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 _gpsi -frac{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 {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 |psi |rightarrow +infty 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–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for Lambda ge 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 {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 |psi |le 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.
Highlights
The Kerr–Anti-de Sitter (Kerr–AdS) black hole spacetimes (M, g) constitute a 2-parameter family of solutions to the celebrated Einstein equations Ricμν (g) 1 2 Rgμν +gμν = 8π Tμν (1.1)in vacuum (Tμν = 0) and with negative cosmological constant < 0
We show in Theorem 1 that√there exists a set√PBlow-up of dimensionless Kerr–AdS parameters m := M − and a := a − which is Baire-generic but Lebesgue-exceptional, such that on all Kerr–AdS black hole whose parameters lie in PBlow-up, generic linear scalar perturbations ψ blow up |ψ| → +∞ at the Cauchy horizon
Diophantine approximation and the small divisors problem are intimately tied to the problem of the stability of the solar system [89] and more generally, the stability of Hamiltonian systems in classical mechanics
Summary
The Kerr–Anti-de Sitter (Kerr–AdS) black hole spacetimes (M, g) constitute a 2-parameter family of solutions to the celebrated Einstein equations. One can view (1.2) as a linear scalar analog of (1.1), and so the linear scalar analog of the C0-formulation of Strong Cosmic Censorship is the statement that for generic black hole parameters, linear scalar perturbations ψ, arising from generic initial data for (1.2), fail to be continuous at the Cauchy horizon (see already Conjecture 3). We√ conjecture√that, if the dimensionless black hole parameters m = M − and a = a − satisfy a Diophantine condition, linear scalar perturbations ψ remain bounded |ψ| ≤ C at the Cauchy horizon. This would hold for Lebesgue-generic but Baire-exceptional black hole parameters If true, this would provide a negative resolution of the linear scalar analog of the C0-formulation of Strong Cosmic Censorship provided that genericity of the parameters is taken in the Lebesgue-generic sense.
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