Static Spherically Symmetric Solutions in Nonconservative Unimodular Gravity

Abstract

The traceless field equations of unimodular gravity determine the source of a given geometry only up to a pure-trace contribution. We study the consequences of this reconstruction freedom for a static, spherically symmetric black-hole geometry containing the leading inverse-cubic deformation of Schwarzschild not fixed by weak-field data. We show that among the standard combinations entering the energy conditions, only the two null projections are independent of the reconstruction, whereas the energy density and the strong and dominant combinations depend on the chosen split between matter and vacuum. This yields a general criterion: conclusions based on the null energy condition are reconstruction invariant, while those based only on the strong or dominant conditions are not. Applied to wormhole throats, this criterion reproduces the null-energy no-go result for every split. For the black-hole family, the inverse-cubic parameter carries the complete trace sector. Within the Rastall restriction of the reconstruction family, nonconservation is controlled by the product of this parameter with the Rastall coupling, and the traceless split is selected by the energy conditions. We also derive strong-field observables and identify a charge-independent combination of photon-sphere and innermost-stable-orbit shifts that isolates the trace-sector parameter, while current observational precision remains insufficient to measure it.