On the Global Temperature of the Schwarzschild–de Sitter Spacetime

It is shown that the Schwarzschild–de Sitter spacetime has the universal temperature. This temperature describes the thermal processes of decay of the composite particles and the other processes, which are energetically forbidden in the Minkowski spacetime, but are allowed in the de Sitter and in Schwarzschild–de Sitter backgrounds. In particular, this temperature describes the probability of ionization of the atom in the Schwarzschild–de Sitter, which is observed by the stationary observer at the point where the shift function (velocity) in the Arnowitt–Deser–Misner formalism changes sign. This activation temperature does not depend on the black hole mass and is fully determined by the Hubble parameter, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{T}_{{{\text{SdS}}}}} = \sqrt 3 H{\text{/}}\pi $$\end{document}. This temperature is twice the Bousso–Hawking temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{T}_{{{\text{BH}}}}}$$\end{document}, which characterizes the limit of degenerate Lorentzian Schwarzschild–de Sitter universe, when the cosmological and black hole horizons are close to each other, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{T}_{{{\text{SdS}}}}} = 2{{T}_{{{\text{BH}}}}}$$\end{document}. The similar doubling of the temperature of Hawking radiation is known in the pure de Sitter spacetime, where the corresponding local temperature describing the ionization of atoms is twice the Gibbons–Hawking temperature, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{T}_{{{\text{dS}}}}} = 2{{T}_{{{\text{GH}}}}} = H{\text{/}}\pi $$\end{document}. We suggest that the activation temperature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{T}_{{{\text{dS}}}}}$$\end{document} can be considered as the thermodynamic temperature of the de Sitter state, which determines the local entropy in this state, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = 3H{\text{/}}4G$$\end{document}.