Consider the AdS metric in D+1 dimensions(adsbygoogle = window.adsbygoogle || []).push({});

[tex]ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right)[/tex]

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

[tex]\Gamma^{t}_{tz}=\frac{L^{2}}{z^{3}}, \quad \Gamma^{x}_{xz}=\Gamma^{y}_{yz}=\Gamma^{z}_{zz}=-\frac{L^{2}}{z^{3}}[/tex]

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

[tex]R^{x}_{zxz}, \quad R^{y}_{zyz},\quad R^{t}_{ztz}[/tex]

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

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# Homework Help: Calculating Ricci Tensor

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