Acta Mathematica

The global non-linear stability of the Kerr–de Sitter family of black holes

Peter Hintz and András Vasy

Full-text: Open access

Abstract

We establish the full global non-linear stability of the Kerr–de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: we develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations. In particular, the iteration scheme used to solve Einstein’s equations automatically finds the parameters of the Kerr–de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.

Article information

Source
Acta Math., Volume 220, Number 1 (2018), 1-206.

Dates
Received: 1 July 2016
First available in Project Euclid: 19 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.acta/1560966510

Digital Object Identifier
doi:10.4310/ACTA.2018.v220.n1.a1

Mathematical Reviews number (MathSciNet)
MR3816427

Zentralblatt MATH identifier
1391.83061

Subjects
Primary: 83C57: Black holes
Secondary: 35B40: Asymptotic behavior of solutions 58J47: Propagation of singularities; initial value problems 83C05: Einstein's equations (general structure, canonical formalism, Cauchy problems) 83C35: Gravitational waves

Keywords
Einstein’s equation black hole stability constraint damping global iteration gauge modification Nash–Moser iteration microlocal analysis

Citation

Hintz, Peter; Vasy, András. The global non-linear stability of the Kerr–de Sitter family of black holes. Acta Math. 220 (2018), no. 1, 1--206. doi:10.4310/ACTA.2018.v220.n1.a1. https://projecteuclid.org/euclid.acta/1560966510


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