The Einstein Theory of Relativity.

by H.A. Lorentz.

NOTE

Whether it is true or not that not more than twelve persons in all the world are able to understand Einstein's Theory, it is nevertheless a fact that there is a constant demand for information about this much-debated topic of relativity. The books published on the subject are so technical that only a person trained in pure physics and higher mathematics is able to fully understand them. In order to make a popular explanation of this far-reaching theory available, the present book is published.

Professor Lorentz is credited by Einstein with sharing the development of his theory. He is doubtless better able than any other man--except the author himself--to explain this scientific discovery.

The publishers wish to acknowledge their indebtedness to the New York Times, The Review of Reviews and The Athenaeum for courteous permission to reprint articles from their pages. Professor Lorentz's article appeared originally in The Nieuwe Rotterdamsche Courant of November 19, 1919.

INTRODUCTION

The action of the Royal Society at its meeting in London on November 6, in recognizing Dr. Albert Einstein's "theory of relativity"

has caused a great stir in scientific circles on both sides of the Atlantic. Dr. Einstein propounded his theory nearly fifteen years ago. The present revival of interest in it is due to the remarkable confirmation which it received in the report of the observations made during the sun's eclipse of last May to determine whether rays of light pa.s.sing close to the sun are deflected from their course.

The actual deflection of the rays that was discovered by the astronomers was precisely what had been predicted theoretically by Einstein many years since. This striking confirmation has led certain German scientists to a.s.sert that no scientific discovery of such importance has been made since Newton's theory of gravitation was promulgated. This suggestion, however, was put aside by Dr. Einstein himself when he was interviewed by a correspondent of the New York Times at his home in Berlin. To this correspondent he expressed the difference between his conception and the law of gravitation in the following terms:

"Please imagine the earth removed, and in its place suspended a box as big as a room or a whole house, and inside a man naturally floating in the center, there being no force whatever pulling him. Imagine, further, this box being, by a rope or other contrivance, suddenly jerked to one side, which is scientifically termed 'difform motion', as opposed to 'uniform motion.' The person would then naturally reach bottom on the opposite side. The result would consequently be the same as if he obeyed Newton's law of gravitation, while, in fact, there is no gravitation exerted whatever, which proves that difform motion will in every case produce the same effects as gravitation.

"I have applied this new idea to every kind of difform motion and have thus developed mathematical formulas which I am convinced give more precise results than those based on Newton's theory. Newton's formulas, however, are such close approximations that it was difficult to find by observation any obvious disagreement with experience."

Dr. Einstein, it must be remembered, is a physicist and not an astronomer. He developed his theory as a mathematical formula. The confirmation of it came from the astronomers. As he himself says, the crucial test was supplied by the last total solar eclipse. Observations then proved that the rays of fixed stars, having to pa.s.s close to the sun to reach the earth, were deflected the exact amount demanded by Einstein's formulas. The deflection was also in the direction predicted by him.

The question must have occurred to many, what has all this to do with relativity? When this query was propounded by the Times correspondent to Dr. Einstein he replied as follows:

"The term relativity refers to time and s.p.a.ce. According to Galileo and Newton, time and s.p.a.ce were absolute ent.i.ties, and the moving systems of the universe were dependent on this absolute time and s.p.a.ce. On this conception was built the science of mechanics. The resulting formulas sufficed for all motions of a slow nature; it was found, however, that they would not conform to the rapid motions apparent in electrodynamics.

"This led the Dutch professor, Lorentz, and myself to develop the theory of special relativity. Briefly, it discards absolute time and s.p.a.ce and makes them in every instance relative to moving systems. By this theory all phenomena in electrodynamics, as well as mechanics, hitherto irreducible by the old formulae--and there are mult.i.tudes--were satisfactorily explained.

"Till now it was believed that time and s.p.a.ce existed by themselves, even if there was nothing else--no sun, no earth, no stars--while now we know that time and s.p.a.ce are not the vessel for the universe, but could not exist at all if there were no contents, namely, no sun, earth and other celestial bodies.

"This special relativity, forming the first part of my theory, relates to all systems moving with uniform motion; that is, moving in a straight line with equal velocity.

"Gradually I was led to the idea, seeming a very paradox in science, that it might apply equally to all moving systems, even of difform motion, and thus I developed the conception of general relativity which forms the second part of my theory."

