# Derivation of Hawking temperature

• gentsagree
In summary, the Hawking temperature for a black hole can be obtained by compactifying the timelike dimension and identifying the time coordinate of the euclidean metric as a periodic coordinate. This can also be derived from the euclideanised AdS Schwarzschild metric, with the result in AdS4 being β=4πb^(2)r+/(b^(2)+3r+^(2)) and can be generalized to higher dimensions. There are various methods to derive this result, such as linearizing the metric element V and solving for the differential equations.
gentsagree
Hi, my first thread. The Hawking temperature for a BH can be derived through compactifying the timelike dimension and hence identifying the time coordinate of the euclidean metric such as a periodic coordinate τ with period β.

Now, very interestingly in the context of AdS/CFT correspondence, it is useful to derive said periodicity from the euclideanised AdS Schwarzschild metric, i.e.

ds^2=Vdτ^(2)+V^(-1)dr^(2)+r^(2)dΩ

where V=1-2M/(m^(2)r)+r^(2)/b^(2).

The result in AdS4 is given by β=4πb^(2)r+/(b^(2)+3r+^(2))
which is easily generalised to higher dimensions.

Anyone knows how to derive the above result from the metric?

I tried to linearise the metric element V, to change to Rindler coordinates and solving for the differential equations, but I only get a "close enough" answer. Perhaps I need a different linearisation of V.

Thanks

nevermind, found the solution

## 1. What is Hawking temperature?

Hawking temperature is the temperature associated with the event horizon of a black hole. It is named after the renowned physicist Stephen Hawking, who first proposed its existence in 1974.

## 2. How is Hawking temperature derived?

Hawking temperature is derived using quantum field theory in curved spacetime. Specifically, it is calculated by analyzing the quantum fluctuations near the event horizon of a black hole.

## 3. What does Hawking temperature tell us about black holes?

Hawking temperature provides insight into the thermodynamics of black holes. It suggests that black holes have a temperature and therefore, can emit thermal radiation, known as Hawking radiation. This contradicts the classical understanding of black holes as objects that do not emit any form of radiation.

## 4. Are there any limitations to the derivation of Hawking temperature?

Yes, there are limitations to the derivation of Hawking temperature. It is based on certain assumptions and approximations, such as the existence of a quantum field theory in curved spacetime and the absence of other matter or fields near the black hole.

## 5. What implications does Hawking temperature have for our understanding of the universe?

Hawking temperature has significant implications for our understanding of the universe, as it challenges traditional notions of black holes and their behavior. It also provides a possible avenue for reconciling quantum mechanics and general relativity, two fundamental theories of physics that have been difficult to unify.

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