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Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds

Yannis Angelopoulos, Stefanos Aretakis, Dejan Gajic

TL;DR

This paper delivers rigorous, fully global late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordström backgrounds, expressing leading coefficients directly in terms of initial data. It develops purely physical-space methods, including novel $r^p$-weighted energy hierarchies and a singular time inversion theory, to disentangle near-horizon and near-infinity contributions to tails. A new horizon charge $H_0^{(1)}[\psi]$ is introduced and linked to time-inverted data, enabling precise descriptions of horizon, null-infinity, and interior dynamics, with results that confirm and extend prior heuristic and numerical analyses. The framework also yields decay properties for scalar invariants and insights into interior behavior relevant to strong cosmic censorship.

Abstract

We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordström black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory, which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti-Murata-Reall-Tanahashi, Ori-Sela and Blaksley-Burko.

Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds

TL;DR

This paper delivers rigorous, fully global late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordström backgrounds, expressing leading coefficients directly in terms of initial data. It develops purely physical-space methods, including novel -weighted energy hierarchies and a singular time inversion theory, to disentangle near-horizon and near-infinity contributions to tails. A new horizon charge is introduced and linked to time-inverted data, enabling precise descriptions of horizon, null-infinity, and interior dynamics, with results that confirm and extend prior heuristic and numerical analyses. The framework also yields decay properties for scalar invariants and insights into interior behavior relevant to strong cosmic censorship.

Abstract

We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner-Nordström black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory, which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti-Murata-Reall-Tanahashi, Ori-Sela and Blaksley-Burko.

Paper Structure

This paper contains 78 sections, 64 theorems, 415 equations, 10 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Introduction and first remarks
  3. Outline of the introduction
  4. Asymptotics for the wave equation on sub-extremal black holes
  5. Physical importance of extremal black holes
  6. The horizon instability of extremal black holes
  7. Conservation laws along the event horizon
  8. The trapping effect on the event horizon
  9. Energy and pointwise boundedness and weak decay
  10. Energy and pointwise blow-up
  11. Physics literature on the dynamics of extremal Reissner--Nordström
  12. The Blaksley--Burko asymptotic analysis
  13. The Lucietti--Murata--Reall--Tanahashi asymptotic analysis
  14. The Ori--Sela asymptotic analysis
  15. The Murata--Reall--Tanahashi spacetimes
  16. ...and 63 more sections

Key Result

Theorem 4.1

Solutions $\psi$ to eq:waveequation on extremal Reissner--Nordström arising from smooth compactly supported data on an extension of $\Sigma_0$ into the black hole interior are extendible across the black hole inner horizon as functions in $C^{0,\alpha}\cap W^{1,2}_{\rm loc}$, with $\alpha<1$. Furthe

Figures (10)

  • Figure 1: The initial value problem for the wave equation on black hole backgrounds.
  • Figure 2: The local redshift effect for sub-extremal horizons.
  • Figure 3: The sections $S_{\tau}$ of $\mathcal{H}^{+}$ and the transversal to $\mathcal{H}^{+}$ vector field $Y$.
  • Figure 4: The Newman--Penrose constant on $\mathcal{I}^{+}$.
  • Figure 5: Time inversion for the spherical mean $\psi_0$ of $\psi$.
  • ...and 5 more figures

Theorems & Definitions (130)

  • Definition 1.1
  • Definition 2.1
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Theorem 5.1: Asymptotics for Type C perturbations
  • Theorem 5.2: Asymptotics for Type A perturbations
  • Theorem 5.3: Asymptotics for Type D perturbations
  • Theorem 5.4: Asymptotics for Type B perturbations
  • Theorem 5.5: Logarithmic corrections for Type C perturbations
  • ...and 120 more