On linear (in)stability of steady states for the free boundary hard phase model in general relativity

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 admits a $1$-parameter family of steady states with spherical symmetry. In this work, for perturbations within spherical symmetry, we study the linear stability and instability of this family. We prove that the linearized operator around steady states with large central density admits a growing mode, while such growing modes do not exist for steady states with small central density