As summarized by an American astronomer, Professor Henry Norris Russell, of Princeton, in the Scientific American for November 29, Einstein's contribution amounts to this:

"The central fact which has been proved--and which is of great interest and importance--is that the natural phenomena involving gravitation and inertia (such as the motions of the planets) and the phenomena involving electricity and magnetism (including the motion of light) are not independent of one another, but are intimately related, so that both sets of phenomena should be regarded as parts of one vast system, embracing all Nature. The relation of the two is, however, of such a character that it is perceptible only in a very few instances, and then only to refined observations."

Already before the war, Einstein had immense fame among physicists, and among all who are interested in the philosophy of science, because of his principle of relativity.

Clerk Maxwell had shown that light is electro-magnetic, and had reduced the whole theory of electro-magnetism to a small number of equations, which are fundamental in all subsequent work. But these equations were entangled with the hypothesis of the ether, and with the notion of motion relative to the ether. Since the ether was supposed to be at rest, such motion was indistinguishable from absolute motion. The motion of the earth relatively to the ether should have been different at different points of its...o...b..t, and measurable phenomena should have resulted from this difference. But none did, and all attempts to detect effects of motions relative to the ether failed. The theory of relativity succeeded in accounting for this fact. But it was necessary incidentally to throw over the one universal time, and subst.i.tute local times attached to moving bodies and varying according to their motion. The equations on which the theory of relativity is based are due to Lorentz, but Einstein connected them with his general principle, namely, that there must be nothing, in observable phenomena, which could be attributed to absolute motion of the observer.

In orthodox Newtonian dynamics the principle of relativity had a simpler form, which did not require the subst.i.tution of local time for general time. But it now appeared that Newtonian dynamics is only valid when we confine ourselves to velocities much less than that of light. The whole Galileo-Newton system thus sank to the level of a first approximation, becoming progressively less exact as the velocities concerned approached that of light.

Einstein's extension of his principle so as to account for gravitation was made during the war, and for a considerable period our astronomers were unable to become acquainted with it, owing to the difficulty of obtaining German printed matter. However, copies of his work ultimately reached the outside world and enabled people to learn more about it. Gravitation, ever since Newton, had remained isolated from other forces in nature; various attempts had been made to account for it, but without success. The immense unification effected by electro-magnetism apparently left gravitation out of its scope. It seemed that nature had presented a challenge to the physicists which none of them were able to meet.

At this point Einstein intervened with a hypothesis which, apart altogether from subsequent verification, deserves to rank as one of the great monuments of human genius. After correcting Newton, it remained to correct Euclid, and it was in terms of non-Euclidean geometry that he stated his new theory. Non-Euclidean geometry is a study of which the primary motive was logical and philosophical; few of its promoters ever dreamed that it would come to be applied in physics. Some of Euclid's axioms were felt to be not "necessary truths," but mere empirical laws; in order to establish this view, self-consistent geometries were constructed upon a.s.sumptions other than those of Euclid. In these geometries the sum of the angles of a triangle is not two right angles, and the departure from two right angles increases as the size of the triangle increases. It is often said that in non-Euclidean geometry s.p.a.ce has a curvature, but this way of stating the matter is misleading, since it seems to imply a fourth dimension, which is not implied by these systems.

Einstein supposes that s.p.a.ce is Euclidean where it is sufficiently remote from matter, but that the presence of matter causes it to become slightly non-Euclidean--the more matter there is in the neighborhood, the more s.p.a.ce will depart from Euclid. By the help of this hypothesis, together with his previous theory of relativity, he deduces gravitation--very approximately, but not exactly, according to the Newtonian law of the inverse square. The minute differences between the effects deduced from his theory and those deduced from Newton are measurable in certain cases. There are, so far, three crucial tests of the relative accuracy of the new theory and the old.

(1) The perihelion of Mercury shows a discrepancy which has long puzzled astronomers. This discrepancy is fully accounted for by Einstein. At the time when he published his theory, this was its only experimental verification.

