On the number of light rings in curved spacetimes of ultra-compact objects

TL;DR

This paper proves that the number of light rings in ultra-compact objects can be odd if degenerate light rings are present, challenging previous theorems that predicted an even number of light rings.

Contribution

It identifies a specific condition for degenerate light rings that allows ultra-compact objects to have an odd number of light rings, providing explicit examples and counterexamples.

Findings

Degenerate light rings can lead to an odd number of light rings.

Spherically symmetric constant density stars with M/R=1/3 have one unstable light ring.

The previously claimed theorem holds except in the case of degenerate light rings.

Abstract

In a very interesting paper, Cunha, Berti, and Herdeiro have recently claimed that ultra-compact objects, self-gravitating horizonless solutions of the Einstein field equations which have a light ring, must possess at least {\it two} (and, in general, an even number of) light rings, of which the inner one is {\it stable}. In the present compact paper we explicitly prove that, while this intriguing theorem is generally true, there is an important exception in the presence of degenerate light rings which, in the spherically symmetric static case, are characterized by the simple dimensionless relation $8\pi r^2_{\gamma}(\rho+p_{\text{T}})=1$ [here $r_{\gamma}$ is the radius of the light ring and $\{\rho,p_{\text{T}}\}$ are respectively the energy density and tangential pressure of the matter fields]. Ultra-compact objects which belong to this unique family can have an {\it odd} number of light rings. As a concrete example, we show that spherically symmetric constant density stars with dimensionless compactness $M/R=1/3$ possess only {\it one} light ring which, interestingly, is shown to be {\it unstable}.