On the geometrical and dynamical distinction between Unimodular and General Relativistic wormholes

We characterize traversable wormholes in Unimodular Gravity (UG) and investigate what distinguishes them from their General Relativistic (GR) counterparts. For a class of static and spherically symmetric solutions, we analyze their embedding and timelike geodesics, showing that the geodesic structure is entirely determined by the metric and is therefore identical in both theories.
To identify the physical origin of their differences, we compare the source sectors required to sustain the same wormhole geometry. We find that preserving a fixed UG geometry while imposing energy-momentum conservation requires restricting the equation of state, whereas relaxing this condition introduces an effective inhomogeneous vacuum contribution. We further show that the degree of exoticity is preserved even when energy-momentum is not conserved, while departures from the GR sector are encoded in an effective inhomogeneous vacuum structure whose asymptotic behavior resembles that of a cosmological constant.
Our results reinforce that UG is geometrically equivalent to GR, while its distinctive features emerge in the dynamical interpretation of the source sector. More generally, our analysis illustrates that different gravitational theories may give rise to identical spacetime geometries while requiring different sources to sustain them.