Diophantine approximation as Cosmic Censor for Kerr-AdS black holes

• 成果类型: Article
• 署名作者: Kehle, Christoph
• 署名单位: University of Cambridge; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01078-6
• 发表日期: 2022
• 页码: 1169-1321
• 关键词: massive wave-equation; quasi-periodic solutions; cauchy horizon; bianchi cosmologies; orthogonal models; systems approach; linear waves; scalar waves; instability; interior

摘要

The 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 and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations psi of Kerr-AdS solving square(g)psi - 2/3 Lambda psi = 0 with reflecting boundary conditions imposed at infinity. Understanding the behavior of psi 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 m = M root-Lambda and angular momentum a = a root-Lambda satisfy a certain non-Diophantine condition, then perturbations psi arising from generic smooth initial data blow up vertical bar psi vertical bar -> +infinity 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-Nordstrom-AdS (Kehie in Commun Math Phys 376(1):145-200, 2020) as well as to previous work on the analogous problem for Lambda >= 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 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 m, a which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon vertical bar psi vertical bar <= C. This suggests that the validity of the C-0-formulation of Strong Cosmic Censorship for Lambda < 0 may change in a spectacular way according to the notion of genericity imposed.