How the expansion of the Universe determines the causal structure of McVittie spacetimes

We present a theorem which allows one to recognize and classify the asymptotic behavior and causal structure of McVittie metrics for different choices of scale factor, establishing whether a black hole or a pair black-white hole appears in the appropriate limit. Incidentally, the theorem also solves an apparent contradiction present in the literature over the causal structure analysis of the McVittie solution. Although the classification we present is not fully complete, we argue that this result covers most if not all physically relevant scenarios.

Figure 1

An example of the McVittie apparent horizons plotted in spacetime coordinates. Some ingoing ($r_{\mathrm{in}}$) and outgoing geodesics ($r_{\mathrm{out}}$) are also shown, as well as some light cones that are represented by the shaded regions, where the arrows indicate the future direction. The singularity at $r=2m$ is represented by a thick dotted line.

Figure 2

An extension of McVittie spacetime with an unbounded image of the inner horizon over the singularity. Ingoing geodesics are represented by lines inclined 45° to the left. The left part of the graph shows the appearance of a black hole whose horizon is given by $r_{-}$ in the limit $t\to \infty$.

Figure 3

A possible null geodesically complete extension of Fig. 2, where a time-reversed McVittie metric with a big crunch also appears [15].

Figure 4

A null geodesically complete extension of a McVittie spacetime characterized by a bounded upper value for $\Phi ({\mathcal{H}}_{-})$, the first-time projector of the inner horizon. Notice that geodesics starting after $t_{\mathrm{e}}$ do not reach the horizon in a finite time. This completion shows the presence of a white hole and a black hole region, separated at the branching surface $b$, as discussed in Ref. 13.

Figure 5

Convergence criterion applied to $\Lambda \mathrm{CDM}$ (16). Values of $\lambda =m{H}_0$ situated to the left of the root correspond to inner horizons which are always reached in finite time. Values situated to the right correspond to metrics with horizons that are only reachable by geodesics which start before a critical first time.