Diophantine approximation as Cosmic Censor for Kerr-AdS black holes

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.

Automated summary

This paper investigates the unexpected connection between Diophantine approximation and waves behavior inside black holes with negative cosmological constant Λ < 0, analyzing the consequences for the Strong Cosmic Censorship conjecture in general relativity. The study of linear scalar perturbations reveals a novel resonance phenomenon between stable trapping on the black hole exterior and interior scattering operator poles, leading to blow-up of perturbations under certain non-Diophantine conditions. The results present a stark contrast to previous findings on black holes with Λ >= 0, suggesting a potential change in the behavior of linear scalar perturbations at the Cauchy horizon based on the notion of genericity.