The modified first laws of thermodynamics of anti-de Sitter and de Sitter space-times

TL;DR

This paper modifies the first laws of thermodynamics for anti-de Sitter black holes and de Sitter space-times by incorporating surface tensions and variable cosmological constants, revealing new thermodynamic relations.

Contribution

It introduces modified first laws of thermodynamics for these space-times, including surface tensions and variable cosmological constants, expanding the understanding of their thermodynamic behavior.

Findings

Modified thermodynamic laws incorporate surface tension effects.

Treating cosmological constants as variables leads to enthalpy-based laws.

Distinct laws derived for black holes and de Sitter space-times.

Abstract

We modify the first laws of thermodynamics of a Reissner-Nordstrom anti-de Sitter black hole and a pure de Sitter space-time by the surface tensions. The corresponding Smarr relations are obeyed. The cosmological constants are first treated as fixed constants, and then as variables associated to the pressures. For the black hole, the law is written as $\delta E = T \delta S - \sigma\delta A$ when the cosmological constant is fixed, where $E$ is the Misner-Sharp mass and $\sigma$ is the surface tension. Adopting the varied constant, we modify the law as $\delta E_0 = T \delta S - \sigma_{eff}\delta A +V\delta P$, where $E_0=M-\frac{Q^2}{2r_+}$ is the enthalpy. The thermodynamical properties are investigated. For the de Sitter space-time, the expressions of the modified laws are different from these of the black hole. The differential way to derive the law is discussed.