The Relation Between the Quasi-localized Energy Complexes and the Thermodynamic Potential for the Schwarzschild-de Sitter Black Hole

TL;DR

This paper explores the connection between quasi-localized energy complexes and thermodynamic potentials in Schwarzschild-de Sitter black holes, revealing a Legendre transformation-like relation that links energy measures to heat flows.

Contribution

It demonstrates that Einstein and Møller energy complexes are related to thermodynamic potentials through a Legendre transformation in Schwarzschild-de Sitter spacetime.

Findings

The energy complexes obey a Legendre transformation relation.

Quasi-localized energy complexes correspond to thermodynamic potentials.

The relation holds between energy measures and heat flows across horizons.

Abstract

The Schwarzschild-de Sitter (SdS) black hole solution, which has two event horizons, is considered to examine the relation between the quasi-localized energy complexes on ${\cal M}$ and the heat flows passing through its boundary $\partial {\cal M}$. Here ${\cal M}$ is the patch between cosmological event horizon and black hole event horizon of the SdS black hole solution. Conclusively, the relation, like the Legendre transformation, between the quasi-localized Einstein and M{\o}ller energy complex and the heat flows passing through the boundary is obeyed, and these two quasi-localized energy complexes could be corresponding to thermodynamic potentials.