Cosmic no-hair in spherically symmetric black hole spacetimes

TL;DR

This paper investigates the geometry of spherically symmetric black hole spacetimes with a positive cosmological constant, demonstrating that they asymptotically resemble de Sitter space and confirming aspects of the Cosmic No-Hair Conjecture.

Contribution

It provides a detailed analysis of the asymptotic behavior of such spacetimes and proves a version of the Cosmic No-Hair Conjecture under weak assumptions.

Findings

The radius blows up along null rays near the cosmological horizon.

The spacetime asymptotically approaches de Sitter space.

Conditions for globalization of results are discussed.

Abstract

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 approaches that of a subextremal Reissner-Nordstr\"om-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" $i^+$), in such a way that $r=+\infty$ 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 those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.