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General Relativity and Quantum Cosmology

arXiv:1804.04297 (gr-qc)
[Submitted on 12 Apr 2018 (v1), last revised 6 Oct 2020 (this version, v3)]

Title:Decay of weakly charged solutions for the spherically symmetric Maxwell-Charged-Scalar-Field equations on a Reissner-Nordström exterior space-time

Authors:Maxime Van de Moortel
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Abstract:We consider the Cauchy problem for the (non-linear) Maxwell-Charged-Scalar-Field equations with spherically symmetric initial data, on a sub-extremal Reissner--Nordström or Schwarzschild exterior space-time. We prove that the solutions are bounded and decay at an inverse polynomial rate towards time-like infinity and along the black hole event horizon, provided the charge of the Maxwell equation is sufficiently small. This condition is in particular satisfied for small data in energy space that enjoy a sufficient decay towards the asymptotically flat end. Some of the decay estimates we prove are arbitrarily close to the conjectured optimal rate in the limit where the charge tends to zero, according the heuristics present in the physics literature. Our result can also be interpreted as a first step towards the stability of Reissner--Nordström black holes for the gravity coupled Einstein--Maxwell-Charged-Scalar-Field model. This problem is closely connected to the understanding of strong cosmic censorship and charged gravitational collapse in this setting.
Comments: Version accepted for publication in Annales Scientifiques de l'Ecole Normale Superieure
Subjects: General Relativity and Quantum Cosmology (gr-qc); Analysis of PDEs (math.AP)
Cite as: arXiv:1804.04297 [gr-qc]
  (or arXiv:1804.04297v3 [gr-qc] for this version)
  https://doi.org/10.48550/arXiv.1804.04297

Submission history

From: Maxime Van de Moortel [view email]
[v1] Thu, 12 Apr 2018 03:14:04 UTC (212 KB)
[v2] Wed, 25 Apr 2018 22:48:18 UTC (213 KB)
[v3] Tue, 6 Oct 2020 02:58:02 UTC (214 KB)
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