Statistical Entropy Based on the Generalized-Uncertainty-Principle-Induced Effective Metric

TL;DR

This paper explores how generalized uncertainty principle (GUP) modifications affect black hole entropy, showing that the area law remains valid and GUP naturally regularizes divergences without artificial cutoffs.

Contribution

It introduces three GUP-induced effective metrics and demonstrates their consistent support for the Bekenstein-Hawking area law, revealing the universality of black hole entropy under quantum gravity corrections.

Findings

GUP-corrected metrics uphold the area law

GUP regularizes ultraviolet divergences

Finite entropy obtained without artificial cutoffs

Abstract

We investigate the statistical entropy of black holes within the framework of the generalized uncertainty principle (GUP) by employing effective metrics that incorporate leading-order and all-orders quantum gravitational corrections. We construct three distinct effective metrics induced by the GUP, which are derived from GUP-corrected temperature, entropy, and all-orders GUP corrections, and analyze their impact on black hole entropy using 't Hooft's brick wall method. Our results show that, despite the differences in the effective metrics and the corresponding ultraviolet cutoffs, the statistical entropy consistently satisfies the Bekenstein-Hawking area law when expressed in terms of an invariant (coordinate-independent) distance near the horizon. Furthermore, we demonstrate that the GUP naturally regularizes the ultraviolet divergence in the density of states, eliminating the need for artificial cutoffs and yielding finite entropy even when counting quantum states only in the vicinity of the event horizon. These findings highlight the universality and robustness of the area law under GUP modifications and provide new insights into the interplay between quantum gravity effects and black hole thermodynamics.