TransverseDiff gravity is to scalar-tensor as unimodular gravity is to General Relativity

Transverse Diffeomorphism (TDiff) theories are well-motivated theories of gravity from the quantum perspective, which are based upon a gauge symmetry principle. The main contribution of this work is to firmly establish a correspondence between TransverseDiff and the better-known scalar-tensor gravity --- in its more general form ---, a relation which is completely analogous to that between unimodular gravity and General Relativity. We then comment on observational aspects of TDiff. In connection with this proof, we derive a very general rule that determines under what conditions the procedure of fixing a gauge symmetry can be equivalently applied before the variational principle leading to the equations of motion, as opposed to the standard procedure, which takes place afterwards; this rule applies to gauge-fixing terms without derivatives.