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

Stefanos Aretakis

TL;DR

This paper analyzes linear scalar perturbations on the exterior of an extreme Reissner–Nordström black hole, focusing on the intricate interplay between horizon redshift degeneracy and photon-sphere trapping. It develops a robust vector-field method, combining Morawetz/X estimates, Hardy inequalities, and carefully constructed currents to prove boundedness, local integrated decay up to the horizon, and pointwise bounds, while also revealing non-decay phenomena along the horizon for certain low-frequency modes. A key outcome is that horizon degeneracy enforces a separation between dispersion and redshift effects, and trapping on the horizon necessitates higher-regularity and transversal commutations to obtain non-degenerate estimates. These results establish a foundation for the full linear stability/instability analysis (in a companion paper) and highlight qualitative differences between extreme and non-extreme black holes relevant to nonlinear stability questions.

Abstract

We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider 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 obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon. The fundamental new aspect of this problem is the degeneracy of the redshift on the event horizon. Several new analytical features of degenerate horizons are also presented.

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

TL;DR

This paper analyzes linear scalar perturbations on the exterior of an extreme Reissner–Nordström black hole, focusing on the intricate interplay between horizon redshift degeneracy and photon-sphere trapping. It develops a robust vector-field method, combining Morawetz/X estimates, Hardy inequalities, and carefully constructed currents to prove boundedness, local integrated decay up to the horizon, and pointwise bounds, while also revealing non-decay phenomena along the horizon for certain low-frequency modes. A key outcome is that horizon degeneracy enforces a separation between dispersion and redshift effects, and trapping on the horizon necessitates higher-regularity and transversal commutations to obtain non-degenerate estimates. These results establish a foundation for the full linear stability/instability analysis (in a companion paper) and highlight qualitative differences between extreme and non-extreme black holes relevant to nonlinear stability questions.

Abstract

We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider 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 obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon. The fundamental new aspect of this problem is the degeneracy of the redshift on the event horizon. Several new analytical features of degenerate horizons are also presented.

Paper Structure

This paper contains 62 sections, 43 theorems, 183 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extreme Black Holes
  4. Linear Scalar Perturbations
  5. Previous Results for Waves on Non-Extreme Black Holes
  6. Overview of Results and Techniques
  7. Zeroth Order Morawetz and $X$ Estimates
  8. Uniform Boundedness of Non-Degenerate Energy
  9. Non-Decay along $\mathcal{H}^{+}$
  10. Local Integrated Energy Decay
  11. Pointwise Uniform Boundedness
  12. Remarks on the Analysis of Extreme Black Holes
  13. Dispersion vs Redshift
  14. Trapping Effect on $\mathcal{H}^{+}$
  15. Geometry of Extreme Reissner-Nordström Spacetime
  16. ...and 47 more sections

Key Result

Theorem 1

(Morawetz and $X$ Estimates) Let $\delta >0$. There exists a constant $C_{\delta}>0$ which depends on $M$, $\delta$ and $\Sigma_{0}$ such that for all solutions $\psi$ of the wave equation the following estimates hold

Figures (11)

  • Figure :
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  • ...and 6 more figures

Theorems & Definitions (82)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 5.2.1
  • proof
  • Proposition 6.0.1
  • proof
  • Proposition 6.0.2
  • ...and 72 more