Einstein-Maxwell Equations on Mass-Centered GCM Hypersurfaces
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| Format: | Preprint |
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2025
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| _version_ | 1866909840561930240 |
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| author | Fang, Allen Juntao Giorgi, Elena Wan, Jingbo |
| author_facet | Fang, Allen Juntao Giorgi, Elena Wan, Jingbo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10814 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Einstein-Maxwell Equations on Mass-Centered GCM Hypersurfaces Fang, Allen Juntao Giorgi, Elena Wan, Jingbo General Relativity and Quantum Cosmology Analysis of PDEs 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. |
| title | Einstein-Maxwell Equations on Mass-Centered GCM Hypersurfaces |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs |
| url | https://arxiv.org/abs/2510.10814 |