Gravitational lensing signature of matter distribution around
  Schwarzschild black hole
  In this work, we focus on the situation where a significant amount of matter
could be located close to the event horizon of the central black hole and how
it affects the gravitational lensing signal. We consider a simple toy model
where the matter is concentrated in the rather small region between the inner
photon sphere associated with the mass of central black hole and outer photon
sphere associated with the total mass outside. If no photon sphere is present
inside the matter distribution, then effective potential displays an
interesting trend with maxima at inner and outer photon sphere, with a peak at
inner photon sphere higher than that at outer photon sphere. In such a case we
get three distinct set of infinitely many relativistic images and Einstein
rings that occur due to the light rays that approach the black hole from a
distant source and get reflected just outside the outer photon sphere, due to
light rays that enter the outer photon sphere slightly above the outer peak and
get reflected off the potential barrier inside the matter distribution and due
to the light rays that get reflected just outside the inner photon sphere. This
kind of pattern of images is quite unprecedented. We show that since
relativistic images are highly demagnified, only three images are prominently
visible from the point of observations in the presence of matter as opposed to
only one prominent image in case of a single isolated black hole and also
compute the time delay between them. This provides a smoking gun signature of
the presence of matter lump around the black hole. We further argue that if the
mass of the black hole inferred from the observation of the size of its shadow
is less than the mass inferred from the motion of objects around it, it signals
the presence of matter in the vicinity of the black hole.
