Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Charged scalar fields on Reissner--Nordström spacetimes I: integrated energy estimates

Dejan Gajic

Abstract

This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordström spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner--Nordström(--de Sitter) spacetimes. These estimates extend to Reissner--Nordström spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner--Nordström spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes.

Charged scalar fields on Reissner--Nordström spacetimes I: integrated energy estimates

Abstract

This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner--Nordström spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner--Nordström(--de Sitter) spacetimes. These estimates extend to Reissner--Nordström spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner--Nordström spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes.

Paper Structure

This paper contains 59 sections, 68 theorems, 548 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Previous results
  3. Integrated energy estimates on sub-extremal black holes
  4. Integrated energy estimates on extremal black holes
  5. Integrated estimates for charged scalar fields
  6. Sketch of the proof
  7. Fourier transform
  8. Step 1: high-frequency regimes
  9. Step 2: bounded frequency regimes
  10. Step 3: Plancherel
  11. Step 4: verifying future integrability
  12. Comparison with wave equation on extremal Kerr
  13. Overview of the remainder of the paper
  14. Acknowledgments
  15. Preliminaries
  16. ...and 44 more sections

Key Result

Theorem 1.1

Let $\epsilon>0$. Assume that $0\leq 1-\frac{|Q|}{M}\ll 1$ or $|\mathfrak{q}Q|\ll 1$. Write: Then there exists a uniform constant $C>0$ such that for all $\tau_1\leq \tau_2$: where: The estimate holds also when $1-\frac{|Q|}{M}\gtrsim 1$ and $|\mathfrak{q}Q|\gtrsim 1$, under the assumption of mode stability on the real axis.

Figures (2)

  • Figure 1: A sketch of the graph of the potential $V_{\ell \omega}$ as a function of $\rho_+$ in the high, $\ell$-dominated frequency regime with $q\widetilde{\omega}<0$.
  • Figure 2: A sketch of the graph of the potential $V$ as a function of $\rho_+$ in the frequency regime $\mathcal{F}_{\sharp,{\rm angular}}$ with $q\widehat{\omega}<0$ and $q\widetilde{\omega}<0$, $\kappa_c$ suitably small and sketches of the graphs of the functions $f$, $h$, $\chi_T$ and $\chi_K$ that appear in the currents $j_2^h[u]$, $j_3^f[u]$, $\chi_Tj^T[u]$ and $\chi_Kj^K[u]$.

Theorems & Definitions (142)

  • Theorem 1.1: Energy boundedness and integrated energy estimates; informal version of Theorem \ref{['thm:main']}
  • Remark 1.1: Coupling between Morawetz and $(\Omega^{-1}r)^{p}$-weighted energy estimates
  • Remark 1.2: Loss of weights in energy boundedness
  • Remark 1.3: Small $|\mathfrak{q}|$
  • Remark 1.4: Mode stability
  • Remark 1.5: The role of $\Lambda$
  • Remark 1.6: Electromagnetic gauge
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 132 more