  Atom interferometry is now reaching sufficient precision to motivate
laboratory tests of general relativity. We begin by explaining the
non-relativistic calculation of the phase shift in an atom interferometer and
deriving its range of validity. From this we develop a method for calculating
the phase shift in general relativity. This formalism is then used to find the
relativistic effects in an atom interferometer in a weak gravitational field
for application to laboratory tests of general relativity. The potentially
testable relativistic effects include the non-linear three-graviton coupling,
the gravity of kinetic energy, and the falling of light. We propose
experiments, one currently under construction, that could provide a test of the
principle of equivalence to 1 part in 10^15 (300 times better than the present
limit), and general relativity at the 10% level, with many potential future
improvements. We also consider applications to other metrics including the
Lense-Thirring effect, the expansion of the universe, and preferred frame and
location effects.
