On stability analysis for steady states of 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 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.