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 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 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 distant source and get reflected back 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 single isolated black hole and also compute the time delay between them. This provides a smoking gun signature of presence of matter lump around black hole. We further argue that if the mass of the black hole inferred from the observation of 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 black hole.