we get the motion of the point with reference to K₁, given by
"(46)"
We now make the very simple assumption that this general covariant system of equations defines also the motion of the point in the gravitational field, when there exists no reference-system K₀, with reference to which the special relativity theory holds throughout a finite region. The assumption seems to us to be all the more legitimate, as (46) contains only the first differentials of gμν, among which there is no relation in the special case when K₀ exists.
If γμντ’s vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field.
§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 gμν’s have certain constant values. This would be the case in a certain finite region with reference to a definite co-ordinate system K₀. With reference to this system, all the components Bρμστ 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 Bρμστ 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 gμν’s.
Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors Bμν deduced from the tensors Bρμστ should vanish. We thus get 10 equations for 10 quantities gμν which are fulfilled in the special case when Bρμστ’s all vanish.
Remembering (44) we see that in absence of matter the field-equations come out as follows; (when referred to the special co-ordinate-system chosen.)