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.
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.
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.