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

In this paper we study the proposal present in Viaggiu (2017) concerning the statistical description of trapped gravitons and applied to derive the semi-classical black hole (BH) entropy SBH. We study the possible configurations depending on physically reasonable expressions for the internal energy U. In particular, we show that expressions for U∼Rk,k≥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 one at 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.