Hyperboloidal Einstein-matter evolution and tails for scalar and Yang–Mills fields
We show how matter can be included in a constrained ADM-like formulation of the Einstein equations on constant mean curvature surfaces. Previous results on the regularity of the equations at future null infinity are unaffected by the addition of matter with a trace-free energy–momentum tensor. Two examples are studied in detail, namely a conformally coupled scalar field and a Yang–Mills field. We first derive the equations under no symmetry assumptions and then reduce them to spherical symmetry. Both sectors (gravitational and sphaleron) of the spherically symmetric Yang–Mills field are included. We implement this scheme numerically in order to study late-time tails of scalar and Yang–Mills fields coupled to the Einstein equations. We are able to evolve spacetimes that disperse to flat space, accrete onto a given black hole or collapse to a black hole from regular initial data. The sphaleron sector of Yang–Mills is found to exhibit some nontrivial gauge dynamics.