  Some exotic compact objects possess evanescent ergosurfaces: timelike
submanifolds on which a Killing vector field, which is timelike everywhere
else, becomes null. We show that any manifold possessing an evanescent
ergosurface but no event horizon exhibits a linear instability of a peculiar
kind: either there are solutions to the linear wave equation which concentrate
a finite amount of energy into an arbitrarily small spatial region, or the
energy of waves measured by a stationary family of observers can be amplified
by an arbitrarily large amount. In certain circumstances we can rule out the
first type of instability. We also provide a generalisation to asymptotically
Kaluza-Klein manifolds. This instability bears some similarity with the
"ergoregion instability" of Friedman, and we use many of the results from the
recent proof of this instability by Moschidis.
