Christoffel Symbol Ansatz for 4D Spacetime

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

 

[HallsofIvy]

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.

 

[darida]

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 spatial 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]

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

 

[darida]

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

 

[George Jones]

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

 

[darida]

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

 

[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.

 

[darida]

Oh okay