Instability of the Kerr Cauchy horizon under linearised gravitational perturbations

TL;DR

The paper establishes a rigorous linear blue-shift instability at Kerr’s Cauchy horizon for gravitational perturbations by proving that a Teukolsky s=+2 field, under horizon decay bounds with p0≥2, generically blows up at the CH when analyzed in a regular frame. The authors implement a low-frequency analysis via Teukolsky separation, analyze the ω=0 Heun equation, and relate horizon data to interior behavior through transmission coefficients, with m≠0 treated by explicit hypergeometric limits and m=0 handled via Teukolsky–Starobinsky conservation. This work provides the first precise, quantitative criterion linking event-horizon decay to interior singularity formation for Kerr perturbations, supporting the strong cosmic censorship program by indicating a weak null singularity at CH and guiding prospects for nonlinear stability/instability results. The results are shown to hold generically for slowly rotating Kerr with compact data (Ma–Zhang), and they lay a foundation for extending the analysis toward nonlinear Einstein dynamics and broader parameter ranges, thereby contributing a key linear milestone toward inextendibility and deterministic breakdown in rotating black hole interiors.

Abstract

This paper establishes a mathematical proof of the blue-shift instability at the sub-extremal Kerr Cauchy horizon for the linearised vacuum Einstein equations. More precisely, we exhibit conditions on the $s=+2$ Teukolsky field, consisting of suitable integrated upper and lower bounds on the decay along the event horizon, that ensure that the Teukolsky field, with respect to a frame that is regular at the Cauchy horizon, becomes singular. The conditions are in particular satisfied by solutions of the Teukolsky equation arising from generic and compactly supported initial data by the recent work [51] of Ma and Zhang for slowly rotating Kerr.

Figures (9)

• Figure 1: The blue-shift effect. For observer $A$ an infinite time passes, while observer $B$ reaches the Cauchy horizon in finite time; signals sent by $A$ are received by $B$ shifted to the blue.
• Figure 2: The statement of Theorem \ref{['ThmInt']}.
• Figure 3: The statement of Theorem \ref{['ThmDafLuk']}
• Figure 4: Spacelike singularity emanating from $i^+$ is ruled out. Picture cannot occur.
• Figure 5: The interior of subextremal Reissner-Nordström

Theorems & Definitions (95)

• Theorem 1.1
• Theorem 1.4: Dafermos-Luk, DafLuk17
• Theorem 1.6: Corollary 4.2 in LukOhShlaForth
• Theorem 1.11: Theorem 4.1 in LukOhShlaForth
• Definition 2.15
• Remark 2.16
• Proposition 2.18
• proof
• Definition 2.22
• Lemma 2.23