McVittie's mass particle in an expanding universe and related solutions of Einstein's equations

Three alternative forms are presented for the Robertson-Walker line element of the k = -1 Riemannian curvature isotropic cosmological models, as well as two alternative forms for the k = 0 model and one form for the k = +1 model; these forms are directly derived by solving Einstein's field equations. It is then noted that McVittie's (1933) embedding of a Schwarzschild field in the Robertson-Walker spacetimes fits naturally into one each of the alternative line elements. Related solutions of Einstein's equations are derived with a perfect fluid source which fit naturally in the remaining two forms of the k = -1 universe and the one remaining form of the k = 0 universe. Three additional perfect fluid solutions of Einstein's equations that are in a different sense analogous to McVittie's original solutions are also presented. 12 refs.