# Euclidean Methods for BTZ black Hole

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1. Aug 16, 2017

### craigthone

This is an exercise from Hartman's lecture 6th. Using the Euclidean method to calculate the BTZ black hole mass entropy. The BTZ metric is given by
$$ds^2=(r^2-8M)d\tau^2 +\frac{dr^2}{r^2-8M}+r^2d\phi^2$$
and $\tau \sim \tau+\beta, \beta=\frac{\pi}{\sqrt{2M}}$.
Then we calculate the Euclidean action
$$S_E=-\frac{1}{16π}\int_M \sqrt{g}(R+2)-\frac {1} {8π} \int_{\partial M} \sqrt { h } K+\frac {a} {8π} \int_{\partial M} \sqrt { h }$$

For the BTZ black hole solution, we can calculate
$$\sqrt{g}=r, \sqrt{h}=r \sqrt{r^2-8M}$$
$$n_{\alpha}=(r^2-8M)^{-1/2}\partial _{\alpha}r$$
$$R=-6, K=\frac{\sqrt{r^2-8M}}{r}+\frac{r}{\sqrt{r^2-8M}}$$
And then we have
$$-\frac{1}{16π}\int_M \sqrt{g}(R+2)=\frac{\beta}{4}r^2_0$$
$$-\frac {1} {8π} \int_{\partial M} \sqrt { h } K=-\frac{\beta}{4}[2r^2_0 -8M ]$$
$$\frac {a} {8π} \int_{\partial M} \sqrt { h }=a \frac{\beta}{4}[2r^2_0 -4M ]$$
where the boundary is at $r=r_0$
In order to cancel the divergent part of the action, we take $a=1$.
Then the Euclidean action is
$$S_E=\beta M=\frac{\pi^2}{2\beta}$$
The black hole energy is $$E=\frac{\partial}{\partial \beta}S_E=-M$$
This is awkard since we know that $E=M$ for the black hole.

Who can help me out. Thanks in advanced.

2. Aug 21, 2017

### PF_Help_Bot

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