Static Spherically Symmetric Solutions in Nonconservative Unimodular Gravity

Abstract

We study static and spherically symmetric black holes in nonconservative unimodular gravity, where the traceless field equations determine the source of a given geometry only up to a pure-trace term. We consider the Reissner-Nordström metric with an additional $1/r^3$ term and show that its coefficient fixes the Ricci scalar and, when the freedom in the source is restricted to the Rastall form, controls the nonconservation of energy and momentum. We find that only the null projections of the source are independent of how the geometry is divided between matter and vacuum. Therefore, no-go results based on the null energy condition hold in unimodular gravity, while those based on the strong or dominant conditions are generally split dependent. We identify the traceless source as the optimal split with respect to the energy conditions and determine the region in which all standard energy conditions can be satisfied. We also identify a combination of the shifts of the photon sphere and of the innermost stable circular orbit that measures the $1/r^3$ coefficient independently of the charge, although its detection requires a precision far beyond current observations.

arXiv comment13 pages, 6 figures, 2 tables. Updated refs, figures