Study of multi-black-hole and ring-singularity apparent horizons

We study critical black-hole separations for the formation of a common apparent horizon in systems of $N$ black holes in a time-symmetric configuration. We study in detail the aligned equal mass cases for $N=2$, 3, 4, 5, 6, and relate them to the unequal mass binary black-hole case. We then study the apparent horizon of the time-symmetric initial geometry of a ring singularity of different radii. The apparent horizon is used as indicative of the location of the event horizon in an effort to predict a critical ring radius that would generate an event horizon of toroidal topology. We found that a good estimate for this ring critical radius is $20/(3\pi )M$. We briefly discuss the connection of this two cases through a discrete black-hole “necklace” configuration.

Figure 1

Boundary conditions for a system of $N$ black holes.

Figure 2

MOTS and apparent horizon for a system of two black holes.

Figure 3

System of three black holes.

Figure 4

System of four black holes.

Figure 5

System of five black holes.

Figure 6

Three black holes represented as two black holes with a mass ratio of $2:1$.

Figure 7

Four black holes represented as two black holes with a mass ratio of $1:1$.

Figure 8

Four black holes represented as two black holes with a mass ratio of $3:1$.

Figure 9

Five black holes represented as two black holes with a mass ratio of $3:2$.

Figure 10

Five black holes represented as two black holes with a mass ratio of $4:1$.

Figure 11

System of six black holes.

Figure 12

Six black holes represented as two black holes with a mass ratio of $1:1$.

Figure 13

Six black holes represented as two black holes with a mass ratio of $2:1$.

Figure 14

Six black holes represented as two black holes with a mass ratio of $5:1$.

Figure 15

Plot of critical separation normalized by mass vs mass of 2nd black hole.

Figure 16

Plot to show $|\mathrm{Re}[\frac{z^2+{\rho }^2+{\rho }_o^2}{\rho {\rho }_o}]|\nless 2$.

Figure 17

First set of boundary conditions for a black-hole ring.

Figure 18

Second set of boundary conditions for a black-hole ring

Figure 19

Plot of axis ratio $C_1/C_2$ as a function of mass $M/\pi$.

Figure 20

Apparent horizon for ring masses $m=0.17$, 0.18, 0.19, 0.20 of radius 1.

Figure 21

Apparent horizon for a ring of radius 1 and masses $m=0.17$, 0.18, 0.19, 0.20.

Figure 22

Apparent horizon for a ring of radius 1 and masses $m=0.30$, 0.40, 0.50, 0.60.

Figure 23

Apparent horizon for a ring of radius 1 and masses $m=0.80$, 1.00.

Figure 24

Data obtained for a system of 4 black holes.

Figure 25

Data obtained for a system of 5 black holes.