Wald entropy in Kaluza-Klein black holes

TL;DR

This paper explores the thermodynamics of 4D Kaluza-Klein black holes using Wald's formalism, revealing geometric interpretations of electric potential and the origin of the electric work term in higher dimensions.

Contribution

It provides a geometric interpretation of electric potential and derives the electric work term from a 5D perspective in Kaluza-Klein black holes.

Findings

Electric work term arises from 5D gravitational framework

Geometric interpretation of electrostatic potential

Compatibility condition leads to momentum map equation

Abstract

We study the thermodynamics of the 4-dimensional electrically charged black-hole solutions of the simplest 5-dimensional Kaluza-Klein theory using Wald's formalism. We show how the electric work term present in the 4-dimensional first law of black-hole thermodynamics arises in the purely gravitational 5-dimensional framework. In particular, we find an interesting geometric interpretation of the 4-dimensional electrostatic potential similar to the angular velocity in rotating black holes. Furthermore, we show how the momentum map equation arises from demanding compatibility between the timelike Killing vector of the black-hole solution and the spatial Killing vector of the 5-dimensional background.