Mechanical First Law of Black Hole Spacetimes with Cosmological Constant and Its Application to Schwarzschild-de Sitter Spacetime

TL;DR

The paper develops a mechanical first-law framework for spacetimes with a cosmological constant and two horizons by extending the Iyer-Wald formalism to treat the mass parameter $M$ and the cosmological constant $Λ$ as two independent variables. It defines horizon entropy $S$ as the total area of BEH and CEH and introduces a system volume $V$ between the horizons, deriving an MFL of the form $δM = T_{ m eff} δS - p_{ m eff} δV$ with explicit, positive $T_{ m eff}$ and $p_{ m eff}$ for SdS spacetime. The analysis demonstrates that treating $Λ$ as a variable yields a mathematically consistent, Noether-charge–based description of SdS thermodynamics and provides a natural state-variable framework for discussing SdS evaporation without invoking the generalized second law. It also opens up a link between black hole thermodynamics and non-equilibrium thermodynamics through the interpretation of $T_{ m eff}$ as an effective temperature for a two-horizon system.

Abstract

The mechanical first law (MFL) of black hole spacetimes is a geometrical relation which relates variations of mass parameter and horizon area. While it is well known that the MFL of asymptotic flat black hole is equivalent to its thermodynamical first law, however we do not know the detail of MFL of black hole spacetimes with cosmological constant which possess black hole and cosmological event horizons. Then this paper aims to formulate an MFL of the two-horizon spacetimes. For this purpose, we try to include the effects of two horizons in the MFL. To do so, we make use of the Iyer-Wald formalism and extend it to regard the mass parameter and the cosmological constant as two independent variables which make it possible to treat the two horizons on the same footing. Our extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and gives the abstract form of MFL of black hole spacetimes with cosmological constant. Then, as a representative application of that formalism, we derive the MFL of Schwarzschild-de Sitter (SdS) spacetime. Our MFL of SdS spacetime relates the variations of three quantities; the mass parameter, the total area of two horizons and the volume enclosed by two horizons. If our MFL is regarded as a thermodynamical first law of SdS spacetime, it offers a thermodynamically consistent description of SdS black hole evaporation process: The mass decreases while the volume and the entropy increase. In our suggestion, the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process.