Global Stability for Charged Scalar Fields in an Asymptotically Flat Metric in Harmonic Gauge

We prove global stability for the Charge-Scalar Field system on a background spacetime which is close to $1+3$-dimensional Minkowski space and whose outward light cones converge to those for the Schwarzschild metric at null infinity. The key technique to this proof is the use of a modified null frame, depending only on the mass $M$ of the metric, which captures the asymptotic behavior of the metric at future null infinity. Our results are analogous to results obtained in Minkowski space by Lindblad and Sterbenz up to a change in coordinates, and will in the sequel be used to prove the full structure of the Einstein-Charge scalar field system in these modified harmonic coordinates.