On a cosmological Oppenheimer-Snyder model: matching McVittie and FLRW spacetimes

We consider the necessary and sufficient conditions for the smooth matching of an expanding McVittie spacetime and a spatially flat Friedmann-Lema tre-Robertson-Walker (FLRW) spacetime across a general hypersurface - that is, a hypersurface of arbitrary causal character, where the character possibly changes from point to point. We identify a number of special cases, and determine some no-go results. In the general case, we show that the full set of matching conditions is equivalent to a 3-dimensional non-linear system of first order ODEs, subject to a single initial value constraint. Taking the perspective that the McVittie region is specified, we prove a semi-global existence result for this system and interpret these solutions in terms of the physical set-up. Solutions exist and are unique globally to the future. In this direction, the required FLRW spacetime exists and is unique and the matching hypersurface exists almost everywhere and is unique. To the past, the matching hypersurface extends to meet the past singularity of the McVittie region. We determine the causal character of the matching hypersurface in these limits, and consider the implications for the existence (or otherwise) of an isotropic source of the McVittie cosmological black hole spacetime. We find that generically, the matching hypersurface has spacelike as well as timelike portions, and so the isotropic region cannot be considered to be an interior for the McVittie exterior. This is in distinction to the Oppenheimer-Snyder model, in which an isotropic dust sphere matches across an everywhere-timelike boundary to the Schwarzschild exterior. On the basis of the sometimes-spacelike nature of the hypersurface, we conclude that an isotropic source of the McVittie spacetime does not exist in a global sense.