EXPANDING SPHERICALLY SYMMETRICAL MODELS WITHOUT SHEAR

The integrability properties of the field equation L(xx) = F(x)L(2) of a spherically symmetric shear-free fluid are investigated. A first in tegral, subject to an integrability condition on F(x), is found, givin g a new class of solutions which contains the solutions of Stephani an d Srivastava as special cases. The integrability condition on F(x) is reduced to a quadrature which is expressible in terms of elliptic inte grals in general. There are three classes of solution and in general t he 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 t he solution of Stephani. A Lie analysis of L(xx) = F(x)L(2) is also pe rformed. If a constant cu vanishes, then the solutions of Kustaanheimo and Qvist and of this paper are regained. For alpha not equal 0 we re duce the problem to a simpler, autonomous equation. The applicability of the Painleve analysis is also briefly considered.