Einstein-æther models III: conformally static metrics, perfect fluid and scalar fields

The asymptotic properties of conformally static metrics in Einstein-ae ther theory with a perfect fluid source and a scalar field are analyzed. In case of perfect fluid, some relativistic solutions are recovered such as: Minkowski spacetime, the Kasner solution, a flat FLRW space and static orbits depending on the barotropic parameter gamma. To analyze locally the behavior of the solutions near a sonic line v2=gamma -1, where v is the tilt, a new shock variable is used. Two new equilibrium points on this line are found. These points do not exist in General Relativity when 1<<2. In the limiting case of General Relativity these points represent stiff solutions with extreme tilt. Lines of equilibrium points associated with a change of causality of the homothetic vector field are found in the limit of general relativity. For non-homogeneous scalar field (t,x) with potential V(phi (t,x)) the symmetry of the conformally static metric restrict the scalar fields to be considered to phi (t,x)=psi (x)-lambda t,V(phi (t,x))=e-2tU(psi (x)), U(psi)=U0e-2 psi lambda. An exhaustive analysis (analytical or numerical) of the stability conditions is provided for some particular cases.