Perihelion precession and shadows near black holes and naked singularities

It is now known that, apart from black holes, some naked singularities can also cast shadows which provide their possible observable signatures. We examine the relevant question here as to how to distinguish then these entities from each other, in terms of further physical signatures. In earlier literature, it has been pointed out that the Schwarzschild black holes always admit timelike bound orbits having positive perihelion precession. It has been also shown, while some naked singularities [e.g., Joshi-Malafarina-Narayan spacetime (JMN)] with a photon sphere can cast a shadow, it could also admit positive perihelion precession for such orbits, thereby mimicking a Schwarzschild black hole. This indicates that nonspinning compact objects with photon spheres may always admit positive perihelion precession of timelike bound orbits around them. In this paper, we construct a static, nonrotating spacetime configuration which has a central naked singularity but no photon sphere, and it can give both shadow and a negative perihelion precession. In the present case, the shadow is cast by a spherically symmetric thin shell of matter, not by a photon sphere. Our results imply that, whereas the presence of a shadow and a positive perihelion precession implies either a Schwarzschild black hole or a naked singularity, the presence of a shadow and a negative perihelion precession simultaneously may imply a naked singularity only. We discuss our results in the context of stellar motions (motion of the S stars) around the Sgr-A* galactic center.