2010/10/15 by Gustav Holzegel, Holzegel, Gustav · 3 citations
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Relativity and Gravitational Theory #gr-qc #math.AP
paper · pdf · doi:10.48550/arxiv.1010.3216
118 pages
arxiv created 2010/10/15 · openalex publication_date 2010/10/15 · arxiv updated 2010/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce a class of spacetimes (M,g) which satisfy the vacuum Einstein equations and dynamically approach a Schwarzschild solution of mass M, a class we shall call 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 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.