# Conditions on Christoffel Symbols?

Tags:
1. Jun 13, 2015

### unscientific

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

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 $ds^2 = -c^2dt^2 +g_{ij}dx^idx^j$.

2. Relevant equations

3. The attempt at a solution

The geodesic equation is:
$$\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu~_{\alpha \beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau}$$

Using the condition given
$$\Gamma^0~_{\alpha \beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$
$$\Gamma^0~_{00} = \Gamma^i~_{00} = 0$$

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

2. Jun 13, 2015

### strangerep

But is that the question? Aren't you simply being asked to show that such a metric does satisfy those conditions on $\Gamma$ ?

3. Jun 14, 2015

### unscientific

If $x^i = constant$, then wouldn't $dx^i = 0$?

4. Jun 14, 2015

### strangerep

Yes, but I'm puzzled why you replied to my question this way. Just write the formula for the Chrisoffel symbols, and substitute the metric components that appear in the given line element $ds^2$.

5. Jun 19, 2015

### unscientific

What do you mean by "substitute the metric components that appear in the given line element $ds^2$"?

6. Jun 19, 2015

### strangerep

Which part of my sentence are you having trouble with? The word "substitute"? The word "metric"? The concept of metric components appearing a line element?

If the last one, then read this Wiki page.