Page:The principle of relativity (1920).djvu/199

 Some special cases of Particular Importance.

''A few auxiliary lemmas concerning the fundamental tensor.'' We shall first deduce some of the lemmas much used afterwards. According to the law of differentiation of determinants, we have

(28) dg = g^{μν} g dg_{μν} = -g_{μν} gdg^{μν}.

The last form follows from the first when we remember that

g_{μν} g^{μ´ν} = δ^{μ´}_{μ}, and therefore g_{μν}g^{μν} = 4,

consequently g_{μν}dg^{μν} + g^{μν} dg_{μν} = 0.

From (28), it follows that

(29) 1/[sqrt](-g) [part][sqrt](-g)/[part]x_{σ} = 1/2 log(-g)/[part]x_{σ} = 1/2 g^{μν} [part]g_{μν}/[part]x_{σ} = -1/2 g_{μν} [part]g^{μν}/[part]x_{σ}.

Again, since g_{μν} g^{νσ} = δ^ν_μ, we have, by differentiation,

{g_{μσ} dg^{νσ} = -g^{νσ} dg_{μσ}

(30) { or

{g_{μσ} [part]g^{νσ}/[part]x_{λ} = -g^{νσ} [part]g_{μσ}/[part]x_{λ}

By mixed multiplication with g^{στ} and g_{νλ} respectively we obtain (changing the mode of writing the indices).