On slowly rotating axisymmetric solutions of the Einstein-Euler equations

Abstract

In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which give mathematical models of slowly uniformly rotating gaseous stars. We try to extend this result to the study of solutions of the Einstein-Euler equations in the framework of the general theory of relativity. Although many interesting studies have been done about axisymmetric metric in the general theory of relativity, they are restricted to the region of the vacuum. Mathematically rigorous existence theorem of the axisymmetric interior solutions of the stationary metric corresponding to the energy-momentum tensor of the perfect fluid with non-zero pressure may be not yet established until now except only one found in the pioneering work by U. Heilig done in 1993. In this article, along a different approach to that of Heilig's work, axisymmetric stationary solutions of the Einstein-Euler equations are constructed near those of the Euler-Poisson equations when the speed of light is sufficiently large in the considered system of units, or, when the gravitational field is sufficiently weak.