’s; but we have also seen that other more complicated types of invariants could be constructed. Had these been substituted in the expression of the action, we should have obtained alternative laws of gravitation. But the point we wish to stress is the following:
Regardless of the precise invariant we may select in order to represent the function of action of the metrical field, there is one property which is common to all these invariants. Their mathematical nature is such that when we apply to them the principle of action, we find that the curvature of space-time which would be present in the interior of matter must be of a type which would remain conserved. Granting that a certain indeterminateness may exist in our choice of the action, and hence of the law of gravitation, all these laws would compel us to recognise that in our universe, permanent entities such as matter cannot help but be present. To be sure, the type of conservation necessitated by the principle of action is not altogether the classical conception of conservation; but, as we saw in a previous chapter, the new principle becomes that of classical conservation when a Galilean reference system is adhered to.
Finally, we may note that by applying the principle of action to the total action formed by the gravitational action and the field-action of electricity, we should obtain the precise type of space-time curvature which would exist in an electromagnetic field free of electrons and magnets; for instance, in an empty region flooded with light. It is then found (as was mentioned in [Chapter XXXIII]) that the electromagnetic field also produces a curvature of space-time, less pronounced than that reigning in the interior of matter, but more pronounced than that existing in the empty space around matter (that is to say, in a gravitational field due to ordinary matter).
CHAPTER XXXVI
THE MYSTERY OF MATTER
ONE of the greatest merits of the theory of relativity has been to allow us to represent gravitation as a direct consequence of the curvature of space-time. It would certainly be the crowning achievement of this superb theory if it could enable us to interpret all the manifestations of the physical universe in terms of the various types of curvature of one sole fundamental continuum, space-time. But at the present stage two separate reasons render this solution highly improbable.
We remember that in the course of our wanderings we came across two foreign tensors:
, the matter-tensor, and
, the electromagnetic tensor. It seems impossible to express these tensors in terms of the basic structural tensors of space-time, namely, the