  Outer boundary conditions for strongly and symmetric hyperbolic formulations
of 3D Einstein's field equations with a live gauge condition are discussed. The
boundary conditions have the property that they ensure constraint propagation
and control in a sense made precise in this article the physical degrees of
freedom at the boundary. We use Fourier-Laplace transformation techniques to
find necessary conditions for the well posedness of the resulting
initial-boundary value problem and integrate the resulting three-dimensional
nonlinear equations using a finite-differencing code. We obtain a set of
constraint-preserving boundary conditions which pass a robust numerical
stability test. We explicitly compare these new boundary conditions to
standard, maximally dissipative ones through Brill wave evolutions. Our
numerical results explicitly show that in the latter case the constraint
variables, describing the violation of the constraints, do not converge to zero
when resolution is increased while for the new boundary conditions, the
constraint variables do decrease as resolution is increased. As an application,
we inject pulses of ``gravitational radiation'' through the boundaries of an
initially flat spacetime domain, with enough amplitude to generate strong
fields and induce large curvature scalars, showing that our boundary conditions
are robust enough to handle nonlinear dynamics.
  We expect our boundary conditions to be useful for improving the accuracy and
stability of current binary black hole and binary neutron star simulations, for
a successful implementation of characteristic or perturbative matching
techniques, and other applications. We also discuss limitations of our approach
and possible future directions.
