# What is the geodesic in this case?

Tags:
1. May 23, 2015

### unscientific

1. The problem statement, all variables and given/known data

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 $ds^2 = -c^2 d\tau^2 + g_{ij}dx^i dx^j$.

2. Relevant equations

3. 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??

2. May 25, 2015

### unscientific

bumpp

3. May 31, 2015

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4. Jun 1, 2015

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