Corrections to the Black hole entropy from a Bose Einstein condensate: a semi-classical phenomenological approach

In this paper we obtain logarithmic corrections to the black hole entropy. Motivated by our recent proposal concerning the nature of the degrees of freedom leading to the black hole entropy in terms of a Bose Einstein (BEC) condensate of gravitons, we study how to introduce logarithmic corrections. In fact we show that, after modifying the internal energy by means of simple by physically sound arguments dictated by ordinary quantum mechanics and possible non-commutative effects at Planckian scales, a logarithmic term does appear in the Bekenstein Hawking entropy law. We also obtain that the entropy $S_{BH}$ of a ball of Planckian areal radius is $2\pi K_B$, i.e. $S_{BH}(R=L_P)=2\pi K_B$. Our approach show that the possibility that the interior of a black hole is composed with a BEC of gravitons is a viable physically motivated possibility.