Internal symmetry to the rescue: well-posed 1+1 evolution of self-interacting vector fields

Abstract

Previous studies have identified potential instabilities in self-interacting vector theories associated with the breakdown of the well-posedness of the initial-value problem. However, these conclusions are restricted to Abelian vector fields, leaving room to explore alternative setups, such as non-Abelian vector fields with internal symmetries. Building on this idea, we study the well-posed 1+1 evolution of self-interacting SU(2) vector fields minimally coupled to gravity within the framework of the 't Hooft-Polyakov magnetic monopole configuration. In this context, we present a counterexample in which self-interacting vector fields retain a well-posed initial value problem formulation. Remarkably, this system exhibits the same characteristic speeds as those found in general relativity (GR) in one spatial dimension. Unlike its Abelian counterpart, we achieve stable numerical evolutions across a wide range of initial conditions within a fully non-linear dynamical background, as evidenced in our time integration algorithm. Although our conclusions are strictly valid for the spherical symmetry case with only magnetic part for the vector field, this study serves as a valuable diagnostic tool for investigating more realistic astrophysical scenarios in three-dimensional settings and under more general background and vector field configurations.