Conditions on Christoffel Symbols?

[unscientific]

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

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

 

[strangerep]

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

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

 

[unscientific]

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

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

 

[strangerep]

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

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 $ds^2$.

 

[unscientific]

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 $ds^2$.

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

 

[strangerep]

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

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.