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The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results

Stefanos Aretakis

TL;DR

The paper analyzes the linear wave equation on extreme Reissner-Nordström spacetimes, addressing boundedness, decay, non-decay, and blow-up phenomena up to the degenerate horizon. It develops a robust vector-field framework combining Morawetz/X estimates, horizon-specific conservation laws, and refined higher-order $L^{2}$ controls to overcome the lack of redshift and horizon trapping. The results reveal a nuanced picture: certain energies decay while higher-order transversal derivatives exhibit non-decay or blow-up along the horizon for generic data, with sharp behavior tied to angular frequency. This work provides a foundation for understanding stability and potential instability mechanisms in degenerate black hole spacetimes, and it introduces analytical tools likely applicable to nonlinear problems and other degenerate horizons.

Abstract

We consider solutions to the linear wave equation on a suitable globally hyperbolic subset of an extreme Reissner-Nordstrom spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We obtain boundedness, decay, non-decay and blow-up results. Our estimates hold up to and including the event 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.

The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results

TL;DR

The paper analyzes the linear wave equation on extreme Reissner-Nordström spacetimes, addressing boundedness, decay, non-decay, and blow-up phenomena up to the degenerate horizon. It develops a robust vector-field framework combining Morawetz/X estimates, horizon-specific conservation laws, and refined higher-order controls to overcome the lack of redshift and horizon trapping. The results reveal a nuanced picture: certain energies decay while higher-order transversal derivatives exhibit non-decay or blow-up along the horizon for generic data, with sharp behavior tied to angular frequency. This work provides a foundation for understanding stability and potential instability mechanisms in degenerate black hole spacetimes, and it introduces analytical tools likely applicable to nonlinear problems and other degenerate horizons.

Abstract

We consider solutions to the linear wave equation on a suitable globally hyperbolic subset of an extreme Reissner-Nordstrom spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We obtain boundedness, decay, non-decay and blow-up results. Our estimates hold up to and including the event 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 91 sections, 77 theorems, 500 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extreme Black Holes
  4. Linear Waves
  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. Conservation Laws on $\mathcal{H}^{+}$
  10. Sharp Higher Order $L^{2}$ Estimates
  11. Energy and Pointwise Decay
  12. Higher Order Pointwise Estimates
  13. Remarks on the Analysis of Extreme Black Holes
  14. Dispersion vs Redshift
  15. Trapping Effect on $\mathcal{H}^{+}$
  16. ...and 76 more sections

Key Result

Theorem 1

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

Figures (26)

  • Figure 1: The coordinate systems $\left(t,r\right)$ and $\left(t,r^{*}\right)$. Each point in this diagram represents a sphere of symmetry.
  • Figure 2: The coordinate system $\left(v,r\right)$
  • Figure 3: The coordinate system $\left(t^{*},r\right)$
  • Figure :
  • Figure :
  • ...and 21 more figures

Theorems & Definitions (144)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Definition 5.1
  • ...and 134 more