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

The asymptotic properties of conformally static metrics in Einstein–æther 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 γ . To analyze locally the behavior of the solutions near a sonic line v 2 = γ - 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 < γ < 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 ϕ ( t , x ) with potential V ( ϕ ( t , x ) ) the symmetry of the conformally static metric restrict the scalar fields to be considered to ϕ ( t , x ) = ψ ( x ) - λ t , V ( ϕ ( t , x ) ) = e - 2 t U ( ψ ( x ) ) , U ( ψ ) = U 0 e - 2 ψ λ . An exhaustive analysis (analytical or numerical) of the stability conditions is provided for some particular cases.