# Statistical mechanics of gravitons in a box and the black hole entropy

## TL;DR

This paper investigates the statistical mechanics of trapped gravitons in a spherical box, deriving a spectrum and entropy formula that aligns with black hole thermodynamics without relying on quantum gravity theories.

## Contribution

It introduces a model of trapped gravitons with a discrete spectrum and derives the Bekenstein-Hawking entropy from classical thermodynamics principles.

## Key findings

- Derived a discrete graviton spectrum dependent on Legendre index
- Obtained a mean energy as a sum of two Planck distributions
- Reproduced the Bekenstein-Hawking entropy without quantum gravity

## Abstract

This paper is devoted to the study of the statistical mechanics of trapped gravitons obtained by 'trapping' a spherical gravitational wave in a box. As a consequence, a discrete spectrum dependent on the Legendre index $\ell$ similar to the harmonic oscillator one is obtained and a statistical study is performed. The mean energy $<E>$ results as a sum of two discrete Planck distributions with different dependent frequencies. As an important application, we derive the semiclassical Bekenstein-Hawking entropy formula for a static Schwarzschild black hole by only requiring that the black hole internal energy $U$ is provided by its ADM rest energy, without invoking particular quantum gravity theories. This seriously suggests that the interior of a black hole can be composed of trapped gravitons at a thermodynamical temperature proportional by a factor $\simeq 2$ to the horizon temperature $T_h$.