Schwarzchild Basis Vectors

Question

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?

Answer 1

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.