(2) Modern physicists were willing to suppose that light might be subject to gravitation--i.e., that a ray of light pa.s.sing near a great ma.s.s like the sun might be deflected to the extent to which a particle moving with the same velocity would be deflected according to the orthodox theory of gravitation. But Einstein's theory required that the light should be deflected just twice as much as this. The matter could only be tested during an eclipse among a number of bright stars. Fortunately a peculiarly favourable eclipse occurred last year. The results of the observations have now been published, and are found to verify Einstein's prediction. The verification is not, of course, quite exact; with such delicate observations that was not to be expected. In some cases the departure is considerable. But taking the average of the best series of observations, the deflection at the sun's limb is found to be 1.98'', with a probable error of about 6 per cent., whereas the deflection calculated by Einstein's theory should be 1.75''. It will be noticed that Einstein's theory gave a deflection twice as large as that predicted by the orthodox theory, and that the observed deflection is slightly larger than Einstein predicted. The discrepancy is well within what might be expected in view of the minuteness of the measurements. It is therefore generally acknowledged by astronomers that the outcome is a triumph for Einstein.

(3) In the excitement of this sensational verification, there has been a tendency to overlook the third experimental test to which Einstein's theory was to be subjected. If his theory is correct as it stands, there ought, in a gravitational field, to be a displacement of the lines of the spectrum towards the red. No such effect has been discovered. Spectroscopists maintain that, so far as can be seen at present, there is no way of accounting for this failure if Einstein's theory in its present form is a.s.sumed. They admit that some compensating cause may be discovered to explain the discrepancy, but they think it far more probable that Einstein's theory requires some essential modification. Meanwhile, a certain suspense of judgment is called for. The new law has been so amazingly successful in two of the three tests that there must be some thing valid about it, even if it is not exactly right as yet.

Einstein's theory has the very highest degree of aesthetic merit: every lover of the beautiful must wish it to be true. It gives a vast unified survey of the operations of nature, with a technical simplicity in the critical a.s.sumptions which makes the wealth of deductions astonishing. It is a case of an advance arrived at by pure theory: the whole effect of Einstein's work is to make physics more philosophical (in a good sense), and to restore some of that intellectual unity which belonged to the great scientific systems of the seventeenth and eighteenth centuries, but which was lost through increasing specialization and the overwhelming ma.s.s of detailed knowledge. In some ways our age is not a good one to live in, but for those who are interested in physics there are great compensations.

THE EINSTEIN THEORY OF RELATIVITY

A Concise Statement by Prof. H. A. Lorentz, of the University of Leyden

The total eclipse of the sun of May 29, resulted in a striking confirmation of the new theory of the universal attractive power of gravitation developed by Albert Einstein, and thus reinforced the conviction that the defining of this theory is one of the most important steps ever taken in the domain of natural science. In response to a request by the editor, I will attempt to contribute something to its general appreciation in the following lines.

For centuries Newton's doctrine of the attraction of gravitation has been the most prominent example of a theory of natural science. Through the simplicity of its basic idea, an attraction between two bodies proportionate to their ma.s.s and also proportionate to the square of the distance; through the completeness with which it explained so many of the peculiarities in the movement of the bodies making up the solar system; and, finally, through its universal validity, even in the case of the far-distant planetary systems, it compelled the admiration of all.

But, while the skill of the mathematicians was devoted to making more exact calculations of the consequences to which it led, no real progress was made in the science of gravitation. It is true that the inquiry was transferred to the field of physics, following Cavendish's success in demonstrating the common attraction between bodies with which laboratory work can be done, but it always was evident that natural philosophy had no grip on the universal power of attraction. While in electric effects an influence exercised by the matter placed between bodies was speedily observed--the starting-point of a new and fertile doctrine of electricity--in the case of gravitation not a trace of an influence exercised by intermediate matter could ever be discovered. It was, and remained, inaccessible and unchangeable, without any connection, apparently, with other phenomena of natural philosophy.

Einstein has put an end to this isolation; it is now well established that gravitation affects not only matter, but also light. Thus strengthened in the faith that his theory already has inspired, we may a.s.sume with him that there is not a single physical or chemical phenomenon--which does not feel, although very probably in an unnoticeable degree, the influence of gravitation, and that, on the other side, the attraction exercised by a body is limited in the first place by the quant.i.ty of matter it contains and also, to some degree, by motion and by the physical and chemical condition in which it moves.

It is comprehensible that a person could not have arrived at such a far-reaching change of view by continuing to follow the old beaten paths, but only by introducing some sort of new idea. Indeed, Einstein arrived at his theory through a train of thought of great originality. Let me try to restate it in concise terms.

