  Ultracompact objects are self-gravitating systems with a light ring. It was
recently suggested that fluctuations in the background of these objects are
extremely long-lived and might turn unstable at the nonlinear level, if the
object is not endowed with a horizon. If correct, this result has important
consequences: objects with a light ring are black holes. In other words, the
nonlinear instability of ultracompact stars would provide a strong argument in
favor of the "black hole hypothesis," once electromagnetic or
gravitational-wave observations confirm the existence of light rings. Here we
explore in some depth the mode structure of ultracompact stars, in particular
constant-density stars and gravastars. We show that the existence of very
long-lived modes -- localized near a second, stable null geodesic -- is a
generic feature of gravitational perturbations of such configurations. Already
at the linear level, such modes become unstable if the object rotates
sufficiently fast to develop an ergoregion. Finally, we conjecture that the
long-lived modes become unstable under fragmentation via a
Dyson-Chandrasekhar-Fermi mechanism at the nonlinear level. Depending on the
structure of the star, it is also possible that nonlinearities lead to the
formation of small black holes close to the stable light ring. Our results
suggest that the mere observation of a light ring is a strong evidence for the
existence of black holes.
