Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric Spacetime

TL;DR

This paper investigates thermodynamic volume products in various black hole spacetimes, revealing conditions under which these products are mass-independent or universal, with specific focus on spherically symmetric, axisymmetric, and BTZ black holes.

Contribution

It introduces new formulations of volume products in extended phase space and identifies cases where these products are universal or mass-dependent.

Findings

Volume products are not always mass-independent in AdS spacetimes.

Reissner Nordström and Kehagias-Sfetsos black holes have universal volume products.

Only the rotating BTZ black hole exhibits a mass-independent, quantized volume product.

Abstract

In this Letter, we have examined the thermodynamic volume products for spherically symmetric and axisymmetric spacetimes in the framework of \emph{extended phase space}. Such volume products usually formulated in terms of the outer horizon~(${\cal H}^{+}$) and the inner horizon~(${\cal H}^{-}$) of black hole ~ (BH) spacetime. Besides volume product, the other thermodynamic formulations like \emph{volume sum, volume minus and volume division} are considered for a wide variety of spherically symmetric spacetime and axisymmetric spacetimes. Like area~(or entropy) product of multihorizons, the mass-independent~(universal) feature of volume products are sometimes also \emph{fail}. In particular for a spherically symmetric AdS spacetimes the simple thermodynamic volume product of ${\cal H}^{\pm}$ is not mass-independent. In this case, more complicated combinations of outer and inner horizon volume products are indeed mass-independent. For a particular class of spherically symmetric cases i.e. Reissner Nordstr\"om BH of Einstein gravity and Kehagias-Sfetsos BH of Ho\v{r}ava Lifshitz gravity, the thermodynamic volume products of ${\cal H}^{\pm}$ is indeed \emph{universal}. For axisymmetric class of BH spacetime in Einstein gravity all the combinations are \emph{mass-dependent}. There has been no chance to formulate any combinations of volume product relation is to be mass-independent. Interestingly, \emph{only the rotating BTZ black hole} in 3D provides the volume product formula is mass-independent i.e. \emph{universal} and hence it is quantized.