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.

分析了具有理想流体源和标量场的爱因斯坦–埃瑟理论中的共形静态度量的渐近性质。在理想流体的情况下，将恢复一些相对论解，例如：Minkowski时空，Kasner解，平坦的FLRW空间以及取决于正压参数\（\ gamma \）的静态轨道。为了局部分析声波线\（v ^ 2 = \ gamma -1 \）附近的解的行为，其中v是倾斜，使用了一个新的“ shock”变量。在这条线上找到两个新的平衡点。当\（1 <\ gamma <2 \）时，这些点在广义相对论中不存在。在广义相对论的极限情况下，这些点代表具有极端倾斜的刚性解。在广义相对论的极限中发现了与同矢量场因果关系的变化相关的平衡点线。对于具有势\（V（\ phi（t，x））\）的非均匀标量场\（\ phi（t，x）\），共形静态度量的对称性将标量场限制为\（ \ 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}} \）。针对某些特定情况，提供了稳定性条件的详尽分析（分析或数值分析）。