What Are the Conditions on Christoffel Symbols for Given Geodesics?

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Homework Statement

Using the geodesic equation, find the conditions on christoffel symbols for $x^\mu(\tau)$ geodesics where $x^0 = c\tau, x^i = constant$.
Show the metric is of the form $ds^2 = -c^2 d\tau^2 + g_{ij}dx^i dx^j$.

Homework Equations

The Attempt at a Solution

The geodesic equation is
$$\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha \beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$
$$\Gamma^\mu_{\alpha \beta} = \frac{1}{2} g^{\mu \gamma} \left( \partial_\alpha g_{\gamma \beta} + \partial_\beta g_{\alpha \gamma} - \partial_\gamma g_{\alpha \beta} \right)$$

For $x^0 = c\tau$, we have that $\Gamma^0_{00} = 0$. This means that $\partial_0 g_{\gamma 0} = \frac{1}{2} \partial_\gamma g_{00}$. How does this help??

 

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