Einstein–æther models III: conformally static metrics, perfect fluid and scalar fields
AbstractThe 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 $$\gamma $$ γ . To analyze locally the behavior of the solutions near a sonic line $$v^2=\gamma -1$$ 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 $$1<\gamma <2 $$ 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 $$\phi (t,x)$$ ϕ ( t , x ) with potential $$V(\phi (t,x))$$ V ( ϕ ( t , x ) ) the symmetry of the conformally static metric restrict the scalar fields to be considered to $$ \phi (t,x)=\psi (x)-\lambda t, V(\phi (t,x))= e^{-2 t} U(\psi (x))$$ ϕ ( t , x ) = ψ ( x ) - λ t , V ( ϕ ( t , x ) ) = e - 2 t U ( ψ ( x ) ) , $$U(\psi )=U_0 e^{-\frac{2 \psi }{\lambda }}$$ U ( ψ ) = U 0 e - 2 ψ λ . An exhaustive analysis (analytical or numerical) of the stability conditions is provided for some particular cases.
