Density-metric unimodular gravity: vacuum spherical symmetry

We analyze an alternative theory of gravity characterized by metrics that are tensor density of rank (0, 2) and weight-1/2. The metric compatibility condition is supposed to hold. The simplest expression for the action of gravitational field is used. Taking the metric and trace of connections as dynamical variables, the field equations in the absence of matter and other kinds of sources are derived. The solutions of these equations are obtained for the case of vacuum static spherical symmetric spacetime. The null geodesics and advance of perihelion of ellipses are discussed. We confirm a subclass of solutions is regular for r > 0 and there is no event horizon while it is singular at r = 0.