In flat spacetime, light moves on straight lines. In curved spacetime, light moves on geodesics, the straightest-possible lines as prescribed by the geometry. Around a black hole, spacetime is so curved that light can go into orbit! Below you can play around with those orbits and see what they look like.
You can also ask the code to try to match some frequency ratio (it might be impossible). Try some small-integer ratios, and drag the spin until a solution is found (the r slider will be disabled).
Everything is in geometric units, , and furthermore the total mass has been scaled out, . The which appears here is , and the is . Everything is in Boyer-Lindquist coordinates.
In the visualization you will see a red curve animating around, a gray ellipsoid, and a blue arrow. The gray ellipsoid represents the event horizon (a constant-r surface in B-L coords), and the blue arrow represents the direction of the spin vector. The red curve is a chunk of the photon’s trajectory. Some history of the trajectory is saved.
Correction 2019-01-19: Many thanks to Charles Gammie for spotting a typo in the equation for (this typo did not affect the code).