Can light of certain wavelengths stably orbit a black hole at the photon sphere? I've recently learned that for a black hole, its photon sphere is an unstable orbit. Light either manages to escape from or fall into the event horizon, depending on how far away from the edge of the photon sphere the light is introduced.
But let's say that the wavelength of a photon is exactly equal to the circumference of the great circle of the photon sphere, and that the photon approaches the vicinity of the black hole at precisely the right angle and distance from the photon sphere such that it ends up tangentially on the surface of the sphere. 


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*Is it possible for this photon to form a standing circular wave at the photon sphere given that its wavelength is precisely equal to the circumference of the great circle of the sphere? 

*If so, would the photon simply stay at the photon sphere, perhaps due to some quantum stability of the standing wave? 

*How can we observe this or test it experimentally?


My question also applies to the case in which the circumference of the great circle of the photon sphere is a natural number multiple (n=1,2,3,...) of the wavelength of the photon. This would be similar to the quantization of electron orbits in the Bohr model of the atom, except for photons, on a much larger scale, and in the context of gravity (i.e. curved space-time).
 A: No, this is not possible.
In the limit that the wavelength of an electromagnetic wave is similar to the size of the black hole, we can no longer use the approximation that electromagnetic waves travel along geodesics. Instead one has to study the solutions of Maxwell's equations on the black hole background.
The solution that is proposed in the question would be a solution (with no waves coming in from infinity or the past horizon) to the vacuum Maxwell equations that does not decay overtime. One can prove that no such solutions exist. (At least, for some known types of black holes. arXiv:1910.02854 proves this statement for Kerr black holes and massless fields of arbitrary spin.)
You can find solutions that correspond (roughly) to waves going around the photon sphere, but these solutions decay exponentially with time. Such solutions are known as the (electromagnetic) quasi-normal modes of the black hole. They occur only at specific frequencies of the electromagnetic field, determined by the mass, angular momentum (and charge) of the black hole.
