Thermodynamics of black and white holes in ensemble of Planckons

TL;DR

This paper models black and white holes as collections of Planck-scale objects, deriving their thermodynamic properties and entropy, and explores their behavior in different charge and stability conditions using a toy model.

Contribution

It introduces a novel toy model representing black holes as ensembles of Planckons with quantized mass and entropy based on their correlations, extending to charged and white hole cases.

Findings

Black hole entropy is proportional to the number of Planckons and their correlations.

Reissner-Nordström black holes share the same entropy as neutral ones in this model.

White holes have negative entropy, indicating different thermodynamic properties.

Abstract

The Tsallis-Cirto non-extensive statistics with $\delta=2$ describes the processes of splitting and merging of black holes and their thermodynamics. Here we consider a toy model, which matches this generalized statistics and extends it by providing the integer valued entropy of the black hole, $S_{\rm BH}(N)=N(N-1)/2$. In this model the black hole consists of $N$ the so-called Planckons -- objects with reduced Planck mass $m_{\rm P}=1/\sqrt{8\pi G}$ -- so that its mass is quantized, $M=Nm_{\rm P}$. The entropy of each Planckon is zero, but the entropy of black hole with $N$ Planckons is provided by the $N(N-1)/2$ degrees of freedom -- the correlations between the gravitationally attracted Planckons. This toy model can be extended to a charged Reissner-Nordstr\"om (RN) black hole, which consists of charged Planckons. Despite the charge, the statistical ensemble of Planckons remains the same, and the RN black hole with $N$ Planckons has the same entropy as the electrically neutral hole, $S_{\rm RNBH}(N)=N(N-1)/2$. This is supported by the adiabatic process of transformation from the RN to Schwarzschild black hole by varying the fine structure constant. The adiabaticity is violated in the extreme limit, when the gravitational interaction between two Planckons is compensated by the repulsion between their electric charges, and the RN black hole loses stability. The entropy of a white hole formed by the same $N$ Planckons has negative entropy, $S_{\rm WH}(N)=-N(N-1)/2$.