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The linear stability of Reissner-Nordström spacetime: the full subextremal range

Elena Giorgi

TL;DR

This work establishes the linear stability of subextremal Reissner–Nordström spacetimes to gravitational and electromagnetic perturbations for the full subextremal range |Q|<M. It introduces a symmetric mixed spin ±1/±2 Regge–Wheeler system built from gauge-invariant perturbation quantities, and defines a combined energy-momentum tensor that enables cancellations without assuming small charge. Through physical-space analysis, the authors prove uniform energy bounds, Morawetz (integrated decay), and an r^p-hierarchy that yields decay rates for the coupled perturbations, thereby completing the linear stability program in this regime. The approach leverages the adjoint-sphere structure of the coupling to obtain a positive, conserved energy and robust decay estimates, integrating prior results on gauge-invariant quantities with a new symmetric formulation. The results are significant for understanding perturbative dynamics around RN black holes and provide a foundation for potential nonlinear stability analyses in the subextremal setting.

Abstract

We prove the linear stability of subextremal Reissner-Nordström spacetimes as solutions to the Einstein-Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations separately derived in previous works, where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range |Q|<M, completely in physical space. Such decay estimates, together with the estimates for the gauge-dependent quantities of the perturbations previously obtained, settle the problem of linear stability to gravitational and electromagnetic perturbations of Reissner-Nordström solution in the full subextremal range |Q|< M.

The linear stability of Reissner-Nordström spacetime: the full subextremal range

TL;DR

This work establishes the linear stability of subextremal Reissner–Nordström spacetimes to gravitational and electromagnetic perturbations for the full subextremal range |Q|<M. It introduces a symmetric mixed spin ±1/±2 Regge–Wheeler system built from gauge-invariant perturbation quantities, and defines a combined energy-momentum tensor that enables cancellations without assuming small charge. Through physical-space analysis, the authors prove uniform energy bounds, Morawetz (integrated decay), and an r^p-hierarchy that yields decay rates for the coupled perturbations, thereby completing the linear stability program in this regime. The approach leverages the adjoint-sphere structure of the coupling to obtain a positive, conserved energy and robust decay estimates, integrating prior results on gauge-invariant quantities with a new symmetric formulation. The results are significant for understanding perturbative dynamics around RN black holes and provide a foundation for potential nonlinear stability analyses in the subextremal setting.

Abstract

We prove the linear stability of subextremal Reissner-Nordström spacetimes as solutions to the Einstein-Maxwell equation. We make use of a novel representation of gauge-invariant quantities which satisfy a symmetric system of coupled wave equations. This system is composed of two of the three equations separately derived in previous works, where the estimates required arbitrary smallness of the charge. Here, the estimates are obtained by defining a combined energy-momentum tensor for the system in terms of the symmetric structure of the right hand sides of the equations. We obtain boundedness of the energy, Morawetz estimates and decay for the full subextremal range |Q|<M, completely in physical space. Such decay estimates, together with the estimates for the gauge-dependent quantities of the perturbations previously obtained, settle the problem of linear stability to gravitational and electromagnetic perturbations of Reissner-Nordström solution in the full subextremal range |Q|< M.

Paper Structure

This paper contains 19 sections, 17 theorems, 209 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The spin $\pm2$ system of equations in Giorgi4
  3. The spin $\pm1$ equation in Giorgi5
  4. The proof of linear stability for small charge in Giorgi6
  5. The mixed spin $\pm1$ and spin $\pm2$ system of equations
  6. The combined energy-momentum tensor
  7. The Reissner-Nordström spacetime
  8. The manifold and the metric
  9. The Killing vector fields
  10. The spherical harmonics and elliptic estimates
  11. The derivation of the mixed spin $\pm1$ and spin $\pm2$ system of equations
  12. Energy quantities and statements of the main theorem
  13. The energy-momentum tensor associated to the system
  14. Boundedness of the energy
  15. Morawetz estimates
  16. ...and 4 more sections

Key Result

theorem 1

[Linear stability of Reissner-Nordström spacetime to gravitational and electromagnetic perturbations for $|Q|<M$ (Rough version)] All solutions to the linearized Einstein--Maxwell equations around a Reissner--Nordström solution $g_{M, Q}$ for $|Q|<M$ in a certain choice of gaugeThe proof is obtained

Figures (1)

  • Figure 1: Penrose diagram of the patch covered by the $U$ and $V$ coordinates.

Theorems & Definitions (32)

  • theorem 1
  • remark 1
  • remark 2
  • theorem 2
  • definition 1
  • proposition 1: Ch-Kl
  • theorem 3
  • definition 2
  • lemma 1
  • proof
  • ...and 22 more