  Cosmic bubble collisions provide an important possible observational window
on the dynamics of eternal inflation. In eternal inflation, our observable
universe is contained in one of many bubbles formed from an inflating
metastable vacuum. The collision between bubbles can leave a detectable imprint
on the cosmic microwave background radiation. Although phenomenological models
of the observational signature have been proposed, to make the theory fully
predictive one must determine the bubble collision spacetime, and thus the
cosmological observables, from a scalar field theory giving rise to eternal
inflation. Because of the intrinsically non-linear nature of the bubbles and
their collision, this requires a numerical treatment incorporating General
Relativity. In this paper, we present results from numerical simulations of
bubble collisions in full General Relativity. These simulations allow us to
accurately determine the outcome of bubble collisions, and examine their effect
on the cosmology inside a bubble universe. We confirm the validity of a number
of approximations used in previous analytic work, and identify qualitatively
new features of bubble collision spacetimes. Both vacuum bubbles and bubbles
containing a realistic inflationary cosmology are studied. We identify the
constraints on the scalar field potential that must be satisfied in order to
obtain collisions that are consistent with our observed cosmology, yet leave
detectable signatures.
