A note on a class of spherically symmetric solutions

We find the general solution of the Einstein field equations in terms of elementary functions for a class of accelerating, expanding and shearing spherically symmetric metrics. We demonstrate that these solutions satisfy the equation of statep = π+ const, which is a generalisation of the stiff equation of state. The properties of the solutions are briefly discussed; in particular we show that our solutions have constant anisotropy. We relate our results to the solutions of Gutman and Bespal’ko, Wesson, Lake, Shaver and Lake, and Hajj-Boutros. We show that the Wesson solution is not new and is equivalent to the metric found earlier by Gutman and Bespal’ko. This equivalence was also noted by Lake. We explicitly find the coordinate transformation that equates these solutions. We also show that solutions given by Hajj-Boutros, for particular choices of a metric function, in terms of Painlevé transcendents can be written completely in terms of elementary functions. This is consistent with the results of Shaver and Lake who have shown the equivalence of the Hajj-Boutros and Lake metrics.