On the nature of inner light-rings

Non-singular horizonless ultracompact objects provide a simple resolution to the black holes singularity problem. It has been shown that, if these objects are compact enough to exhibit the presence of the light-ring required to mimic the phenomenology of general relativity black holes, they must have at least one additional light-ring. The stability of the inner light ring has been proven under the assumption of Einstein equations and the validity of the null energy condition. Since this can have important repercussions on the instability of a horizonless ultracompact object and the existence of the latter requires some modified gravitational dynamics and/or exotic matter, it is desirable to obtain a model-independent proof of the stability of the additional light-ring. In this paper, we prove the stability of the inner light-ring without any assumption on the dynamics of the theory, while assuming that the outer light-ring has the same properties as the Kerr light-ring. Given the stringent observational constraints on the geometry at the outer light-ring scale, our result now rests solely on geometric considerations and applies to any theory of gravity with any matter content that cannot be ruled out by observations.