Making sense of the bizarre behavior of horizons in the McVittie spacetime

The bizarre behavior of the apparent (black hole and cosmological) horizons of the McVittie spacetime is discussed using, as an analogy, the Schwarzschild-de Sitter-Kottler spacetime (which is a special case of McVittie anyway). For a dust-dominated “background” universe, a black hole cannot exist at early times because its (apparent) horizon would be larger than the cosmological (apparent) horizon. A phantom-dominated background universe causes this situation, and the horizon behavior, to be time-reversed.

Figure 1

The behavior of the McVittie apparent horizons versus time in a dust-dominated background universe. We arbitrarily fix $m=1$, hence time $t$ and radius $r$ are measured in units of $m$ (see text for details).

Figure 2

The total horizon entropy $S$ (in units $\frac{k_B{c}^3}{\hslash G}$, where $k_B$ is the Boltzmann constant) associated with the apparent horizons as a function of time.

Figure 3

The behavior of the McVittie apparent horizons versus time in a phantom-dominated background universe for the parameter values $w=-1.5$ and $t_{\mathrm{rip}}=0$.