Geometry of Higher-Dimensional Black Hole Thermodynamics

TL;DR

This paper explores the thermodynamic geometries of higher-dimensional Kerr and Reissner-Nordström black holes, revealing curvature behaviors related to stability and extremality across different spacetime dimensions.

Contribution

It provides a detailed analysis of thermodynamic geometries for higher-dimensional black holes, highlighting differences in curvature and stability indicators compared to four-dimensional cases.

Findings

Ruppeiner curvature diverges at extremality in 5D Kerr black holes.

In higher dimensions, Kerr black holes show divergence indicating potential instability.

Reissner-Nordström black holes have flat Ruppeiner geometry in all dimensions.

Abstract

We investigate thermodynamic curvatures of the Kerr and Reissner-Nordstr\"om (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for $d=5$ Kerr black hole is curved and divergent in the extremal limit. For $d \geq 6$ Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In $d \geq 5$ the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in $d=5$ with double angular momenta.