Chapter VI: completes the preliminary course in the fundamentals of (7)

There can in fact be only one interpretation. If we each find that the light has moved the same number of miles in the same number of seconds, then we must be meaning something different when we speak of miles and seconds. We are speaking in different languages. Some subsidence has occurred in the foundations of our systems of measurement. We are each referring to one and the same objective fact; but since we describe it quite differently, and at first sight incompatibly, some profound alteration must have occurred in our perceptions--all unsuspected by ourselves. It has been shown precisely what this alteration is. A body moving at high velocity must become flattened in the direction of its motion; all its measuring apparatus, when turned in that direction, is shortened, so that no hint of the flattening can be obtained from it. Furthermore, the standards of time are lengthened out, and clocks go slower. The extent of this alteration in standards of space and time is stated in the equations of the so-called Lorentz transformation.

Objection might be urged to the above paragraph on the ground that the connection of the observer with the variability of measured lengths and times is not sufficiently indicated, and that this variability therefore might be taken as an intrinsic property of the observed body--which of course it is not.--Editor.

We are accustomed to describe space as being of three dimensions, and time as being of one dimension. As a matter of fact, both space and time are "ideas," and not immediate sense-perceptions. We perceive matter; we then infer a universal continuum filled by it, which we call space. If we had no knowledge of matter, we should have no conception of space. Similarly in the case of time: we perceive one event following another, and we then invent a continuum which we call time, as an abstraction based on the sequence of events. We do not see space, and we do not see time. They are not real things, in the sense that matter is real, and that events are real. They are products of imagination: useful enough in common life, but misleading when we try to look on the universe as a whole, free from the artificial divisions and landmarks which we introduce into it for practical convenience. Hence it is perhaps not so surprising after all that in certain highly transcendental investigations, these artificial divisions should cease to be a convenience, and become a hindrance.

Take for instance our conception of time. It differs from our conception of space in that it has only one dimension. In space, there is a right and left, an up and down, a before and after. But in time there is only before and after. Why should there be this limitation of the time-factor? Merely because that is the verdict of all our human experience. But is our human experience based on a sufficiently broad foundation to enable us to say that, under all conditions and in all parts of the universe, there can be only one time-direction? May not our belief in the uniformity of time be due to the uniformity of the motion of all observers on the earth? Such in fact is the postulate of relativity. We now believe that, at velocities very different from our own, the standard of time would also be different from ours. From our point of view, that different standard of time would not be confined to the single direction fore and aft, as we know it, but would also have in it an element of what we might call right and left. True, it would still be of only one dimension, but its direction would differ from the direction of our time. It would still run like a thread through the universe, but not in the direction which we call straight forward. It would have a slant in it, and the angle of the slant depends upon the velocity of motion. It does not follow that because we are all traveling in the same direction down the stream of time, therefore that stream can only flow in the direction which we know. "Before" and "after" are expressions which, like right and left, depend upon our personal situation. If we were differently situated, if to be precise we were moving at very high velocity, we should, so to speak, be facing in a new direction and "before" and "after" would imply a different direction of progress from that with which we are now familiar.

THE WORLD OF REALITY

But, after all, the objective universe is the same old universe however fast we are moving about in it, and whatever way we are facing. These details merely determine the way we divide it up into space and time. The universe is not affected by any arbitrary lines which we draw through it for our personal convenience. For practical purposes, we ascribe to it four dimensions, three in space and one in time. Clearly if the time direction is altered, all dimensions both of space and time must have different readings. If, for instance, the time direction slopes away to the left, as compared with ours, then space measurements to right and left must be correspondingly altered. An analogy will simplify the matter.

Suppose we desire to reach a point ten miles off in a roughly northeasterly direction. We might do so by walking six miles due east and then eight miles due north. We should then be precisely ten miles from where we started. But suppose our compass were out of order, so that its north pole pointed somewhat to the west of north. Then in order to get to our destination, we might have to walk seven miles in the direction which we thought was east, and a little more than seven miles in the direction which we thought was north. We should then reach the same point as before. Both observers have walked according to their lights, first due east and then due north, and both have reached the same point: the one observer is certain that the finishing point is six miles east of the starting-point, while the other is sure it is seven miles.

Now we on the earth are all using a compass which points in the same direction as regards time. But other observers, on bodies moving with very different velocity, have a compass in which the time-direction is displaced as compared with ours. Hence our judgments of distances will not be alike. In our analogy, the northerly direction corresponds to time, and the easterly direction to space; and so long as we use the same compass we do not differ in our measurements of distances. But for any one who has a different notion of the time-direction, not only time intervals but space distances will be judged differently.

