The mechanical first law of black hole spacetimes with a cosmological constant and its application to the Schwarzschild–de Sitter spacetime

Abstract

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 Schwarzschildde 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.

Horndeski gravity and the violation of reverse isoperimetric inequality

“…By contrast, the entropy can be determined locally as 1/4 of the area of the horizon in Einstein gravity, or more generally by the Wald entropy formula [5,6], which requires only the near horizon geometry. An integral form of calculating the volume can also be derived from the Wald formalism [7,8], which necessarily requires exact solutions. Interestingly the integration is more naturally performed from horizon to asymptotic infinity [8].…”

Black holes and Boyle's law — The thermodynamics of the cosmological constant

“…But it was shown in [39], using the Smarr relation (15), that there exists a negative norm vector of the bi-linear form U AB when P V ≤ 0, regardless of the values of J and Q, and this vector was explicitly constructed as a linear combination of the X A . For fixed J and Q the norm of the vector becomes positive only if…”

Thermodynamics of phase transition in higher-dimensional Reissner–Nordström–de Sitter black hole

“…(3.7) into Eq. (3.3), one can obtain the thermodynamic equation for the thermodynamic quantities of higher-dimensional RN-dS black holes [45]:…”

“…The thermodynamic quantities on both horizons satisfy the first law of thermodynamics, and the corresponding entropy fulfills the area formula [22,43,44]. In recent years, the research on the thermodynamic properties of de Sitter spacetime has drawn much attention [16,22,[43][44][45][46][47][48]. In the inflation epoch of the early universe, the universe is a quasi-de Sitter spacetime.…”

“…Firstly, we expect the thermodynamic entropy to satisfy the Nernst theorem [45,47,51]. At present a satisfactory explanation to the problem that the thermodynamic entropy of the horizon of the extreme de Sitter spacetime does not fulfill the Nernst theorem is still lacking.…”