Horizons, singularities and causal structure of the generalized McVittie space-times

The generalized McVittie space-times, which are the natural extensions of a class of metrics introduced by McVittie in 1933 to model the space-time outside a spherical object embedded in an expanding universe, have recently been proposed by Faraoni and Jacques as candidate space-times for cosmological black holes. In this paper I analyze the singularities and horizon structure of the generalized McVittie space-times, and show that any expanding space-time of this type that satisfies the null energy condition and develops apparent horizons must asymptote to a standard McVittie solution. Furthermore, if the scale factor is asymptotically exponential then the space-time is future-incomplete and can be joined smoothly to an eternally-inflating Kottler (Schwarzschild-de Sitter) solution. I argue that no generalized McVittie space-time, apart from the Schwarzschild solution, can adequately represent a black hole, because all singular points are surrounded by anti-trapped regions rather than trapped regions.