In short, the universe is regarded as a space-time continuum of four dimensions. A "point" in space-time is called an "event"--that which occurs at a specified moment and at a specified place. The distance between two points in space-time is called their "interval." All observers will agree as to the magnitude of any interval, since it is a property of the objective universe; but they will disagree as to its composition in space and time separately. In short, space and time are relative conceptions; their relativity is a necessary consequence of the relativity of motion. The paradox named at the outset is overcome; for the two observers measuring the velocity of the light produced as they passed one another, were using different units of space and time. And hence emerges triumphant the Special Principle of Relativity, which states that the laws of nature are the same for all observers, whether they are in a state of rest or of uniform motion in a straight line.

ACCELERATED MOTION

Uniform motion in a straight line is however a very special kind of motion. Our experience in ordinary life is of motions that are neither uniform nor in a straight line; both speed and direction of motion are altering. The moving body is then said to undergo "acceleration": which means either that its speed is increasing or diminishing, or that its direction of motion is changing, or both. If we revert to our former supposition of a universe in which there is only a single body in "empty" space, we clearly cannot say whether it has acceleration any more than whether it is moving, there being no outside standard of comparison; and the General Principle of Relativity asserts the invariance of the laws of nature for all states of motion of the observer. In this case, however, a difference might be detected by an observer on the moving body itself. It would be manifested to him as the action of a force; such for instance as we feel when a train in which we are traveling is increasing or reducing speed, or when, without changing speed, it is rounding a corner. The force dies away as soon as the velocity becomes uniform. Thus acceleration reveals itself to us under the guise of action by a force. Force and acceleration go together, and we may either say that the acceleration is due to the force, or the impression of force to the acceleration.

Now when we are traveling with accelerated motion, we have quite a different idea of what constitutes a straight line from that which we had when at rest or in uniform motion. If we are moving at uniform velocity in an airplane and drop a stone to the earth it will appear to us in the airplane to fall in a straight line downward, while to an observer on the earth it will appear to describe a parabola. This is due to the fact that the stone gathers speed as it falls; it is subject to the acceleration associated with gravity. Acceleration obliterates the fundamental difference between a straight and curved line. Unless we know what is the absolute motion of the stone, and the two observers, we cannot say whether the line is "really" a straight or a curved line. Since absolute motion is an illegitimate conception, it follows that there is no such thing as "really" straight or "really" curved. These are only appearances set up as a consequence of our relative motions with respect to the bodies concerned. If there were no such thing as acceleration--if the stone fell to the earth at uniform velocity--then an observer on the earth or anywhere else would agree that it fell in a straight line; and straight lines would always be straight lines.

Under these circumstances, Euclidean geometry would be absolutely true. But if we are in a state of acceleration, then what we think are straight lines are "really" curved lines, and Euclidean geometry, based on the assumption that its lines are straight, must founder when tested by more accurate measurements. And in point of fact we are in a state of acceleration: for we are being acted upon by a force--namely, the force of gravitation. Wherever there is matter, there is gravitation; wherever there is gravitation there is acceleration; wherever there is acceleration Euclidean geometry is inaccurate. Hence in the space surrounding matter a different geometry holds the field; and bodies in general move through such space in curved lines.

Different parts of space are thus characterized by different geometrical properties. All bodies in the universe proceed on their established courses through space and time. But when they come to distorted geometrical areas, their paths naturally seem to us different from when they were moving through less disturbed regions. They exhibit the difference by acquiring an acceleration; and we explain the acceleration by alleging the existence of a force, which we call the force of gravitation. But their motions can in fact be perfectly predicted if we know the geometry of the space through which they are traveling. The predictions so based have in fact proved more accurate than those based on the law of gravitation.

X

SPACE, TIME AND GRAVITATION

An Outline of Einstein's Theory of General Relativity

BY W. DE SITTER PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF LEYDEN

"Henceforth space by itself and time by itself shall sink to mere shadows, and only a union of the two shall preserve reality."

The prophecy contained in the above-quoted words, spoken by Minkowski at the meeting of German "Naturforscher und Aerzte" at Cologne in 1908, has, however, only been completely fulfilled by Einstein's "Allgemeine Relativitäts-theorie" of 1915, which incorporated gravitation into the union. In the following pages an attempt is made to set forth, without using any technical language, the leading ideas of that theory: I will confine myself to the theory as published by Einstein in November, 1915, which forms a consistent whole, complete in itself; and I will not refer to later developments, which are still more or less tentative, and not necessary for the understanding of the theory. The mathematics used by Einstein is the so-called Absolute Differential Calculus. It is not more difficult or recondite than that used in other branches of theoretical physics, but it is somewhat unfamiliar to most of us, because it is not generally taught in the regular university courses. I will, however, in this essay abstain from using any mathematics at all, at least, I will not be using it openly. It is of course unavoidable to use at least the results of the mathematical reasoning, if not the reasoning itself; but so long as they are not put into formulas they will, it is hoped, not look so formidable to the reader.

