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