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 $2πK_B$, i.e. $S_{BH}(R=L_P)=2πK_B$. 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.

💡 Research Summary

The paper proposes a semi‑classical, phenomenological route to obtain the well‑known logarithmic corrections to the Bekenstein‑Hawking black‑hole entropy by modelling the interior of a black hole as a Bose‑Einstein condensate (BEC) of trapped gravitons. The authors start from their earlier work in which a gas of massless excitations confined within a spherical box of radius (R) (identified with the Schwarzschild radius) obeys the radiation equation of state (PV=U/3). The internal temperature of the box is taken to be (T_i=\alpha T_{BH}), where (\alpha) is a real parameter; (\alpha=2) reproduces the usual radiation temperature relation.

In the original model the internal energy is identified with the ADM mass, (U(R)=Mc^2=c^4R^2/G). This choice reproduces the area law only in the limit (\alpha\to0) (i.e. (T_i\to0)), which is unsatisfactory. To generate a genuine logarithmic term the authors modify the internal energy by adding a quantum‑gravity motivated correction: \