Einstein-Maxwell Equations on Mass-Centered GCM Hypersurfaces
Allen Juntao Fang, Elena Giorgi, Jingbo Wan
TL;DR
This work constructs and analyzes mass-centered General Covariant Modulated (GCM) hypersurfaces in electrovacuum spacetimes to solve the Einstein–Maxwell constraint equations on a spacelike slice. By introducing a mass-centered renormalization of the center-of-mass quantity ${\bm C}$ and enforcing ${\bm C}_{\ell=1}=0$ on every leaf, the authors overcome the loss of favorable transport in the charged setting and reduce the full nonlinear constraint problem to a parametrization by gauge-invariant radiation seeds ${\rm seed} = \{\alpha, \underline{\alpha}, \mathfrak{f}, \mathfrak{b}, \underline{\mathfrak{b}}, \mathfrak{p}, \mathfrak{q}^{\mathbf{F}}\}$. The main theorem shows that, under bootstrap and dominance conditions, all geometric perturbations along ${\Sigma}_*$ are controlled by minimal seed data via transport, flux, and elliptic estimates, enabling modular integration with hyperbolic evolutions for the gauge-invariant fields. This provides a rigorous, quantitative path toward nonlinear stability analyses for Reissner–Nordström and Kerr–Newman black holes by decoupling constraint control from hyperbolic dynamics. The construction and estimates establish precise decay rates for both $\ell=0$ and $\ell=1$ modes and yield strong control of the canonical one-forms, yielding a robust gauge-fixed framework for the coupled gravitational–electromagnetic system.
Abstract
The resolution of the nonlinear stability of black holes as solutions to the Einstein equations relies crucially on imposing the right geometric gauge conditions. In the vacuum case, the use of Generally Covariant Modulated (GCM) spheres and hypersurfaces has been successful in the proof of stability for slowly rotating Kerr spacetime. For the charged setting, our companion paper introduced an alternative mass-centered GCM framework, adapted to the additional difficulties of the Einstein-Maxwell system. In this work, we solve the Einstein-Maxwell equations on such a mass-centered spacelike GCM hypersurface, which is equivalent to solving the constraint equations there. We control all geometric quantities of the solution in terms of some seed data, corresponding to the gauge-invariant fields describing coupled gravitational-electromagnetic radiation in perturbations of Reissner-Nordström or Kerr-Newman, first identified by the second author and expected to be governed by favorable hyperbolic equations. This provides the first step toward controlling gauge-dependent quantities in the nonlinear stability analysis of the Reissner-Nordström and Kerr-Newman families.