Isotropy, shear, symmetry and exact solutions for relativistic fluid spheres

TL;DR

This paper employs symmetry methods to derive and extend solutions of Einstein's equations for relativistic fluid spheres, including shear and non-comoving frames, providing new exact solutions and insights into their physical properties.

Contribution

It introduces new exact solutions for relativistic fluid spheres with shear using classical and non-classical symmetry approaches, expanding the known solution space.

Findings

Derived classical symmetry solutions for shear-free fluid spheres.

Extended solution set using non-classical symmetries with non-zero shear.

Analyzed kinematics, pressure, energy density, and mass functions of solutions.

Abstract

The symmetry method is used to derive solutions of Einstein's equations for fluid spheres using an isotropic metric and a velocity four vector that is non-comoving. Initially the Lie, classical approach is used to review and provide a connecting framework for many comoving and so shear free solutions. This provides the basis for the derivation of the classical point symmetries for the more general and mathematicaly less tractable description of Einstein's equations in the non-comoving frame. Although the range of symmetries is restrictive, existing and new symmetry solutions with non-zero shear are derived. The range is then extended using the non-classical direct symmetry approach of Clarkson and Kruskal and so additional new solutions with non-zero shear are also presented. The kinematics and pressure, energy density, mass function of these solutions are determined.