Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

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-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 + ), in such a way that r = +∞ 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.

Decay of solutions of the wave equation in expanding cosmological spacetimes

“…Moreover, if one regards the wave equation as a proxy for the Einstein equations, then time derivatives are akin to second fundamental forms of time slices, and their decay rates may be useful in formulating and proving cosmic no-hair theorems (in the spirit of [2]). In fact, the results of the present paper provided important insights for the recent analysis of the cosmic no-hair conjecture in spherically symmetric black hole spacetimes in [7].…”

The formation of trapped surfaces in the gravitational collapse of spherically symmetric scalar fields with a positive cosmological constant

“…In view of the recent developments concerning the global analysis of black hole spacetimes with a positive cosmological constant -from the remarkable proofs of the non-linear stability of the local/exterior regions of Kerr-Newman de Sitter [23,24], to the new results concerning the structure of black hole interiors [13][14][15][16], as well as the improved understanding of the geometry of cosmological regions [19,22,29] -it seems relevant to study the formation of trapped surfaces in the presence of Λ > 0. In this paper we start this study by revisiting Christodoulou's original criterion and extending it to the context of solutions to the Einstein-scalar field system with a positive cosmological constant.…”

The formation of trapped surfaces in the gravitational collapse of spherically symmetric scalar fields with a positive cosmological constant

“…In view of the recent developments concerning the global analysis of black hole spacetimes with a positive cosmological constant-from the remarkable proofs of the non-linear stability of the local/exterior regions of Kerr-Newman de Sitter [27,28], to the new results concerning the structure of black hole interiors [16][17][18][19], as well as the improved understanding of the geometry of cosmological regions [22,25,33]-it seems relevant to study the formation of trapped surfaces in the presence of Λ > 0. In this paper we start this study by revisiting Christodoulou's original criterion and extending it to the context of solutions to the Einsteinscalar field system with a positive cosmological constant.…”