# Diagonal connection problem

1. Sep 22, 2007

### ehrenfest

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

2. Relevant equations

3. The attempt at a solution

Last edited: Sep 22, 2007
2. Sep 22, 2007

### Dick

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

3. Sep 22, 2007

### ehrenfest

It comes right out of my book!

#### Attached Files:

• ###### hobson exercise 36.jpg
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4. Sep 23, 2007

### Hurkyl

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

5. Sep 23, 2007

### 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.

6. Sep 23, 2007

### dextercioby

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

7. Sep 23, 2007

### 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?

8. Sep 23, 2007

### Hurkyl

Staff Emeritus
Why do you think, for all d, that gad is a constant with respect to the b, c, and d-th variables?

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.

Last edited: Sep 23, 2007
9. Sep 23, 2007

### ehrenfest

I see. You just replace d with a.

10. Sep 23, 2007

### 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?

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• ###### hobson page 66.jpg
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11. Sep 23, 2007

### Hurkyl

Staff Emeritus
Did you try differentiating it?