Gravitational red-shift measurements as tests of nonmetric theories of gravity

Gravitational red-shift measurements can be powerful tools for testing nonmetric theories of gravity. We show that some nonmetric gravitation theories may predict the "wrong" first-order gravitational red shift even while possessing the "correct" Newtonian limit, whereas metric theories with the "correct" Newtonian limit automatically predict the "correct" red shift. We also argue that all nonmetric theories ultimately predict a red shift which depends on the nature of the atomic clock whose red shift is being measured. These statements can be worded in the form of a conjecture: Any complete, self-consistent, and relativistic theory of gravity that embodies the universality of gravitational red shift is necessarily a metric theory. Restricting attention to a certain subclass of nonmetric theories and to electromagnetic atomic clocks residing in static, spherically symmetric gravitational fields, we give a detailed proof of this conjecture. Red-shift experiments can thus distinguish between metric and nonmetric theories of gravity. For weak gravitational fields appropriate to the solar system, we compare the sensitivity of red-shift experiments and E\"otv\"os experiments to nonmetric effects. Particularly important are null red-shift measurements (comparison of the red shifts of different clocks at the same location in a gravitational field) designed to be sensitive to second-order gravitational red shifts.