# Hawking Temperature of the BTZ Black Hole

The metric of the BTZ Black Hole is given by $$ds^2 = - N^2 dt^2 + N^{-2} dr^2 +r^2(d\phi + N^\phi dt)^2$$ with $$N^2 = -M+ \frac{r^2}{l^2} + \frac{J^2}{4 r^2}, \ \ \ \ \ \ N^\phi = -\frac{J}{2r}$$ The $g_{rr}$ component of the metric is singular at points where $N^2=0$, yielding the horizons $r_\pm$ $$r_\pm = \sqrt{ \frac{Ml^2}{2}\left( 1 \pm \sqrt{1-\left(\frac{J}{Ml}\right)^2} \right)}$$ Then for these $r_\pm$, the metric component $g_{tt}$ does not vanish but becomes $$g_{tt} =\frac{J^2}{4r_\pm^2}$$ Now the perscription I learned to find a Hawking Temperature at a horizon (e.g. Schwarzschild BH) is you expand the Wick rotated metric ($t\to i\tau$) around the solution where $g_{\tau\tau}$ vanishes, find the metric is actually flat at this point, and impose $\tau$-periodicity such that there is no conical singularity at the horizon. This period is then the inverse Hawking temperature $T_H^{-1}$ .

I don't see any singularities in the $g_{\tau\tau}$ component right now so no conical singularity will appear, and I don't know how to interpret this. Does this mean there is no restriction on $T_H$ and the $\tau$ periodicity is free? Or is my way to calculate it somehow not applicable to BTZ horizons?

I think I found a coordinate transformation that shines more light on this. Instead of looking for a conical singularity in $g_{tt}$ alone I must look at conical singularities in any part of the metric (apart from $g_{rr}$) First of all the metric can be expressed in terms of $r_\pm$ and Wick rotated $t\to it_E$ such that $$ds_{E}^2 = \frac{ (r^2-r_+^2)(r^2+r_-^2)}{l^2r}dt_E^2 + \frac{l^2 r^2}{(r^2-r_+^2)(r^2+r_-^2)} dr^2 + r^2(d\phi + \frac{i r_+ r_-}{l r^2} dt_E)^2$$ which under the coordinate transformation $$t'_E = r_+ t_E + r_-\phi, \ \ \ \phi' = r_+ \phi - r_-t_E, \ \ \ r'^2= \frac{r^2-r_+^2}{r_+^2+r_-^2}$$ becomes $$ds_E^2 = \frac{r'^2}{l^2} dt_E'^2 +\frac{l^2}{1+r'^2} dr'^2 + (1+r'^2)d\phi'^2$$ which for $r\to r_+$ ($r'\to 0$) becomes $$ds_E^2 = r'^2 dt_E'^2+dr'^2 +d\phi'^2$$ which represents flat polar coordinates iff $t_E' \sim t_E' +2\pi$. Furthermore $\phi'$ is not periodic. So the periodicity of $t_E'$ is $\Delta t_E'=2\pi$, and of $\phi'$ is $\Delta \phi'=0$. Combining this yields $$2\pi = r_+ \Delta t_E + r_-\Delta \phi, \ \ \ \ 0 = r_+\Delta \phi - r_- \Delta t_E$$ So the time periodicity becomes $\beta =\Delta t_E = \frac{2\pi l r_+}{r_+^2+r_-^2}$ whilst also setting a fixed periodicity for $\phi$. Obviously the Hawking temperature is then $$T_H = \frac{r_+^2+r_-^2}{2\pi l r_+}$$
• Ah, I was not complete enough in my euclidean description but in fact it does. In euclidean coordinates the $r_-$ is still the value given above but purely imaginary. So when this BH is extremal @ $Ml=J$, the numerator cancels out. Is this vanishing of the hawking temperature something one always sees when taking a BH to exremal limits? Is there any intuition on why this is the case? – 123hoedjevan Jun 11 '15 at 0:00