McVittie solution with a negative cosmological constant

Whereas current cosmological observations suggest that the Universe is dominated by a positive cosmological constant ($\Lambda >0$), the AdS/CFT correspondence tells us that the case $\Lambda <0$ is still worthy of consideration. In this paper we study the McVittie solution with $\Lambda <0$. Following a related study, the solution is understood here by way of a systematic construction of conformal diagrams based on detailed numerical integrations of the null geodesic equations. As in the pure Robertson-Walker case, we find that $\Lambda <0$ ensures collapse to a Big Crunch, a feature which completely dominates the global structure.

Figure 1

Numerical integrations of (29) and the locus $f=0$. The ingoing geodesics (“$-$”) are shown dashed and reach their maximal value of $R/m$ on the left-hand branch of the locus $f=0$. The outgoing geodesics (“$+$”) are shown solid and reach their maximal value of $R/m$ on the right-hand branch of the locus $f=0$. The future orientation is $T$ increasing as shown. All geodesics begin and end at $R=2m$ which has two distinct parts, separated by the exceptional point $T=\pi$, $R=2m$. This exceptional point is not part of the spacetime. An enlarged view near $R=2m$ is shown in Fig. 2.

Figure 2

As in Fig. 1 in the neighborhood of $R=2m$.

Figure 3

Numerical integrations of (31) and the locus $f=0$. The ingoing geodesics (“$-$”) are now shown solid and reach their maximal value of $z$ on the left-hand branch of the locus $f=0$. The outgoing geodesics (”$+$”) are now shown dashed and reach their maximal value of $z$ on the right-hand branch of the locus $f=0$. The future orientation is $T$ increasing as above. All geodesics begin at $z=0$ for some value of $0<T<\pi$ and terminate again at $z=0$ for $\pi <T<2\pi$.

Figure 4

Conformal representation of the outgoing null geodesics (dashed) and ingoing null geodesics (solid). The locus $f=0$ is also shown. This locus is globally spacelike. Note that $f>0$ between the two branches of the locus. The boundaries are at $R=2m$. The point at the lower right is $R\to \infty$, $0\le T\le 2\pi$. The point at the lower left is $T=\pi$, $R=2m$.

Figure 5

The trajectories show surfaces of constant $R$. These are timelike within the locus $f=0$ and spacelike outside the locus. The values of $z$ used to generate these curves are 0.15, 0.3, 0.5 and 0.7.

Figure 6

The trajectories show surfaces of constant $T$. These are globally spacelike. The values of $T$ used to generate these curves (bottom up) are $\pi /3$, $\pi /2$, $\pi$ and $3\pi /2$.

Figure 7

Trajectories of constant $r$ characterized by the constant $ϵ$ as explained in the text. We note that $ϵ\to 1/2\sqrt[3]{2}$ to the left and $ϵ\to \infty$ to the right.