Referring to the quoted words of Minkowski, we may ask what is meant by "reality." Physical science, like common sense, takes for granted that there is a reality behind the phenomena, which is independent of the person by whom, and the particular methods by which it is observed, and which is also there when it is not observed. Strictly speaking, all talk about what is not observed is metaphysics. Nevertheless the physicist unhesitatingly believes that his laws are general, and that the phenomena continue to happen according to them when nobody is looking. And since it would be impossible to prove that they did not, he is fully entitled to his belief. The observed phenomena are the effects of the action of this reality, of which we assume the existence, on the observer's senses--or apparatus, which are extended and refined sense-organs. The laws governing the phenomena therefore must convey some information regarding this reality. We shall never by any means be able to know anything else about it but just these laws. To all intents and purposes the laws are the reality, if we eliminate from them all that refers to the observer alone. What refers to the reality is called "absolute," and what involves reference to the observer "relative." The elimination of the relative is one of the things the theory of relativity has set out to do.

THE EXTERNAL WORLD AND ITS GEOMETRY

To describe the phenomena and derive laws from them, we locate them in space and time. To do this we use geometry. Here it is that the part contributed by the observer comes in. There are an infinite number of geometries, and a priori there seems to be no reason to choose one rather than the other. Taking geometry of two dimensions as an example, we can draw figures on a piece of paper, and discuss their properties, and we can also do so on the shell of an egg. But we cannot draw the same figures on the egg as on the paper. The ones will be distorted as compared with the others: the two surfaces have a different geometry. Similarly it is not possible to draw an accurate map of the earth on a sheet of paper, because the earth is spherical and its representation on the flat paper is always more or less distorted. The earth requires spherical geometry, which differs from the flat, or Euclidean, geometry of the paper.

Up to a few years ago Euclidean (i.e. flat) geometry of three dimensions had been exclusively used in physical theories. Why? Because it is the true one, is the one answer generally given. Now a statement about facts can be true or false, but a mathematical discipline is neither true nor false; it can only be correct--i.e. consistent in itself--or incorrect, and of course it always is correct. The assertion that a certain geometry is the "true" one can thus only mean, that it is the geometry of "true" space, and this again, if it is to have any meaning at all, can only mean that it corresponds to the physical "reality." Leaving aside the question whether this reality has any geometry at all, we are confronted with the more immediately practical consideration how we shall verify the asserted correspondence. There is no other way than by comparing the conclusions derived from the laws based upon our geometry, with observations. It thus appears that the only justification for the use of the Euclidean geometry is its success in enabling us to "draw an accurate map" of the world. As soon as any other geometry is found to be more successful, that other must be used in physical theories, and we may, if we like, call it the "true" one.

Accurate observations always consist of measures, determining the position of material bodies in space. But the positions change, and for a complete description we also require measures of time. An important remark must be made here. Nobody has ever measured a pure space-distance, nor a pure lapse of time. The only thing that can be measured is the distance from a body at a certain point of space and a certain moment of time, to a body (either the same or another) at another point and another time. We can even go further and say that time cannot be measured at all. We profess to measure it by clocks. But a clock really measures space, and we derive the time from its space-measures by a fixed rule. This rule depends on the laws of motion of the mechanism of the clock. Thus finally time is defined by these laws. This is so, whether as a "clock" we use an ordinary chronometer, or the rotating earth, or an atom emitting light-waves, or anything else that may be suggested. The physical laws, of course, must be so adjusted that all these devices give the same time. About the reality of time, if it has any, we know nothing. All we know about time is that we want it. We cannot adequately describe nature with the three space-coordinates alone, we require a fourth one, which we call time. We might thus say with some reason that the physical world has four dimensions. But so long as it was found possible adequately to describe all known phenomena by a space of three dimensions and an independent time, the statement did not convey any very important information. Only after it had been found out that the space-coordinates and the time are not independent, did it acquire a real meaning.

As is well known the observation by which this was found out is the famous experiment of Michelson and Morley. It led to the "special" theory of relativity, which is the one referred to by Minkowski in 1908. In it a geometry of four dimensions is used, not a mere combination of a three-dimensional space and a one-dimensional time, but a continuum of truly fourfold order. This time-space is not Euclidean, since the time-component and the three space-components are not on the same footing, but its fundamental formula has a great resemblance to that of Euclidean geometry. We may call it "pseudo-Euclidean."

This theory, which we need not explain here, was very satisfactory so far as the laws of electromagnetism, and especially the propagation of light, were concerned, but it did not include gravitation, and mechanics generally. We then had this curious state of affairs, that physicists actually believed in two different "realities." When they were thinking of light they believed in Minkowski's time-space; when they were thinking of gravitation they believed in the old Euclidean space and independent time. This, of course, could not last. Attempts were made so to alter Newton's law of gravitation that it would fit into the four-dimensional world of the special relativity-theory, but these only succeeded in making the law, which had been a model of simplicity, extremely complicated, and, what was worse, it became ambiguous.

It is Einstein's great merit to have perceived that gravitation is of such fundamental importance, that it must not be fitted into a ready-made theory, but must be woven into the space-time geometry from the beginning. And that he not only saw the necessity of doing this, but actually did it.

