Page:The principle of relativity (1920).djvu/211

 We thus get 10 equations for 10 quantities g_{μν} which are fulfilled in the special case when B^{ρ}_{μστ}'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.)

(47) [part]Γ^{α}_{μυ}/[part]x_{α} + Γ^{α}_{μβ} Γ^{β}_{υα} = 0;

[sqrt](-g) = 1; Γ^{α}_{μυ} = -{^{μυ} _{α}}.

It can also be shown that the choice of these equations is connected with a minimum of arbitrariness. For besides B_{μυ}, there is no tensor of the second rank, which can be built out of g_{μυ}'s and their derivatives no higher than the second, and which is also linear in them.

It will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity, together with equations (46), give us the Newtonian law of attraction as a first approximation, and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier (the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors). My view is that these are convincing proofs of the physical correctness of my theory.