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We consider the equations for the coefficients of stationary rotatingaxisymmetric metrics governed by the Einstein-Euler equations, that is, theEinstein equations together with the energy-momentum tensor of a barotropicperfect fluid. Although the reduced system of equations for the potentials inthe co-rotating co-ordinate system is known, we derive the system of equationsfor potentials in the so called zero angular momentum observer co-ordinatesystem. We newly give a proof of the equivalence between the reduced system andthe full system of Einstein equations. It is done under the assumption that theangular velocity is constant on the support of the density. Also theconsistency of the equations of the system is analyzed. On this basic theory weconstruct on the whole space the stationary asymptotically flat metricgenerated by a slowly rotating compactly supported perfect fluid with vacuumboundary.

On the axisymmetric metric generated by a rotating perfect fluid with the vacuum boundary