i.e., the extension of a four-vector.
Thus we get (by slightly changing the indices) the tensor of the third rank
We use these expressions for the formation of the tensor Aμστ - Aμτσ. Thereby the following terms in Aμστ cancel the corresponding terms in Aμτσ; the first member, the fourth member, as well as the member corresponding to the last term within the square bracket. These are all symmetrical in σ, and τ. The same is true for the sum of the second and third members. We thus get
"(43)"
The essential thing in this result is that on the right hand side of (42) we have only Aρ, but not its differential co-efficients. From the tensor-character of Aμστ - Aμτσ, and from the fact that Aρ is an arbitrary four vector, it follows, on account of the result of §7, that Bρμστ 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 gμν’s are constants, Bρμστ all vanish.
If we choose instead of the original co-ordinate system any new one, so would the gμν’s referred to this last system be no longer constants. The tensor character of Bρμστ 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 gμν’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