Generating anisotropic fluids from vacuum Ernst equations

TL;DR

This paper develops a method to generate anisotropic fluid solutions from vacuum Ernst equations in stationary axisymmetric spacetimes, providing new interior solutions that satisfy energy conditions and match vacuum exterior metrics.

Contribution

It introduces a novel approach using Ernst equations to construct anisotropic fluid solutions in different coordinate systems, including explicit matching with vacuum solutions.

Findings

Derived basic equations for anisotropic fluids from vacuum Ernst equations.

Presented interior solutions satisfying energy conditions.

Matched interior solutions with exterior vacuum metrics in oblate spheroidal coordinates.

Abstract

Starting with any stationary axisymmetric vacuum metric, we build anisotropic fluids. With the help of the Ernst method, the basic equations are derived together with the expression for the energy-momentum tensor and with the equation of state compatible with the field equations. The method is presented by using different coordinate systems: the cylindrical coordinates $\rho, z$ and the oblate spheroidal ones. A class of interior solutions matching with stationary axisymmetric asymptotically flat vacuum solutions is found in oblate spheroidal coordinates. The solutions presented satisfy the three energy conditions.