  The hard phase model describes a relativistic barotropic and irrotational
fluid with sound speed equal to the speed of light. In the framework of general
relativity, the fluid, as a matter field, affects the geometry of the
background spacetime. Therefore the motion of the fluid must be coupled to the
Einstein equations which describe the structure of the underlying spacetime. In
this work we prove a priori estimates and well-posedness in Sobolev spaces for
this model with free boundary. Estimates for the curvature are derived using
the Bianchi equations in a frame that is parallel transported by the fluid
velocity. The fluid velocity is also decomposed with respect to this parallel
frame, and its components are estimated using a coupled interior-boundary
system of wave equations.
