On the health of a vector field with (R A^2)/6 coupling to gravity

The coupling (R A^2)/6 of a vector field to gravity was proposed as a mechanism for generating a primordial magnetic field, and more recently as a mechanism for generating a statistically anisotropic contribution to the primordial curvature perturbation. In either case, the vector field's perturbation has both a transverse and a longitudinal component, and the latter has some unusual features which call into question the health of the theory. We calculate for the first time the energy density generated by the longitudinal field perturbations, and go on to argue that the theory may well be healthy in at least some versions.