Diophantine approximation as Cosmic Censor for Kerr-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 Λ < 0 and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity.We study linear scalar perturbations ψ of Kerr-AdS solving g ψ− 2 3 Λψ = 0 with reflecting boundary conditions imposed at infinity. Understanding the behavior of ψ 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 √ −Λ and angular momentum a = a √ −Λ satisfy a certain non-Diophantine condition, then perturbations ψ arising from generic smooth initial data blow up |ψ| → +∞ 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 [61] as well as to previous work on the analogous problem for Λ ≥ 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 |ψ| ≤ C. This suggests that the validity of the C 0 -formulation of Strong Cosmic Censorship for Λ < 0 may change in a spectacular way according to the notion of genericity imposed.

Echoes of Compact Objects in Scalar-Tensor Theories of Gravity

“…It is worthy to note that further analytical investigations of perturbations in AdS BHs led to the conclusion that solutions of the Klein-Gordon equation with fixed angular quantum number , indeed decay exponentially [63]. However, an accumulation of all solutions, possessing finite energy, achieves a logarithmic decay rate, due to the presence of stable trapping [64] (for a discussion of the wave equation in the interior of AdS BHs see [65,66]). In Fig.…”

Blowup of the local energy of linear waves at the Reissner-Nordström-AdS Cauchy horizon

“…Reminiscent of small divisors problem are certain Diophantine condition which turn out (see [26]) to play a crucial role in addressing Conjecture 1 for Kerr-AdS. To state the following theorem we denote with P the parameter space of dimensionless masses and angular momenta (m, • P Blowup ⊂ P is dense,…”

“…On the other hand the proof of Theorem 2 in [26] and the heuristics outlined in [26] strongly suggest that for the complement of P Blowup , a set which is Lebesgue-generic (but Baire-exceptional), the amplitude of ψ remains uniformly bounded at the Cauchy horizon. • P Bounded ⊂ P is dense,…”

A scattering theory approach to Cauchy horizon instability and applications to mass inflation

“…(Note that such a point of view for the black hole interior is not new, and has been used in [9,18,25]. See also [17,19] in different settings. The results in [18] are especially relevant to our paper.)…”