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$$ 0=\frac{d}{d \tau} (g_{\alpha \beta} \frac{dx^\beta}{d \tau}) - \frac{1}{2} \partial_\alpha g_{\mu\nu} \frac{dx^\mu}{d \tau} \frac{dx^\nu}{d \tau} $$

as well as

$$ 0= \frac{d^2x^\gamma}{d \tau^2} + g^{\gamma\alpha} (\partial_\mu g_{\alpha\nu}-\frac{1}{2}\partial_\alpha g_{\mu\nu}) \frac{dx^\mu}{d \tau} \frac{dx^\nu}{d \tau}.

$$

However, I have a friend writing down the geodesic equation as

$$ g_{\alpha j}\frac{d^2x^j}{d \tau^2} + (\partial_i g_{\alpha j}-\frac{1}{2}\partial_\alpha g_{ij}) \frac{dx^j}{d \tau} \frac{dx^i}{d \tau}.

$$

and getting the right answers.

And I losing my mind? Is that equation the geodesic equation?!