Page:The principle of relativity (1920).djvu/210

 §14. The Field-equation of Gravitation in the absence of matter.

In the following, we differentiate gravitation-field from matter in the sense that everything besides the gravitation-field will be signified as matter; therefore the term includes not only matter in the usual sense, but also the electro-dynamic field. Our next problem is to seek the field-equations of gravitation in the absence of matter. For this we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motion for material points. A special case in which the field-equations sought-for are evidently satisfied is that of the special relativity theory in which g_{μυ}'s have certain constant values. This would be the case in a certain finite region with reference to a definite co-ordinate system K_{0}. With reference to this system, all the components B^{ρ}_{μστ} of the Riemann's Tensor [equation 43] vanish. These vanish then also in the region considered, with reference to every other co-ordinate system.

The equations of the gravitation-field free from matter must thus be in every case satisfied when all B^{ρ}_{μστ} vanish. But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed away by any choice of axes, i.e., it cannot be transformed to a case of constant g_{μυ}'s.

Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors B_{μν} deduced from the tensors B^{ρ}_{μστ} should vanish.