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shinobi20

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- 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.

##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.

**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.