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Annals of PDE

, 4:11 | Cite as

Non-linear Stability of the Kerr–Newman–de Sitter Family of Charged Black Holes

  • Peter Hintz
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Abstract

We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein–Maxwell equations: if one perturbs the initial data of a slowly rotating Kerr–Newman–de Sitter (KNdS) black hole, then in a neighborhood of the exterior region of the black hole, the metric and the electromagnetic field decay exponentially fast to their values for a possibly different member of the KNdS family. This is a continuation of recent work of the author with Vasy on the stability of the Kerr–de Sitter family for the Einstein vacuum equations. Our non-linear iteration scheme automatically finds the final black hole parameters as well as the gauge in which the global solution exists; we work in a generalized wave coordinate/Lorenz gauge, with gauge source functions lying in a suitable finite-dimensional space. We include a self-contained proof of the linear mode stability of Reissner–Nordström–de Sitter black holes, building on work by Kodama–Ishibashi. In the course of our non-linear stability argument, we also obtain the first proof of the linear (mode) stability of slowly rotating KNdS black holes using robust perturbative techniques.

Keywords

Black hole stability Einstein–Maxwell equations Global iteration Microlocal analysis Constraint damping 

Mathematics Subject Classification

Primary 83C57 Secondary 83C22 35B40 83C35 

Notes

Acknowledgements

I would like to thank András Vasy, Maciej Zworski, Jim Isenberg, Sergiu Klainerman, and Yakov Shlapentokh-Rothman for valuable discussions and for their interest and support. I am very grateful to Mihalis Dafermos, and to an anonymous referee, for valuable comments on both content and exposition. I would also like to thank the Miller Institute at the University of California, Berkeley, for support.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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