Gravitational lensing by naked singularities

We model massive dark objects in galactic nuclei as spherically symmetric static naked singularities in the Einstein massless scalar field theory and study the resulting gravitational lensing in detail. Based on whether or not a naked singularity is covered within a photon sphere we classify naked singularities into two kinds: weakly naked (those contained within at least one photon sphere) and strongly naked (those not contained within any photon sphere). The qualitative features of gravitational lensing due to a weakly naked singularity are similar to those due to a Schwarzschild black hole (these give rise to one Einstein ring but no radial critical curve). However, the gravitational lensing due to a strongly naked singularity is qualitatively different from that due to a Schwarzschild black hole; a strongly naked singularity gives rise to either two or nil Einstein ring(s) and one radial critical curve. A light ray passing close to a photon sphere of a black hole or a weakly naked singularity goes around its photon sphere once, twice, or many times (before reaching an observer) depending upon the impact parameter and thus gives rise to a sequence of theoretically infinite number of relativistic images, which are highly demagnified. A strongly naked singularity produces no relativistic images.