Geometric horizons in the Kastor-Traschen multi-black-hole solutions

We investigate the existence of invariantly defined quasilocal hypersurfaces in the Kastor-Traschen solution containing $N$ charge-equal-to-mass black holes. These hypersurfaces are characterized by the vanishing of particular curvature invariants, known as Cartan invariants, which are generated using the frame approach. The Cartan invariants of interest describe the expansion of the outgoing and ingoing null vectors belonging to the invariant null frame arising from the Cartan-Karlhede algorithm. We show that the evolution of the hypersurfaces surrounding the black holes depends on an upper-bound on the total mass for the case of two and three equal mass black holes. We discuss the results in the context of the geometric horizon conjectures.

Figure 1

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=500t_0$ (left) and $t=180t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=2$ subcritical case.

Figure 2

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=130t_0$ (left) and $t=70t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=2$ subcritical case.

Figure 3

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=18.6t_0$ (left) and $t=t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=2$ subcritical case.

Figure 4

The surfaces defined by the vanishing of $\rho$ (top-left) and $-\mu$ (top-right) in 3D space, at $t=-0.914/H$ for the $N=2$ subcritical case. Due to scale, the surfaces surrounding the black holes are not visible in the graph of $\rho =0$; one of these surfaces defined by $\rho =0$ centered on the black hole location $x=-0.1,y=0,z=0$ is depicted (below). A similar surface is formed around the other black hole location.

Figure 5

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=t_0/4$ (left), $t=t_0/4.1$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=2$ subcritical case.

Figure 6

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t_0/4.5$ (left) and $t={10}^{-10}{t}_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=2$ subcritical case.

Figure 7

The surfaces defined by the vanishing of $\rho$ (left) and $-\mu$ (right) in 3D space, at $t={10}^{-10}{t}_0$ for the $N=2$ subcritical case.

Figure 8

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=180t_0$ (left) and $t=70t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=2$ supercritical case.

Figure 9

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=t_0=-0.914/H$ (left), $t={10}^{-10}{t}_0$ (right) in the $z=0$ plane for the $N=2$ supercritical case.

Figure 10

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=1000t_0$ (left), $t=500t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=3$ subcritical case.

Figure 11

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=130t_0$ (left), $t=40t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=3$ subcritical case.

Figure 12

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=35t_0$ (left), $t=25t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=3$ subcritical case.

Figure 13

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=t_0$ (left) and $t={10}^{-10}{t}_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane.

Figure 14

The surfaces surrounding the 3 black holes defined by the vanishing of $\rho$ (left) and $-\mu$ (right) at $t_0=-0.914/H$ viewed from above.

Figure 15

The surfaces surrounding the 3 black holes defined by the vanishing of $\rho$ (left) and $-\mu$ (right) in 3D space, at $t={10}^{-10}{t}_0$.

Figure 16

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=500t_0$ (left), $t=70t_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=3$ supercritical case.

Figure 17

Slices of the zeroes of $\rho$ (black) and $-\mu$ (red) at time $t=0.25t_0$ (left) and $t={10}^{-10}{t}_0$ (right) with $t_0=-0.914/H$ in the $z=0$ plane for the $N=3$ supercritical case.