Schwarzschild black hole lensing

We study strong gravitational lensing due to a Schwarzschild black hole. Apart from the primary and the secondary images we find a sequence of images on both sides of the optic axis; we call them relativistic images. These images are formed due to large bending of light near $r=3M$ (the closest distance of approach ${r}_{o}$ is greater than $3M).$ The sources of the entire universe are mapped in the vicinity of the black hole by these images. For the case of the Galactic supermassive ``black hole'' they are formed at about $17$ microarcseconds from the optic axis. The relativistic images are not resolved among themselves, but they are resolved from the primary and secondary images. However the relativistic images are very much demagnified unless the observer, lens and source are very highly aligned. Because of this and some other difficulties the observation of these images does not seem to be feasible in the near future. However, it would be a great success of the general theory of relativity in a strong gravitational field if they ever were observed and it would also give an upper bound, ${r}_{o}=3.21M,$ to the compactness of the lens, which would support the black hole interpretation of the lensing object.