A Note on the Axisymmetric Stationary Metric in the General Theory of Relativity

TL;DR

This paper proves the equivalence and consistency of equations governing stationary rotating axisymmetric metrics with Einstein-Euler equations, focusing on constant angular velocity cases in general relativity.

Contribution

It establishes the equivalence and consistency of derived equations for axisymmetric stationary metrics with Einstein-Euler equations, extending known results beyond vacuum cases.

Findings

Proves the equivalence of derived equations with Einstein equations.

Establishes the consistency of these equations under certain assumptions.

Focuses on the case of constant angular velocity near the fluid's support.

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 derived equations are not already known except for the case of the constant angular velocity described in the corotating coordinate system, the main content of this article is not to derive the equations, but to prove the equivalence of the derived equations with the full set of the Einstein equations, and to prove the consistency of the derived equations. These affairs have not yet been discussed except for the vacuum case, and are far from being self-evident, requiring tedious careful calculations and some tricks. The proof is done under the assumption that the angular velocity is constant on a neighborhood of the support of the density. The conclusions seem to be doubtful if this assumption does not hold.