  We prove the following theorem: axisymmetric, stationary solutions of the
Einstein field equations formed from classical gravitational collapse of matter
obeying the null energy condition, that are everywhere smooth and ultracompact
(i.e., they have a light ring) must have at least two light rings, and one of
them is stable. It has been argued that stable light rings generally lead to
nonlinear spacetime instabilities. Our result implies that smooth, physically
and dynamically reasonable ultracompact objects are not viable as observational
alternatives to black holes whenever these instabilities occur on
astrophysically short time scales. The proof of the theorem has two parts: (i)
We show that light rings always come in pairs, one being a saddle point and the
other a local extremum of an effective potential. This result follows from a
topological argument based on the Brouwer degree of a continuous map, with no
assumptions on the spacetime dynamics, and hence it is applicable to any metric
gravity theory where photons follow null geodesics. (ii) Assuming Einstein's
equations, we show that the extremum is a local minimum of the potential (i.e.,
a stable light ring) if the energy-momentum tensor satisfies the null energy
condition.
