Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem
Gustav Holzegel
TL;DR
The paper develops a nonlinear stability framework for spacetimes that asymptotically approach Schwarzschild, aiming toward Kerr stability. It couples decay of Ricci-rotation coefficients to decay of the Weyl curvature through the Bianchi equations, using Bel–Robinson currents, an approximately Killing vector field $T$, and a redshift vector field to control horizon-near regions, while invoking $r^p$-weighted energies to capture infinity decay. A key innovation is renormalizing the problematic curvature components $ ho$ and $ ilde{ ho}$ (and $ ilde{ ho}$, $ ilde{eta}$), forming a coupled Regge–Wheeler-like system that links interior decay to asymptotic decay. The results establish uniform boundedness and a degenerate integrated decay for $k$-derivatives of the curvature under smallness and decay assumptions on the Ricci coefficients, and they provide a detailed, hierarchical treatment of commuted Bianchi equations, redshift effects, and interior/infinity estimates, laying groundwork toward the nonlinear stability of the Kerr family.
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.