Duke Mathematical Journal

Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations

Jonathan Luk and Sung-Jin Oh

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Abstract

It has long been suggested that solutions to the linear scalar wave equation

gϕ=0 on a fixed subextremal Reissner–Nordström spacetime with nonvanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to Wloc1,2. This instability is related to the celebrated blue-shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein–Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner–Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price’s law decay is generically sharp along the event horizon.

Article information

Source
Duke Math. J., Volume 166, Number 3 (2017), 437-493.

Dates
Received: 11 February 2015
Revised: 5 April 2016
First available in Project Euclid: 24 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1477321009

Digital Object Identifier
doi:10.1215/00127094-3715189

Mathematical Reviews number (MathSciNet)
MR3606723

Zentralblatt MATH identifier
1373.35306

Subjects
Primary: 35Q75: PDEs in connection with relativity and gravitational theory
Secondary: 83C57: Black holes

Keywords
black holes Reissner–Nordström wave equation instability of Cauchy horizon strong cosmic censorship conjecture

Citation

Luk, Jonathan; Oh, Sung-Jin. Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations. Duke Math. J. 166 (2017), no. 3, 437--493. doi:10.1215/00127094-3715189. https://projecteuclid.org/euclid.dmj/1477321009


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