Spatial variations of fundamental constants

We show how observational limits on possible time variations of fundamental constants of nature are influenced by spatial variations in the density and gravitational potential. Bekenstein's generalization of Maxwell's equations, incorporating variations of the fine structure 'constant', is used as a simple example. These features are expected to be present in a wide class of theories with varying 'constants' and will affect attempts to coordinate cosmological and geophysical bounds on allowed time variations.