Page:The principle of relativity (1920).djvu/201

 Divergence of the contravariant four-vector.

Let us multiply (26) with the contravariant fundamental tensor g^{μν} (inner multiplication), then by a transformation of the first member, the right-hand side takes the form

[part]/[part]x_{ν} (g^{μν} A_{μ}) - A_{μ} [part]g^{μν}/[part]x_{ν} - 1/2 g^{τα} ([part]g_{μα}/[part]x_{ν} + [part]g_{να}/[part]x_{μ} - [part]g_{μν}/[part]x_{α}) g^{μν} A_{τ} (A)

According to (31) and (29), the last member can take the form

1/2 [part]g^{τν}/[part]x_{ν} A_{τ} + 1/2 [part]g^{μτ}/[part]x_{μ} A_{τ} + 1/[sqrt](-g) [part][sqrt](-g)/[part]x_{α} g^{μα} A_{τ} (B)

Both the first members of the expression (B), and the second member of the expression (A) cancel each other, since the naming of the summation-indices is immaterial. The last member of (B) can then be united with first of (A). If we put

g^{μν} A_{μ} = A^{ν},

where A^{ν} as well as A_{μ} are vectors which can be arbitrarily chosen, we obtain finally

Φ = 1/[sqrt](-g) [part]/[part]x_{ν} ([sqrt](-g) A^{ν}).

This scalar is the Divergence of the contravariant four-vector A^{ν}.