Signatures of quantum geometry from exponential corrections to the black hole entropy

TL;DR

This paper explores how exponential corrections to black hole entropy reveal signatures of quantum geometry, reconstructs related spacetime geometries, and connects these findings to quantum Einstein equations and black hole thermodynamics.

Contribution

It introduces a novel method to derive black hole spacetime geometry from entropy corrections and links quantum corrections to Einstein's equations and thermodynamics.

Findings

Reconstructed black hole geometries resemble noncommutative-inspired Schwarzschild black holes.

Derived matter density functions are similar to those in noncommutative models.

Established a connection between quantum Einstein equations and black hole thermodynamics.

Abstract

It has been recently shown in [Phys. Rev. Lett. 125 (2020) 041302] that microstate counting carried out for quantum states residing on the horizon of a black hole leads to a correction of the form $\exp(-A/4l_p^2)$ in the Bekenstein-Hawking form of the black hole entropy. In this paper, we develop a novel approach to obtain the possible form of the spacetime geometry from the entropy of the black hole for a given horizon radius. The uniqueness of this solution for a given energy-momentum tensor has also been discussed. Remarkably, the black hole geometry reconstructed has striking similarities to that of noncommutative-inspired Schwarzschild black holes [Phys. Lett. B 632 (2006) 547]. We also obtain the matter density functions using Einstein field equations for the geometries we reconstruct from the thermodynamics of black holes. These also have similarities to that of the matter density function of a noncommutative-inspired Schwarzschild black hole. The conformal structure of the metric is briefly discussed and the Penrose-Carter diagram is drawn. We then compute the Komar energy and the Smarr formula for the effective black hole geometry and compare it with that of the noncommutative-inspired Schwarzschild black hole. We also discuss some astrophysical implications of the solutions. Finally, we propose a set of quantum Einstein vacuum field equations, as a solution of which we obtain one of the spacetime solutions obtained in this work. We then show a direct connection between the quantum Einstein vacuum field equations and the first law of black hole thermodynamics.