Is the Christoffel Symbol Zero for Diagonal Metrics?

[ehrenfest]

Homework Statement

I am trying to show that the connection $$\Gamma^a_{bc}$$ is equal to 0 when the metric g_ab is diagonal. Will the formula
$$\Gamma^a_{bc} = 1/2 g^{ad}(\partial_bg_{dc} + \partial_cg_{bd} - \partial_dg_{bc})$$ be of use? How can I manipulate that equation and use the fact that the metric is diagonal?

Homework Equations

The Attempt at a Solution

 

[Dick]

Why do you think the connection is zero if the metric is diagonal? It's not even true. Look at spherical coordinates.

 

[ehrenfest]

It comes right out of my book!

 

[Hurkyl]

No it doesn't; your book doesn't ask you to prove all of the components are zero: just some of them.

 

[ehrenfest]

OK. So it only wants me to prove that the off-diagonal components are zero (that statement in parentheses was pretty important). Anyway, I still have the same two questions as in the first post.

 

[dextercioby]

If $\Gamma$ is meant to be the Riemann - Christoffel (or the metric) connection, then, yes, that's the formula you should use.

 

[ehrenfest]

Yes. It is the Christoffel symbol of the second kind. So am allowed to do this:

$$\Gamma^a_{bc} = 1/2 (\partial_bg^{ad}g_{dc} + \partial_cg^{ad}g_{bd} - \partial_dg^{ad}g_{bc})$$
?

Then the first term becomes the Kronecker delta, I think.

If not, how should I use the fact that the metric is diagonal?

 

[Hurkyl]

Yes. It is the Christoffel symbol of the second kind. So am allowed to do this:

$$\Gamma^a_{bc} = 1/2 (\partial_bg^{ad}g_{dc} + \partial_cg^{ad}g_{bd} - \partial_dg^{ad}g_{bc})$$
?

Why do you think, for all d, that g^ad is a constant with respect to the b, c, and d-th variables?

If not, how should I use the fact that the metric is diagonal?

Any way you can imagine. You could substitute zero for the off-diagonal terms. You could decompose the metric into a linear combination of simpler tensors. Et cetera.

 

[ehrenfest]

I see. You just replace d with a.

 

[ehrenfest]

Now, can someone explain to me where in the world this natural log comes from at the bottom of attached page of the Hobson book?

 

[Hurkyl]

Did you try differentiating it?