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I'm trying to derive the thermodynamics of the black hole solution presented at A New Class of Exact Hairy Black Hole Solutions by Kolyvaris et al.

I'm asking for resource recommendations on this matter, $3+1$ decomposition, boundary terms etc. Is there any paper that derives black hole thermodynamics, using the Euclidean method, for a simpler solution and describes the procedure in detail?

An answer presenting the calculation and procedure in detail is also appreciated!

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  • $\begingroup$ I will point out that resource recommendations cannot usually be mixed with an actual physics question, as per site policy. $\endgroup$ Jun 24, 2021 at 13:46
  • $\begingroup$ @NiharKarve should i then remove the resourse reccomendation tag? $\endgroup$
    – Noone
    Jun 24, 2021 at 13:47
  • $\begingroup$ I'm not sure: your post won't automatically become non-community wiki. Perhaps you can remove the res-rec tag, make your request for resources a supplementary part of the actual question, and then a mod can make the Q&A non-community wiki. That way you won't invalidate existing answer(s) either. $\endgroup$ Jun 24, 2021 at 13:53
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    $\begingroup$ @NiharKarve Thank you! I'll wait for a moderator to tell me what to do to be sure! $\endgroup$
    – Noone
    Jun 24, 2021 at 13:58
  • $\begingroup$ In that paper in your post, the conserved charged has been computed using the Hamiltonian approach (In this method the on-shell Euclidean action is evaluated by use of the Euclidean continuation of the action in Hamiltonian form, instead of the action in Lagrangian form). The approach explained in the answer is called the subtraction method which works with the Euclidean action in Lagrangian form. Do you want to learn the method in that paper or do you just want to calculate Euclidean action in any way? There are several approaches for this purpose. It is better to clarify the question. $\endgroup$
    – SG8
    Jun 24, 2021 at 15:00

1 Answer 1

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As a fundamental tool in black hole thermodynamics, the Euclidean method has been applied to both complicated and simple solutions. The sources I would read are the following.

  1. Thermodynamics of black holes in anti-de Sitter space by Hawking and Page. This is the paper which first developed the method and it is very readable. The calculation starts in section 2 and one cannot help but notice how streamlined it is compared to Hawking's earlier tour de force which derived evaporation for black holes in Minkowski space.
  2. Anti de-Sitter space, thermal phase transition and confinement in gauge theories by Witten. This starts off by reviewing the Hawking Page calculation for AdS black holes and generalizing it to $n$ dimensions. Later on, we get to see it applied in a situation which still receives much attention, obtained by compactiying AdS on a circle in order to give the dual gauge theory a scale.
  3. Previous SE answers from 2013 and 2017.

As discussed in the latter, a major point of the derivation should already be familiar if we think about converting between polar and Cartesian co-ordinates for Euclidean space. \begin{equation} ds^2 = dr^2 + r^2 d\theta^2 \to ds^2 = dx^2 + dy^2 \end{equation} The polar form shows a vanishing component at $r = 0$. This is shown to be merely a co-ordinate singularity IF we can write it in the Cartesian form. However, writing it in the Cartesian form is only valid if $\theta$ is $2\pi$ periodic because we need to be able to set \begin{equation} (x, y) = (r\cos \theta, r\sin \theta). \end{equation} In other words, \begin{equation} ds^2 = dr^2 + r^2 d\theta^2, \;\;\; \theta \in [0, \alpha] \end{equation} is a different manifold from Euclidean space when $\alpha \neq 2\pi$. It is situations like these which show us that the Einstein Hilbert action is a functional which acts on manifolds. Writing it as something which acts on the metric alone is abuse of notation. This is similar to how tori $\mathbb{T}^2$ with different complex structures $z \sim z + 2\pi \tau$ are also different manifolds. Because even though tilting the axes is innocuous in $\mathbb{R}^2$, it becomes meaningful when there are periodicities involved.

Let's see how this works for the Euclidean Schwarzschild metric \begin{equation} ds^2 = \left ( 1 - \frac{2M}{r} \right )d\tau^2 + \left ( 1 - \frac{2M}{r} \right )^{-1} dr^2 \end{equation} by forcing it to pick up a simple $d\rho^2$ term in a new co-ordinate system. This means we need to solve \begin{equation} dr = \sqrt{1 - \frac{2M}{r}} d\rho \end{equation} which is separable. The solution is \begin{equation} \rho = \mathrm{const} + \sqrt{8M(r - 2M)} + O(r - 2M). \end{equation} We can now plug this into the metric and again drop higher powers of $r - 2M$ to find \begin{equation} ds^2 = d\rho^2 + (\rho / 4M)^2 d\tau^2 \end{equation} near the horizon. Then in precise analogy with the polar co-ordinate example above, $\tau / 4M$ needs to have period $2\pi$. This implies that the inverse temperature (the period for $\tau$ itself) is given by \begin{equation} \beta = 8\pi M \end{equation} which is the well known result.

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