Gedanken experiments to destroy a BTZ black hole

We consider Gedanken experiments to destroy an extremal or near-extremal Banados-Teitelboim-Zanelli (BTZ) black hole by throwing matter into the horizon. These black holes are vacuum solutions to ($2+1$)-dimensional gravity theories, and are asymptotically ${\mathrm{AdS}}_3$. Provided the null energy condition for the falling matter, we prove the following—(i) in a Mielke-Baekler model without ghost fields, when torsion is present, an extremal BTZ black hole can be overspun and becomes a naked conical singularity; (ii) in three-dimensional Einstein gravity and chiral gravity, which both live in the torsionless limits of the Mielke-Baekler model, an extremal BTZ black hole cannot be overspun; and (iii) in both Einstein gravity and chiral gravity, a near-extremal BTZ black hole cannot be overspun, leaving the weak cosmic censorship preserved. To obtain these results, we follow the analysis of Sorce and Wald on their Gedanken experiments to destroy a Kerr-Newman black hole, and calculate the second order variation of the black hole mechanics. Furthermore, Wald’s type of Gedanken experiments provide an operational procedure of proving the third law of black hole dynamics. Through the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, our results on BTZ black holes also indicate that a third law of thermodynamics holds for the holographic conformal field theories dual to three-dimensional Einstein gravity and chiral gravity.

Figure 1

Carter-Penrose diagram of an extremal BTZ black hole. The shaded region consists of the falling matter which all goes into the black hole. The perturbation $\delta \varphi$ vanishes in a neighborhood of ${\mathrm{\Sigma }}_0\cap \mathcal{H}$.

Figure 2

Carter-Penrose diagram of a near-extremal BTZ black hole. The shaded region consists of the falling matter which all goes into the black hole. The perturbation $\delta \varphi$ and ${\delta }^2\varphi$ vanishes in a neighborhood of $B$.

Figure 3

The parameter space of BTZ black holes in chiral gravity. The black solid line corresponds to extremal BTZ black holes. Any point above this line corresponds to a nonextremal BTZ black hole, while any point below the line is a naked conical singularity. The orange dashed line is one of the lines of constant entropy, which is parallel to the line for extremal BTZ black holes. Starting with some point on the constant entropy line, any tangent vector will always be parallel to the extremal BTZ line. That is, there is no Hubeny-type violation that can overspin a near-extremal BTZ black hole in chiral gravity.

Figure 4

The parameter space of BTZ solutions in the three-dimensional Einstein gravity. The black solid line corresponds to extremal BTZ black holes. Any point above this line corresponds to a nonextremal BTZ black hole, while any point below the line is a naked conical singularity. The orange dashed curve is one of the curves of constant entropy, which meets the extremal BTZ line tangentially. The tangent vector at the point of an extremal BTZ black hole will always bring it to another extremal BTZ black hole. However, starting from a slightly nonextremal BTZ black hole, to linear order, the tangent vector can perturb the spacetime to become a naked conical singularity.