  We consider the initial boundary value problem for free-evolution
formulations of general relativity coupled to a parametrized family of
coordinate conditions that includes both the moving puncture and harmonic
gauges. We concentrate primarily on boundaries that are geometrically
determined by the outermost normal observer to spacelike slices of the
foliation. We present high-order-derivative boundary conditions for the gauge,
constraint violating and gravitational wave degrees of freedom of the
formulation. Second order derivative boundary conditions are presented in terms
of the conformal variables used in numerical relativity simulations. Using
Kreiss-Agranovich-Metivier theory we demonstrate, in the frozen coefficient
approximation, that with sufficiently high order derivative boundary conditions
the initial boundary value problem can be rendered boundary stable. The precise
number of derivatives required depends on the gauge. For a choice of the gauge
condition that renders the system strongly hyperbolic of constant multiplicity,
well-posedness of the initial boundary value problem follows in this
approximation. Taking into account the theory of pseudo-differential operators,
it is expected that the nonlinear problem is also well-posed locally in time.
