears that it is possible to frame laws which are absolute in the sense that, if they are true at all, they are true independently of the particular frame of reference which we may happen to choose. If these laws are verified by experiment, we shall have succeeded in dispensing with absolute space and with all the difficulties to which the introduction of this concept into our scientific work has given rise.
As so often happens in scientific research, Einstein's efforts to clarify our fundamental ideas of mechanics led to an important extension of knowledge. He was able for the first time to bring gravitation into relation with other physical phenomena. Let us return for a moment to the view of gravitation which we have already considered. In the Solar system, and in the fall of heavy bodies towards the earth, we observe the same essential feature, namely, that any body placed in a particular position experiences an acceleration which depends in no way upon itself, but only upon the position in which it is placed. In his determination to confine himself to the description of phenomena, Einstein accordingly regards gravitation as a property of space varying from place to place, leaving open for the time being the question as to whether this property can be expressed in terms of the influence of attracting bodies. In this sense Einstein's space, unlike that of Newton, is not homogeneous, but differs in its properties from place to place.
We can best explain Einstein's discovery by means of a simple, if somewhat fanciful, illustration. Imagine a lift working in a deep well, and let it be one of the kind which is operated, not from within the cage, but by a man at the bottom. Suppose that within the lift is the ghost of Galileo. He will be unconscious of the mechanism of modern lifts, but he might well return to his old task of the investigation of the laws of falling bodies. This he might do by allowing a marble to fall through a measured height to the floor of the lift and timing its fall. To avoid complications, we will allow him a stop-watch in place of his water-clock. The ghost sits there all day long, condemned to time the fall of this marble over and over again. So long as the lift remains stationary he will get the same answer every time. But suddenly he finds that the marble is falling more quickly, and he will say that gravity has increased. The man at the bottom knows better. He is sending the lift upwards with an accelerating speed. The floor of the lift is rising to meet the marble, and thus the latter accomplishes its measured journey more quickly. The man has only to make the lift go upwards or downwards with the right acceleration in order to make the ghost's gravity anything he pleases, downwards or upwards, or nil. If the man chooses to play tricks by sending the lift now up and now down, the poor ghost will find that gravity is fluctuating wildly, and will think that some kind of gravity storm is in progress. But the man on solid earth at the bottom knows that it is all due to the motion of the lift. If only the ghost would realise that he is being fooled, and that his frame of reference is being accelerated upwards and downwards, he would see that gravity has remained the same all the time. Einstein is inclined to make allowances for the ghost. He claims the liberty to take any frame of reference he pleases, and he is prepared to allow that the ghost was perfectly reasonable in taking himself and his immediate surroundings as a frame of reference. This fact, however, emerges, that if the same phenomenon is described from the point of view of different frames of reference, the gravitation inferred will, in general, be different. This is the essence of Einstein's equivalence hypothesis. He describes a physical phenomenon in the absence of gravitation by means of an accelerated frame of reference, and thus obtains a description of the same phenomenon in the presence of gravitation. It was by this method that he was able to establish the influence of gravitation on the propagation of light.
It is interesting to note how near Newton got to this idea. He lived much closer to the Copernican controversy than we do. Men had only just given up the belief of centuries that the stars revolved in their courses once a day. In his System of the World we find him facing the problem that the choice between the Ptolemaic and Copernican systems could not be settled by observation alone, but he points out that, whereas on the Copernican system gravitation can be expressed in terms of forces directed towards definite bodies which may be regarded as the sources of the gravitation, on the Ptolemaic theory the forces would be directed, not to the earth, but to points on the axis of the earth. He says:-
"That forces should be directed to no body on which they physically depend, but to innumerable imaginary points on the axe of the earth, is an hypothesis too incongruous. Tis more incongruous still that those forces should increase exactly in proportion of the distance from this axe. For this is an indication of an increase to immensity, or rather infinity; whereas the forces of natural things commonly decrease in receding from the fountain from which they flow."Newton adopted the Copernican frame of reference, not on observational grounds, but because that frame of reference possessed the peculiar convenience that it enabled him to express gravitation in a simple way as arising from the influence of attracting bodies.
