Ruppeiner thermodynamic geometry for the Schwarzschild-AdS black hole

TL;DR

This paper addresses the singularity issues in Ruppeiner thermodynamic geometry for Schwarzschild-AdS black holes by proposing a scheme that uses enthalpy as the fundamental characteristic, revealing insights into black hole interactions.

Contribution

It introduces a natural approach using enthalpy for thermodynamic geometry of spherically symmetric AdS black holes, resolving singularity problems and comparing different phase space geometries.

Findings

Thermodynamic curvature is negative, indicating attractive interactions.

Curvature in different phase spaces is equivalent.

Black hole interactions vary from repulsive at low temperatures to attractive at high temperatures.

Abstract

Due to the non-independence of entropy and thermodynamic volume for spherically symmetric black holes in the AdS spacetime, when applying the Ruppeiner thermodynamic geometry theory to these black holes, we often encounter an unavoidable problem of the singularity about the line element of thermodynamic geometry. In this paper, we propose a basic and natural scheme for dealing with the thermodynamic geometry of spherically symmetric AdS black holes. We point out that enthalpy, not internal energy, is the fundamental thermodynamic characteristic function for the Ruppeiner thermodynamic geometry. Based on this fact, we give the specific forms of the line element of thermodynamic geometry for Schwarzschild AdS (SAdS) black hole in different phase spaces and the results show that the thermodynamic curvatures obtained in different phase spaces are equivalent. It is shown that the thermodynamic curvature is negative which implies that the attractive interaction dominates between black hole molecules for the SAdS black hole. Meanwhile we also give an approximate expression of the thermodynamic curvature of the Schwarzschild black hole which indicates that the black hole is dominated by repulsion on low temperature region and by attraction on high temperature region.