In general relativity (GR), gravity influences the motion of particles by changing the local flow of time. This gravitational redshift has been tested to an accuracy of 7x10-5 by clock comparisons; matter wave interferometers confirm it with an accuracy of 7x10-9, as a particle’s wave function can be interpreted as a clock ticking at the Compton frequency mc2/h. We will demonstrate exact correspondence between matter wave and classical clock comparison experiments. We also show how these experiments, along with tests of the universality of free fall, place stringent and comprehensive bounds on leading order GR violations, using the Standard Model Extension as a theoretical framework. In addition, matter wave interferometers may be able to detect higher order effects that arise through the nonlinearity of GR. These effects are known to exist from solar system observations, but have never been confirmed in experiments under controlled laboratory conditions.