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

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 $\mathrm{\ell }$ 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 $T_h$.