TITLE: Schwarzchild Basis Vectors
QUESTION [0 upvotes]: In Schwarzschild metric, are the basis vectors $e_r$ and $e_t$ always orthogonal? If not, we would get non diagonal $g_{tr}$ terms, which we don't.
But if we differentiate $e_r$ with respect to time and take the time component of the derivative we get a non zero value, i.e $\Gamma^0_{01}\neq 0$. Therefore, $e_r$ changes in the direction of time - does that mean after a while $e_r$ and $e_t$ won't remain orthogonal, and their dot product will produce a non zero value? Or, $e_t$ will change accordingly to make sure this does not happen and they remain always orthogonal?

REPLY [2 votes]: If not, we would get not $g_{tr}$ terms, which we don't.

That's your answer.  $g(\partial_t,\partial_r) \equiv g_{tr}=0$, so $\partial_t$ and $\partial_r$ are orthogonal. Furthermore, the Schwarzschild metric has no dependence on the Schwarzschild time coordinate, so this fact does not change with time.
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