Can light-rings self-gravitate?

In a spherically symmetric and static spacetime of a compact object, such as that of a Schwarzschild black hole, the light-ring is a 2-sphere where photons experience the only possible circular orbits. As a "Gedankenexperiment", we imagine an advanced civilisation able to populate the light-ring of a nonrotating black hole of mass $M$ with photons having a fine-tuned impact parameter that allows their orbits to be exactly circular with radius $r=3M$. As the number of photons in the light-ring increases in time, its mass will no longer be negligible and hence it will impact on the background spacetime, that is, it will "self-gravitate". We here consider two different routes to assign a nonzero mass to the light-ring that are either based on a discrete concentration of photons on a specific radial location or on a suitable distribution of photons in a given region. In both cases, and using the Einstein equations, we find that the inclusion of the energy from the accumulated photons leads to the generation of new light-rings. Such new light-rings can either appear at well-defined but discrete locations, or be fused in a well-defined region. In either case, we show that such light-ring configurations are dynamically unstable and a small perturbation, either via the inclusion of an additional photon onto the light-ring or via the absorption of a photon by the black hole, leads to a catastrophic destruction of the light-ring structures.
Comment: 6 pages, 1 figure