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

摘要

We modify the first laws of thermodynamics of a Reissner-Nordstrom anti-deSitter black hole and a pure de Sitter space-time by the surface tensions. Thecorresponding Smarr relations are obeyed. The cosmological constants are firsttreated as fixed constants, and then as variables associated to the pressures.For the black hole, the law is written as $\delta E = T \delta S - \sigma\deltaA$ when the cosmological constant is fixed, where $E$ is the Misner-Sharp massand $\sigma$ is the surface tension. Adopting the varied constant, we modifythe 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 areinvestigated. For the de Sitter space-time, the expressions of the modifiedlaws are different from these of the black hole. The differential way to derivethe law is discussed.