dc.contributor.author Kehle, Christoph dc.date.accessioned 2022-02-22T16:00:55Z dc.date.available 2022-02-22T16:00:55Z dc.date.issued 2022-03 dc.date.submitted 2020-08-02 dc.identifier.issn 0020-9910 dc.identifier.other s00222-021-01078-6 dc.identifier.other 1078 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/334323 dc.description.abstract AbstractThe purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant $$\Lambda <0$$ Λ < 0 and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations $$\psi$$ ψ of Kerr–AdS solving $$\Box _g\psi -\frac{2}{3}\Lambda \psi =0$$ g ψ - 2 3 Λ ψ = 0 with reflecting boundary conditions imposed at infinity. Understanding the behavior of $$\psi$$ ψ at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass $${\mathfrak {m}} = M \sqrt{-\Lambda }$$ m = M - Λ and angular momentum $${\mathfrak {a}} = a \sqrt{-\Lambda }$$ a = a - Λ satisfy a certain non-Diophantine condition, then perturbations $$\psi$$ ψ arising from generic smooth initial data blow up $$|\psi |\rightarrow +\infty$$ | ψ | + at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for $$\Lambda \ge 0$$ Λ 0 —in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters $${\mathfrak {m}}, {\mathfrak {a}}$$ m , a for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking–Reall bound. On the other hand, we conjecture that for a set of parameters $${\mathfrak {m}}, {\mathfrak {a}}$$ m , a which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon $$|\psi |\le C$$ | ψ | C . This suggests that the validity of the $$C^0$$ C 0 -formulation of Strong Cosmic Censorship for $$\Lambda <0$$ Λ < 0 may change in a spectacular way according to the notion of genericity imposed. dc.language en dc.publisher Springer Science and Business Media LLC dc.subject Article dc.title Diophantine approximation as Cosmic Censor for Kerr–AdS black holes dc.type Article dc.date.updated 2022-02-22T16:00:53Z prism.endingPage 1321 prism.issueIdentifier 3 prism.publicationName Inventiones mathematicae prism.startingPage 1169 prism.volume 227 dc.identifier.doi 10.17863/CAM.81736 dcterms.dateAccepted 2021-09-24 rioxxterms.versionofrecord 10.1007/s00222-021-01078-6 rioxxterms.version VoR rioxxterms.licenseref.uri http://creativecommons.org/licenses/by/4.0/ dc.identifier.eissn 1432-1297 cam.issuedOnline 2021-11-24
﻿