Now the tensor on the left possesses the remarkable mathematical property of expressing a form of curvature which is preserved regardless of our mesh-system. It follows that in virtue of the equations of gravitation written above, the same conservation must hold for
, hence for mass, energy and momentum.
The mathematical identity that expresses this property, let it be noted, does not depend on Einstein’s theory; it is a purely mathematical property between the
’s of a continuum, which had been discovered by the mathematician Ricci before the theory of relativity was born. But in view of the physical relationships which Einstein had established between the curvatures of the space-time continuum and the presence of matter, it now followed as a necessary consequence of the theory that the right-hand side of the gravitational equations must present the same characters of conservation as the left-hand side. If we interpret this result physically, we see that it implies the conservation of mass, momentum and energy.
We thus reach the remarkable conclusion that the principles of conservation, which in classical science were of an empirical nature and were totally unconnected with the law of gravitation, now appear as immediate and necessary consequences of Einstein’s law of gravitation. In other words, granting the correctness of the law of gravitation, conservation cannot help but exist in this universe. To many thinkers this is one of the most beautiful aspects of Einstein’s theory, showing as it does the wonderful unity and relatedness of nature.
Nevertheless we must be careful to understand the deeper significance of conservation in relativity. We shall see that the type of conservation we are now discussing is somewhat different from the classical conception of conservation, under which matter, for instance, disappeared from here only to reappear there. In the present case conservation is more ethereal and abstract. We may explain these points better by examining conservation in classical science. Here, when fields of force are present, the principles of the conservation of energy and momentum would appear at first sight to be contradicted by experiment. Thus, a body falling freely towards the earth can certainly not be said to possess a constant vis viva, or energy, since, originally starting from rest, it acquires an increasing speed as it falls towards the ground. Rather than recognise the breakdown of the principle of the conservation of energy, classical science preferred to assume that the body possessed two types of energy: a kinetic energy given by
, and a potential energy depending on the position of the body in the field of gravitation. It was the sum total of these two types of energy that remained constant and was therefore conserved during the fall of the body.