Page:The principle of relativity (1920).djvu/207

 The essential thing in this result is that on the right hand side of (42) we have only A_{ρ}, but not its differential co-efficients. From the tensor-character of A_{μστ} - A_{μτσ}, and from the fact that A_{ρ} is an arbitrary four vector, it follows, on account of the result of §7, that B^ρ_{μστ} is a tensor (Riemann-Christoffel Tensor).

The mathematical significance of this tensor is as follows; when the continuum is so shaped, that there is a co-ordinate system for which g_{μν}'s are constants, B^ρ_{μστ} all vanish.

If we choose instead of the original co-ordinate system any new one, so would the g_{μν}'s referred to this last system be no longer constants. The tensor character of B^ρ_{μστ} shows us, however, that these components vanish collectively also in any other chosen system of reference. The vanishing of the Riemann Tensor is thus a necessary condition that for some choice of the axis-system g_{μν}'s can be taken as constants. In our problem it corresponds to the case when by a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region. By the reduction of (43) with reference to indices to τ and ρ, we get the covariant tensor of the second rank

{B_{μυ} = R_{μυ} + S_{μυ} should be ν?] { (44){R_{μυ} = -[part]/[part]x_{α}{μυ α} + {μα β}{υβ α} {   {S_{μυ} = [part] log [sqrt](-g)/([part]x_{μ} [part]x_{υ}) - {μυ α} [part] log [sqrt](-g)/[part]x_{α}.