  The Galactic center offers us a unique opportunity to test General Relativity
(GR) with the orbits of stars around a supermassive black hole. Observations of
these stars have been one of the great successes of adaptive optics on 8-10 m
telescopes, driving the need for the highest angular resolution and astrometric
precision. New tests of gravitational physics in the strong gravity regime with
stellar orbits will be made possible through the leap in angular resolution and
sensitivity from the next generation of extremely large ground-based
telescopes. We present new simulations of specific science cases such as the
detection of the GR precession of stars, the measurement of extended dark mass,
and the distance to the Galactic center. We use realistic models of the
adaptive optics system for TMT and the IRIS instrument to simulate these
science cases. In additions, the simulations include observational issues such
as the impact of source confusion on astrometry and radial velocities in the
dense environment of the Galactic center. We qualitatively show how
improvements in sensitivity, astrometric and spectroscopic precision, and
increasing the number of stars affect the science with orbits at the Galactic
center. We developed a tool to determine the constraints on physical models
using a joint fit of over 100 stars that are expected to be observable with
TMT. These science cases require very high astrometric precision and stability,
thus they provide some of the most stringent constraints on the planned
instruments and adaptive optics systems.
