Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein–Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approach that of a subextremal Reissner–Nordström-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon (“near” i+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i^+$$\end{document}), in such a way that r=+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=+\infty $$\end{document} is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the cosmic no-hair conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote to those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.