Photon sphere

[3] The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole, where G is the gravitational constant, M is the mass of the black hole, c is the speed of light in vacuum, and rs is the Schwarzschild radius (the radius of the event horizon); 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 (a black hole or possibly an "ultracompact" neutron star[4]).

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 is emitted (or reflected) from the back of one's head and, following an orbit of the black hole, is then intercepted by the person's eye, allowing one to see the back of the head, see e.g.[2] 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 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 the photon sphere is centrifugal force (note: not centripetal) reversal.

[5] 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. an object weighs the same no matter how fast it orbits, and becomes negative inside it.

Inside the photon sphere, faster orbiting leads to greater weight or inward force.

This derivation involves using the Schwarzschild metric, given by For a photon traveling at a constant radius r (i.e. in the φ-coordinate direction),

Setting ds, dr and dθ to zero, we have Re-arranging gives To proceed, we need the relation

, therefore Substituting it all into the radial geodesic equation (the geodesic equation with the radial coordinate as the dependent variable), we obtain Comparing it with what was obtained previously, we have where we have inserted

radians (imagine that the central mass, about which the photon is orbiting, is located at the centre of the coordinate axes.

-coordinate line, for the mass to be located directly in the centre of the photon's orbit, we must have

There are two circular photon orbits in the equatorial plane (prograde and retrograde), with different Boyer–Lindquist radii: where

[7] There exist other constant-radius orbits, but they have more complicated paths which oscillate in latitude about the equator.