  We study spherically symmetric solutions to the Einstein-Euler equations
which model an idealized relativistic neutron star surrounded by vacuum. These
are barotropic fluids with a free boundary, governed by an equation of state
which sets the speed of sound equal to the speed of light. We demonstrate the
existence of a 1-parameter family of static solutions, or ''hard stars,'' and
describe their stability properties:
  First, we show that small stars are a local minimum of the mass energy
functional under variations which preserve the total number of particles. In
particular, we prove that the second variation of the mass energy functional
controls the ''mass aspect function.''
  Second, we derive the linearisation of the Euler-Einstein system around small
stars in ''comoving coordinates,'' and prove a uniform boundedness statement
for an energy, which is exactly at the level of the variational argument.
Finally, we exhibit the existence of time periodic solutions to the linearised
system, which shows that energy boundedness is optimal for this problem.
