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

TL;DR

This paper establishes the global stability of charged scalar fields on a spacetime close to Minkowski space, with asymptotic behavior similar to Schwarzschild, using a novel null frame technique.

Contribution

It introduces a modified null frame depending only on the mass to prove stability, extending Minkowski results to a more general asymptotically flat spacetime.

Findings

Proves global stability of the Charge-Scalar Field system in near-Minkowski spacetime.

Develops a new null frame capturing asymptotic metric behavior.

Sets groundwork for analyzing Einstein-Charge scalar field system.

Abstract

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.