Black hole temperature in an asymptotically de Sitter spacetime

Question

I am trying to calculate the Hawking temperature of a Schwarzschild black hole in a spacetime which is asymptotically dS. Ignoring the 2-sphere, the metric is given by $ds^2=\left(1-\frac{2M}{r}-\frac{r^2}{L^2}\right)d\tau^2+\left(1-\frac{2M}{r}-\frac{r^2}{L^2}\right)^{-1}dr^2$

where $\tau=it$ the Euclidean time and $L^2=\frac{3}{\Lambda}$.

In asymptotically flat space ($\Lambda=0$), one has to require that $\tau$ be periodic in the inverse temperature $\beta$ in order to prevent a conical singularity at the event horizon, from which the temperature follows.

In asymptotically de Sitter spacetime however, there are 2 positive roots of $g_{\tau\tau}$: the event horizon of the black hole $r_h$, but also the cosmological horizon $r_c>r_h$. As in the flat case, we can deduce the period of $\tau$ that is needed to prevent a conical singularity at $r_h$, but then we're still left with a conical singularity at $r_c$. Similarly we could make $\tau$ periodic in a way such that the conical singularity at $r_c$ disappears.

However, we cannot make both conical singularities disappear! Then how can we derive the black hole's Hawking temperature in this case? Should I just ignore the singularity at the cosmological horizon? Or should I use different coordinate patches?

Answer 1

See

L. Rodriguez and T. Yildirim, Class. Quantum Grav. 27, 155003 (2010), arXiv:1003.0026.

Section 2.3 has the Schwarzschild-dS calculation.

Lets define $f(r)=1-\frac{2M}{r}-\frac{r^2}{L^2}$

Radius of the horizon is given by the largest real root of f(r)=0

But of course $L\rightarrow \infty$ is still important. Once you obtain the energy momentum tensor for the fields near the horizon, you need to force Unruh boundary conditions which includes taking the limit $L\rightarrow \infty$.

In the light cone coordinates,

$T_{++}=0$ for $r\rightarrow \infty$, $L\rightarrow \infty$

$T_{--}=0$ for $r\rightarrow r_+$

This fixes the integration constants. The anomaly is cancelled by the Hawking flux

$\langle T_{++}=0 \rangle= \frac{\pi}{12}T_H^2$

where $T_H$ is the Hawking temperature.