Are they? Right at the point of the event horizon, wouldn’t there be many photons that hit the black hole at just the right angle to be trapped in an “orbit” around it forever? Well, as forever as the black hole retains its current size and mass. Also, wouldn’t the “shield” be sort of fuzzy, with photons just outside of the event horizon that will orbit for zillions of years until they manage to break free, and photons just within the event horizon that will orbit for zillions of years until they finally fall in?
I do not know how this would be at all significant, it just seemed like it my be an interesting quirk of black holes.
No, it’s an unstable orbit. The slightest disruption will either send the photon spiraling into the black hole, or out and away from it. A “stable” orbit is one with a corrective force that pushes things back into the orbit. It’s the difference between balancing a ball on the top of a hill and on the bottom of a valley - both places are flat, but only one is stable.
It’s true that for any single photon, it’s an unstable orbit, but when you consider how many photons pass near something the size of a black hole over the course of it’s life cycle, there’s got to be some percentage of photons that pass close enough that they get traped in shallowly decaying orbits that take long times to decay below the event horizon. Sure it might only be some small fraction of a percent that gets trapped in an orbit that takes a century or longer to decay, but it’s a small fraction of a percentage of a large number of photons to start with. So I understood it.
Classically, something which is at the escape speed will not enter into a stable circular orbit; the escape speed and the orbital speed differ by a factor of the square root of 2. So, if you were at a place where the escape speed is 7 km/s, the speed of a stable circular orbit would be more like 5 km/s. So even classically speaking, photons would not orbit at the event horizon.
Of course, that’s classically speaking. General Relativistically speaking, it’s completely different. In classical physics, there are stable orbits at any distance; you just have to be going the right speed. In GR, as scr4 rightly points out, though, at R = R[sub]S[/sub] = 2GM/c[sup]2[/sup], no stable orbit exists. In the Schwarzschild metric, there is a stable orbit at R = 3R[sub]S[/sub] = 6GM/c[sup]2[/sup].
As for slowly-decaying orbits, I don’t know. I know that classically, there’s no such thing. If something is not in a stable orbit, it will either escape or fall in within one orbit. None of this slowly falling in business.
Achernar, it’s true that classical orbits in gravitational potentials are stable, and in particular any orbiting particle with nonzero angular momentum is bounded away from the center of the gravitational well. This is not true in general relativity; the “pseudopotential” for the orbiting particle (the potential in the corotating frame, basically) has a maximum and is unbounded below near r = 0. A particle which approaches too closely is captured forever. A particle with finely-tuned orbital parameters may hover near the turning point for an arbitrarily long time before either being captured or escaping.
As scr4 says, the circular photon orbit in the Schwarzschild geometry (at R[sub]crit[/sub] = 3 G M / c[sup]2[/sup], the location of the maximum of the Schwarzschild pseudopotential) is unstable. In fact it’s very unstable: a small perturbation will grow exponentially with distance (measured along the photon’s trajectory). A photon initially having r - R[sub]crit[/sub] = d << R[sub]crit[/sub] will find d growing by a factor of exp(2 pi) = 535.49… in each revolution (if I’ve done the math right), independent of the size of the black hole. This means, intuitively, that only about 1 in 500 photons released near the circular orbit will make it around more than once. So although there is an “amplification” of photons near the critical orbit, the amplification factor is something like 1 + 1/500 + 1/500[sup]2[/sup] + …, or about 0.2% brighter than you’d expect.
A good starting reference for this is (of course) Misner, Thorne, and Wheeler’s Gravitation, section 25.6, giving differential equations for photon orbits in Schwarzschild geometry.