# Christoffel Symbol

1. May 20, 2014

### darida

Ansatz metric of the four dimensional spacetime:

$ds^2=a^2 g_{ab}dx^a dx^b - du^2$

where:

$a,b=0,1,2$

$a(u)=$warped factor

Christoffel symbol of a three dimensional AdS spacetime:

$\Gamma^{c}_{ab}= \frac{1}{2} g^{cd}(∂_b g_{da} + ∂_a g_{bd} - ∂_d g_{ba})$

Now how to find $\Gamma^{a}_{b}$?

2. May 20, 2014

### HallsofIvy

Staff Emeritus
First, what are the $g_{ab}$? Are they functions of the "x"s only or also of u? Clearly the Christoffel symbols depend on exactly how they depend on the coordinates. Second, what do you mean by "a(u)"? Is it that the "a" in "$a^2$"? You are already using "a" as an index. Surely "a(u)" is not an index so it would be better to use a different symbol.

3. May 21, 2014

### darida

Ok I fix them:

4. May 21, 2014

### George Jones

Staff Emeritus
What does $\Gamma^{i}_{j}$ mean, i.e., what is a $\Gamma$ with two indices?

5. May 21, 2014

### darida

I don't know that's why I asked

*edit:

Well, one said that

$\Gamma^{\rho}_{\mu\nu} = ........\Gamma^{i}_{j}$

$R_{\mu\nu} = ........R_{ij}$

where:

$\mu, \nu, \rho = (D + 1)$ dimensional curved spacetime indices

$R_{ij} = \Lambda_{D} g_{ij}$

$\Lambda_{D} =$ cosmological constant

Last edited: May 21, 2014
6. May 21, 2014

### George Jones

Staff Emeritus
Can you give specific references? In what articles or books have you seen this notation?

7. May 21, 2014

### darida

It's not in any articles/books. I just met someone who told me that :(

8. May 21, 2014

### Matterwave

Just giving my two cents, but I have never seen, in my study of differential geometry, a Christoffel symbol with 2 indices. The closest I can think of would be the the connection one-forms which are a set of 6 one-forms, and so they are sometimes written with two (anti-symmetric) indices...but...that's usually written in a way different notation.

9. May 22, 2014

Oh okay