  We calculate the first dynamical evolutions of merging black hole-neutron
star binaries that construct the combined black hole-neutron star spacetime in
a general relativistic framework. We treat the metric in the conformal flatness
approximation, and assume that the black hole mass is sufficiently large
compared to that of the neutron star so that the black hole remains fixed in
space. Using a spheroidal spectral methods solver, we solve the resulting field
equations for a neutron star orbiting a Schwarzschild black hole. The matter is
evolved using a relativistic, Lagrangian, smoothed particle hydrodynamics (SPH)
treatment. We take as our initial data recent quasiequilibrium models for
synchronized neutron star polytropes generated as solutions of the conformal
thin-sandwich (CTS) decomposition of the Einstein field equations. We are able
to construct from these models relaxed SPH configurations whose profiles show
good agreement with CTS solutions. Our adiabatic evolution calculations for
neutron stars with low compactness show that mass transfer, when it begins
while the neutron star orbit is still outside the innermost stable circular
orbit, is more unstable than is typically predicted by analytical formalisms.
This dynamical mass loss is found to be the driving force in determining the
subsequent evolution of the binary orbit and the neutron star, which typically
disrupts completely within a few orbital periods. The majority of the mass
transferred onto the black hole is accreted promptly; a significant fraction
(~30%) of the mass is shed outward as well, some of which will become
gravitationally unbound and ejected completely from the system. The remaining
portion forms an accretion disk around the black hole, and could provide the
energy source for short-duration gamma ray bursts.
