Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem

摘要

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