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

I am wondering if there is a different counter-term that I need to add to the AdS Schwarzschild case.In summary, the conversation discusses the attempt to calculate the on shell action for an AdS Schwarzschild black hole. The person mentions trying to use a counter-term similar to the one used for the flat Schwarzschild black hole, but it did not result in a finite answer. They are asking if there is a different counter-term that should be used for the AdS case.
  • #1
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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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  • #2
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.
 

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

1. What is an AdS black hole on-shell action?

An AdS black hole on-shell action is a mathematical formula that describes the dynamics of a black hole in Anti-de Sitter (AdS) space. It is a key concept in the study of black holes and is used to understand the thermodynamics and behavior of these objects.

2. How is the AdS black hole on-shell action related to the holographic principle?

The AdS black hole on-shell action is closely related to the holographic principle, which states that the information contained in a region of space can be described by the information on the boundary of that region. The on-shell action is used to calculate the entropy of a black hole, which is related to the amount of information it contains.

3. How is the AdS black hole on-shell action calculated?

The AdS black hole on-shell action is calculated using a mathematical technique called the holographic renormalization. This involves mapping the AdS black hole solution onto a simpler, asymptotically AdS space, and then calculating the action using this simplified space.

4. What is the significance of the AdS black hole on-shell action in string theory?

In string theory, the AdS black hole on-shell action is used to study the AdS/CFT correspondence, which relates a theory of gravity in AdS space to a theory of quantum field theory on its boundary. This allows for a better understanding of the nature of black holes and their behavior.

5. Can the AdS black hole on-shell action be used to study other types of black holes?

Yes, the concept of the AdS black hole on-shell action has been extended to other types of black holes, such as charged black holes and rotating black holes. It has also been used to study the thermodynamics and behavior of these black holes in various dimensions and theories of gravity.

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