Diophantine Approximation as Cosmic Censor for AdS Black Holes
type: CreativeWork
description: The present thesis reveals a novel connection of Diophantine approximation arising from small divisors to general relativity, more precisely, the Strong Cosmic Censorship conjecture. The main results provide theorems which resolve a linear scalar analog of the Strong Cosmic Censorship conjecture in general relativity for Λ < 0. The proofs are intimately tied to small divisors and the resolution crucially depends on suitable Diophantine conditions. A further ingredient is the novel scattering theory on black hole interiors established at first. The thesis consists of three parts. In the first part we develop a scattering theory for the linear wave equation on the interior of Reissner–Nordström black holes. The main result shows the existence, uniqueness and asymptotic completeness of finite energy scattering states on the interior of Reissner–Nordström. The past and future scattering states are represented as suitable traces of the solution ψ on the bifurcate event and Cauchy horizons. Finally, we prove that, in contrast to the above, on the Reissner–Nordström–(Anti-)de Sitter interior, there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass. This part is joint work with Yakov Shlapentokh-Rothman (Princeton University). The second and third parts are motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes. We consider smooth linear perturbations governed by the conformal wave equation on Reissner–Nordström–AdS and Kerr–AdS black holes, respectively. We prescribe initial data on a spacelike hypersurface of a Reissner–Nordström–AdS and Kerr–AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously by work of Holzegel–Smulevici that such waves only decay at a sharp logarithmic rate (in contrast to the polynomial rate in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, th…
abstract: The present thesis reveals a novel connection of Diophantine approximation arising from small divisors to general relativity, more precisely, the Strong Cosmic Censorship conjecture. The main results provide theorems which resolve a linear scalar analog of the Strong Cosmic Censorship conjecture in general relativity for Λ < 0. The proofs are intimately tied to small divisors and the resolution crucially depends on suitable Diophantine conditions. A further ingredient is the novel scattering theory on black hole interiors established at first. The thesis consists of three parts. In the first part we develop a scattering theory for the linear wave equation on the interior of Reissner–Nordström black holes. The main result shows the existence, uniqueness and asymptotic completeness of finite energy scattering states on the interior of Reissner–Nordström. The past and future scattering states are represented as suitable traces of the solution ψ on the bifurcate event and Cauchy horizons. Finally, we prove that, in contrast to the above, on the Reissner–Nordström–(Anti-)de Sitter interior, there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass. This part is joint work with Yakov Shlapentokh-Rothman (Princeton University). The second and third parts are motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes. We consider smooth linear perturbations governed by the conformal wave equation on Reissner–Nordström–AdS and Kerr–AdS black holes, respectively. We prescribe initial data on a spacelike hypersurface of a Reissner–Nordström–AdS and Kerr–AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously by work of Holzegel–Smulevici that such waves only decay at a sharp logarithmic rate (in contrast to the polynomial rate in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, th…
author: Kehle, Christoph
language: eng
keywords: Strong Cosmic Censorship, Diophantine Approximation, General Relativity, AdS, Wave equation, black hole, PDE, Kerr-AdS
license: https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f28f7083-27cb-40a7-a35d-2321ddf588a3/download
license: http://purl.org/NET/rdflicense/allrightsreserved
date published: 2020-07-18
date modified: 2026-07-22T22:24:21Z
provider: Cambridge University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Cambridge

PropertyValue
property ID: DOI
value: 10.17863/CAM.54843