  The hard phase model describes a relativistic barotropic fluid with sound
speed equal to the speed of light. In the framework of general relativity, the
motion of the fluid is coupled to the Einstein equations which describe the
structure of the underlying spacetime. This model with free boundary admits a
$1$-parameter family of steady states with spherical symmetry. In this work,
for perturbations within spherical symmetry, we study the stability and
instability of this family. We prove that the linearized operator around steady
states with large central densities admits a growing mode, while such growing
modes do not exist for steady states with small central densities. Based on the
linear analysis, we further demonstrate a dynamical nonlinear instability for
steady states with large central densities. The proof relies on a spectral
analysis of the linearized operator and an a priori estimate on the full
nonlinear free boundary problem.
