2023/06/12 by Peter Hintz, Hintz, Peter · 1 citation
Physics and Astronomy · #35B25 #35B40 #35C20 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Astrophysical Phenomena and Observations #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Primary: 83C05 #Secondary: 83C57
paper · pdf · doi:10.48550/arxiv.2306.07409
openalex publication_date 2023/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a smooth globally hyperbolic (3+1)-dimensional spacetime satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic, we construct a family of metrics depending on a small parameter ε>0 with the following properties. (1) They solve the Einstein vacuum equations modulo O(ε^∞). (2) Away from the geodesic they tend to the original metric as ε→ 0. (3) Their ε-1-rescalings near every point of the geodesic tend to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes without nontrivial Killing vector fields in a neighborhood of a point on the geodesic. If (M,g) is a neighborhood of the domain of outer communications of subextremal or extremal Kerr(-anti de~Sitter) spacetime, our metrics model extreme mass ratio mergers if we choose the timelike geodesic to cross the event horizon. The metrics which we construct here depend on ε and the (rescaled) coordinates on the original spacetime in a log-smooth fashion. This in particular justifies the formal perturbation theoretic setup in work of Gralla-Wald on gravitational self-force in the case of small black holes.