Robertson-Walker fluid sources endowed with rotation characterised by quadratic terms in angular velocity parameter

TL;DR

This paper derives Einstein's equations for a rotating Robertson-Walker fluid source including quadratic angular velocity terms, providing analytic solutions and analyzing velocity components and density distributions.

Contribution

It introduces a family of analytic solutions for rotating Robertson-Walker fluids with quadratic angular velocity terms, including special cases like dust and smooth merging with non-rotating sources.

Findings

Explicit solutions for rotating fluids with quadratic angular velocity terms.

Density and pressure depend sinusoidally on polar angle.

Radial velocity component exists and vanishes at poles.

Abstract

Einstein's equations for a Robertson-Walker fluid source endowed with rotation Einstein's equations for a Robertson-Walker fluid source endowed with rotation are presented upto and including quadratic terms in angular velocity parameter. A family of analytic solutions are obtained for the case in which the source angular velocity is purely time-dependent. A subclass of solutions is presented which merge smoothly to homogeneous rotating and non-rotating central sources. The particular solution for dust endowed with rotation is presented. In all cases explicit expressions, depending sinusoidally on polar angle, are given for the density and internal supporting pressure of the rotating source. In addition to the non-zero axial velocity of the fluid particles it is shown that there is also a radial component of velocity which vanishes only at the poles. The velocity four-vector has a zero component between poles.