In deriving Schwarzschild solution one assumes many constraints on the metric, in particular parity invariance (invariance of $g _{\mu \nu}$ under $t \rightarrow-t, \phi \rightarrow-\phi, \theta \rightarrow-\theta$).
I have been told (both in class and on Carroll's book) that this implies that the non-diagonal Einstein equations $R_{\mu \nu} = 0$ are identically null and so the calculation is significant only for diagonal components. So, for example, $R_{tr}$ vanishes automatically.
What i have tried to do is write explicitely $R_{tr}$, for example, and prove that it is equal to $-R_{tr}$ meaning that it is zero. However this seems not to work assuming only the invariance of $g_{\mu \nu}$ under $t-$parity leaving the $\Gamma$'s totally general(i.e. not calculating them for Schwarzschild but writing them in terms of a general parity-invariant metric; i wanted to prove this for as a general property of Einstein equations).
Any tip for the proof of this statement?