GRAVITATION AND ITS PLACE IN THE UNIVERSE

To see the necessity we must go back to Newton's system of mechanics. Newton did two things (amongst others). He canonised Galileo's system of mechanics into his famous "laws of motion," the most important of which is the law of inertia, which says that:

a body, that is not interfered with, moves in a straight line
with constant velocity.

The velocity, of course, can be nil, and the body at rest. This is a perfectly general law, the same for all material bodies, whatever their physical or chemical status. Newton took good care exactly to define what he meant by uniform motion in a straight line, and for this purpose he introduced the absolute Euclidean space and absolute time as an essential part of his system of laws at the very beginning of his great work. The other thing Newton did was to formulate the law of gravitation. Gravitation was in his system considered as an interference with the free, or inertial, motion of bodies, and accordingly required a law of its own.

But gravitation has this in common with inertia, and in this it differs from all other interferences, that it is perfectly general. All material bodies are equally subjected to it, whatever their physical or chemical status may be. But there is more. Gravitation and inertia are actually indistinguishable from each other, and are measured by the same number: the "mass". This was already remarked by Newton himself, and from his point of view it was a most wonderful accidental coincidence. If an apple falls from the tree, that which makes it fall is its weight, which is the gravitational attraction by the earth, diminished by the centrifugal force due to the earth's rotation and the apple's inertia. In Newton's system the gravitational attraction is a "real" force, whereas the centrifugal force is only "fictitious". But the one is as real as the other. The most refined experiments, already begun by Newton himself, have not succeeded in distinguishing between them. Their identity is actually one of the best established facts in experimental physics. From this identity of "fictitious," or inertial, and "real," or gravitational, forces it follows that locally a gravitational field can be artificially created or destroyed. Thus inside a closed room which is falling freely, say a lift of which the cable has been broken, bodies have no weight: a balance could be in equilibrium with different weights in the two scales.

Having thus come to the conclusion that gravitation is not an interference, but is identical with inertia, we are tempted to restate the law of motion, so as to include both, thus:

Bodies which are not interfered with--do not move in straight
lines, but--fall.

Now this is exactly what Einstein did. Only the "falling" of course requires a precise mathematical definition (like the uniform motion in a straight line), and the whole gist of his theory is the finding of that definition. In our earthly experience the falling never lasts long, very soon something--the floor of the room, or the earth itself--interferes. But in free space bodies go on falling forever. The motion of the planets is, in fact, adequately described as falling, since it consists in nothing else but obeying Newton's law of gravitation together with his law of inertia. A body very far removed from all other matter is not subjected to gravitation, consequently it falls with constant velocity in a straight line according to the law of inertia. The problem was thus to find a mathematical definition of "falling," which would embrace the uniform straight-line motion very far from all matter as well as the complex paths of the planets around the sun, and of an apple or a cannon-ball on earth.

GRAVITATION AND SPACE-TIME

For the definition of the uniform rectilinear motion of pure inertia Newton's Euclidean space and independent time were sufficient. For the much more complicated falling under the influence of gravitation and inertia together, evidently a more complicated geometry would be needed. Minkowski's pseudo-Euclidean time-space also was insufficient. Einstein accordingly introduced a general non-Euclidean four-dimensional time-space, and enunciated his law of motion thus:

Bodies which are not interfered with move in geodesics.

A geodesic in curved space is exactly the same thing as a straight line in flat space. We only call it by its technical name, because the name "straight line" would remind us too much of the old Euclidean space. If the curvature gets very small, or zero, the geodesic becomes very nearly, or exactly, a straight line.

The problem has now become to assign to time-space such curvatures that the geodesics will exactly represent the tracks of falling bodies. Space of two dimensions can just be flat, like a sheet of paper, or curved, like an egg. But in geometry of four dimensions there are several steps from perfect flatness, or "pseudo-flatness," to complete curvature. Now the law governing the curvature of Einstein's time-space, i.e., the law of gravitation, is simply that it can never, outside matter, be curved more than just one step beyond perfect (pseudo-)flatness.

Since I have promised not to use any mathematics I can hardly convey to the reader an adequate idea of the difficulty of the problem, nor do justice to the elegance and beauty of the solution. It is, in fact, little short of miraculous that this solution, which was only adopted by Einstein because it was the simplest he could find, does so exactly coincide in all its effects with Newton's law. Thus the remarkably accurate experimental verification of this law can at once be transferred to the new law. In only one instance do the two laws differ so much that the difference can be observed, and in this case the observations confirm the new law exactly. This is the well known case of the motion of the perihelion of Mercury, whose disagreement with Newton's law had puzzled astronomers for more than half a century.

Since Einstein's time-space includes Minkowski's as a particular case, it can do all that the other was designed to do for electro-magnetism and light. But it does more. The track of a pulse of light is also a geodesic, and time-space being curved in the neighborhood of matter, rays of light are no longer straight lines. A ray of light from a star, passing near the sun, will be bent round, and the star consequently will be seen in a different direction from where it would be seen if the sun had not been so nearly in the way. This has been verified by the observations of the eclipse of the sun of 1919 of May 29.

