Determining the Minimal Length Scale of the Generalized Uncertainty Principle from the Entropy-Area Relationship

TL;DR

This paper derives a black hole entropy formula incorporating a minimal length scale from the generalized uncertainty principle (GUP), applying it to various black holes to determine the GUP parameter via the entropy-area relationship.

Contribution

It introduces a method to determine the GUP parameter by linking the entropy-area law to a minimal length scale in black hole physics.

Findings

GUP modifies black hole entropy calculations.

The GUP parameter can be fixed by the entropy-area relation.

Universal near-horizon entropy formula derived.

Abstract

We derive the formula of the black hole entropy with a minimal length of the Planck size by counting quantum modes of scalar fields in the vicinity of the black hole horizon, taking into account the generalized uncertainty principle (GUP). This formula is applied to some intriguing examples of black holes - the Schwarzschild black hole, the Reissner-Nordstrom black hole, and the magnetically charged dilatonic black hole. As a result, it is shown that the GUP parameter can be determined by imposing the black hole entropy-area relationship, which has a Planck length scale and a universal form within the near-horizon expansion.