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

TL;DR

This paper derives and analyzes the equations for stationary axisymmetric metrics generated by rotating perfect fluids in general relativity, proving equivalence of formulations and constructing asymptotically flat solutions with vacuum boundaries.

Contribution

It introduces a new formulation of the Einstein-Euler equations in zero angular momentum coordinates and constructs explicit solutions for slowly rotating fluids with vacuum boundaries.

Findings

Established equivalence between reduced and full Einstein equations systems.

Constructed asymptotically flat metrics for slowly rotating fluids.

Analyzed the consistency of the governing equations.

Abstract

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.