Does a photon lose energy or redshift as it circles?
Will it's wavelength be the rest wavelength with centripetal and gravitational forces exactly cancelling?
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A photon, being a quantum of EM radiation, is not only a particle but also a wave. Basic solutions of wave equation around the black hole are quasinormal modes, which are exponentially decaying in time. They correspond to a wave either being absorbed by a black hole horizon or escaping to infinity. So no, a photon could not be orbiting black hole indefinitely even in principle. For a given value of angular momentum number $\ell$ there would be a quasinormal mode with the lowest imaginary part of an eigenvalue (this is a generalization of a circular orbit on a photon sphere), but this imaginary part (which is a decay constant of that mode) would approach a constant value for a large $\ell$.
Additionally, if a photon has high enough energy, its back-reaction on the gravitational field would cause it to lose energy via emission of gravitons. Conceptually the process is similar to an electron in excited state in an atom losing energy via emission of a photon.
Conceptually yes. There is an orbit called the photon sphere with a 1.5 gravitational radius. A photon can circle around a black hole on this orbit indefinitely without losing energy. In reality however this is not possible, because the orbit is unstable. Any deviation of the direction of light from this exact orbit (e.g. due to dispersion) would send the photon either to the black hole or away from it. So in reality, light may make a number of circles, but eventually will all disperse away from the photon sphere.
The wavelength is not important, but it cannot be longer than the circumference of the orbit (very long radio waves).