Topology changing process of coalescing black holes on Eguchi-Hanson space

We numerically study the event horizons of two kinds of five-dimensional coalescing black hole solutions with different asymptotic structures: the five-dimensional Kastor-Traschen solution (5DKT) and the coalescing black hole solution on Eguchi-Hanson space (CBEH). Topologies of the spatial infinity are ${\mathrm{S}}^3$ and $L(2;1)={\mathrm{S}}^3/{\mathbb{Z}}_2$, respectively. We show that the crease sets of event horizons are topologically ${\mathrm{R}}^1$ in 5DKT and ${\mathrm{R}}^1\times {\mathrm{S}}^1$ in CBEH, respectively. If we choose the time slices that respect space-time symmetry, the first contact points of the coalescing process is a point in the 5DKT case but a ${\mathrm{S}}^1$ in the CBEH case. We also find that in CBEH, time slices can be chosen so that a black ring with ${\mathrm{S}}^1\times {\mathrm{S}}^2$ topology can be also formed during a certain intermediate period unlike the 5DKT.

Figure 1

Event horizon of five-dimensional Kastor-Traschen solution ($m_{+}=m_{-}=1$, $a=1$, $\lambda =1/(2\sqrt{2})$).

Figure 2

Coordinate value of event horizon of each time slices in the five-dimensional Kastor-Traschen solution.

Figure 3

Event horizon of coalescing black holes on Eguchi-Hanson space ($M_{+}=M_{-}=2$, $a=1$, $\lambda =1/(2\sqrt{2})$).

Figure 4

Coordinate value of event horizon of each time slices in coalescing black holes on Eguchi-Hanson space.

Figure 5

Schematic figure of the event horizon of 5DKT focused on one black hole in some time slice before the coalescence occur. The topology of the spatial cross section of the crease set is a point.

Figure 6

Schematic figure of the event horizon of CBEH focused on one black hole in some time slice before the coalescence occur. The topology of the spatial cross section of the crease set is ${\mathrm{S}}^1$, which corresponds to a great circle of ${\mathrm{S}}^3$.

Figure 7

Schematic figure of the side view of the event horizon of 5DKT and CBEH. Typical time slices (i) and (ii) that are different from the time slice $\tau$ are depicted. We assume time slices (i) and (ii) respect $SO(3)$ symmetry for 5DKT and $SO(2)\times U(1)$ symmetry for CBEH.