There is one other new phenomenon predicted by the theory, which falls within the reach of observation with our present means. Gravitation chiefly affects the time-component of the four-dimensional continuum, in such a way that natural clocks appear to run slower in a strong gravitational field than in a weak one. Thus, if we make the hypothesis--which, though extremely probable, is still a hypothesis--that an atom emitting or absorbing light-waves is a natural clock, and the further hypothesis--still very probable, though less so than the former--that there is nothing to interfere with its perfect running, then an atom on the sun will give off light-waves of smaller frequency than a similar atom in a terrestrial laboratory emits. Opinions as yet differ as to whether this is confirmed or contradicted by observations.

The great strength and the charm of Einstein's theory do however not lie in verified predictions, nor in the explanation of small outstanding discrepancies, but in the complete attainment of its original aim: the identification of gravitation and inertia, and in the wide range of formerly apparently unconnected subjects which it embraces, and the broad view of nature which it affords.

Outside matter, as has been explained, the law of gravitation restricts the curvature of time-space. Inside continuous matter the curvature can be of any arbitrary kind or amount; the law of gravitation then connects this curvature with measurable properties of the matter, such as density, velocity, stress, etc. Thus these properties define the curvature, or, if preferred, the curvature defines the properties of matter, i.e. matter itself.

From these definitions the laws of conservation of energy, and of conservation of momentum, can be deduced by a purely mathematical process. Thus these laws, which at one time used to be considered as the most fundamental ones of mechanics, now appear as simple corollaries from the law of gravitation. It must be pointed out that such things as length, velocity, energy, momentum, are not absolute, but relative, i.e. they are not attributes of the physical reality, but relations between this reality and the observer. Consequently the laws of conservation are not laws of the real world, like the law of gravitation, but of the observed phenomena. There is, however one law which, already before the days of relativity, had come to be considered as the most fundamental of all, viz: the principle of least action. Now action is absolute. Accordingly this principle retains its central position in Einstein's theory. It is even more fundamental than the law of gravitation, since both this law, and the law of motion, can be derived from it. The principle of least action, so far as we can see at present, appears to be the law of the real world.

XI

THE PRINCIPLE OF GENERAL RELATIVITY

How Einstein, to a Degree Never Before Equalled, Isolates the External Reality from the Observer's Contribution

BY E. T. BELL UNIVERSITY OF WASHINGTON SEATTLE

Einstein's general relativity is of such vast compass, being coextensive with the realm of physical events, that in any brief account a strict selection from its numerous aspects is prescribed. The old, restricted principle being contained in the general, we shall treat the latter, its close relations with gravitation, and the significance of both for our knowledge of space and time. The essence of Einstein's generalization is its final disentanglement of that part of any physical event which is contributed by the observer from that which is inherent in the nature of things and independent of all observers.

The argument turns upon the fact that an observer must describe any event with reference to some framework from which he makes measurements of time and distance. Thus, suppose that at nine o'clock a ball is tossed across the room. At one second past nine the ball occupies a definite position which we can specify by giving the three distances from the centre of the ball to the north and west walls and the floor. In this way, refining our measurements, we can give a precise description of the entire motion of the ball. Our final description will consist of innumerable separate statements, each of which contains four numbers corresponding to four measurements, and of these one will be for time and three for distances at the time indicated.

Imagine now that a man in an automobile looks in and observes the moving ball. Suppose he records the motion. To do so, he must refer to a timepiece and some body of reference. Say he selects his wrist-watch, the floor of his auto and two sides meeting in a corner. Fancy that just as he begins his series of observations his auto starts bucking and the main-spring of his watch breaks, so that he must measure "seconds" by the crazy running-down of his watch, and distances with reference to the sides of his erratic auto. Despite these handicaps he completes a set of observations, each of which consists of a time measured by his mad watch and three distances reckoned from the sides of his bucking machine. Let us assume him to have been so absorbed in his experiment that he noticed neither the disorders of his watch nor the motion of his auto. He gives us his sets of measurements. We remark that his seconds are only small fractions of ours, also his norths and wests are badly mixed. If we interpret his sets in terms of our stationary walls and sober clock we find the curious paradox that the ball zigzagged across the room like an intoxicated bee. He obstinately argues that we know no more than he about how the ball actually moved. For we got a smooth description, he asserts, by choosing an artificially simple reference framework, having no necessary relations whatever to the ball. The crooked path plotted from his observations proves, he declares, that the ball was subject to varying forces of which we in the room suspected nothing. He contends that our room was being jarred by a system of forces which exactly compensated and smoothed out the real jaggedness of path observed by himself. But if we know all about his watch and auto we can easily apply necessary corrections to his measurements, and, fitting the corrected set to our reference-framework of walls and clock, recover our own smooth description.

