The BTZ Black Hole

The BTZ black hole has horizons and conserved mass and angular momentum even though every regular point is locally anti-de Sitter space. Three-dimensional vacuum Einstein gravity has no local Weyl curvature; the black hole is created by a discrete global identification. It is therefore the canonical example in which local curvature invariants cannot distinguish a black hole from its maximally symmetric covering space.

Required background. The Einstein Equation supplies the three-dimensional $\Lambda$-vacuum equation.

Helpful background. The Riemann Curvature Tensor supplies constant-curvature identities; Exact Solutions of Einstein’s Equations places the family; Global Structure develops quotient horizons and causal boundaries; and Killing Fields and Isometry Groups supplies the quotient generator.

Metric, charges, and horizons

Take $\Lambda=-1/\ell^2$ and $\phi\sim\phi+2\pi$. In a normalization where $M$ and $J$ are the physical conserved charges, the rotating BTZ metric is

$$ds^2=-N^2dt^2+\frac{dr^2}{N^2} +r^2(d\phi+N^\phi dt)^2,$$ $$N^2=-8G_3M+\frac{r^2}{\ell^2} +\frac{16G_3^2J^2}{r^2}, \qquad N^\phi=-\frac{4G_3J}{r^2}.$$

The horizon radii satisfy

$$r_\pm^2=4G_3\ell \left(M\ell\pm\sqrt{M^2\ell^2-J^2}\right).$$

A black hole with real horizons requires $M\ge0$ and $\lvert J\rvert\le M\ell$. The extremal case has $\lvert J\rvert=M\ell$ and $r_+=r_-$. For $J=0$,

$$r_+^2=8G_3M\ell^2, \qquad r_-=0.$$

The horizon generator is

$$\chi=\partial_t+\Omega_H\partial_\phi, \qquad \Omega_H=-N^\phi(r_+)=\frac{4G_3J}{r_+^2}.$$

Alternative literature conventions absorb $8G_3$ and $4G_3$ into dimensionless parameters also called $M$ and $J$. The formula for $N^2$ must be stated before comparing horizon or charge equations.

Locally AdS₃, globally a black hole

The vacuum equation in three dimensions gives

$$R_{ab}=-\frac{2}{\ell^2}g_{ab}.$$

Because the three-dimensional Weyl tensor vanishes identically, the Riemann tensor is fixed algebraically:

$$R_{abcd}=-\frac1{\ell^2} (g_{ac}g_{bd}-g_{ad}g_{bc}).$$

Consequently

$$R=-\frac6{\ell^2}, \qquad R_{ab}R^{ab}=\frac{12}{\ell^4}, \qquad R_{abcd}R^{abcd}=\frac{12}{\ell^4},$$

independent of $M$ and $J$. There is no curvature blow-up at $r=0$ of the kind found in Schwarzschild. The excluded or singular set is instead associated with the quotient identification and the onset of causally pathological orbits.

The quotient can be constructed explicitly. Embed AdS$_3$ as

$$-v^2-u^2+x^2+y^2=-\ell^2$$

in $\mathbb R^{2,2}$ with metric $-dv^2-du^2+dx^2+dy^2$, and then pass to the universal cover in the timelike direction. For a nonextremal hole, $r_+>r_-$, choose $J\geq0$ and $r_-\geq0$ for the representative below. The $J<0$ geometry follows by $\phi\mapsto-\phi$, equivalently by reversing the sign of the $J_{03}$ coefficient. The lifted AdS Killing field is

$$\xi_{\mathrm{ne}}=\frac{r_+}{\ell}J_{12} -\frac{r_-}{\ell}J_{03},$$

where

$$J_{12}=x\,\partial_u+u\,\partial_x, \qquad J_{03}=y\,\partial_v+v\,\partial_y.$$

The cyclic quotient identifies every point $p$ with

$$p\sim \exp(2\pi n\xi_{\mathrm{ne}})\,p, \qquad n\in\mathbb Z.$$

In adapted coordinates $\xi_{\mathrm{ne}}=\partial_\phi$ and

$$\xi_{\mathrm{ne}}^2 =\frac{r_+^2-r_-^2}{\ell^2}(u^2-x^2)+r_-^2 =r^2,$$

so the group action reduces to $\phi\sim\phi+2\pi$. Form the quotient only on an invariant connected region with $\xi_{\mathrm{ne}}^2>0$. The region with $\xi_{\mathrm{ne}}^2\le0$ is excluded: identification there would close null or timelike generator orbits. The causal-singular quotient boundary is therefore $r=0$, not the event horizon. The event horizon $r=r_+$ lies inside the allowed region. For rotating nonextremal holes $r_->0$, the inner horizon $r=r_-$ also lies there, with quotient-generator norm $r_-^2$. When $r_-=0$, only $r_+$ is a horizon; the formal zero root cancels from $N^2$ and coincides with the excluded $r=0$ boundary. In this nonrotating case the boost generator also has a fixed line on that boundary, producing the additional failure of proper discontinuity and manifold structure described in Appendix B of the source below.

