Self-Gravitating Bjorken Flow

I present a solution to the full Einstein-fluid equations representing a self-gravitating Bjorken flow. The motion and the geometry become inhomogeneous in the plane transversal to the flow and the energy density profile acquires, due to gravity, corrections in terms of proper time as compared to the original test hydrodynamics. The transverse distribution of energy density, for example, becomes $\epsilon(\tau,r)/\epsilon(\tau,0)\,=\, \cosh^{-4}{(3ar)}$