Under such conditions, it would be difficult to see what argument we could invoke for maintaining that the curved course of the parabola was a geodesic, hence constituted the shortest spatial distance between two points on the trajectory. In short, we should be faced with a dualism subsisting as between rod measurements and explorations with moving bodies. Inasmuch as it is the constant endeavour of science to obviate dualism wherever possible and to establish unity, we may anticipate that an interpretation of gravitation along the preceding lines would have had no chance of being accepted by scientists. Rather would they have retained Newton’s Euclidean space, which would account perfectly for the results of rod measurements and light-ray triangulations, and then appealed to an additional force of gravitation whose action would be to tear bodies away from the straight geodesics they would normally have followed.
Furthermore, there is still another aspect to the problem. If we assume that moving points describe geodesics in a non-Euclidean space which presents a high degree of non-Euclideanism, we must recognise that our material bodies (which behave Euclideanly) must diverge widely from standard when displaced. If, then, we consider the translational motion of a large non-rotating mass, all its molecules should describe geodesics. But this would be impossible, since, owing to the enormous variations in the shape of the body, as contrasted with the geometry of space, the majority of the molecules would be torn from their natural courses. Enormous forces of reaction would therefore arise, so that the body would be either torn apart, crushed or unable to move at all.
With space-time, however, all these drawbacks are removed, since no dualism arises as between measurements with rods and the courses of free bodies. In Einstein’s theory it is the geodesics of space-time, not those of space, which are followed by free bodies. This curvature of space-time is sufficient to account for the motions of projectiles and planets; and when space-time is split up into space and time, we find that over limited areas the geodesics of space remain Euclidean straight lines (owing to the enormous value of
), as in classical science. For this reason, it is fully in order that in spite of the space-time curvature, measurements with rods and stretched strings should yield Euclidean results.
Thus we see that in Einstein’s theory the metrics of space-time permits us to account both for gravitation and for the results of ordinary measurements with rods. It also enables us to co-ordinate temporal processes as exhibited in the Einstein shift-effect. In short, with the rejection of space-time, the unity of nature revealed by the relativity theory would be lost.
Let us now consider a second objection to the space-curvature hypothesis. The law of motion for falling bodies is not concerned solely with spatial courses. For instance, projectiles describe parabolas, but their motions along these trajectories are just as characteristic as are the shapes of the trajectories themselves. Again, in the case of the planets, we are telling only half the tale when we state that they describe ellipses round the sun, situated at one of the foci. It was not from an incomplete statement of this sort that Newton could ever have obtained his law of the inverse square. Kepler’s second and third laws complete the description of the planetary motions by defining the nature of the motions and velocities of the planets along their elliptical orbits. Only when all three of Kepler’s laws are taken into consideration can Newton’s law of gravitation be deduced without ambiguity. For example, if the planets pursued their elliptical orbits with constant speeds, Newton’s law would be unable to account for their motions.
Turning, then, to our tentative scheme of gravitation in terms of a curved space, we see that there is nothing in it to suggest the peculiar accelerated motions of planets and projectiles, so that we are unable to account for all the effects that Newton ascribed to gravitation. Worse still, with a theory of space-curvature the motions of bodies along their trajectories would be uniform, since geodesics are described with constant velocity. And this would be in utter conflict with the accelerated motions of falling bodies and of the planets.
We might, however, seek an avenue of escape by claiming that the curvature of space would account for the accelerated appearance of these motions, much as the apparent velocity of a body depends on its orientation with respect to our line of vision. To be sure, a curvature of space would entail a visual distortion of this sort, since the light rays emanating from a luminous source would now reach us along curved routes. As a result, we should misjudge the position of the source, much as occurs in the case of refraction. This effect has to be taken into consideration in Einstein’s cylindrical universe, for example. The distances of stars, as deduced from a measurement of their parallaxes, have to be decreased somewhat, owing to the spherical curvature of space. It follows that a star leaving us radially with uniform speed would appear to possess an increasing acceleration as it moved away from us, as though repelled from the point where we happened to be standing.[168]
But it is obvious that if separate space and time are retained, our belief in the accelerated motions of bodies falling in a gravitational field can in no wise be attributed to effects due to optical distortions varying with the relative position of the percipient. These accelerations are objective in that they exist to the same degree for all observers in the objective space and time world of science. To take a simple case, if we allow a stone to fall on our foot, its velocity of impact will vary with the height whence it was released. We cannot say that these variations in velocity are to be ascribed to a mere optical distortion of the light rays reflected from the stone. We know, indeed, that quite aside from any visual image of motion, the pain we experience will tell the tale. Added to all these considerations, it would be quite impossible to account for the precise accelerations of falling bodies and of the planets, even were we to interpret them by means of optical distortions.