# Formation of a focal point by gravitational lensing effect

Can a black hole form a focal point of a nearby star at a certain distant point by gravitational lensing? Also, can the BH make a virtual picture of that star to a distant observer so the star looks distorted due to the same effect of gravitational lensing? But as I understand grav. lensing there should also be an aberration of the light beams at the focal point...

• Gravitational lenses do not have a focal point. Parallel rays converge at different places along the optic axis depending on their distance from the axis. Sep 21 at 6:58
• Possible duplicate: Do gravitational lenses have a focus point? Sep 21 at 13:03

Actually, the image of an object due to the gravitational lensing of a black hole (or whatever the massive lensing object may be), appears as a ring.

To get an idea why the image formed is circular, consider the following diagram: As one can see, the star behind the lens will appear in two separate locations, in two dimensions. Now consider that in three dimensions, the star will appear at all points forming a circle going into and coming out of the page, as can be seen in this image taken by the Hubble telescope: This shows the lensing object and ring image as concentric. In this image, the gravitational lens itself is a galaxy pictured in the center with the image surrounding it. These images are termed "Einstein rings". The image does not form at a single focal point as can be seen, but rather around the lens forming a ring. There are many smaller images of the distant galaxy located at all points on the ring.

In fact, the equation $$\tag 1\Delta\theta = \frac{4GM}{r c^2}$$ tells us how much massive objects deflect light in terms of the distance of the rays from the optical axis, which leads to the kinds of images we see above.
For a standard convex lens, the further we move from the center $$r$$, rays parallel to the axis will deflect by a greater angle, and so $$\Delta\theta\propto r$$ or $$\Delta\theta=kr$$ where in the thin lens approximation, the constant $$k=\frac{1}{f}$$.
But note in equation (1) we now have $$r$$ in the denominator. For a gravitational lens (like a blackhole), rays of light coming in parallel will not converge at the same points as we move further from the axis (the equation suggests the magnitude of the deflection is inversely proportional to $$r$$, or the angle decreases for parallel rays further from the center of the axis). The net effect is the formation of a ring with the lens at the center (annulus).