We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein–Euler equations, that is, the Einstein equations together with the energy–momentum tensor of a barotropic perfect fluid. Although the reduced system of equations for the potentials in the co-rotating co-ordinate system is known, we derive the system of equations for potentials in the so called zero angular momentum observer co-ordinate system. We newly give a proof of the equivalence between the reduced system and the full system of Einstein equations. It is done under the assumption that the angular velocity is constant on the support of the density. Also the consistency of the equations of the system is analyzed. On this basic theory we construct on the whole space the stationary asymptotically flat metric generated by a slowly rotating compactly supported perfect fluid with vacuum boundary.

我们考虑由爱因斯坦-欧拉方程控制的静止旋转轴对称度量的系数方程，即爱因斯坦方程与正压完美流体的能量-动量张量。尽管同向旋转坐标系中势能的简化方程组是已知的，但我们推导出了所谓的零角动量观测器坐标系中势能的方程组。我们新给出了简化系统和爱因斯坦方程全系统之间的等价性证明。它是在假设密度支持下角速度恒定的情况下完成的。还分析了系统方程的一致性。