Quantum BTZ black hole

TL;DR

The paper develops a holographic construction of quantum rotating BTZ black holes (quBTZ) by solving exactly a four-dimensional AdS bulk with a black hole localized on a brane, thereby incorporating the full backreaction of strongly coupled conformal fields and higher-curvature corrections in the 3D effective theory. The authors derive the quantum-corrected geometry and the renormalized stress tensor, and they show that the generalized entropy $S_{gen}=A_{bulk}/(4G_{bulk})$ satisfies the first law $T dS_{gen}=dM-\Omega dJ$ exactly, while the Wald entropy on the brane does not, providing a nontrivial holographic consistency check. They map out multiple holographic branches of quBTZ, analyze static and rotating cases, compare to free-CFT results, and discuss the thermodynamics, stability, and entropy contributions from entanglement. The work offers insights into quantum gravity on braneworlds, the role of higher-curvature corrections, and the entanglement structure of quantum black holes, with implications for cosmic censorship, extended thermodynamics, and entanglement island scenarios.

Abstract

We study a holographic construction of quantum rotating BTZ black holes that incorporates the exact backreaction from strongly coupled quantum conformal fields. It is based on an exact four-dimensional solution for a black hole localized on a brane in AdS$_4$, first discussed some years ago but never fully investigated in this manner. Besides quantum CFT effects and their backreaction, we also investigate the role of higher-curvature corrections in the effective three-dimensional theory. We obtain the quantum-corrected geometry and the renormalized stress tensor. We show that the quantum black hole entropy, which includes the entanglement of the fields outside the horizon, satisfies the first law of thermodynamics exactly, even in the presence of backreaction and with higher-curvature corrections, while the Bekenstein-Hawking-Wald entropy does not. This result, which involves a rather non-trivial bulk calculation, shows the consistency of the holographic interpretation of braneworlds. We compare our renormalized stress tensor to results derived for free conformal fields, and for a previous holographic construction without backreaction effects, which is shown to be a limit of the solutions in this article.

Figures (6)

• Figure 1: Bulk geometry in a slice at constant $t$ and $\phi$. Left: C-metric coordinates $(x,r)$ in the spatial Poincaré disk of empty global AdS$_4$ ($\mu=0$, $\kappa=+1$). Lines of constant $x\in [-1,1]$ are blue arcs; lines of constant $r\in [-\infty,-\ell]\cup [0,\infty]$ are red arcs (full circles for $0<r\leq \ell$). The asymptotic boundary (black circle) is at $xr=-\ell$. The $\phi$ axis of rotation is $x=\pm 1$. Right: Sketch of braneworld construction with a black hole in it. The bulk is cut off at a brane at $x=0$ and only the (gray) region $0\leq x\leq x_1$ is retained; the root $x=x_1$ of $G(x)$ is now the $\phi$ axis. A second copy of this region, not shown, is glued at the brane to make a $\mathbb{Z}_2$-symmetric two-sided braneworld. A bulk black hole with event horizon at $r=r_+$ is attached to the brane. Dual three-dimensional fields satisfy transparent boundary conditions at the junction between the dynamical brane (thick blue) and the non-dynamical AdS$_4$ boundary (black).
• Figure 2: The holographic stress-energy function $F(M)$\ref{['FM']} for the three branches $1a$, $1b$ and 2, of quantum black hole solutions. We also include a branch $3$ of bulk BTZ black strings, which give BTZ on the brane with $F=0$ for all $M\geq 0$.
• Figure 3: The stress-energy function $F(M)$, \ref{['FMBTZ']} and \ref{['FMns']}, for a free conformal scalar.
• Figure 4: Temperature of the quantum black holes and the classical BTZ black hole for given mass $M$.
• Figure 5: Quantum and classical entropies of quantum black holes with given mass. The entropy of black holes in branch 3 is $S_\text{BTZ}$.