Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000

by J. Malcolm Bird

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.

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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.