  General Relativity (GR) and Unimodular Gravity (UG) provide two equivalent
descriptions of gravity that differ in the nature of the cosmological constant.
While GR is based on the group of diffeomorphisms that permits the cosmological
constant in the action, UG is based on the subgroup of volume-preserving
diffeomorphisms together with Weyl transformations which forbid the presence of
the cosmological constant. However, the cosmological constant reappears in UG
as an integration constant so it arises as a global degree of freedom. Since
gauge symmetries are simply redundancies in our description of physical
systems, a natural question is whether there exists a "parent theory" with the
full diffeomorphisms and Weyl transformations as gauge symmetries so that it
reduces to GR and UG respectively by performing suitable (partial) gauge
fixings. We will explore this question by introducing Stueckelberg fields both
in GR and UG to complete the gauge symmetries in each theory to that of the
would-be parent theory. Despite the dynamical equivalence of the two theories,
we find that precisely the additional global degree of freedom provided by the
cosmological constant in UG obstructs the construction of the parent theory.