For consistency we must carry our readjustments farther. The path mapped from our measurements is a curve. Perhaps the curvature was introduced by some peculiarity of our reference framework? Possibly our own room is being accelerated upward, so that it makes the ball's true path--whatever that may be--appear curved downward, just as the autoist's zigzags made the path he mapped appear jagged. Tradition attributes the downward curving to the tug of gravity. This force we say accelerates the ball downward, producing the curved path. Is this the only possible explanation? Let us see.

GRAVITATION AND ACCELERATION

Imagine a man in a room out of which he cannot see. He notices that when he releases anything it falls to the floor with a constant acceleration. Further he observes that all his objects, independently of their chemical and physical properties, are affected in precisely the same way. Now, he previously has experimented with magnets, and has remarked that they attract certain bodies in essentially the same way that the things which he drops are "attracted" to whatever is beneath the floor. Having explained magnetic attraction in terms of "forces," he makes his first hypothesis: (A) He and his room are in a strong "field of force," which he designates gravitational. This force pulls all things downward with a constant acceleration. Here he notes a singular distinction between magnetic and gravitational "forces": magnets attract only a few kinds of matter, notably iron; the novel "force," if indeed a force at all, acts similarly upon all kinds of matter. He makes another hypothesis: (B) His room and he are being accelerated upward.

Either (A) or (B) describes the facts perfectly. By no experiment can he discriminate between them. So he takes the great step, and formulates the Equivalence Hypothesis:

A gravitational field of force is precisely equivalent in its effects to an artificial field of force introduced by accelerating the framework of reference, so that in any small region it is impossible to distinguish between them by any experiment whatever.

Next reconsidering his magnetic "forces," he extends the equivalence hypothesis to cover all manifestations of force: The effects attributed to forces of any kind whatever can be described equally well by saying that our reference frameworks are accelerated; and moreover there is possible no experiment which will discriminate between the descriptions.

If the accelerations are null, the frameworks are at rest or in uniform motion relatively to one another. This special case is the "restricted" principle of relativity, which asserts that it is impossible experimentally to detect a uniform motion through the ether. Being thus superfluous for descriptions of natural phenomena, the ether may be abandoned, at least temporarily. The older physics sought this absolute ether framework to which all motions could be unambiguously referred, and failed to find it. The most exacting experiments, notably that of Michelson-Morley, revealed no trace of the earth's supposed motion through the ether. Fitzgerald accounted for the failure by assuming that such motion would remain undetected if every moving body contracted by an amount depending upon its velocity in the direction of motion. The contraction for ordinary velocities is imperceptible. Only when as in the case of the beta particles, the velocity is an appreciable fraction of the velocity of light, is the contraction revealed. This contraction follows immediately from Einstein's generalization constructed upon the equivalence hypothesis and the restricted relativity principle. We shall see that the contraction inevitably follows from the actual geometry of the universe. [2]

Let us return for a moment to the moving ball. Four measures, three of distances and one of time, are required in specifying its position with reference to some framework at each point and at each instant. All of these measures can be summed up in one compendious statement--the equations of motion showed how in changing from our room to his accelerated auto we found a new summary, "transformed equations," which seemed to indicate that the ball had traversed a strong, variable field of force. Is there then in the chaos of observational disagreements anything which is independent of all observers? There is, but it is hidden at the very heart of nature.

PATHS THROUGH THE WORLD OF FOUR DIMENSIONS

To exhibit this, we must recall a familiar proposition of geometry: the square on the longest side of a right-angled triangle is equal to the sum of the squares on the other two sides. It has long been known that from this alone all the metrical properties of Euclidean space--the space in which for 2,000 years we have imagined we were living--can be deduced. Metrical properties are those depending upon measurement. Now, in the geometry of any space, Euclidean or not, there is a single proposition of a similar sort which tells us how to find the most direct distance between any two points that are very close together. This small distance is expressed in terms of the two sets of distance measurements by which the end-points are located, just as two neighboring positions of our ball were located by two sets of four measurements each. We say by analogy that two consecutive positions of the ball are separated by a small interval of time-space. From the formula for the very small interval of time-space we can calculate mathematically all the metrical properties of the time and space in which measurements for the ball's motion must be made. So in any geometry mathematical analysis predicts infallibly the truth about all facts depending upon measurements from the simple formula of the interval between neighboring points. Thus, on a sphere the sum of the angles of any triangle formed by arcs of great circles exceeds 180°, and this follows from the formula for the shortest ("geodesic") distance between neighboring points on the spherical surface.

We saw that it takes four measurements, one for time and three for distances, to fix an elementary event, viz., the position of the centre of our ball at any instant. A system of all possible such sets of four measurements each, constitutes what mathematicians call a four-dimensional space. The study of the four-dimensional time-space geometry, once its shortest-distance proposition is known, reveals all those relations in nature which can be ascertained by measurements, that is, experimentally. We have then to find this indispensable proposition.

