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The hard phase model describes a relativistic barotropic fluid with soundspeed equal to the speed of light. In the framework of general relativity, themotion of the fluid is coupled to the Einstein equations which describe thestructure of the underlying spacetime. This model admits a $1$-parameter familyof steady states with spherical symmetry. In this work, for perturbationswithin spherical symmetry, we study the linear stability and instability ofthis family. We prove that the linearized operator around steady states withlarge central density admits a growing mode, while such growing modes do notexist for steady states with small central density.

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