Einstein\u2013\xe6ther models III: conformally static metrics, perfect fluid and scalar fields

Abstract

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 gamma . To analyze locally the behavior of the solutions near a sonic line v^2=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<gamma <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 phi (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^{-2 t} U(psi (x)), U(psi )=U_0 e^{-frac{2 psi }{lambda }}. An exhaustive analysis (analytical or numerical) of the stability conditions is provided for some particular cases.