I'm not sure how to approach this question.
So I start off with the fact the path taken is space-like,
$$ds^2>0$$
Input the Schwarzschild metric,
$$−(1−\frac{2GM}{r})dt^2+(1−\frac{2GM}{r})^{−1}dr^2>0$$
Where I assume the mass doesn't move in angular direction.
How should I continue?
This question wasn't particularly hard, so I assume metric compatibility and input Ricci tensor to the left side of Einstein's equation.
$$R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=Cg_{\mu\nu}-\frac{1}{2} (4C)g_{\mu\nu}=-Cg_{\mu\nu}$$
Then apply covariant derivative on both side...
I see! Thank you.
I could prove ##U_{\alpha}\partial_{\beta}U^{\alpha}=0## and eliminate second term.
As for the first term, I don't think it could proceed further. Am I done?
In Minkowski spacetime, calculate ##P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}##.
I had calculated previously that ##P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}##
When I subsitute it back into the expression...
Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group
Generator of translation: ##P_{\rho}=-i\partial_{\rho}##
Generator of rotation...