# How to calculate the Hawking temperature of a "constant curvature black hole" by the Euclidean method?

+ 4 like - 0 dislike
593 views

The constant curvature black hole (Bañados black hole) in $AdS_{d+1}$ is

$${\small ds^2=\frac{H^2}{\rho_+^2}(\rho^2-\rho_+^2)\left(-(1-H^2r^2)dt^2+\frac{dr^2}{1-H^2r^2}+r^2d\Omega_{d-2}^2\right)+\frac{d\rho^2}{\rho^2-\rho_+^2}+\rho^2d\phi^2.}$$

The AdS boundary is $dS_{d-1}\times S^1$. More information about this geometry can be found in [1,2]. How to calculate its temperature at the horizon $\rho=\rho_+$ by the Euclidean method? We can see that $g_{tt}$ depends on both coordinates $\rho$ and $r$, but its temperature should be constant.

The following example shows the Euclidean method to calculate the Hawking temperature of a black hole described by the metric

$$ds^2=-f(r)dt^2+\frac{dr^2}{g(r)}+r^2d\Omega_{2}^2,$$

where $f=f_0(r-r_h)+\cdots$, and $g=g_0(r-r_h)+\cdots$ near the horizon. Write the metric near the horizon as

$$ds^2=\kappa^2\rho^2d\tau^2+d\rho^2+\cdots.$$

To avoid conical singularity, the period of $\kappa\tau$ must be $2\pi$. The temperature is the inverse of the period of $\tau$:

$$T=\frac{\kappa}{2\pi}=\frac{\sqrt{f_0g_0}}{4\pi}.$$​

Another way to calculate the temperature is by the following formula for surface gravity:

$$\kappa^2=-\frac{1}{2}(\nabla^a\xi^b)(\nabla_a\xi_b),$$

where $\xi^a=(\partial_t)^a$. For the constant curvature black hole described above, we have

$$\kappa|_{\rho=\rho_+}=\kappa|_{r=1/H}=H.$$

So $T=H/2\pi.$ But I want to know whether the Euclidean method still works for the constant curvature black hole.

[1] M. Bañados, Constant Curvature Black Holes, gr-qc/9703040.
[2] D. Marolf, M. Rangamani, and M.V. Raamsdonk, Holographic models of de Sitter QFTs, arXiv:1007.3996.

edited Oct 25, 2017

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.