Thermodynamic Product Formula for Ho\v{r}ava Lifshitz Black Hole

TL;DR

This paper analyzes the thermodynamic properties of Hořava Lifshitz black holes, identifying universal and non-universal horizon products, and explores their thermodynamic laws, stability, and phase transitions.

Contribution

It introduces the computation of various horizon products for Hořava Lifshitz black holes and examines their universality and thermodynamic relations, which differ from classical black hole thermodynamics.

Findings

Surface area and entropy products are universal quantities.

First law and Smarr relations do not hold due to scale invariance.

Black holes can undergo second order phase transitions.

Abstract

We examine the thermodynamic properties of inner and outer horizons in the background of Ho\v{r}ava Lifshitz black hole. We compute the \emph{horizon radii product, the surface area product, the entropy product, the surface temperature product, the Komar energy product and the specific heat product} for both the horizons of said black hole. We show that surface area product, entropy product and irreducible mass product are \emph{universal} quantities, whereas the surface temperature product, Komar energy product and specific heat product are \emph{not universal} quantities because they all are depends on mass parameter. We also observe that the \emph{First law} of black hole thermodynamics and \emph {Smarr-Gibbs-Duhem } relations do not hold for this black hole. The underlying reason behind this failure due to the scale invariance of the coupling constant. We further derive the \emph{Smarr mass formula} and \emph{Christodolou-Ruffini mass formula} for such black hole spacetime. Moreover we study the 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 second order phase transition.