Start with an event horizon:
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible.Outside the event horizon of black hole is another interesting boundary - the photon sphere:
A photon sphere is a spherical region of space where gravity is strong enough 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:
which is one and half times the Schwarzschild radius.
Also important about the photon sphere:
As photons travel near the event horizon of a black hole they can escape being pulled in by the gravity of a black hole by traveling at a nearly vertical direction known as an exit cone. A photon on the boundary of this cone will not completely escape the gravity of the black hole. Instead it orbits the black hole.And finally:
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. 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.
Putting the above together, I have drawn this diagram:
In this figure each lettered arrow represents a possible initial photon position and initial trajectory, and a statement of how I understand it will behave. Photon A starts arbitrarily close to, but outside of the photon sphere, with an initial trajectory heading in. Photon B starts arbitrarily close to, but outside of the event horizon with an initial trajectory directly away from the center. Photon C starts arbitrarily close to, but inside of the event horizon with an initial trajectory directly away from the center..
What is unclear to me is what is the intermediate behavior of C? It is fairly well supported that photon C will not escape from the black hole, but my question is whether it will cross the event horizon and travel through the region between the event horizon and the photon sphere before re-crossing the event horizon on its way back to the singularity. This statement was my original hypothesis, but working through the next example convinced me of the opposite. Basically, I wanted to devise a reasonable "story" about the paths of photons B & C such that there would be a smooth transition in the behavior regardless of how close to the event horizon one was (neglecting quantum effects).
Consider the hypothesis in which C never crosses the event horizon - despite being arbitrarily close and with an initial trajectory directly at / normal to the even horizon. In order to not cross the event horizon photon C's path must be sharply curved. Now consider Photon B which is an arbitrarily small distance outside the event horizon. In order for spacetime to be continuous, photon B must also have a sharply curved path. Here is a rough schematic:
Since the initial position of photon B and photon C are arbitrarily close, their initial trajectories are also arbitrarily similar - but they are different. After making the sharp turn, Photon B is spiraling outward, while photon C after making the sharp turn is spiraling inward.