# Calculating Ricci Tensor

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1. Oct 14, 2017

### HamOnRye

Consider the AdS metric in D+1 dimensions
$$ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right)$$
I wanted to calculate the Ricci tensor for this metric for D=3. ([\eta_{\mu\nu} is the Minkowski metric in D dimensions)
I have found the following Christoffel symbols
$$\Gamma^{t}_{tz}=\frac{L^{2}}{z^{3}}, \quad \Gamma^{x}_{xz}=\Gamma^{y}_{yz}=\Gamma^{z}_{zz}=-\frac{L^{2}}{z^{3}}$$
From this point I wanted to determine the Riemann tensor in order to finally determine the Ricci tensor.
What I've got the following contributing Riemann tensors
$$R^{x}_{zxz}, \quad R^{y}_{zyz},\quad R^{t}_{ztz}$$
I also noticed that if I have a z-coordinate in the upper index for the Riemann tensor it will be zero no matter what I choose for the lower indices.
My problem is as follows, based on symmetry, the above Riemann tensors should also be zero but I can't see how. Did I make a mistake with my Christoffel symbols or anywhere else?
Any help is appreciated!

Tim

2. Oct 20, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Nov 24, 2017

### egourgoulhon

It seems that you made an error in the computation of the Christoffel symbols. They should be either zero or equal to +/- 1/z. See this CoCalc worksheet for the computation, as well as the expression of the Riemann and Ricci tensors.