Linear stability of slowly rotating Kerr black holes
- PDF / 1,997,680 Bytes
- 180 Pages / 439.37 x 666.142 pts Page_size
- 99 Downloads / 225 Views
Linear stability of slowly rotating Kerr black holes Dietrich Häfner1 · Peter Hintz2 · András Vasy3
Received: 15 July 2019 / Accepted: 4 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the lineariza-
B Peter Hintz
[email protected] Dietrich Häfner [email protected] András Vasy [email protected]
1
Institut Fourier, Université Grenoble Alpes, 100 rue des maths, 38402 Gières, France
2
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
3
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA
123
D. Häfner et al.
tion of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equations have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications. Mathematics Subject Classification Primary 83C05 · 58J50; Secondary 83C57 · 35B40 · 83C35 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Ingredients of the proof . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Mode stability . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Zero energy modes; resolvent near zero . . . . . . . . . . . 1.1.3 Perturbation to Kerr metrics . . . . . . . . . . . . . . . . . 1.2 Further related work . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 2 b- and scattering structures . . . . . . . . . . . . . . . . . . . . . . . . 3 The spacetime manifold and the Kerr family of metrics . . . . . . .
Data Loading...
