Global exterior stability of the Minkowski space: coupled Einstein-Yang-Mills perturbations

Abstract

Here we prove a global gauge-invariant radiation estimates for the perturbations of the $3+1$ dimensional Minkowski spacetime in the presence of Yang-Mills sources. In particular, we obtain a novel gauge invariant estimate for the Yang-Mills fields coupled to gravity in a double null framework in the Causal complement of a compact set of a Cauchy slice. A consequence of our result is the global exterior stability of the Minkowski space under coupled Yang-Mills perturbations. A special structure present both in the null Bianchi equations and the null Yang-Mills equations is utilized crucially to obtain the dispersive estimates necessary to conclude the global existence property. Direct use of Bel-Robinson and Yang-Mills stress-energy tensor to obtain the energy estimates is avoided in favor of weighted integration by parts taking advantage of the manifestly symmetric hyperbolic characteristics of null Bianchi and null Yang-Mills equations. Our result holds for any compact semi-simple gauge group. This is the first stability result of Minkowski space including a non-linear source.