Mass and entropy of asymptotically flat eternal quantum black holes in 2D

TL;DR

In this work, we analyze mass and thermodynamics of eternal quantum black holes in a 1+1D dilaton gravity setting, leveraging exact semi-classical solutions that interpolate between RST and BPP models and extending to a numerical treatment of semi-classical CGHS black holes. The authors develop a quasi-local formalism to define energy and entropy, showing the semi-classical Wald entropy coincides with the generalized entropy that includes the von Neumann entropy of quantum matter. They derive exact solutions for Hartle–Hawking and Boulware vacua, study energy conditions and singularity structure, and reveal thermally stable regimes as well as naked singularities in a Boulware background at large mass. Complementing analytics, they construct numerical quantum black holes in semi-classical CGHS, confirm the large-$| extphi_H|$ expansion as a good semiclassical approximation, and demonstrate that the mass–entropy relation differs from the RST limit in the fully quantum-corrected theory.

Abstract

Semi-classical dilaton gravity in (1+1)-dimensions remains one of the only arenas where quantum black holes can be exactly constructed, fully accounting for backreaction due to quantum matter. Here we provide a comprehensive analysis of the mass and thermodynamic properties of static asymptotically flat quantum black holes both analytically and numerically. First, we analytically investigate eternal quantum black hole solutions to a one-parameter family of analytically solvable models interpolating between Russo-Susskind-Thorlacius and Bose, Parker, and Peleg gravities. Examining these models in a semi-classically allowed parameter space, we find naked singularities may exist for quantum fields in the Boulware state. Using a quasi-local formalism, where we confine the black hole to a finite sized cavity, we derive the conserved energy and analyze the system's thermal behavior. Specifically, we show the semi-classical Wald entropy precisely equals the generalized entropy, accounting for both gravitational and fine grained matter entropies, and we find a range where the quantum black holes are thermally stable. Finally, we numerically construct eternal black hole solutions to semi-classical Callan-Giddings-Harvey-Strominger gravity and find their thermal behavior is qualitatively different from their analytic counterparts. In the process, we develop an analytic expansion of the solutions and find it accurately approximates the full numerical solutions in the semi-classical limit.

Figures (10)

• Figure 1: Penrose diagram for a static two-dimensional dilatonic black hole.
• Figure 2: NEC for the Hartle-Hawking (left) and Boulware (right) vacua as a function of $\lambda x^-$ for $N=100$, $M/\lambda=1000$ and $a=1/2$ at $\lambda x^+=1$.
• Figure 3: Critical mass $M^{\ast}/\lambda$. Left. Hartle-Hawking; $N=100,200,300$ (top, blue; middle orange; bottom black). Right. Boulware; $N=100,200,300$ (bottom left, blue; middle left; top left, black).
• Figure 4: $\phi^{(\text{B})}(x)$ as a function of $x$ for $N=100$, $a=1/2$ and masses $M/\lambda=1000$ and $M^{(*,\text{B})}/\lambda$.
• Figure 5: Temperature $T$ as a function of $A_B$ based on the lower bound \ref{['eq:scvalidtemp']} (regions in red) and the upper bound \ref{['eq:heatcapvalid']} (regions in blue). Overlapping regions indicate the parameter space where the quantum black holes are thermally stable and are semi-classical valid. Here $a=1/2$ (RST), $N=120$ and $T_H/\lambda=1$.