  We model massive dark objects at centers of many galaxies as Schwarzschild
black hole lenses and study gravitational lensing by them in detail. We show
that the ratio of mass of a Schwarzschild lens to the differential time delay
between outermost two relativistic images (both of them either on the primary
or on the secondary image side) is extremely insensitive to changes in the
angular source position as well as the lens-source and lens-observer distances.
Therefore, this ratio can be used to obtain very accurate values for masses of
black holes at centers of galaxies. Similarly, angular separations between any
two relativistic images are also extremely insensitive to changes in the
angular source position and the lens-source distance. Therefore, with the known
value of mass of a black hole, angular separation between two relativistic
images would give a very accurate result for the distance of the black hole.
Accuracies in determination of masses and distances of black holes would
however depend on accuracies in measurements of differential time delays and
angular separations between images. Deflection angles of primary and secondary
images as well as effective deflection angles of relativistic images on the
secondary image side are always positive. However, the effective deflection
angles of relativistic images on the primary image side may be positive, zero,
or negative depending on the value of angular source position and the ratio of
mass of the lens to its distance. We show that effective deflection angles of
relativistic images play significant role in analyzing and understanding strong
gravitational field lensing.
