’s in a given mesh-system with the gravitational potentials in the frame corresponding to this mesh-system, we find ourselves in the presence of as many different gravitational potentials as there are separate
’s. That is, we have ten different potentials. Newton knew of only one of these, the Newtonian potential, which turns out to be
—the one directed along the time-axis of the mesh-system. It is the presence of these other potentials ignored by Newton and discovered by Einstein which is responsible for the major differences distinguishing Newton’s law of planetary motions from that of Einstein. We see, therefore, that this natural generalisation of the geometry of space-time, by compelling us to recognise forces of gravitation and inertia as of the same nature, leads us to exactly the same conclusions as were obtained by Einstein when he started from the postulate of equivalence. So although we laid stress on the postulate of equivalence as a separate discovery in order to abide by the historical sequence of Einstein’s investigations, we see that when we view the problem from a loftier standpoint the postulate does not add anything to what would normally have been discovered, or at least suspected, as a result of the more philosophical generalisation we have discussed.
Of course, if the postulate of equivalence should prove to be unjustified, as it would were experiment to detect a difference between inertial and gravitational mass, the generalisation would also be unjustified and gravitation could never be ascribed to a curvature of space-time. However, as the most refined experiments have invariably disclosed the complete identity of the two types of masses, the space-time generalisation appears to be in order.
CHAPTER XXVI
TENSORS AND THE LAWS OF NATURE
WE have now reached a point where further progress would appear to require at least a few elementary notions on tensors and the absolute calculus. We shall endeavour to present these matters in as clear a way as possible by considering, first of all, tensors from the standpoint of pure mathematics, without any reference to their applications in Einstein’s theory.
Let us suppose that we are dealing with the ordinary three-dimensional Euclidean space of classical science. We remember that it is often convenient to refer the positions of objects situated in space to some definite system of reference or system of co-ordinates. This procedure is equivalent to drawing an imaginary three-dimensional mesh-system with respect to which we locate the points of space. The precise shape of this mesh-system is arbitrary. It may split up space into equal cubes (in which case it is called Cartesian), or again it may be curvilinear and split up space into irregularly shaped volumes. In all cases, however, we shall assume that these mesh-systems remain fixed and do not vary in shape or orientation once they have been drawn.