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McVittie spacetimes represent an embedding of the Schwarzschild field in isotropic cosmological backgrounds.
The mass-particle in an expanding universe
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On spherically symmetric shear-free perfect fluid configurations (neutral and charged). III. Global view
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We note that both [ 8 ] and [ 9 ] identify solutions of the Einstein equations that do for the k = − 1 k=-1 isotropic spacetimes what the McVittie spacetime does for the k = 0 k=0 class. In comoving coordinates, these solutions are given in terms of elliptic integrals, and so are not readily amenable to the analytic studies that have been done in the k = 0 k=0 case. As far as the author is aware, the possibility that these solutions have an elementary form in area radial coordinates has not been investigated: this may make the solutions more tractable
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