McVittie’s legacy: Black holes in an expanding universe

We prove that a class of solutions to Einstein’s equations—originally discovered by McVittie in 1933—includes regular black holes embedded in Friedmann-Robertson-Walker cosmologies. If the cosmology is dominated at late times by a positive cosmological constant, the metric is regular everywhere on and outside the black hole horizon and away from the big-bang singularity, and the solutions asymptote in the future and near the horizon to the Schwarzschild-de Sitter geometry. For solutions without a positive cosmological constant the would-be horizon is a weak null singularity.

Figure 1

The section of the Schwarzschild-de Sitter conformal diagram, which is covered by our coordinates $r,t$ when $H=H_0=\mathrm{const}$. Black solid lines are the surfaces of constant $t$, and the green arrow represents an ingoing observer, which crosses these surfaces en route to the bold red dashed line $r=r_{-},t=\infty$, which is the null apparent horizon. This means that the coordinate $t$ is a good time coordinate for ingoing observers.

Figure 2

The conformal diagram of the McVittie solutions which asymptote the Schwarzschild-de Sitter geometry in the future. The red dashed line is the surface $r=r_{-},t=\infty$, and the blue dashed line is the cosmological horizon through $r=r_{+}$. The broken curve at the bottom is the inhomogeneous big bang at $r=2m$ and finite times $t$, and the thin green dashed line is the union of the two branches of apparent horizons for finite $t$. Geodesic expansion is indicated by the now-standard “Bousso wedges”.