A photon sphere or photon circle is an area or region of space where gravity is so strong that photons are forced to travel in orbits. The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole: where is the gravitational constant, is the black hole mass, and is the speed of light in vacuum and is the Schwarzschild radius - see below for a derivation of this result. This equation entails that photon spheres can only exist in the space surrounding an extremely compact object. The photon sphere is located farther from the center of a black hole than the event horizon. Within a photon sphere, it is possible to imagine a photon that's emitted from the back of one's head, orbiting the black hole, only then to be intercepted by the person's eyes, allowing one to see the back of the head. For non-rotating black holes, the photon sphere is a sphere of radius 3/2 rs. There are no stable free fall orbits that exist within or cross the photon sphere. Any free fall orbit that crosses it from the outside spirals into the black hole. Any orbit that crosses it from the inside escapes to infinity or falls back in and spirals into the black hole. No unaccelerated orbit with a semi-major axis less than this distance is possible, but within the photon sphere, a constant acceleration will allow a spacecraft or probe to hover above the event horizon. Another property of a photon sphere is centrifugal force reversal. Outside the photon sphere, the faster one orbits the greater the outward force one feels. Centrifugal force falls to zero at the photon sphere, including non-freefall orbits at any speed, i.e. you weigh the same no matter how fast you orbit, and becomes negative inside it. Inside the photon sphere the faster you orbit the greater your felt weight or inward force. This has serious ramifications for the fluid dynamics of inward fluid flow. A rotating black hole has two photon spheres. As a black hole rotates, it drags space with it. The photon sphere that is closer to the black hole is moving in the same direction as the rotation, whereas the photon sphere further away is moving against it. The greater the angular velocity of the rotation of a black hole, the greater the distance between the two photon spheres. Since the black hole has an axis of rotation, this only holds true if approaching the black hole in the direction of the equator. If approaching at a different angle, such as one from the poles of the black hole to the equator, there is only one photon sphere. This is because approaching at this angle the possibility of traveling with or against the rotation does not exist.
Since a Schwarzschild black hole has spherical symmetry, all possible axes for a circular photon orbit are equivalent and all circular orbits have the same radius. This derivation involves using the Schwarzschild metric, given by: For a photon traveling at a constant radius r,. Since it is a photon . We can always rotate the coordinate system such that is constant, . Setting ds, dr and dθ to zero, we have: Re-arranging gives: To proceed we need the relation. To find it, we use the radial geodesic equation Non vanishing -connection coefficients are, where. We treat photon radial geodesics with constant r and, therefore Substituting it all into the radial geodesic equation, we obtain Comparing it with what was obtained previously, we have: where we have inserted radians. Hence, rearranging this final expression gives: which is the result we set out to prove.
In contrast to a Schwarzschild black hole, a Kerr black hole does not have spherical symmetry, but only an axis of symmetry, which has profound consequences for the photon orbits, see e.g. Cramer for details and simulations of photon orbits and photon circles. A circular orbit can only exist in the equatorial plane, and there are two of them, with different Boyer–Lindquist-radii, where is the angular momentum per unit mass of the black hole. There exist other constant coordinate-radius orbits, but they have more complicated paths which oscillate in latitude about the equator.
