Alternative Action for Generalized Unimodular Gravity

We present an alternative formulation of generalized unimodular gravity (GUMG), a class of modifications to general relativity characterized by a special partial breaking of general coordinate covariance. The action for this formulation is derived constructively through a sequence of equivalent representations, starting from the original GUMG setup and extending the configuration space and gauge structure by introducing time parametrization. Our approach, based on a canonical formalism, parallels the method used by Henneaux and Teitelboim to covariantize the action of unimodular gravity (UMG), which was generalized to consistently accommodate a parameterization via local fields for the entire GUMG family. For completeness, we explore the dynamical structure of the theory and provide a detailed account of its gauge properties. A notable consequence of the consistent parametrization is the emergence of explicit spatial delocalization in the action, manifestations of which we carefully examine. Within the GUMG family, we identify a subfamily of models described by a local action. While preserving the essential structural features, these models avoid the complications of spatial nonlocality allowing a clearer analysis. The dynamical and constraint structures of GUMG family models are different from those of UMG, however the latter appears as an exceptional special case within the family, providing valuable comparative insights.