Entropy Product Formula for spinning BTZ Black Hole

TL;DR

This paper explores the thermodynamic properties of spinning BTZ black holes, revealing that the entropy product is universal while other products depend on mass, and discusses stability and phase transitions.

Contribution

It provides a detailed analysis of horizon thermodynamics, deriving relations and mass formulas, and identifies conditions for phase transitions in spinning BTZ black holes.

Findings

Entropy product is mass-independent (universal).

Surface temperature, Komar energy, and specific heat products depend on mass.

Black hole can undergo second order phase transition under certain conditions.

Abstract

We investigate the thermodynamic properties of inner and outer horizons in the background of spinning BTZ(Ba\~{n}ados,Teitelboim and Zanelli) black hole. We compute the \emph{horizon radii product, the entropy product, the surface temperature product, the Komar energy product and the specific heat product} for both the horizons. We observe that the entropy product is \emph{universal}(mass-independent), whereas the surface temperature product, Komar energy product and specific heat product are \emph{not universal} because they all depends on mass parameter. We also show that the \emph{First law} of black hole thermodynamics and \emph {Smarr-Gibbs-Duhem } relations hold for inner horizon as well as outer horizon. The Christodoulou-Ruffini mass formula is derived for both the horizons. We further study the \emph{stability} of such black hole by computing the specific heat for both the horizons. It has been observed that under certain condition the black hole possesses \emph{second order phase transition}.