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

TL;DR

This work addresses the origin and form of logarithmic corrections to black-hole entropy by embedding a Bose–Einstein condensate of trapped gravitons inside the horizon within a semi-classical framework. By introducing Planck-scale quantum corrections to the internal energy, motivated by non-commutative spacetime and Casimir-like reasoning, the authors derive a corrected entropy formula that naturally yields a negative logarithmic term: ${S_{BH}=K_B\left[ \frac{A_h}{4L_P^2} - \pi \ln\left(\frac{A_h}{4L_P^2}\right) + \pi\left(1+\ln\pi\right) \right]}$ after fixing the model parameter ${D=1/2}$. They show that in the limit ${\alpha\to 0}$ (vanishing interior temperature fluctuations) the standard area law is recovered, while Planck-scale fluctuations generate the logarithmic correction with a Planckian minimum entropy ${S_{BH}(R=L_P)=2\pi K_B}$. The work also introduces the parameter ${\alpha}$ as a control for thermal versus quantum fluctuations and discusses extensions to rotating black holes. Overall, the paper provides a physically motivated, semi-classical mechanism to obtain logarithmic corrections to black-hole entropy within a BEC graviton interior picture.

Abstract

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.