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

shinobi20
Messages
277
Reaction score
20
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.
 
Physics news on Phys.org
OK, so this has bugged me for a while about the equivalence principle and the black hole information paradox. If black holes "evaporate" via Hawking radiation, then they cannot exist forever. So, from my external perspective, watching the person fall in, they slow down, freeze, and redshift to "nothing," but never cross the event horizon. Does the equivalence principle say my perspective is valid? If it does, is it possible that that person really never crossed the event horizon? The...
From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
Back
Top