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Mass-Centered GCM Framework in Perturbations of Kerr(-Newman)

Allen Juntao Fang, Elena Giorgi, Jingbo Wan

TL;DR

The paper addresses nonlinear stability for charged black holes by extending the GCM gauge to perturbations of Reissner–Nordström and Kerr–Newman spacetimes. It replaces the vacuum transport-based control of the center-of-mass mode with a sphere-by-sphere vanishing condition on a renormalized $m{C}_{\ell=1}$ quantity, yielding mass-centered GCM spheres and a mass-centered GCM hypersurface. The construction relies on fixing an effective canonical $\,\ell=1$ basis via uniformization and solving a determined elliptic-transport system to propagate the gauge consistently from spheres to a hypersurface. This gauge design preserves the structural advantages of GCM methods while accommodating electromagnetic coupling, providing a foundation for nonlinear stability results in the electrovacuum regime and potential applicability to broader matter models.

Abstract

The nonlinear stability problem for black hole solutions of the Einstein equations critically depends on choosing an appropriate geometric gauge. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In this work, we develop an alternative GCM framework, that we call mass-centered, designed to overcome the breakdown of the standard GCM construction in the charged case, where electromagnetic-gravitational coupling destroys the exceptional behavior of the $\ell=1$ mode of the center-of-mass quantity used in the vacuum analysis. This construction is aimed at the nonlinear stability of Reissner-Nordström and Kerr-Newman spacetimes. Our approach replaces transport-based control of the center-of-mass quantity with a sphere-wise vanishing condition on a renormalized $\ell=1$ mode, yielding mass-centered GCM hypersurfaces with modified gauge constraints. The resulting elliptic-transport system remains determined once an $\ell=1$ basis is fixed via effective uniformization and provides an alternative construction in vacuum in the uncharged limit.

Mass-Centered GCM Framework in Perturbations of Kerr(-Newman)

TL;DR

The paper addresses nonlinear stability for charged black holes by extending the GCM gauge to perturbations of Reissner–Nordström and Kerr–Newman spacetimes. It replaces the vacuum transport-based control of the center-of-mass mode with a sphere-by-sphere vanishing condition on a renormalized quantity, yielding mass-centered GCM spheres and a mass-centered GCM hypersurface. The construction relies on fixing an effective canonical basis via uniformization and solving a determined elliptic-transport system to propagate the gauge consistently from spheres to a hypersurface. This gauge design preserves the structural advantages of GCM methods while accommodating electromagnetic coupling, providing a foundation for nonlinear stability results in the electrovacuum regime and potential applicability to broader matter models.

Abstract

The nonlinear stability problem for black hole solutions of the Einstein equations critically depends on choosing an appropriate geometric gauge. In the vacuum setting, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has played a central role in the proof of stability for slowly rotating Kerr spacetime. In this work, we develop an alternative GCM framework, that we call mass-centered, designed to overcome the breakdown of the standard GCM construction in the charged case, where electromagnetic-gravitational coupling destroys the exceptional behavior of the mode of the center-of-mass quantity used in the vacuum analysis. This construction is aimed at the nonlinear stability of Reissner-Nordström and Kerr-Newman spacetimes. Our approach replaces transport-based control of the center-of-mass quantity with a sphere-wise vanishing condition on a renormalized mode, yielding mass-centered GCM hypersurfaces with modified gauge constraints. The resulting elliptic-transport system remains determined once an basis is fixed via effective uniformization and provides an alternative construction in vacuum in the uncharged limit.

Paper Structure

This paper contains 36 sections, 18 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The nonlinear stability of black hole spacetimes
  3. The GCM framework of Klainerman, Szeftel, Shen
  4. GCM spheres
  5. GCM hypersurfaces
  6. Tracking the center of mass via GCM spheres
  7. The charged case and the breakdown of the exceptional structure
  8. Mass-centered GCM spheres and hypersurfaces
  9. Main ideas behind the mass-centered GCM construction
  10. Organization of the paper
  11. Acknowledgments
  12. Preliminaries
  13. Background spacetime and adapted coordinates
  14. Basis of modes
  15. Definition of mass and charge parameters
  16. ...and 21 more sections

Key Result

Theorem 1.1

Vacuum initial data sets, sufficiently close to Kerr initial data for $|a| \ll M$, have a maximal development with a complete future null infinity and with domain of outer communication which approaches (globally) a nearby Kerr solution.

Figures (2)

  • Figure 1: The GCM admissible spacetime of klainermanKerrStabilitySmall2023.
  • Figure 2: The blue spheres represent the original foliation spheres $S$, while the red spheres ${\bf S}$ denote the deformed ones satisfying the vanishing center-of-mass condition.

Theorems & Definitions (54)

  • Theorem 1.1: klainermanConstructionGCMSpheres2022klainermanEffectiveResultsUniformization2022shenConstructionGCMHypersurfaces2023klainermanKerrStabilitySmall2023giorgiWaveEquationsEstimates2024
  • Conjecture 1.2: Stability of Kerr-Newman conjecture
  • Theorem 1.3: Existence of mass-centered GCM spheres and hypersurfaces, First version
  • Definition 2.1
  • Theorem 2.2: Effective Uniformization; Corollary 3.8 in klainermanEffectiveResultsUniformization2022
  • Definition 2.3: Basis of canonical $\ell=1$ modes on $S$
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 44 more