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General Relativity

Schwarzschild Lens Simulator

If we could look at a nearby black hole, we would see that it wraps around itself an infinite series of annuli, each containing a complete and distorted image of the sky. This is quite different from the image of the sky produced by the gravitational lens of a star, which creates at most two images of the sky immediately behind the star. Passing starlight can orbit a black hole many times before escaping to an observer, which creates an infinite number of images of any object in the sky. The simulator on this page calculates our view of the sky around a black hole.

The simulator shows two sets of maps. The map on the left-hand side of the page, which is labeled “Lens Image,” shows the appearance of the sky around the black hole. The two maps on the right-hand side, which as a pair is labeled ”Source Positions,” show the appearance of the full sky in the absence of a black hole. The buttons below the maps permit the reader to rotate the sky to any desired position.

The Lens Image map is centered on the black hole, which is located at the zenith of the sky. The two red circles on this plot show the Einstein rings. The outer ring corresponding to the Einstein ring of a star, marking the image on the sky of light traveled around the black hole from the zenith. The inner ring marks the image on the sky of light traveling from the nadir. There are an infinite number of Einstein rings nested inside the inner red circle that alternate between the zenith image and the nadir image. Each pair of Einstein rings mark the boundaries of an annulus that contains a complete image of the sky. Because each annulus is exponentially smaller than the annulus that surrounds it, most annuli are too close together to show on the Lens Image map.

The black circle on the Lens Image map shows the region bounded by the Einstein rings. The boundary of this region is the black hole's last stable orbit; light at precisely this boundary can orbit the black hole indefinitely. The region inside the last stable orbit is black because light from a distant star that passes inside the last stable orbit inevitably falls onto the black hole's event horizon. The event horizon is smaller that the black region on the map, but it is not visible through any effect on passing starlight.

The Source Positions map shows the full sky. The half hemisphere map centered on the zenith is plotted above the half hemisphere map centered on the nadir. The zenith map shows the two Einstein rings and the disk inside the last stable orbit. The sources are shown as they would appear in the absence of the black hole.

Simulator Notes

Technical Details

The sky is mapped onto the plane in all of the plots through the stereographic projection. This projection has the properties that a circle on a sphere projects to a circle on the plane and angles on the sphere are preserved when projected to the plane.

The observer is situated at 12.5 Schwarzschild radii in the simulation. This distance defines the size of the Einstein rings and the last stable orbit on the sky. The angle by which light is deflected is approximated under the assumption that the observer is at infinity.

While only the two images resulting from less than a full rotation of the black hole are readily visible on the Lens Image, the two images resulting from more than a full rotation but less than two full rotations of the black hole are also calculated and plotted. These extra images lie within the inner Einstein ring, and at times appear as slight smudges of color inside the inner red circle.


The orientation of the sky relative to the black hole can be changed using the four buttons at the bottom of the simulator. These buttons can be activated with either the mouse or the keyboard. The buttons labeled “Left” and “Right” rotate the sky to the left and the right, changing the zenith angle between the black hole and the image sources. The buttons labeled “Clockwise” and “Counterclockwise” rotate the sky clockwise and counterclockwise around the zenith, changing the azimuthal angle of the image sources.

When a button is clicked using the right mouse button, the map rotates a fixed interval. If the mouse button is held for more than half a second, the map rotates continuously until the mouse button is released. To increase the rate of rotation, press the control key along with the mouse button.

The space key controls the rotation of the sky in precisely the same manner as the mouse button. If the space key is pressed for less than half a second, the sky rotates a fixed interval. If the space key is held for more than a half second, the sky rotates continuously. Pressing the control key along with the space key increases the rate of rotation.

I would appreciate hearing from you if you encounter an error while running the simulator or if you have suggestions for improvement. Send your e-mail to the editor of the website.

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