Entropy, temperature and internal energy of trapped gravitons and corrections to the Black Hole entropy

TL;DR

This paper investigates the statistical properties of trapped gravitons to derive black hole entropy, revealing logarithmic corrections, phase transitions, and evaporation stopping at Planck scale, thus enhancing understanding of quantum gravity effects.

Contribution

It introduces a semi-classical model of trapped gravitons with specific internal energy configurations, deriving known entropy corrections and phase transition phenomena.

Findings

Logarithmic corrections to black hole entropy are obtained.

A phase transition with positive specific heat occurs at Planckian lengths.

Black hole evaporation halts at a radius near the Planck length.

Abstract

In this paper we study the proposal present in \cite{1} concerning the statistical description of trapped gravitons and applied to derive the semi-classical black hole (BH) entropy $S_{BH}$. We study the possible configurations depending on physically reasonable expressions for the internal energy $U$. In particular, we show that expressions for $U\sim R^k, k\geq 1$, with $R$ the radius of the confining spherical box, can have a semi-classical description, while behaviors with $k<1$ derive from thermodynamic or quantum fluctuations. There, by taking a suitable physically motivated expression for $U(R)$, we obtain the well known logarithmic corrections to the BH entropy, with the usual behaviors present in the literature of BH entropy. Moreover, a phase transition emerges with a positive specific heat $C$ at Planckian lengths instead of the usual negative oneat non-Planckian scales, in agreement with results present in the literature. Finally, we show that evaporation stops at a radius $R$ of the order of the Planck length.