Page:Eddington A. Space Time and Gravitation. 1920.djvu/106

then the conditions $G_{\mu \nu }=0$ ...........................(5), will satisfy our requirements for a general law of nature.

This law is independent of the mesh-system, though this can only be proved by elaborate mathematical analysis. Evidently, when all the $B$'s vanish, equation (5) is satisfied; so, when flat space-time occurs, this law of nature is not violated. Further it is not so stringent as the condition for flatness, and admits of the occurrence of a limited variety of non-Euclidean geometries. Rejecting duplicates, it comprises 10 equations; but four of these can be derived from the other six, so that it gives six conditions, which happens to be the number required for a law of gravitation^[1].

The suggestion is thus reached that $G_{\mu \nu }=0$ may be the general law of gravitation. Whether it is so or not can only be settled by experiment. In particular, it must in ordinary cases reduce to something so near the Newtonian law, that the remarkable confirmation of the latter by observation is accounted for. Further it is necessary to examine whether there are any exceptional cases in which the difference between it and Newton's law can be tested. We shall see that these tests are satisfied.

What would have been the position if this suggested law had failed? We might continue the search for other laws satisfying the two conditions laid down; but these would certainly be far more complicated mathematically. I believe too that they would not help much, because practically they would be indistinguishable from the simpler law here suggested—though this has not been demonstrated rigorously. The other alternative is that there is something causing force in nature not comprised in the

1. Isolate a region of empty space-time; and suppose that everywhere outside the region the potentials are known. It should then be possible by the law of gravitation to determine the nature of space-time in the region. Ten differential equations together with the boundary-values would suffice to determine the ten potentials throughout the region; but that would determine not only the kind of space-time but the mesh-system, whereas the partitions of the mesh-system can be continued across the region in any arbitrary way. The four sets of partitions give a four-fold arbitrariness; and to admit of this, the number of equations required is reduced to six.