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A note on instabilities of extremal black holes under scalar perturbations from afar

Stefanos Aretakis

TL;DR

The paper shows that instabilities for scalar waves on extremal black holes arise not only from data on the horizon but also from initial data supported arbitrarily far away from it, with higher-order transverse derivatives blowing up along the horizon. The authors construct a time integral φ and define ψ=Tφ to produce compactly supported initial data away from the horizon that nevertheless generate growing higher derivatives along H^+ for extremal Kerr (and RN). The results extend the horizon-based instability picture to data far from the horizon, providing a rigorous link to numerical findings and underscoring the robust, generic nature of extremal-horizon instabilities in this setting.

Abstract

In previous work of the author it was shown that instabilities of solutions to the wave equation develop asymptotically along the event horizon of extremal Kerr provided a certain expression H of the initial data is non-trivial on the horizon. In this note we remove this restriction by showing that instabilities develop even from initial data supported arbitrarily far away from the horizon (for which, in particular, H=0). The latter instabilities concern one order higher derivatives compared to the case where H is non-zero. The result also applies to extremal Reissner-Nordstrom. This note was motivated by numerical analysis of Lucietti, Murata, Reall and Tanahashi.

A note on instabilities of extremal black holes under scalar perturbations from afar

TL;DR

The paper shows that instabilities for scalar waves on extremal black holes arise not only from data on the horizon but also from initial data supported arbitrarily far away from it, with higher-order transverse derivatives blowing up along the horizon. The authors construct a time integral φ and define ψ=Tφ to produce compactly supported initial data away from the horizon that nevertheless generate growing higher derivatives along H^+ for extremal Kerr (and RN). The results extend the horizon-based instability picture to data far from the horizon, providing a rigorous link to numerical findings and underscoring the robust, generic nature of extremal-horizon instabilities in this setting.

Abstract

In previous work of the author it was shown that instabilities of solutions to the wave equation develop asymptotically along the event horizon of extremal Kerr provided a certain expression H of the initial data is non-trivial on the horizon. In this note we remove this restriction by showing that instabilities develop even from initial data supported arbitrarily far away from the horizon (for which, in particular, H=0). The latter instabilities concern one order higher derivatives compared to the case where H is non-zero. The result also applies to extremal Reissner-Nordstrom. This note was motivated by numerical analysis of Lucietti, Murata, Reall and Tanahashi.

Paper Structure

This paper contains 13 sections, 3 theorems, 51 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Geometric setup
  3. The Kerr metric
  4. The initial hypersurface $\Sigma$
  5. The well-posedness of the wave equation
  6. Construction of the time integral $\phi$
  7. Construction of $\psi$ with compactly supported initial data
  8. Instabilities for $\psi$ along the horizon
  9. Remarks
  10. Genericity of instabilities
  11. Initial perturbations on $t=0$
  12. Extremal Reissner--Nordström
  13. Acknowledgements

Key Result

Theorem 1

Consider the extremal Kerr black hole with parameters $a,M$ such that $|a|=M$. Let $\Sigma$ be a spacelike hypersurface which either crosses the event horizon and satisfies the assumptions of Section sec:TheInitialHypersurfaceSigma or coincides with $t=0$, where $t$ denotes the Boyer--Lindquist time along $\mathcal{H}^{+}$ as the advanced time $v\rightarrow +\infty$. Here $S_{v}$ denotes the (sphe

Figures (8)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof