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

Abstract

The asymptotic properties of conformally static metrics in Einstein-aether 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 point 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 $\varphi (t,x)$ with potential $V(\varphi (t,x))$ the symmetry of the conformally static metric restrict the scalar fields to be considered to $\varphi (t,x)=\psi (x)-\lambda t,V(\varphi (t,x))=e^{-2t}U(\psi (x))$, $U(\psi )=U_0{e}^{-\frac{2\psi }{\lambda }}$. An exhaustive analysis (analytical or numerical) of the stability conditions is provided for some particular cases.

39 pages, 15 compound figures