Many black hole solutions of the Einstein Field Equations of General Relativity admit a photon sphere or equivalent(s) where light can orbit the black hole, but

Many black hole solutions of the Einstein Field Equations of General Relativity admit a photon sphere or equivalent(s) where light can orbit the black hole, but not usually stably, and in physically reasonable black hole solutions there is no stable circular orbit for light at all, so no eternal orbit. (The orbit can be quite long on human scales though, before the light eventually leaves the area around the black hole, or spirals into the black hole). As a first step, you could can check out <https://en.wikipedia.org/wiki/Photon_sphere https://en.wikipedia.org/wiki/Photon_sphere> and <https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit https://en.wikipedia.org/wiki/Innermost_stable_circular_orbi...>