I Calculate Ricci Scalar & Cosm. Const of AdS-Schwarzschild Metric in d-Dimensions

[shinobi20]

TL;DR Summary

How do you calculate the Ricci scalar and cosmological constant of an AdS-Schwarzschild black hole in $d$-dimensions?

I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar.

Question. How do you calculate the Ricci scalar $R$ and cosmological constant $\Lambda$ of an AdS-Schwarzschild black hole metric in $d$-dimensions?

$ds^2 = \frac{L^2_{\rm{AdS}}}{z^2} \left( -f(z) dt^2 + \frac{dz^2}{f(z)} + \sum_{i=1}^d dx_i^2 \right)$

where $L_{\rm{AdS}}$ is the AdS radius.

I'm reading the article AdS CFT Duality User Guide by Makoto Natsuume and I'm just wondering how to find those quantities since there is a factor of $f(z)$ already present as opposed to the pure AdS case. The Riemann tensor and Ricci scalar for the maximally symmetric spaces are listed in p.98 of the article.

 

[haushofer]

Write down the metric components, calculate the christoffel symbols, and from there the Riemann tensor. See

https://en.wikipedia.org/wiki/Riemann_curvature_tensor

"Coordinate Expression".