Jean-Luc Movie: /NCSA1999/BlackHoles/bs_384_0.2/AppHorizonC

AppHorizon

The Apparent Horizon is a `static equivalent' to the event horizon. It can be computed using an single time slice of some spacetime evolution. In contrast, the even horizon is defined as the surface, out of which nothing can escape. To determine this surface, the entire evolution of the spacetime has to be known. Only in the far future one can say that somewhere in the past there was an event horizon at a given time step.

The Apparent Horizon can however be computed at each time step independently of the former and later timesteps. Its relationship to the event horizon is that the Apparent Horizon is within the Event Horizon. In the case of a static spacetime, the Apparent Horizon and the Event Horizon are identical.

The Movie shows the Apparent Horizons of the two initial Black Holes, which are essentially two spheres, whereby the colors indicate the gaussian curvature along the surfaces. Gaussian curvature of a surface is equal to 1/r², where r is the local radius of curvature at the surface point. Of course, in general relativity the local radius has to be measured using metric distances, not coordinate distances. The movie shows the coordinate appearance of the Apparent Horizon, but the metric gaussian curvature. The gaussian curvature is normalized here to the area of the surface, such that a sphere would appear red independently of its size. Yellow stands for flat areas, blue to negative curvature (hyperbolic curvature, saddle-like surfaces) while green and finally rose represent highly positivly curved surface areas.

Due to its local definition, the Apparent Horizon is not a causal structure. Therefore it may jump into existence at specific times. This is the case with the colliding black holes, too. At T=10.36 the common Apparent Horizon of the two black holes jumps into existence (in truth, the Apparent Horizon was not computed at each evolution time step, so it might already have existed somewhat longer, but at least it is for sure that there was none at the beginning).

At first, the enveloping outer Apparent Horizon has some negative curvature at the equator areas. General Relativity predicts that any black hole must become a Schwarzschild or, in the case of rotation, a Kerr Black Hole (no-hair-theorem). So any perturbation of another kind of black hole must be radiated away and the distorted black hole must become a Kerr Black Hole after sufficient time. So does this newly generated black hole. The outer Apparent Horizon become more spherical (better: elliptic, to approach the Kerr-Black Hole appearance) during the evolution.

Through the transparently rendered surface of the Apparent Horizon one may still see the two inner parts of the Apparent Horizon, corresponding to the two initial black holes. The upper one tends to move forward into x-direction, indicating its linear momentum. Due to this linear momentum the newly created black hole which originates from this black hole merger, has an angular momentum around the y-axis (it becomes a Kerr Black Hole). When concentrating on the gaussian curvature structures along the outer apparent horizon surface, one can follow this kind of rotation of the structures in counter-clockwise rotation around the y-axis.