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15 February 2017 Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations
Jonathan Luk, Sung-Jin Oh
Duke Math. J. 166(3): 437-493 (15 February 2017). DOI: 10.1215/00127094-3715189


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.


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Jonathan Luk. Sung-Jin Oh. "Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations." Duke Math. J. 166 (3) 437 - 493, 15 February 2017.


Received: 11 February 2015; Revised: 5 April 2016; Published: 15 February 2017
First available in Project Euclid: 24 October 2016

zbMATH: 1373.35306
MathSciNet: MR3606723
Digital Object Identifier: 10.1215/00127094-3715189

Primary: 35Q75
Secondary: 83C57

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

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 3 • 15 February 2017
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