A new symmetric hyperbolic formulation and the local Cauchy problem for the Einstein--Yang--Mills system in the temporal gauge

Abstract

Motivated by the future stability problem of solutions of the Einstein-Yang-Mills (EYM) system with arbitrary dimension, we aim to (1) construct a tensorial symmetric hyperbolic formulation for the (n + 1)-dimensional EYM system in the temporal gauge; (2) establish the local well-posedness for the Cauchy problem of EYM equations in the temporal gauge using this tensorial symmetric hyperbolic system. By introducing certain auxiliary variables, we extend essentially the (n + 1)-dimensional Yang-Mills system to a tensorial symmetric hyperbolic system. On the contrary, this symmetric hyperbolic system with data satisfying some constraints (extending the Yang-Mills constraints) reduces to the Yang-Mills system. Consequently, an equivalence between the EYM and the tensorial symmetric hyperbolic system with a class of specific data set is concluded. Furthermore, a general symmetric hyperbolic system over tensor bundles is studied, with which, we conclude the local well-posedness of the EYM system. It turns out the idea of symmetric hyperbolic formulation of the Yang-Mills field is very useful in prompting a tensorial Fuchsian formalism and proving the future stability for the EYM system with arbitrary dimension (i.e., this new symmetric hyperbolic formulation of EYM manifests well behaved lower order terms for long time evolution), see our companion article [21] with Todd A. Oliynyk.

Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions $n\geq 4$

“…We first recall the local wellposedness result from the companion paper [18] and assume that the Einstein-Yang-Mills initial data, see (1.19), is chosen as in the statement of Theorem 1.1, which in particular, implies that it satisfies Einstein-Yang-Mills constraints (1.20)-(1.21) and the gauge constraints (1.22)-(1.23) on Σ 0 = {0} × Σ. Then by Theorem 1.1 from [18] there exists a τ * > 0 and a unique solution (g ab , Ã⋆ a ) to the Einstein-Yang-Mills equations satisfying the temporal and wave gauge conditions (1.17) and (1.18), respectively, and with the regularity gab ∈ Step 2: A local solution of the reduced conformal EYM system. By Lemma 6.0.2 and Proposition 6.0. for some constant C > 0 that is independent of U and U.…”

“…Instead, we show in §3, see, in particular, Theorems 3.1 and 3.2, that solutions of Einstein-Yang-Mills equations that satisfy a temporal and wave gauge condition yield solutions to a Fuchsian equation of the form (1.24); see (5.1) for the actual equation. We then appeal to the local-in-time existence theory for the Einstein-Yang-Mills equations from the companion article [18] to obtain local-in-time solutions to the Fuchsian (1.24) on a spacetime domain of the form t…”

“…In this way, we obtain uniform bounds on the local solution. Because of these bounds, we can then appeal to the continuation principle from [18] to extend the local-in-time solution past t * , and in particular, all the way to t * = 0, which yields the global existence of solutions to the Einstein-Yang-Mills equations.…”