The two-boost expression above is a convenient nonextremal representative, not a formula to continue by setting both roots equal. A representative that retains the extremal and massless sectors before the nonextremal conjugation is

$$\Xi=\frac{r_+}{\ell}J_{12} -\frac{r_-}{\ell}J_{03} -J_{13}+J_{23},$$

with

$$J_{13}=y\,\partial_u+u\,\partial_y, \qquad J_{23}=y\,\partial_x-x\,\partial_y.$$

For $r_+>r_-$, $\Xi$ is related to $\xi_{\mathrm{ne}}$ by an $SO(2,2)$ isometry. At $r_+=r_->0$ its conjugacy type changes, and at $r_+=r_-=0$ it remains a nonzero parabolic generator. Naively inserting zero roots into $\xi_{\mathrm{ne}}$ would therefore erase the massless BTZ quotient. The null boundaries separating exterior, between-horizon, and inner coordinate regions become the BTZ horizons; they are not curvature singularities or fixed points of the quotient. Bañados et al. give the embedding, generators, all-sector and nonextremal representatives, BTZ coordinates, causal excision, and the nonrotating fixed-set subtlety in Bañados et al. 1993, §§3.1.1–3.2.5 and Appendix B, Eqs. (3.1)–(3.2), (3.8), (3.13)–(3.19), and (3.23)–(3.39), Open HTML.

Bañados, Teitelboim, and Zanelli introduced the solution and its charge interpretation in Bañados, Teitelboim, and Zanelli 1992, pp. 1849–1851; its quotient geometry and maximal extension are developed in Bañados et al. 1993, pp. 1506–1525.

Vacuum choices and limiting cases

The state $M=J=0$ is the massless BTZ geometry, not global AdS$_3$. In the charge normalization used here, global AdS$_3$ has

$$M=-\frac{1}{8G_3}, \qquad J=0.$$

This finite offset reflects the chosen asymptotic vacuum. Forgetting it leads to inconsistent comparisons of energy and horizon thresholds.

For $M>0$, $J=0$, the exterior $r>r_+$ is static. Rotation adds an ergoregion because $g_{tt}=8G_3M-r^2/\ell^2$ changes sign outside or at the horizon, while the generator $\chi$ remains null at $r_+$. In the extremal limit the two roots merge, and a near-horizon scaling is required to retain the throat.

BTZ is not an asymptotically flat three-dimensional black hole. Negative cosmological constant and the quotient boundary conditions are essential. Nor do constant curvature and absence of local graviton degrees of freedom make global dynamics trivial: boundary excitations, matter perturbations, and topology still carry information.

The angular identification is indispensable in another practical way. If $\phi$ is unwrapped, the resulting region belongs to the AdS$_3$ covering geometry and the closed horizon section disappears; if its period is changed, the asymptotic charge normalization changes with it. Thus the local equation $R_{ab}=-2g_{ab}/\ell^2$ cannot determine $M$ and $J$. Those charges compare the quotient geometry with a declared asymptotic reference background and a declared normalization of $\partial_t$ and $\partial_\phi$.

Common pitfalls

Searching for a curvature singularity at the horizon. All local polynomial curvature invariants are constant. Horizons and quotient singularities are global causal and identification phenomena.

Calling $M=0$ the AdS vacuum. In the physical-charge convention above, global AdS has $M=-1/(8G_3)$. Always state the reference background.

Exercises

Factor $N^2$ in terms of $r_\pm$ and recover the angular velocity in root variables.

Solution

The sums and products of the roots give

$$r_+^2+r_-^2=8G_3M\ell^2, \qquad r_+r_-=4G_3\ell\lvert J\rvert.$$

Therefore

$$N^2=\frac{(r^2-r_+^2)(r^2-r_-^2)}{\ell^2r^2}.$$

For a chosen sign of $J$,

$$\Omega_H=\frac{4G_3J}{r_+^2} =\operatorname{sgn}(J)\frac{r_-}{\ell r_+}.$$

The bound $\lvert\Omega_H\rvert\le1/\ell$ follows from $r_-\le r_+$.

References

• Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli, “Black Hole in Three-Dimensional Spacetime,” Physical Review Letters 69, 1849–1851 (1992). DOI.
• Máximo Bañados, Marc Henneaux, Claudio Teitelboim, and Jorge Zanelli, “Geometry of the 2+1 Black Hole,” Physical Review D 48, 1506–1525 (1993), with erratum Physical Review D 88, 069902 (2013). Article DOI; erratum DOI.