论文标题
双苯胺的近似值作为Kerr-Ads黑洞的宇宙审查员
Diophantine approximation as Cosmic Censor for Kerr-AdS black holes
论文作者
论文摘要
本文的目的是显示二磷酸近似和在具有负宇宙常数$λ<0 $的黑洞内部的波浪行为之间的意外联系，并探索了这种后果在一般相对论中对强宇宙审查的后果。我们研究Kerr-ads的线性标量扰动$ \box_gψ-\ frac {2} {3}λψ= 0 $，并反映了无穷大的边界条件。了解Cauchy Horizon上$ψ$的行为对应于强宇宙审查问题的线性类似物。我们的主要结果表明，如果无量纲的黑洞参数质量$ \ mathfrak m = m \ sqrt {-λ} $和角动量$ \ mathfrak a = a \ sqrt { - λ} $满足某些非diophantine条件，则可以使一般平稳的初始数据出现在普通的初始数据中。证明至关重要的是，黑洞外部稳定捕获与内部散射算子的极点之间的新型共振现象，这导致了一个小的除数问题。我们的结果与Reissner-Nordström-Ads（Arxiv：1812.06142）的结果以及对类似问题的先前工作的结果形成鲜明对比。由于非毒液条件，参数集$ \ mathfrak m，\ mathfrak a $ a $我们显示的爆炸形成了baire-generic-generic，但lebesgue除外的所有参数的子集，霍金 - 真实界限下方的所有参数。另一方面，我们猜想，对于一组参数$ \ mathfrak m，\ mathfrak a $ a $是baire-excefialitiational，但lebesgue-generic，所有线性标量扰动都保持在Cauchy Horizon的边界。这表明$ C^0 $ - 对$λ<0 $的强宇宙审查制度的有效性可能会根据所施加的仿制药的概念以壮观的方式改变。
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 $\Box_gψ-\frac{2}{3}Λψ=0$ with reflecting boundary conditions 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 $\mathfrak m = M \sqrt{-Λ}$ and angular momentum $\mathfrak a = a \sqrt{-Λ}$ satisfy a certain non-Diophantine condition, then perturbations $ψ$ 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-Nordström-AdS (arxiv:1812.06142) as well as to previous work on the analogous problem for $Λ\geq 0$. 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 $Λ<0$ may change in a spectacular way according to the notion of genericity imposed.