THE EARTH AS A MOVING CAR

Everyone knows that a person may be sitting in any kind of a vehicle without noticing its progress, so long as the movement does not vary in direction or speed; in a car of a fast express train objects fall in just the same way as in a coach that is standing still. Only when we look at objects outside the train, or when the air can enter the car, do we notice indications of the motion. We may compare the earth with such a moving vehicle, which in its course around the sun has a remarkable speed, of which the direction and velocity during a considerable period of time may be regarded as constant. In place of the air now comes, so it was reasoned formerly, the ether which fills the s.p.a.ces of the universe and is the carrier of light and of electro-magnetic phenomena; there were good reasons to a.s.sume that the earth was entirely permeable for the ether and could travel through it without setting it in motion. So here was a case comparable with that of a railroad coach open on all sides. There certainly should have been a powerful "ether wind" blowing through the earth and all our instruments, and it was to have been expected that some signs of it would be noticed in connection with some experiment or other. Every attempt along that line, however, has remained fruitless; all the phenomena examined were evidently independent of the motion of the earth. That this is the way they do function was brought to the front by Einstein in his first or "special" theory of relativity. For him the ether does not function and in the sketch that he draws of natural phenomena there is no mention of that intermediate matter.

If the s.p.a.ces of the universe are filled with an ether, let us suppose with a substance, in which, aside from eventual vibrations and other slight movements, there is never any crowding or flowing of one part alongside of another, then we can imagine fixed points existing in it; for example, points in a straight line, located one meter apart, points in a level plain, like the angles or squares on a chess board extending out into infinity, and finally, points in s.p.a.ce as they are obtained by repeatedly shifting that level spot a distance of a meter in the direction perpendicular to it. If, consequently, one of the points is chosen as an "original point" we can, proceeding from that point, reach any other point through three steps in the common perpendicular directions in which the points are arranged. The figures showing how many meters are comprized in each of the steps may serve to indicate the place reached and to distinguish it from any other; these are, as is said, the "co-ordinates" of these places, comparable, for example, with the numbers on a map giving the longitude and lat.i.tude. Let us imagine that each point has noted upon it the three numbers that give its position, then we have something comparable with a measure with numbered subdivisions; only we now have to do, one might say, with a good many imaginary measures in three common perpendicular directions. In this "system of co-ordinates" the numbers that fix the position of one or the other of the bodies may now be read off at any moment.

This is the means which the astronomers and their mathematical a.s.sistants have always used in dealing with the movement of the heavenly bodies. At a determined moment the position of each body is fixed by its three co-ordinates. If these are given, then one knows also the common distances, as well as the angles formed by the connecting lines, and the movement of a planet is to be known as soon as one knows how its co-ordinates are changing from one moment to the other. Thus the picture that one forms of the phenomena stands there as if it were sketched on the canvas of the motionless ether.

EINSTEIN'S DEPARTURE

Since Einstein has cut loose from the ether, he lacks this canvas, and therewith, at the first glance, also loses the possibility of fixing the positions of the heavenly bodies and mathematically describing their movement--i.e., by giving comparisons that define the positions at every moment. How Einstein has overcome this difficulty may be somewhat elucidated through a simple ill.u.s.tration.

On the surface of the earth the attraction of gravitation causes all bodies to fall along vertical lines, and, indeed, when one omits the resistance of the air, with an equally accelerated movement; the velocity increases in equal degrees in equal consecutive divisions of time at a rate that in this country gives the velocity attained at the end of a second as 981 centimeters (32.2 feet) per second. The number 981 defines the "acceleration in the field of gravitation,"

and this field is fully characterized by that single number; with its help we can also calculate the movement of an object hurled out in an arbitrary direction. In order to measure the acceleration we let the body drop alongside of a vertical measure set solidly on the ground; on this scale we read at every moment the figure that indicates the height, the only co-ordinate that is of importance in this rectilinear movement. Now we ask what would we be able to see if the measure were not bound solidly to the earth, if it, let us suppose, moved down or up with the place where it is located and where we are ourselves. If in this case the speed were constant, then, and this is in accord with the special theory of relativity, there would be no motion observed at all; we should again find an acceleration of 981 for a falling body. It would be different if the measure moved with changeable velocity.