And so we see that the best that could be hoped from a curvature of space alone would be an explanation of the precise paths of falling bodies, but not of their motions along these paths. Hence we should not have succeeded in interpreting gravitation, since, once more, the motion along the paths is as essential as the spatial shape of the paths. Newton’s law, on the other hand, accounts both for paths and for motions.
But we may consider another means of escape. We might assume that the spatial geodesics which free bodies follow act not only as grooves, but more like arteries which propel the blood along by their expansions and contractions. We might then assume that these pulsations of the geodesics would give the planets and projectiles those precise variations in motion which are observed. Needless to say, the hypothesis would be absurd and useless, and, so far as the writer is aware, has fortunately never been suggested. At all events, even with this hypothesis ad hoc, we should have failed to account for gravitation in terms of space-curvature, since we should have been compelled to introduce this additional influence. Furthermore, as in the absence of attracting masses free bodies are not accelerated, we should have to assume that in the absence of attracting masses these pulsations of the geodesics would cease; hence it would be the action of matter which would produce the pulsations. In other words, what we formerly called the force of gravitation would now be called the force of pulsation, and we should be thrown back on the Newtonian law of gravitation under a new name, in spite of our absurd hypothesis ad hoc.
With space-time, again, all these difficulties are obviated, for the geodesics of space-time, in contradistinction to those of space, account both for the paths of bodies and for their precise motions along these paths. This is due to the fact that a space-time geodesic deals with both time and space.
Thus far we have seen that one of the main obstacles to a space-curvature theory of gravitation resided in its inability to account for the precise motions of bodies along their orbits or trajectories. Yet, on the other hand, were it not for the other difficulties mentioned previously in this chapter, it might not have appeared impossible to account for the spatial shapes of the orbits of bodies in a gravitational field. We now propose to show that, quite independently of the reasons already given, even this partial success can never be realised. Here, however, we are compelled to consider a more mathematical aspect of the problem.
First, let us consider ordinary three-dimensional Euclidean space. If we select a point
at random, we know that there exists a triple infinity of straight lines passing through
. But if we impose the restriction that these lines shall all be perpendicular to some arbitrary direction
