Diophantine approximation as Cosmic Censor for Kerr–AdS black holes

Abstract

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Lambda &lt;0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations <jats:inline-formula><jats:alternatives><jats:tex-math>$$\psi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> of Kerr–AdS solving <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Box _g\psi -\frac{2}{3}\Lambda \psi =0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>□</mml:mo> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>ψ</mml:mi> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mi>Λ</mml:mi> <mml:mi>ψ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> with reflecting boundary conditions imposed at infinity. Understanding the behavior of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\psi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {m}} = M \sqrt{-\Lambda }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>M</mml:mi> <mml:msqrt> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and angular momentum <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {a}} = a \sqrt{-\Lambda }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:msqrt> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Λ</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> satisfy a certain non-Diophantine condition, then perturbations <jats:inline-formula><jats:alternatives><jats:tex-math>$$\psi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> arising from generic smooth initial data blow up <jats:inline-formula><jats:alternatives><jats:tex-math>$$|\psi |\rightarrow +\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>→</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Lambda \ge 0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>—in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {m}}, {\mathfrak {a}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathfrak {m}}, {\mathfrak {a}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon <jats:inline-formula><jats:alternatives><jats:tex-math>$$|\psi |\le C$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. This suggests that the validity of the <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-formulation of Strong Cosmic Censorship for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Lambda &lt;0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> may change in a spectacular way according to the notion of genericity imposed.</jats:p>