# Oddly-Shaped Event Horizons?

1. Aug 2, 2006

### Quaoar

I'm wondering if event horizons can exist in shapes other than perfect spheres (for instance, if the object has angular momentum). If two black holes collide, are the event horizons distorted, or are they simply two intersecting spheres?

Last edited: Aug 2, 2006
2. Nov 22, 2006

### Chris Hillman

Interesting shapes for horizons?

You didn't mention what theory you have in mind, so I'll assume you mean gtr. Here is a succession of short answers:

1. The "equilibrium state" of an isolated rotating black hole is modeled in gtr using the Kerr vacuum solution (or its charged generalization, the Kerr-Newman electrovacuum solution), and in these solutions, the horizons (actually a nested pair) are shaped somewhat like flattened spheres. In the popular Boyer-Lindquist coordinates, the outer horizon appears to be flattened equatorially, but this is somewhat misleading; it is more accurate to picture the horizon as being flattened at the poles, pretty much as the Earth's surface is on average a good approximation to an "oblate spheroid". (Strictly speaking, I should avoid leaving the impression that the geometry of a "constant time slice" through the horizon can be realized by an embedding in flat three dimensional space, but one can look at how its Gaussian curvature of such a two-dimensional slice varies.)

2. A spacetime model of a black hole which is "perturbed" by adding some infalling matter or radiation will in general be distorted, but in such cases the system will emit gravitational radiation, which has the effect of restoring the Kerr geometry.

3. When two black holes collide, their horizon merges to form one horizon. It will briefly be highly non-Kerr, but the system will emit a lot of gravitational radiation which quickly smooths out the new merged horizon into the expected Kerr-shape. Roughly speaking. The new horizon will have a "surface area" exceeding the sum of the original two surface areas. In a crude spacetime diagram, you can picture this as a "pair of pants", in which at the bottom the cross sections show two disjoint closed almost circular curves, and at the top, one closed almost circular curve with circumference a bit longer than the sum of the original two circumferences.

4. Some numerical simulations of the merger of two black holes suggest that sometimes the event horizon can briefly maintain a toroidal topology, but if this can really happen, it seems to require unusual conditions.

5. Event horizons have a "teleological character" which is best appreciated by studying certain "Vaidya thought experiments". Advanced students can see the book Black Hole Physics by Frolov and Novikov for a nicely if microscopically illustrated discussion.

6. There are numerous exact electrovacuum solutions other than the Kerr-Newman electrovacuum which exhibit distorted horizons; the ergoregions are typically more highly distorted. Advanced students can see for example http://www.arxiv.org/abs/gr-qc/0109086 (to tell the truth, in this solution the "ergoregion shaping parameter" may be an artifact of inappropriate boundary conditions).

Chris Hillman