Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in $AdS_5\times{S^5}$ spacetime

TL;DR

This paper investigates the thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in $AdS_5\times S^5$ spacetime, revealing relationships between scalar curvature divergences and specific heat behaviors.

Contribution

It introduces a novel analysis of black hole thermodynamics by treating the cosmological constant as a variable related to the boundary gauge theory's color number and explores associated thermodynamic geometries.

Findings

Chemical potential is negative in stable black hole branches.

Scalar curvature divergences correspond to specific heat divergences in different thermodynamic metrics.

Divergences in scalar curvature relate to phase transition points.

Abstract

We study thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in $AdS_5\times{S^5}$ spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the chemical potential is always negative in the stable branch of black hole thermodynamics and it has a chance to be positive, but appears in the unstable branch. We calculate scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedo metric, respectively and we find that the divergence of scalar curvature is related to the divergence of specific heat with fixed chemical potential in the Weinhold metric and Ruppeiner metric, while in the Quevedo metric the divergence of scalar curvature is related to the divergence of specific heat with fixed number of colors and the vanishing of the specific heat with fixed chemical potential.