# Christoffel Symbol

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 $\Gamma^{a}_{b}$?

HallsofIvy
Homework Helper
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.

1 person
Ok I fix them:
Ansatz metric of $D+1$ dimensional spacetime:

$ds^2=a^2 g_{ij}dx^{i} dx^{j} + du^2$

where:

Signature: $- + + +$

Metric $g_{ij} \equiv g_{ij} (x^{i})$ describes $D$ dimensional AdS spacetime

$i,j = 0,1,...,(D-1) = D$ dimensional curved spacetime indices

$a(u) =$ warped factor

$u = x^{D}$

$D =$ number of spacial dimensional

Christoffel symbol of $D$ dimensional AdS spacetime:

$\Gamma^{k}_{ij}= \frac{1}{2} g^{kl}(∂_{j} g_{li} + ∂_{i} g_{jl} - ∂_{l} g_{ji})$

Now how to find $\Gamma^{i}_{j}$?

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

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

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:
George Jones
Staff Emeritus
Gold Member
Can you give specific references? In what articles or books have you seen this notation?

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

Matterwave