Page:The principle of relativity (1920).djvu/205

 If instead of A^{ρσ}, we introduce in a similar way the symmetrical co-variant tensor A_{ρσ} = g_{ρα} g_{σβ} A^{αβ}, then owing to (31) the last member can take the form

1/2 [sqrt](-g) ([part]g^{ρσ}/[part]x_{μ}) A_{ρσ}.

In the symmetrical case treated, (41) can be replaced by either of the forms

(41a) [sqrt](-g) A_{μ} = [part]([sqrt](-g) A^{σ}_{μ})/[part]x_{σ}

-(1/2) ([part]g_{ρσ}/[part]x_{μ}) [sqrt](-g) A^{ρσ}

or

(41b) [sqrt](-g) A_{μ} = [part]([sqrt](-g) A^{σ}_{μ})/[part]x_{σ}

+ 1/2 ([part]g^{ρσ}/[part]x_{μ}) [sqrt](-g) A_{σρ}

which we shall have to make use of afterwards.

§12. The Riemann-Christoffel Tensor.

We now seek only those tensors, which can be obtained from the fundamental tensor g^{μν} by differentiation alone. It is found easily. We put in (27) instead of any tensor A^{μν,} the fundamental tensor g^{μν} and get from