It should be pointed out that Einstein does not say that it is a matter of indifference as to which frame of reference we adopt. A man who attempted to conduct experiments in an unsprung cart, and who took the body of his cart for his frame of reference, would be asking for trouble, for he would have to deal with a hopelessly complicated gravitational field. The importance of the principle lies in this - that while one frame of reference may be more convenient than another for the discussion of some particular problem, all frames of reference are theoretically admissible.
Another consequence of the denial of absolute motion has been to destroy the independence of space and time. There has been so much misunderstanding on this point that it may be well to state exactly what relativity has to say on the matter. It may be stated very simply thus:
At two distant points there is no definite unique instant of time at the second which may be regarded as simultaneous with a given instant at the first.For example, a new star bursts suddenly into view, and the astronomers tell us that, owing to its great distance and the time that it takes light to travel from it to us, the cataclysm which has made it visible must have occurred in the time of Newton. Such a statement would necessarily be approximate, for we have only the roughest notion of the distances of stars so remote as this one would have to be. But let us in imagination concede the astronomer all the accuracy of his wildest dreams. Could he even then assure us, for example, that the cataclysm occurred at the precise instant at which the famous apple struck the ground? No, for perchance the solar system is moving in the direction of this star with a speed which we may very appropriately call . If so, we are rushing to meet the light on its journey towards us, and we shall receive it sooner. How much sooner will depend upon , and has no meaning apart from a frame of reference. Adopt a frame of reference in which we are moving in the direction of the star, and the apple fell before the star burst into flame. Simultaneous and the words before and after, as applied to two instants of time at different points of space, have no precise scientific meaning apart from a specified frame of reference. Thus the time of one frame of reference depends upon the time and the space of another frame of reference. In the words of Minkowski:-
Time of itself, and space of itself, fade into shadows, and only a kind of union of the two shall maintain an independent reality.Thus the new theory has worked a fundamental change in the concepts of space and time. With Newton they were independent, homogeneous, absolute, and infinite; with Einstein they are but different aspects of the same continuum, space-time - heterogeneous, relative, and possibly finite.
It is often objected that relativity, purports to disprove the existence of the ether, and that without the ether phenomena such as the propagation of light are inconceivable. It is not certain that relativity does do this. What has been shown is that the ether cannot be made to provide a standard of rest, and that the idea of motion of the ether is self-contradictory. This may mean no more than that the ether is a reality to which the idea of motion cannot be applied. It may be helpful to remember that precisely the same criticism was directed against Newton by the Cartesians. Because he refused to be drawn into discussions as to the plenum and its vortices, he was made to appear to say that the forces of gravitation were transmitted through emptiness from one heavenly body to another. Now it is quite clear that Newton's space was more than mere nothingness, in that it acted as the medium for the transmission of gravitational influences. Yet Newton was right in regarding the nature of this space, except in so far as it was susceptible to physical measurement, as a problem for philosophy rather than for science. The present position of the problem of spacetime and the ether is, I think, very similar.
Time prevents us from referring to the practical achievements of the new theory or from exploring its possibilities in the regions in which the Newtonian mechanics have never yet shed light the regions of atomic and sub-atomic structure. The formal beauty of the theory can only be exhibited by means of mathematical analysis.
I said at the beginning of the lecture that the record of the past would provide the key to the solution of much that is difficult and perplexing in our present knowledge, and I hope that, by attempting to put Einstein's work into its proper historical setting, I have perhaps made some aspects of the theory of relativity a little clearer. But I also suggested that the record of the past would point us on the way to further advance. The work of Newton was carried on by the great French school of the Revolution period. He laid down the principles, but it was Lagrange, Laplace, Poisson, and others who reduced them to a form in which they could readily be applied to the solution of physical problems. Again Einstein has given us the principles, but it is not always easy to see how to apply them to all those problems of modern physics which are so urgently with us today. That is the task which now lies before mathematics. Einstein has given us the Principia, but La Mécanique Analytique has yet to be written.
Last Updated November 2020