Expanding Spherically Symmetric Models without Shear

TL;DR

This paper investigates the integrability of a key differential equation in spherically symmetric shear-free fluids, deriving new solutions, classifying them, and connecting to previous models through Lie and Painlevé analyses.

Contribution

It introduces a new class of solutions for the field equation, generalizes previous models, and provides a detailed integrability and symmetry analysis.

Findings

Derived a new integrability condition for the field equation.

Identified three classes of solutions, including known special cases.

Reduced the problem to simpler autonomous equations for certain parameters.

Abstract

The integrability properties of the field equation $L_{xx} = F(x)L^2$ of a spherically symmetric shear--free fluid are investigated. A first integral, subject to an integrability condition on $F(x)$, is found, giving a new class of solutions which contains the solutions of Stephani (1983) and Srivastava (1987) as special cases. The integrability condition on $F(x)$ is reduced to a quadrature which is expressible in terms of elliptic integrals in general. There are three classes of solution and in general the solution of $L_{xx} = F(x)L^2$ can only be written in parametric form. The case for which $F=F(x)$ can be explicitly given corresponds to the solution of Stephani (1983). A Lie analysis of $L_{xx} = F(x) L^2$ is also performed. If a constant $\alpha$ vanishes, then the solutions of Kustaanheimo and Qvist (1948) and of this paper are regained. For $\alpha \neq 0$ we reduce the problem to a simpler, autonomous equation. The applicability of the Painlev\'e analysis is also briefly considered.