I'm having issues with computation of Killing equation. I'm using Mathematica to check if the given vectors are Killing vectors or not, and by hand for simple vector like $\xi=\partial_t$ I get the correct result (0=0 when plugging it in the equation), but when I try to check it wiht Mathematica I get weird results.
I don't know if I made my code right, but it should be working, since everything else is working. So I am starting to doubt my by hand method, even tho I'm getting good results :\
The equation is
$$\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0$$
And say my vector is $\xi=\partial_t$, if it is a vector, that means that $\xi^{\mu}=\delta^{\mu}_0$, right? In that case, do I need to raise index in my Killing equation?
And if so is this the correct form:
I'm raising index with metric tensor:
$$\nabla_\mu(g_{\nu\alpha}\xi^\alpha)+\nabla_\nu(g_{\mu\beta}\xi^\beta)$$
Since it applies that my metric is covariantly constant, and I can put my metric components out, and calculate the covariant derivative normally.
$$g_{\nu\alpha}\nabla_\mu\xi^\alpha+g_{\mu\beta}\nabla_\nu\xi^\beta$$
First things first: Is this correct? I mean, when calculating if the expression is true, do I need to raise index?
And do I need to put different indices in metric ($\alpha, \beta$)? Or could I write
$$g_{\nu\alpha}\nabla_\mu\xi^\alpha+g_{\mu\alpha}\nabla_\nu\xi^\alpha?$$