  The extreme conditions found near black holes and neutron stars provide a
unique opportunity for testing physical theories. Observations of both types of
compact objects can be used to probe regions of strong gravity, allowing for
tests of General Relativity. Furthermore, a determination of the properties of
matter at the remarkably high densities that exist within neutron stars would
have important implications for nuclear and particle physics. While many of
these objects are in binary systems ("X-ray binaries"), where accreting matter
from the stellar companion provides a probe of the compact object, a main
difficulty in making measurements that lead to definitive tests has been
uncertainty about basic information such as distances to sources, orientation
of their binary orbits, and masses of the compact objects. Optical astrometry
at the microarcsecond level will allow for accurate determinations of distances
and proper motions with precisions of a few percent and will provide an
opportunity to directly map X-ray binary orbits, leading to measurements of
orbital inclination and compact object masses accurate to better than 4%. Using
astrometry at this unprecedented accuracy, which would be enabled by the Space
Interferometry Mission ("SIM Lite"), will lead to breakthroughs in the
investigations of the physics of black holes and neutron stars
