Two non-comoving stiff fluids in radial motion and spherical symmetry
  The problem of two stiff fluids (energy density = pressure) moving radially in spherical symmetry is treated. The metric ansatz is chosen spherically symmetric, conformally static with a multiplicative separation of variables. The first fluid is described mathematically via a massless scalar field. The coordinate system is chosen comoving with the second fluid which the separation of variables requires to be stiff too. The fluids are interacting only gravitationally and their energy momentum tensors are separately conserved. The Einstein equations are reduced to a single nonlinear ODE of second order which is shown to lead to an Abel ODE. A few particular exact solutions were found using a polynomial ansatz. The two non-comoving gravitational sources in the solutions can be interpreted either as scalar fields or stiff fluids. A complete analysis is performed on the range of parameters for which the stiff fluid interpretation is physically acceptable. General formulas are derived for the conformal vectors of the solutions. By making the second fluid vanish, a few single scalar field solutions are generated some of which appear to be new. All solutions considered in this paper have a time-like singularity at the origin (except the trivial FRW one) and are not asymptotically flat (except the static one with k=0). 
Valentin Kostov