The Equivalence Hypothesis

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THE EQUIVALENCE HYPOTHESIS

The Discussion of This, With Its Difficulties and the Manner in Which Einstein Has Resolved Them, from the Essay by

PROF. E. N. DA C. ANDRADE, ORDNANCE COLLEGE, WOOLWICH, ENGLAND

Having shown that, of several systems all moving with reference to one another with uniform motion, no one is entitled to any preference over the others, and having deduced the laws for such systems, Einstein was confronted with a difficulty which had long been felt. A body rotating, which is a special case of an accelerated body, can be distinguished from one at rest, without looking outside it, by the existence of the so-called centrifugal forces.

This circumstance, which gives certain bodies an absolute or preferential motion, is unpalatable to the relativist; he would like there to be no difference as regards forces1 between the case when the earth rotates with reference to outside bodies (the stars) considered as fixed, and the case when the earth is considered fixed and all outside bodies rotate around it. This point cannot be investigated by direct [335]experiment; we can spin a top but we cannot keep a top at rest and spin the world round it, to see if the forces are same.

In considering the problem of how to devise laws which should make all rotations relative, Einstein conceived the brilliant yet simple idea that gravitation could be brought into the scheme as an acceleration effect, since both ordinary accelerational forces and gravitational forces are proportional to the same thing, the mass of a body. The impossibility of separating the two kinds of effect can be easily seen by considering the starting of an elevator. When the elevator is quickly accelerated upwards we feel a downward pull, just as if the gravitational pull had been increased, and if the acceleration continued to be uniform, bodies tested with a spring balance would all weigh more in the elevator than they did on firm ground. In a similar way the whole of the gravitational pull may be considered to be an accelerational effect, the difficulty being to devise laws of motion which will give the effects that we find by actual observation.

But it is obvious that we cannot, by ordinary mechanics, consider the earth as being accelerated in all directions, which we should have to do, apparently, to account for the fact that the gravitational pull is always toward the center. [It is obvious that we cannot explain gravitation by assuming that the earth’s surface is continually moving outward with an accelerated velocity.]227 So Einstein found that, as long as we treat the problem by Euclid’s geometry, we cannot reach a satisfactory solution. But he found that to the four-dimensional [336]space made up of the three ordinary dimensions of space, together with the time-dimension which we have already mentioned in discussing the special theory, may be attributed a peculiar geometry, the nature of which departs more and more from Euclidean geometry as we approach a gravitational body, and the net result of which is to make possible the universal correspondence of gravitation and acceleration.

This modification of the geometry of space is often spoken of as the “curvature of space,” an expression which is puzzling, especially as the space which is “curved” is four-dimensional time-space. But we can get an idea of what is meant by considering figures, triangles say, drawn on the surface, of a sphere. These triangles, although drawn on a surface, will not have the same properties as triangles drawn on flat paper—their three angles will not together equal right angles. They will be non-Euclidean. This is only a rough analogy, but we can see that the curvature of the surface causes a departure from Euclidean geometry for plane figures, and consequently the departure from Euclidean laws extended to four dimensions may be referred to as caused by “curvature of space.”

It is difficult to imagine a lump of matter affecting the geometry of the space round it. Once more we must use a rough illustration. Imagine a very hot body, and that, knowing nothing of its properties, we have to measure up the space round it with metal measuring-rods. The nearer we are to the body, the longer the rods will become, owing to the expansion of the metal. When we measure out a [337]square, one side of which is nearer the body than the opposite side, its angles will not be right angles. If we knew nothing of the laws of heat we should say that the body had made the space round it non-Euclidean.

Einstein found, then, that by taking the properties of space, as given by measurement, to be modified in the neighborhood of masses of matter, he could devise general laws according to which gravitational effects would be produced, and there would be no absolute rotation. All forces will be the same whether a body rotates with everything outside it fixed, or the body is fixed, and everything rotates round it. All motion is then relative, and the theory is one of “general relativity.” The velocity of light is, however, no longer constant, and its path is not a straight line, if it is passing near gravitating matter. This does not contradict the special theory, which did not allow for gravitation. Rather, the special theory is a particular case to which the generalised theory reduces when there is no matter about, just as the Newtonian dynamics is a special case of the special theory, which we obtain when all velocities are small compared to that of light.

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