# The Banados-Teitelboim-Zanelli (BTZ) black hole The BTZ black hole is a solution of [[0002 3D gravity|3d Einstein gravity]]. In three dimensions, the metric is locally the same everywhere, so there cannot be a curvature singularity like in higher dimensions. Nevertheless, black holes can still exist because there can be causal horizons. The horizon here is a global feature of the spacetime. Consequently, it has many differences from the higher-dimensional cases. For example, there can be only one black hole in any asymptotic region, i.e., the event horizon as viewed from any asymptotic region is connected. Furthermore, any exterior region of any [[0002 3D gravity|3D black hole solution in AdS]] is identical to a BTZ solution. A rotating BTZ solution is given by $d s^2=-\frac{\left(r^2-r_{+}^2\right)\left(r^2-r_{-}^2\right)}{r^2} d t^2+\frac{r^2 d r^2}{\left(r^2-r_{+}^2\right)\left(r^2-r_{-}^2\right)}+r^2\left(d \phi-\frac{r_{+} r_{-}}{r^2} d t\right)^2,$where $\phi$ has periodicity $2\pi$. The mass, angular momentum, and left and right moving temperatures are given by$M=\frac{r_{+}^2+r_{-}^2}{8 G}, \quad J=\frac{r_{+} r_{-}}{4 G}, \quad T_L=\frac{r_{+}-r_{-}}{2 \pi}, \quad T_R=\frac{r_{+}+r_{-}}{2 \pi}.$Via a change of coordinates, $\begin{align}&r^2=r_{+}^2 \cosh ^2 \rho-r_{-}^2 \sinh ^2 \rho,\\ &T+X=\left(r_{+}-r_{-}\right)(t+\phi), \\ &T-X=\left(r_{+}+r_{-}\right)(t-\phi),\end{align}$this becomes$d s^2=-\sinh ^2 \rho d T^2+\cosh ^2 \rho d X^2+d \rho^2.$ The non-rotating case can be found be setting $r_-=0$, reducing the metric to$d s^2=-{\left(r^2-r_{+}^2\right)} d t^2+\frac{d r^2}{\left(r^2-r_{+}^2\right)}+r^2d \phi^2.$ ## Refs - [[BanadosTeitelboimZanelli1993]] - [[BanadosHenneauxTeitelboimZanelli1993]] - reviews - [[Rsc0004 CompereFiorucci Ch2 3d gravity]] - [[BricenoMartinezZanelli2021]][](https://arxiv.org/abs/2105.06488) ## The metric - $d s^{2}=-N^{2}(r) d t^{2}+\frac{d r^{2}}{N^{2}(r)}+r^{2}\left(d \phi+N^{\phi}(r) d t\right)^{2}$ - $N^{2}(r)=-8 M G+\frac{r^{2}}{\ell^{2}}+\frac{16 G^{2} J^{2}}{r^{2}}$ - $N^{\phi}(r)=-\frac{4 G J}{r^{2}}$ - recovers global AdS if $J=0$ and $M=-1/8G$. **Not when $M=0$**. ## Symmetry - only $\partial_\phi$ and $\partial_t$ - identification kills 4 of the 6 Killing fields in pure AdS ## Horizon - $r_{\pm}=\ell \sqrt{4 G M} \sqrt{1 \pm \sqrt{1-\left(\frac{J}{M \ell}\right)^{2}}}$ - for it to exist, need: $|J| \leqslant M \ell ; M>0$ ## Singularity - causal singularity: $\partial_\phi$ becomes timelike at $r=0$ ## Penrose - static ![[0086_static.png|150]] - non-extremal rotating ![[0086_rotating.png|150]] ## Obtain from identification - $\xi=\frac{r_{+}}{\ell} J_{12}-\frac{r_{-}}{\ell} J_{03}$ - $J_{ab}$ belongs to $so(2,2)$ - (see [[BanadosHenneauxTeitelboimZanelli1993]]) - need to first remove a region where $\xi^2\le 0$ => geodesically complete - becomes a feature of the black hole ## Entropy - [[1997#Strominger]]: BTZ entropy from [[0406 Cardy formula|Cardy formula]] ## Geodesics in closed form - review in [[BricenoMartinezZanelli2021]][](https://arxiv.org/abs/2105.06488) ## Overspinning ones - contain naked singularities, but are good classical geometries - see [[BricenoMartinezZanelli2021]] ## Phase transition - see [Hamanaka](http://www2.yukawa.kyoto-u.ac.jp/~masashi.hamanaka/kurita_y04.pdf) fig.6 (p.25)