Quantum Statistical Entropy and Minimal Length of 5D Ricci-flat Black String with Generalized Uncertainty Principle

TL;DR

This paper investigates the quantum statistical entropy of a 5D Ricci-flat black string with a Schwarzschild-de Sitter black hole, incorporating the generalized uncertainty principle to derive entropy without cut-offs and determine a minimal length scale.

Contribution

It introduces a method to compute entropy in higher-dimensional black strings using the generalized uncertainty principle, avoiding traditional cut-offs and constraints.

Findings

Entropy is a linear sum of horizon areas without cut-offs.

Density of states and free energy are convergent near horizons.

Minimal length is constrained by surface gravities and layer thickness.

Abstract

In this paper, we study the quantum statistical entropy in a 5D Ricci-flat black string solution, which contains a 4D Schwarzschild-de Sitter black hole on the brane, by using the improved thin-layer method with the generalized uncertainty principle. The entropy is the linear sum of the areas of the event horizon and the cosmological horizon without any cut-off and any constraint on the bulk's configuration rather than the usual uncertainty principle. The system's density of state and free energy are convergent in the neighborhood of horizon. The small-mass approximation is determined by the asymptotic behavior of metric function near horizons. Meanwhile, we obtain the minimal length of the position $\Delta x$ which is restrained by the surface gravities and the thickness of layer near horizons.