  We derive a general formalism for constructing a four-dimensional effective
action for the radion field in 5 dimensions, taking into account the possible
dependence of the scale factor on the internal coordinate. First and second
variations of the action with respect to the size of the extra dimension
reveals two types of algebraic constraints on the combination of brane and bulk
Lagrangians that extremize the action, and provide an effective mass for the
radion field. We derive a compact formula for the radion mass squared in terms
of brane and bulk potentials for the case of a bulk scalar field. We further
specialize these constraints for several, five-dimensional realizations of
brane world models.