Imagine the path taken by a particle moving solely under the influence of gravitation. This being the simplest possible motion of an actual particle in the real world, it is natural to guess that its path will be such that the particle moves from one point of time-space to another by the most direct route. This in fact is verified by forming the equations of the free particle's motion, which turn out to be precisely those that specify a geodesic (most direct line) joining the two points. On the (two-dimensional) surface of a sphere such a line is the position taken by a string stretched between two points on the surface, and this is the shortest distance on the surface between them. But in the time-space geometry we find a remarkable distinction: the interval between any two points of the path taken is the longest possible, and between any two points there is only one longest path. Translated into ordinary space and time this merely asserts that the time taken between any two points on the natural path is the longest possible.

Recall now that when the line-formula for any kind of space is known all the metrical properties of that space are completely determined, and combine with this what we have just found, namely, the equations of motion of a particle subject only to gravitation are the same equations as those which fix the line-formula for the four-dimensional time-space. Since gravitation alone determines the motion of the particle, and since this motion is completely described by the very equations which fix all the metrical properties of time-space, it follows that the metrical (experimentally determinable) properties of time-space are equivalent to those of gravitation, in the sense that each set of properties implies the other.

THE UNIVERSE OF SPACE-TIME

We have found the thing in nature which is independent of all observers, and it turns out to be the very structure of time-space itself. The motion of the free particle obviously is a thing unconditioned by accidents of observation; the particle under the influence of gravitation alone must go a way of its own. And if some observer in an artificial field of force produced by the acceleration of his reference framework describes the path as knotted, he merely is foisting eccentricities of his own motion upon the direct path of the particle. The conclusion is rational, for we believe that time-space exists independently of any man's way of perceiving it.

Incidentally note that this space is that of the physical world. For only by measurements of distances and times can we become aware of our extension in time and space. If beyond this time-space geometry of measurements there is some "absolute geometry," science can have no concern with it, for never can it be revealed by the one exploring device we possess--measurement.

We have followed a single particle. Let us now form a picture of several. Any event can be analyzed into a multitude of coincidences in time-space. For consider two moving particles--say electrons. If they collide they both are in very approximately one place at the same time. We imagine the path of an electron through time-space plotted by a line (in four-dimensional space), which will deviate from a "most direct" (geodesic) path if the electron is subjected to forces. This is the "world-line" of the electron. If the world lines of several electrons intersect at one point in time-space, the intersection pictures the fact of their coincidence somewhere and somewhen; for all their world-lines having a time-space point in common, at some instant they must have been in collision. Each point of a world-line pictures the position at a certain place at a certain time; and it is the intersections of world-lines which correspond to physical events. Of what lies between the intersections we have no experimental knowledge.

Imagine the world-lines of all the electrons in the universe threading time-space like threads in a jelly. The intersections of the tangle are a complete history of all physical events. Now distort the jelly. Clearly the mutual order of the intersections will be unchanged, but the distances between them will be shortened or lengthened. To a distortion of the jelly corresponds a special choice (by some observer) of a reference framework for describing the order of events. He cannot change the natural sequence of events. Again we have found something which is independent of all observers.

We can now recapitulate our conclusions and state the principle of relativity in its most general form.

(1) Observers describe events by measures of times and distances made with regard to their frameworks of reference.

(2) The complete history of any event is summarized in a set of equations giving the positions of all the particles involved at every instant.

(3) Two possibilities arise. (A) Either these equations are the same in form for all space-time reference frameworks, persisting formally unchanged for all shifts of the reference scheme; or (B), they subsist only when some special framework is used, altering their form as they are referred to different frameworks. If (B) holds, we naturally assume that the equations, and the phenomena which they profess to represent, owe their existence to some peculiarity of the reference framework. They do not, therefore, describe anything which is inherent in the nature of things, but merely some idiosyncrasy of the observer's way of regarding nature. If (A) holds, then obviously the equations describe some real relation in nature which is independent of all possible ways of observing and recording it.

(4) In its most general form the principle of relativity states that those relations, and those alone, which persist unchanged in form for all possible space-time reference frameworks are the inherent laws of nature.

To find such relations Einstein has applied a mathematical method of great power--the calculus of tensors--with extraordinary success. This calculus threshes out the laws of nature, separating the observer's eccentricities from what is independent of him, with the superb efficiency of a modern harvester. The residue is a physical geometry--or geometrical physics--of time-space, in which it appears that the times and spaces contributed by the several observers' reference frameworks are shadows of their own contrivings; while the real, enduring universe is a fourfold order of time and space indissolubly bound together. One observer separates this time-space into his own "time" and "space" in one way, determined by his path through the world of events; another, moving relatively to the first, separates it differently, and what for one is time shades into space for another.

This time-space geometry is non-Euclidean. It is "warped" (curved), the amount of warping at any place being determined by the intensity of the gravitational field there. Thus again gravitation is rooted in the nature of things. In this sense it is not a force, but a property of space. Wherever there is matter there is a gravitational field, and hence a warping of space. Conversely, as long ago imagined by Clifford, wherever there is a warping of space, there is matter; and matter is resolved ultimately into wrinkles in time-space.

