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Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II

Stefanos Aretakis

TL;DR

This work provides a complete description of linear scalar perturbations on extreme Reissner–Nordström spacetimes, establishing rigorous energy and pointwise decay, non-decay, and blow-up phenomena up to the horizon. A key technical advance is the introduction of a horizon-friendly vector field $P$ that yields degeneracy-aware energy decay near $ ext{H}^+$, complemented by a hierarchy of conservation laws $H_l[]$ on degenerate horizons for each angular frequency $l$. The results reveal a sharp separation by angular frequency: transversal derivatives up to order $l$ decay while order $l+1$ typically converges to a nonzero limit along $ ext{H}^+$ and higher orders blow up, signaling dynamical instability in the extreme case. The paper also develops sharp higher-order $L^{2}$ estimates, precise commutator analyses, and dyadic decay arguments, culminating in detailed energy decay, pointwise decay, and blow-up statements for both degenerate and non-degenerate energies. These findings refine the stability/instability picture for extreme black holes and set the stage for extensions to more general spacetimes and coupled systems.

Abstract

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon. A hierarchy of conservations laws on degenerate horizons is also derived.

Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II

TL;DR

This work provides a complete description of linear scalar perturbations on extreme Reissner–Nordström spacetimes, establishing rigorous energy and pointwise decay, non-decay, and blow-up phenomena up to the horizon. A key technical advance is the introduction of a horizon-friendly vector field that yields degeneracy-aware energy decay near , complemented by a hierarchy of conservation laws on degenerate horizons for each angular frequency . The results reveal a sharp separation by angular frequency: transversal derivatives up to order decay while order typically converges to a nonzero limit along and higher orders blow up, signaling dynamical instability in the extreme case. The paper also develops sharp higher-order estimates, precise commutator analyses, and dyadic decay arguments, culminating in detailed energy decay, pointwise decay, and blow-up statements for both degenerate and non-degenerate energies. These findings refine the stability/instability picture for extreme black holes and set the stage for extensions to more general spacetimes and coupled systems.

Abstract

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon. A hierarchy of conservations laws on degenerate horizons is also derived.

Paper Structure

This paper contains 41 sections, 35 theorems, 174 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of Results and Techniques
  4. Conservation Laws on $\mathcal{H}^{+}$
  5. Sharp Higher Order $L^{2}$ Estimates
  6. Energy and Pointwise Decay
  7. Energy and Pointwise Decay
  8. Higher Order Estimates: Energy and Pointwise Decay, Non-decay and Blow-up
  9. Open Problems
  10. The Main Theorems
  11. Notation
  12. Summary of Results of Part I
  13. The Statements of the Main Theorems
  14. Conservation Laws on Degenerate Event Horizons
  15. Applications
  16. ...and 26 more sections

Key Result

Theorem 1

(Conservation Laws along $\mathcal{H}^{+}$) For all $l\in\mathbb{N}$ there exist constants $\beta_{i},i=0,1,...,l$, which depend on $M$ and $l$ such that for all solutions $\psi$ which are supported on the (fixed) angular frequency $l$ the quantity is conserved along the null geodesics of $\mathcal{H}^{+}$.

Figures (6)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (70)

  • Theorem 1
  • Theorem 2
  • Remark 2.1
  • Theorem 3
  • Remark 2.2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 3.0.1
  • proof
  • ...and 60 more