Covariant Derivative: What Is $\nabla^0 A_{\alpha}$?

[S.P.P]

just a quick query, I know that,

$\nabla_0 A_{\alpha}= \partial_0 A_{\alpha} - \Gamma^{\beta}_{0 \alpha} A_{\beta}$

But what does
$\nabla^0 A_{\alpha}$ equal?

 

[shoehorn]

Since $\nabla^0A_\alpha = g^{0\beta}\nabla_\beta A_\alpha$ you have

$$\nabla^0A_\alpha = g^{0\beta}\partial_\beta A_\alpha - g^{0\beta}\Gamma^{\gamma}_{\phantom{\gamma}\alpha\beta}A_\gamma$$

 

[shoehorn]

You're welcome.

 

[StatusX]

You're welcome.

Don't take it personally, it's rare to get thanked for help here.