2023/10/12 by Sari Ghanem, Ghanem, Sari
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)
paper · pdf · doi:10.48550/arxiv.2310.08196
openalex publication_date 2023/10/12 · openalex created_date 2023/10/14 · openalex updated_date 2026/07/28
We prove the exterior stability of the Minkowski space-time, ℝ1+3, solution to the Einstein-Yang-Mills system in both the Lorenz and harmonic gauges, where the Yang-Mills fields are valued in any arbitrary Lie algebra \calG, associated to any compact Lie group G. We start with an arbitrary sufficiently small initial data, defined in a suitable energy norm for the perturbations of the Yang-Mills potential and of the Minkowski space-time, and we show the well-posedness of the Cauchy development in the exterior of the fully coupled Einstein-Yang-Mills equations in the Lorenz gauge and in wave coordinates, and we prove that this leads to solutions converging to the zero Yang-Mills curvature and to the Minkowski space-time. Furthermore, we obtain dispersive estimates in wave coordinates on the Yang-Mills potential in the Lorenz gauge and on the metric, as well as on the gauge invariant norm of the Yang-Mills curvature. This provides a new proof to the exterior stability result by P. Mondal and S. T. Yau, based on an alternative approach, by using a null frame decomposition that was first used by H. Lindblad and I. Rodnianski for the case of the Einstein vacuum equations. In this third paper of a series, we detail all the new material concerning our proof so as to provide lecture notes for Ph.D. students wanting to learn non-linear hyperbolic differential equations and stability problems in mathematical General Relativity.