Diophantine Approximation as Cosmic Censor for AdS Black Holes

Abstract

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 $\Box_g\psi-\frac{2}{3}\Lambda \psi=0$ with reflecting boundary conditions 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 $\mathfrak m = M \sqrt{-\Lambda}$ and angular momentum $\mathfrak a = a \sqrt{-\Lambda}$ satisfy a certain non-Diophantine condition, then perturbations $\psi$ arising from generic smooth initial data blow up 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 (arXiv:1812.06142) as well as to previous work on the analogous problem for $\Lambda \geq 0$. As a result of the non-Diophantine condition, the set of parameters $\mathfrak m, \mathfrak a$ 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 $ which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon. 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.