  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.
