Gauge Symmetry and Consistent Spin-Two Theories

We study Lagrangians with the minimal amount of gauge symmetry required to propagate spin-two particles without ghosts or tachyons. In general, these Lagrangians also have a scalar mode in their spectrum. We find that, in two cases, the symmetry can be enhanced to a larger group: the whole group of diffeomorphisms or a enhancement involving a Weyl symmetry. We consider the non-linear completions of these theories. The intuitive completions yield the usual scalar-tensor theories except for the pure spin-two cases, which correspond to two inequivalent Lagrangians giving rise to Einstein's equations. A more constructive self-consistent approach yields a background dependent Lagrangian.