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unscientific

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## Homework Statement

Write down the geodesic equation. For ##x^0 = c\tau## and ##x^i = constant##, find the condition on the christoffel symbols ##\Gamma^\mu~_{\alpha \beta}##. Show these conditions always work when the metric is of the form ##ds^2 = -c^2dt^2 +g_{ij}dx^idx^j##.

## Homework Equations

## The Attempt at a Solution

The geodesic equation is:

[tex]\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu~_{\alpha \beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau}[/tex]

Using the condition given

[tex]\Gamma^0~_{\alpha \beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0 [/tex]

[tex]\Gamma^0~_{00} = \Gamma^i~_{00} = 0 [/tex]

How do I show the metric is of the form ##ds^2 = -c^2dt^2 +g_{ij}dx^idx^j##?