Expanding Spherically Symmetric Models without Shear

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 $\alpha \ne 0$ we reduce the problem to a simpler, autonomous equation. The applicability of the Painlevé analysis is also briefly considered.

14 pages LaTeX, to appear General Relativity and Gravitation

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