Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem

Abstract

In this paper, we introduce a class of spacetimes $\left(\mathcal{M},g\right)$ which satisfy the vacuum Einstein equations and dynamically approach a Schwarzschild solution of mass $M$, a class we shall call \emph{ultimately Schwarzschildean spacetimes}. The approach is captured in terms of boundedness and decay assumptions on appropriate spacetime-norms of the Ricci-coefficients and spacetime curvature. Given such assumptions at the level of $k$ derivatives of the Ricci-coefficients (and hence $k-1$ derivatives of curvature), we prove boundedness and decay estimates for $k$ derivatives of \emph{curvature}. The proof employs the framework of vectorfield multipliers and commutators for the Bel-Robinson tensor, pioneered by Christodoulou-Klainerman in the context of the stability of the Minkowski space. We provide multiplier analogues capturing the essential decay mechanisms (which have been identified previously for the scalar wave equation on black hole backgrounds) for the Bianchi equations. In particular, a formulation of the redshift-effect near the horizon is obtained. Morever, we identify a certain hierarchy in the Bianchi equations, which leads to the control of strongly $r$-weighted spacetime curvature-norms near infinity. This allows to avoid the use the classical conformal Morawetz multiplier $K$, therby generalizing recent work of Dafermos and Rodnianski in the context of the wave equation. Finally, the proof requires a detailed understanding of the structure of the error-terms in the interior. This is particularly intricate in view of both the phenomenon of trapped orbits and the fact that, unlike in the stability of Minkowski space, not all curvature components decay to zero.

118 pages

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• The $r^p$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications
• The non-linear stability of the Schwarzschild family of black holes
• Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior
• Formation of Trapped Surfaces from Past Null Infinity
• Trapped surfaces in vacuum arising dynamically from mild incoming radiation
• A Scale-Critical Trapped Surface Formation Criterion: A New Proof via Signature for Decay Rates
• Local energy decay for Maxwell fields part I: Spherically symmetric black-hole backgrounds
• On global behavior of solutions of the Maxwell-Klein-Gordon equations
• A scale-critical trapped surface formation criterion for the Einstein-Maxwell system