Can light change the shape of a black hole?

Question

If light orbits a black hole without ever getting closer or further from it, and the black hole pulls on the light from all directions nearly simultaneously, does the black hole extend at its equator? Would more light make this change more extreme?

Answer 1

When we talk about a (non-spinning) black hole we normally mean the spacetime geometry called the Schwarzschild metric. This is one of the simpler solutions to the Einstein equations, but it's important to understand that the solution is based on a number of assumptions and as a result it is only an approximation to a real black hole. The assumption that is important here is that the black hole is the only thing that exists i.e. the Schwarzschild metric describes a universe containing just the black hole and nothing else.

In the real universe black holes aren't isolated. Many of them will have accretion disks, and even those black holes that don't have (significant) accretion disks are likely to form in proximity to other matter. For example the supermassive black hole in our galaxy probably doesn't have much of an accretion disk because it seems pretty quiet. However it has lots of stars that orbit near it, and of course it's surrounded by the whole of the Milky Way.

The presence of other matter perturbs the spacetime geometry so it is no longer exactly described by the Schwarzschild metric. Where this is a small effect, e.g. where the black hole is much more massive than the matter around it, we call the perturbation back-reaction.

In your question you describe a black hole surrounded by (I assume) a ring of light orbiting at the photon sphere. As the comments have mentioned this woudn't happen in practice because the orbit is unstable, but there's no reason we can't ask in principle what would happen. And the answer is that the presence of the light would indeed change the shape of the event horizon so it was no longer perfectly spherical.

However I have to confess I don't know how the shape would change. There is no analytic solution to the field equations for a black hole surrounded by a ring, and I haven't seen any attempts at a perturbative calculation (though they may well exist). What we can say is that the effect is likely to be vanishingly small. The spacetime curvature is related to the energy density, where we consider matter to be equivalent to an amount of energy given by Einstein's famous equation $E=mc^2$. Because $c^2$ is so large this means that unless the light is ridiculously intense its gravitational field is negligible when compared to even small amounts of matter.