To visualize a warped space, consider a simple analogy. A man walks away from a polished globe; his image recedes into the mirror-space, shortening and thinning as it goes, and thinning (in the direction of motion) faster than it shortens. Everything around him experiences a like effect. If he tries to discover this by a footrule it automatically shortens faster as he turns it into the horizontal position, so his purpose eludes him. The mirror-space is warped in the direction of the image's motion. So is our own. For all bodies, as evidenced by the Fitzgerald contraction, shorten in the direction of motion. And just as the image can never penetrate the mirror-space a greater distance than half its radius, so probably time-space is curved in such a way that our universe, like the surface of a sphere, is finite in extent, but unbounded.

XII

FORCE VS. GEOMETRY

How Einstein Has Substituted the Second for the First in Connection With the Cause of Gravitation

BY SAUL DUSHMAN GENERAL ELECTRIC LABORATORIES SCHENECTADY, N. Y.

The theory of relativity represents a most strikingly original conception of time and space, which was suggested by Einstein in order to correlate with all our past experience certain observations made in recent years. It is therefore extremely comprehensive in its scope; it demands from us a radical revision in our notions of time and space; it throws new light on the nature of mass and energy, and finally, it furnishes a totally new conception of the old problem of gravitation.

The starting point of the theory is the familiar observation that motion is always relative: that is, to define the motion of any object we must always use some point of reference. Thus we speak of the velocity of a train as 40 miles per hour with respect to the earth's surface, but would find it impossible to determine its absolute speed, or motion in space, since we know of no star whose position can be spoken of as absolutely fixed. These and similar considerations have led to the conclusion, pointed out by Newton and others, that it is impossible by any mechanical experiments on the earth to measure its velocity in space.

However, the results of observations on the phenomena of light and electricity led to the revival of the same problem under another form. As well known, there was evolved from these discoveries, the theory that light and electrical energy are of the same nature, and are in each case manifestations of wave-disturbances propagated through a hypothetical medium, the ether, with a velocity of 186,000 miles per second.

The problem therefore arose as to whether the earth and all stellar bodies move through this ether. In that case it ought to be possible to measure the velocity of the earth with respect to this medium, and under these conditions we could speak, in a sense, of absolute motion.

A large number of experiments has been tried with this end in view. The most famous of these, and the one which stimulated the subsequent development of the theory of relativity, was that carried out by Michelson and Morley in 1887. To understand the significance of this experiment we shall refer briefly to an analogous observation which is quite familiar.

Does it take longer to swim to a point 1 mile up a stream and back or to a point 1 mile across stream and back? The experienced swimmer will answer that the up-and-down journey takes longer. If we assume that the swimmer has a speed of 5 miles an hour in still water and that the current is 3 miles an hour, we find that, while it requires five-eighths hour to make the up-and-down journey, it takes only one-half hour for the trip across stream and back. The ratio between the times required for the two journeys is thus five-fourths, and if this is written in the form

$$\frac{1}{\sqrt{1 - (\frac{3}{5})^2}}$$

it shows how the result depends upon the square of the ratio of the speeds of the swimmer and the current.

Now the earth is moving in its orbit about the sun with a velocity of 18 miles per second. If the earth moves through the ether and a light-beam passes from one mirror to another and back again, the time taken for this journey ought to be longer when the light-path is in the direction of the earth's motion than when it is at right angles to this direction. For we can consider the light as a swimmer having a speed of 186,000 miles per second and travelling in a stream whose current is 18 miles per second.

When Michelson and Morley tried the experiment they could not observe any difference in the velocity of light in the two directions. The experiment has since been repeated under various conditions, but always with negative results.

Einstein's contribution to science consists in interpreting this result as being in accord with Newton's ideas on mechanical relativity in that it demonstrates the impossibility of measuring absolute motion, not only by mechanical, but also by optical or electrical experiments. Consequently the velocity of light must be regarded as constant and independent of the motion of either source or observer.

THE RELATIVITY OF UNIFORM MOTION

Let us consider some of the consequences which follow from this principle. An observer travelling with say one-half the velocity of light in the same direction as a ray of light would find that the latter has the usual velocity of 186,000 miles per second. Similarly an observer travelling in the opposite direction to that of the light-ray, with one-half the velocity of light, would obtain the same result.

Einstein has shown that these conclusions can be valid only if the units of time and space used by the two observers depend upon their relative motions. A careful calculation shows that the unit of length used by either observer appears to the other observer contracted when placed in the direction of their relative motion (but not, when placed at right angles to this direction), and the unit of time used by either observer appears to the other too great. Moreover, the ratio of the units of length or of time varies with the square of the relative speed of the two observers, according to a relation which is similar to that mentioned above for the swimmer in the current. This relation shows that as the relative speed approaches that of light the discrepancy between the units increases.