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

TL;DR

This paper investigates the asymptotic behavior of conformally static metrics in Einstein-aether theory with perfect fluids and scalar fields, revealing new equilibrium points and stability conditions that extend understanding beyond General Relativity.

Contribution

It introduces new equilibrium points in Einstein-aether models, analyzes their stability, and explores the effects of scalar fields with specific potentials, extending previous relativistic solutions.

Findings

Discovery of new equilibrium points not present in General Relativity for certain parameters.

Analysis of stability conditions for scalar fields with exponential potentials.

Identification of lines of equilibrium points related to causality changes in the models.

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 $\phi(t,x)$ 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.