2024/08/13 by Peter Hintz, Hintz, Peter · 3 citations
Physics and Astronomy · #35B25. Secondary: 83C57 #58K55 #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) #Primary: 35L05 #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.2408.06712
openalex publication_date 2024/08/13 · openalex created_date 2024/09/11 · openalex updated_date 2026/07/28
Given a smooth globally hyperbolic (3+1)-dimensional spacetime (M,g) satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic C, we constructed in Part I a family of metrics gε on the complement Mε⊂ M of an ε-neighborhood of C with the following behavior: away from C one has gε→ g as ε→ 0, while the ε-1-rescaling of gε around every point of C tends to a fixed subextremal Kerr metric; and gε solves the Einstein vacuum equation modulo O(ε^∞) errors. The ultimate goal, achieved in Part III, is to correct gε to a true solution on any fixed precompact subset of M by addition of a size O(ε^∞) metric perturbation which needs to satisfy a quasilinear wave equation (the Einstein vacuum equations in a suitable gauge). The present paper lays the necessary analytical foundations. We develop a framework for proving estimates for solutions of (tensorial) wave equations on (Mε,gε) which, on a suitable scale of Sobolev spaces, are uniform on ε-independent precompact subsets of the original spacetime M. These estimates are proved by combining two ingredients: the spectral theory for the corresponding wave equation on Kerr; and uniform microlocal estimates governing the propagation of regularity through the small black hole, including radial point estimates reminiscent of diffraction by conic singularities and long-time estimates near perturbations of normally hyperbolic trapped sets. As an illustration of this framework, we construct solutions of a toy nonlinear scalar wave equation on (Mε,gε) for uniform timescales and with full control in all asymptotic regimes as ε→ 0.