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Precise late-time asymptotics of scalar field in the interior of a subextreme Kerr black hole and its application in Strong Cosmic Censorship conjecture

Siyuan Ma, Lin Zhang

Abstract

In this work, we compute the precise late-time asymptotics for the scalar field in the interior of a non-static subextreme Kerr black hole, based on recent progress on deriving its precise asymptotics in the Kerr exterior region. This provides a new proof of the generic $H^1_{\text{loc}}$-inextendibility of the Kerr Cauchy horizon against scalar perturbations that is first shown by Luk--Sbierski (J. Func. Anal., 2016). The analogous results in Reissner--Nordström spacetimes are also discussed.

Precise late-time asymptotics of scalar field in the interior of a subextreme Kerr black hole and its application in Strong Cosmic Censorship conjecture

Abstract

In this work, we compute the precise late-time asymptotics for the scalar field in the interior of a non-static subextreme Kerr black hole, based on recent progress on deriving its precise asymptotics in the Kerr exterior region. This provides a new proof of the generic -inextendibility of the Kerr Cauchy horizon against scalar perturbations that is first shown by Luk--Sbierski (J. Func. Anal., 2016). The analogous results in Reissner--Nordström spacetimes are also discussed.
Paper Structure (21 sections, 11 theorems, 173 equations, 7 figures)

This paper contains 21 sections, 11 theorems, 173 equations, 7 figures.

Table of Contents

  1. Main statement and introduction
  2. Literature on the Strong Cosmic Censorship conjecture
  3. Sketch of the proof
  4. Overview of the paper
  5. Geometry of Kerr spacetimes and preliminaries
  6. Scalar functions in the Kerr interior
  7. Operators and derivatives
  8. Eddington--Finkelstein coordinates
  9. Double-null-like coordinate systems
  10. Scalar wave equation in Kerr
  11. Mode decomposition
  12. Precise late-time asymptotics in the region $\mathcal{D}^+_{\text{init}}\cap\{r_{\mathfrak{b}}\leq r\leq r_+\}$
  13. Energy decay for scalar field in region $\mathrm{\mathbf{I}}$
  14. Energy decay for $\psi_{\ell=0}$ and $\psi_{\ell\geqslant 1}$ in $\mathrm{\mathbf{I}}$
  15. Precise late-time asymptotics in region $\mathrm{\mathbf{I}}$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $0<\lvert a\rvert<M$. Let $\psi$ be a solution to the scalar wave equation eq:wave arising from smooth, compactly supported initial data on $\Sigma_{\text{init}}$. Let $j_1\in \mathbb{N}$, $s_1>0$ be suitably largeBy examining the proof, we find that $j_1=4$ and $s_1=9$ are sufficient. The reaso Assume moreover that there are real constants $c_0$ and $c_0'$ and an $\epsilon>0$Without loss of g

Figures (7)

  • Figure 1: Kerr interior and Kerr exterior regions
  • Figure 2: Two-ended initial value problem
  • Figure 3: Region ${_{r}}\mathcal{D}^{+}_{\text{init}}$ and its subregions
  • Figure 4: Region ${_{l}}\mathcal{D}^{+}_{\text{init}}$ and its subregions
  • Figure 5: Different asymptotic regions in the exterior and interior of a Kerr black hole
  • ...and 2 more figures

Theorems & Definitions (24)

  • Theorem 1.1: Global precise late-time asymptotics for the scalar field and its derivatives in Kerr interior
  • Remark 1.2: Global late-time precise asymptotics in ${_{l}}\mathcal{D}^{+}_{\text{init}}\cup{_{r}}\mathcal{D}^{+}_{\text{init}}$
  • Remark 1.3: Global late-time precise asymptotics for $\psi$ and its principal null derivatives in $\mathcal{D}^+_{\text{init}}$
  • Remark 1.4: Precise late-time asymptotics for nondegenerate transverse derivative near event horizon
  • Theorem 1.5: Global precise late-time asymptotics for scalar field in the interior of a Reissner--Nordström spacetime
  • Theorem 1.6: Precise asymptotics for the scalar field on the Kerr event horizon hintz2022sharpangelopoulos2021lateMZ21PLKerr
  • Remark 1.7: Generic inextendibility for scalar field in Kerr interior
  • Theorem 3.1: Precise late-time asymptotics in the region $\mathcal{D}^+_{\text{init}}\cap\{r_{\mathfrak{b}}\leq r\leq r_+\}$
  • Proposition 3.2
  • proof
  • ...and 14 more