Is a volume defined in manifold with spacetime-signature?

Question

A volume in $R^n$ is easily calculated by multiplying the lengths of the $n$ dimensions.

I'm wondering how the different sign of time acts on the volume in a manifold with spacetime signature (1,3). Is a volume defined at all in such a manifold?

Answer 1

The definition of the volume form $\omega$ is essentially the same as in the Riemannian case: If $\{e_1,...,e_n\}$ is an oriented basis in $T_pM$, then $$ \omega(e_1,...,e_n)= \sqrt{|\det(g)|}, $$ where $g=(g_{ij})$ is the semi-Riemannian metric on $T_pM$ in terms of the above basis. Or, if you prefer local (oriented) coordinates, then $$ \omega= \sqrt{|\det(g(x))|}dx^1 ... dx^n. $$ My favorite reference is

O’Neill, Barrett, Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, 103. New York-London etc.: Academic Press. xiii, 468 p. (1983). ZBL0531.53051.