More on McVittie’s legacy: A Schwarzschild–de Sitter black and white hole embedded in an asymptotically $\Lambda \mathrm{CDM}$ cosmology

Recently Kaloper, Kleban, and Martin reexamined the McVittie solution and argued, contrary to a very widely held belief, that the solution contains a black hole in an expanding universe. Here we corroborate their main conclusion but go on to examine, in some detail, a specific solution that asymptotes to the $\Lambda \mathrm{CDM}$ cosmology. We show that part of the boundary of the solution contains the inner bifurcation two-sphere of the Schwarzschild—de Sitter spacetime, and so both the black and white hole horizons together form a partial boundary of this McVittie solution. We go on to show that the null and weak energy conditions are satisfied and that the dominant energy condition is satisfied almost everywhere in the solution. The solution is understood here by way of a systematic construction of a conformal diagram based on detailed numerical integrations of the null geodesic equations. We find that the McVittie solution admits a degenerate limit in which the bifurcation two-sphere disappears. For solutions with a zero cosmological constant, we find no evidence for the development of a weak null singularity. Rather, we find that in this case there is either a black hole to the future of an initial singularity or a white hole to its past.

Figure 1

The $R-t$ plane and the locus $f=0$. The locus is timelike above and to the right of $n$ where $\dot{H}=2H(3m-R)/R^2$. The locus is spacelike below $n$. $R=3m$ is also shown. It passes through $f=0$ at $o$, where $t=T$ such that $27m^{2}H(T{)}^2=1$. $R=3M$ is spacelike to the left of the locus and timelike to the right. The trajectories $R_{\pm }$ are spacelike. The asymptotic points are defined by $a\mathrm{\text{:}}(t\to \infty ,R=2m)$, $b\mathrm{\text{:}}(t\to \infty ,R=R_{-})$, and $c\mathrm{\text{:}}(t\to \infty ,R=R_{+})$. These are explained below.

Figure 2

The function $\psi$ given by (19). The curve has been constructed with $H_0={2.310}^{-18}{s}^{-1}$. $T$ is defined by $t={10}^T$, where $[t]=s$. The current epoch is shown as a dot.

Figure 3

The outgoing solutions of the null geodesic equation (12) under the condition (16). The values taken are $H_0=1/3$ and $m=(1/(H_0{l}_0))=958041/2000000\sim 0.479$ (where $l_0$ is defined below). These values are of no consequence, as we are simply interested in the qualitative behavior of the solutions to (12). The locus $f=0$ is also shown. The values of the roots $R_{\pm }$ are $R_{+}\sim 2.29$ and $R_{-}\sim 1.11$. The cone shows the “leg” of the null cone under consideration.

Figure 4

The ingoing solutions of the null geodesic equation (12) under the same conditions as Fig. 3.

Figure 5

Integration of (30) for $l_0=2000000/319347\sim 6.263$ so that $z_{-}=37/100$ and $z_{+}=-37/200+35{893}^{1/2}/200\sim 0.76$ over the range $0<l<l_0$. The limiting geodesics ${\eta }_1$ and ${\eta }_2$ are shown, as is the locus $f=0$ (which connects $\mathcal{H}$ with $i^{+}$). The dashed curves connecting $z=0$ and $\mathcal{H}$ and $i^{+}$ with $z=1$ indicate $d^{2}z/d{l}^2=0$.

Figure 6

Locating $t_1$ and $t_2$ for any event via null geodesics. The locus $f=0$ is shown. Note that $t$ increases to the right.

Figure 7

Conformal representation of the event $(t,R)$. The locus $f=0$ is shown. Note that $t$ increases to the left.

Figure 8

Right diagram: The conformal representation of the outgoing null geodesics as given in Fig. 3. The locus $f=0$ is shown. The bottom horizontal line represents the singularity $R=2m$. Left diagram: The conformal representation of the ingoing null geodesics as given in Fig. 4.

Figure 9

Left diagram: The conformal representation of surfaces of constant $t$. The locus $f=0$ is also shown. Moving from the bottom right to the upper left we have $t_0<t<T$ [where $27m^{2}H(T{)}^2=1$], $t=T$, $t=t_1>T$, and $t_2>t_1$. The boundary has $t\to \infty$. Right diagram: The conformal representation of surfaces of constant $R$. The locus $f=0$ is also shown. Moving from the bottom left to the upper right we have $2m<R_0<R<R_{-}$, $R=R_{-}$, $R_{-}<R<3m$, $R=3m$, $R=R_{+}$, $R_1>R_{+}$. The right boundary has $R\to \infty$ and the left boundary has $R=R_{-}$.

Figure 10

The fluid streamlines $r=\mathrm{\text{constant}}>0$ in the conformal diagram. The streamlines are timelike and $r$ increases to the right. Note that the streamlines do not cross $\mathcal{H}$, which is the limit $r\to 0$.

Figure 11

The conformal representation of the boundary to the spacetime (4) given (16). The boundary is defined as follows: The singularity $R=2m$, ${\mathcal{I}}^{+}\equiv (R=\infty ,t=\infty )$, $i^{+}\equiv (R_{-}<R<\infty ,t=\infty )$, $\mathcal{H}\equiv (R=R_{-},t=\infty )$, $-i^0\equiv (2m\le R<R_{-},t\to \infty )$, and $+i^0\equiv (2m\le R<\infty ,t>0)$.

Figure 12

An extension of the McVittie spacetime which is null geodesically complete. The inner bifurcate two-sphere of the Schwarzschild–de Sitter spacetime ($b$) is identified with that of the McVittie spacetime.

Figure 13

As in Fig. 5 but for $l_0=1/H_{0}m=3\sqrt{3}$ (again we take $H_0=1/3$). The limiting geodesics ${\eta }_1$ and ${\eta }_2$ now coincide and give $\eta$. There is no locus $f=0$. The dashed curves connecting $z=0$ and $\mathcal{H}$ and $\mathcal{H}$ with $z=1$ indicate $d^{2}z/d{l}^2=0$.

Figure 14

As in Fig. 12 but now for the degenerate case.

Figure 15

As in Fig. 5 but for $H_0=0$. Note that $\mathcal{H}$ is now given by $(R=2m,t=\infty )$.

Figure 16

Conformal representation of surfaces of constant $R$ (dashed lines) and constant $t$ (dotted lines) for the case $H_0=0$. The locus $f=0$ from $p$ to $i^{+}$ is shown (solid line).

Figure 17

An extension of the McVittie spacetime for $H_0=0$ which is null geodesically complete. Note that all trajectories terminate on the spacelike singularity $R=2m$. The loci $f=0$ are shown.