How to Calculate the On-Shell Action for an AdS Schwarzschild Black Hole?

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SUMMARY

The discussion focuses on calculating the on-shell action for an AdS Schwarzschild black hole. The user attempts to apply a Gibbons-Hawking boundary term, similar to that used for flat Minkowski spacetime, but encounters issues with obtaining a finite result as the limit approaches infinity. The need for an appropriate counter-term to achieve a finite answer is emphasized, indicating that the standard approach may not suffice for this specific black hole geometry.

PREREQUISITES
  • Understanding of AdS (Anti-de Sitter) space and Schwarzschild black holes
  • Familiarity with Gibbons-Hawking boundary terms
  • Knowledge of on-shell action in the context of general relativity
  • Basic concepts of limits in mathematical analysis
NEXT STEPS
  • Research the appropriate counter-terms for AdS black holes
  • Study the derivation of the Gibbons-Hawking boundary term in detail
  • Explore the implications of different geometries on the on-shell action
  • Learn about regularization techniques in quantum field theory
USEFUL FOR

The discussion is beneficial for theoretical physicists, particularly those specializing in general relativity, quantum gravity, and black hole thermodynamics.

Jesus
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Homework Statement


I want to calculate the on shell action for the case of a AdS Schwarzschild black hole.

Homework Equations


Following the case of the flat Schwarzschild black hole I tried to add a counter-term of the type of a Gibbons-Hawking boundary term for flat Minkowski spacetime, however this doesn't seem to lead to a finite answer when I take the limit r to infinity.

Is there a method to find out what is the right counter-term that I need to add?
 
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The Attempt at a SolutionI have tried to use the same method as for the flat Schwarzschild black hole, i.e. adding a counter-term of the type of a Gibbons-Hawking boundary term for flat Minkowski spacetime. However this doesn't seem to lead to a finite answer when I take the limit r to infinity.
 

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