Diophantine approximation as Cosmic Censor for Kerr–AdS black holes
Authors
Kehle, Christoph
Publication Date
2022-03Journal Title
Inventiones mathematicae
ISSN
0020-9910
Publisher
Springer Science and Business Media LLC
Volume
227
Issue
3
Pages
1169-1321
Language
en
Type
Article
This Version
VoR
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Kehle, C. (2022). Diophantine approximation as Cosmic Censor for Kerr–AdS black holes. Inventiones mathematicae, 227 (3), 1169-1321. https://doi.org/10.1007/s00222-021-01078-6
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>$$\Lambda <0$$</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> 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>$$\Lambda <0$$</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> may change in a spectacular way according to the notion of genericity imposed.</jats:p>
Keywords
Article
Identifiers
s00222-021-01078-6, 1078
External DOI: https://doi.org/10.1007/s00222-021-01078-6
This record's URL: https://www.repository.cam.ac.uk/handle/1810/334323
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Licence:
http://creativecommons.org/licenses/by/4.0/
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