Global evolution in spherical symmetry for self-gravitating massive fields

TL;DR

This paper proves the global existence of spherically symmetric solutions to Einstein's equations coupled with a massive scalar field, extending previous results for massless fields by addressing additional challenges posed by mass.

Contribution

It develops a new framework for analyzing massive scalar fields in spherical symmetry, including regularization at null infinity and establishing decay estimates.

Findings

Proved global existence of solutions with low regularity.

Established positivity and monotonicity of the Bondi mass.

Developed new energy and decay estimates for solutions.

Abstract

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.