Tuesday, June 22, 2021

The Photon Sphere

For a non-spinning (spherically symmetric) black hole, the photon sphere is a spherical region of space where gravity is strong enough that photons (light particles) are forced to travel in orbits. This means that the photons travel around the black hole, until ultimately they fall in or spiral out. They must do one or the other, because this is an unstable orbit; there are always small perturbations from other masses and gravitational waves etc., and photons diffract instead of following an exact classical-particle trajectory.
Photon spheres exist around black holes, where the “impact parameter” (offset distance from the black hole) is just on the borderline of sufficient for photons to get captured.
 As photons approach the event horizon of a black hole (the region beyond which light cannot escape), the light particles with sufficient angular momentum avoid being pulled into the black hole by traveling in a nearly tangential direction known as an exit cone (orange path).
A photon on the boundary of this cone doesn’t have enough angular momentum to escape the gravity well of the black hole, and instead orbits the black hole temporarily (blue path). These orbits are unstable, meaning that after bending through a finite angle around the black hole, the photon will either fall in or bend back out. 
The photon sphere is located at 1.5 times the Schwarzschild radius (the radius that defines the size of the black hole event horizon). The further you are from the black hole, the weaker the gravitational force, and it’s at the photon sphere that there is just enough gravitational force to sustain semi-stable orbits.
What’s interesting about photons in orbit is that they can pass the same location as in the previous orbit, which, if you were somehow holding yourself above a black hole at 1.5 times the Schwarzschild radius, could lead to some pretty interesting effects. (There are no stable orbits inside of 3 Schwarzschild radii.)
For example, photons that are reflected from the back of your head will travel around the black hole, and will potentially approach your location and enter your eyes. As such, you are effectively looking at the back of your own head!
Of course, in practice individual photons will follow slightly different orbits from each other, and so even if any of them pass the exact same location as you, rather than a complete image of the backside of your head you would see only a speck of it (from the very few photons that enter your eyes), which is not actually discernible as any portion of a head.

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