Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes
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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” \(i^+\)), 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 to those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.
We thank Anne Franzen for sharing and allowing us to use Fig. 1. This work was partially supported by FCT/Portugal through UID/MAT/04459/2013 and Grant (GPSEinstein) PTDC/MAT-ANA/1275/2014. Pedro Oliveira was supported by FCT/Portugal through the LisMath scholarship PD/BD/52640/2014.
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