, Maxwell’s constant, though there might still be an absolute four-dimensional world, the physical laws such as we know them could no longer be conceived of as expressing relations between its tensors and vectors. We might conceive of space-time entities, in this new type of space-time, as entering into laws that would remain covariant to any change of mesh-system, but these entities and laws would not be represented by the physical entities and laws that experiment has revealed. From the standpoint of the physicist, they would be mere fictions, foreign to the world of our experience. It would be as though the entities and laws revealed to us by experience differed from the laws that would normally be expected had the world a unity of structure and of content. In this event the great beauty of Einstein’s space-time theory would be lost, and with it the wonderful unity and harmony it has shown to exist in nature. Thus, although the absolute viewpoint is the one we have stressed in this chapter, we should bear in mind that it was the disclosures of relativity that rendered this absolute viewpoint possible.
Furthermore we may very easily confer a relativistic aspect on the theory if we recall that by selecting curvilinear or Gaussian mesh-systems in space-time, we are merely placing ourselves in the position of an accelerated observer, who refers his measurements to some frame of reference, squirming like an octopus and who times events with clocks running wild. So far as can be gathered, Einstein was guided originally by a desire to prove that all motion was relative; it was thus the relativistic view that preceded the absolute viewpoint in the elaboration of the theory. There is no cause to be surprised at this precedence, seeing that we are referring to days when space-time was as yet unknown. But when space-time was discovered by Minkowski, the theory received an added impetus (though we can realise full well that even before this discovery a theory which aspired to prove that certain permanencies endured regardless of the observer’s motion would obviously lead us to suspect the underlying presence of some absolute world in which these permanencies would be situated). This absolute world could not be one of separate space and time; for in the absolute space of classical science the laws changed in form when our motion was modified. Obviously something more was needed, and space-time was the answer.
It is imperative, however, not to misinterpret the significance of the covariance of natural laws. The mere fact that all observers, whatever their motion, will discover the same laws, does not mean that all observers are alike in every respect. Even in Einstein’s theory, a rotating observer will realise full well, owing to centrifugal forces, that he does not stand in a Galilean frame. The complete relativity of all motion from every point of view is not by any means embodied in the covariance of the laws; and we shall see that this complete relativity of motion, leading to Mach’s mechanics, is far from being accepted by all scientists. The discussion of this subject will have to be left aside for the present, as it is intimately connected with the difficult problem of the shape of the universe, concerning which we shall have something to say in a later chapter.
CHAPTER XXVIII
THE DISCOVERY OF THE EINSTEINIAN LAW OF GRAVITATION
IN the two preceding chapters we have discussed in a very general way the subject of tensors and of the laws of nature. It now remains to be seen how Einstein, guided by the requirements that accompanied all general laws, succeeded in formulating with mathematical precision the true law of gravitation.
In order to understand what is to follow we must revert to Newton’s law. This law defines the intensity and distribution of the field of gravitational force in the empty space which surrounds matter. Also, it enables us in turn to ascertain how the field of force would be distributed in the interior of matter, in the interior of the earth, for instance. Thus, in its most general aspect, Newton’s law of gravitation defines the intensity and distribution of the gravitational field both inside and outside of matter.
Now we have seen that the presence of a field of force connotes the existence of an uneven distribution of potentials in the space where the field exists,[92] since any perfectly even and constant distribution of potentials would correspond to the complete vanishing of the field. Hence, Newton’s law of gravitation can also be written as a law of potential distribution inside and outside of matter. Expressed in this form, it is known as Poisson’s equation and is written
In this celebrated equation,