- #1
Jim Kata
- 197
- 10
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?
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?
