Diophantine Approximation as Cosmic Censor for AdS Black Holes
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, the question of uniform boundedness or blow-up at the Cauchy horizon in the black hole interior (and thus the validity of the linear scalar analog of the C0-formulation of the Strong Cosmic Censorship conjecture) has remained up to now open. In the second part of the thesis, we answer the question of uniform boundedness in the affirmative for Reissner–Nordström–AdS: We show that |ψ| ≤ C in the black hole interior. In this setting, this corresponds to the statement that the linear scalar analog of the C0-formulation of Strong Cosmic Censorship is false. The proof follows a new approach, combining physical space estimates with Fourier based estimates exploited in the scattering theory developed in the first part. In the third part of the thesis, we show that |ψ| → ∞ at the Cauchy horizon of Kerr–AdS if the dimensionless black hole parameters mass and angular momentum satisfy certain Diophantine properties. This is in stark contrast to the second part as well as previous works on Strong Cosmic Censorship for Λ ≥ 0. In particular, as a result of the Diophantine conditions, we show that these resonant black hole parameters form a Baire-generic but Lebesgue-exceptional subset of parameters below the Hawking–Reall bound. On the other hand, we conjecture that, as is the case for Reissner–Nordström–AdS, linear waves remain bounded at the Cauchy horizon |ψ| ≤ C for a set of black hole parameters which is Baire-exceptional but Lebesgue-generic. This means that the answer to the above question concerning uniform boundedness or blow-up on the Kerr–AdS interior is either negative or affirmative depending on the parameters considered. Thus, in this setting, the validity of the linear scalar analog of the C0 -formulation of Strong Cosmic Censorship depends in an unexpected way on the notion of genericity imposed.
Strong Cosmic Censorship, Diophantine Approximation, General Relativity, AdS, Wave equation, black hole, PDE, Kerr-AdS
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