Integral over 4D Metrics: Volume Element?

[Jim Kata]

I'm going to be working in 4D, and I assume my metric is symmetric. So what I am interested in is an integral of the form $$\int {d^{10} \eta \delta (\eta - \bar \eta } )$$. My question is what is the invariant volume element for an integral of this type? My guess is that it is $$\frac{{d^{10} \eta }}{{J^2 }}$$ where $$J$$ is the jacobian, but I don't know. This my logic:
$$d\bar x^\alpha = \frac{{\partial \bar x^\alpha }}{{\partial x^\beta }}dx^\beta$$ and $$d^4 \bar x = \frac{{d^4 x}}{J}$$. So since:$$\bar g^{\alpha \beta } = g^{\mu \tau } \frac{{\partial \bar x^\alpha }}{{\partial x^\mu }}\frac{{\partial \bar x^\beta }}{{\partial x^\tau }}$$ I figured $$d^{10} \bar g = \frac{{d^{10} g}}{{J^2 }}$$. I would also like the requirements that $$g_{00} < 0$$, $$\det [g_{\alpha \beta } ] < 0$$, and $$\det [\gamma _{ij} ] > 0$$ where $$\det [g_{\alpha \beta } ] = g_{00} \det [\gamma _{ij} ]$$. Any suggestions or guesses?

 

[matt grime]

I'm going to be working in 4D, and I assume my metric is symmetric.

A metric, by definition, satisfies d(x,y)=d(y,x).