Dynamics of black hole apparent horizons

Abstract

Dynamic black hole spacetimes are studied by examining the evolution of apparent horizons surrounding the holes. We performed numerical evolutions of three different initial data sets: nonrotating black holes distorted by time symmetric (Brill) gravitational waves, distorted rotating black holes, and the time symmetric two black hole Misner data. Although the initial data sets represent different physical problems, the results for these systems are strikingly similar. At early times in the evolution, the apparent horizons may be very distorted and nonspherical (or disjoint in the case of two black holes), but the systems quickly settle down to a nearly spherical or oblate (in the case of rotating holes) configuration and the horizons are then seen to oscillate at the quasinormal frequency of the final black hole. In the case of two black holes with disjoint horizons, we see the appearance of a larger horizon surrounding both holes as they collide. From this point the horizon dynamics is very similar to the single distorted black hole systems. The wavelength and damping time of the quasinormal modes and the rotation parameter in the rotating cases can be read off directly from oscillations in the geometry of the black hole horizons. The apparent horizon is thus shown to be a powerful tool in the study of black hole spacetimes.PACS number(s): 04.30. Db, 04.20.Ex, 04.25.Dm, 04.70.B~

Evolution of distorted rotating black holes. I. Methods and tests

“…In this first paper in this series we present details of a numerical code designed to evolve rotating black hole initial data sets, such as Kerr, Bowen and York [3], and a new class of data sets we have developed [8]. With such a code we can now study the dynamics of highly distorted, rotating black holes, paralleling recent work of the NCSA group on non-rotating holes [9,10]. The first studies of the physics of these evolving systems are presented in a companion paper [11], referred to henceforth as Paper II, and a complete study of the initial data sets will be discussed in Ref.…”

“…As in Ref. [9], the conformal factor Ψ is determined on the initial slice and, since we do not use it as a dynamical variable, it remains fixed in time afterwards. The introduction of Ψ into the extrinsic curvature variables simplifies the evolution equations somewhat.…”

“…[9,10] in defining the variables used in our code. Additional metric and extrinsic curvature variables must be introduced to allow for the odd-parity modes present now that the system allows for rotation.…”

Isometric embeddings of black-hole horizons in three-dimensional flat space

“…However, since the location of such horizons is obtained only in coordinate space, one typically has little information about the real geometry of those surfaces. One can, for example, obtain very similar shapes in coordinate space for horizons that are in fact very different (see for example the family of distorted black holes studied in [1]; their coordinate locations are very similar, but their geometries are quite different). The most natural way to visualize the geometry of a black-hole horizon, or of any other surface computed in some abstract curved space is to find a surface in ordinary flat space that has the same intrinsic geometry as the original surface.…”

“…It was also shown that for rapidly rotating black holes, with a/m > √ 3/2, the Gaussian curvature becomes negative near the poles, and the surface is not embeddable in Euclidean space, as it is 'too flat'! Extending on this paper, using a direct constructive embedding method described below, a number of studies were made of distorted, rotating and colliding black-hole horizons in axisymmetry [1,[6][7][8][9][10] where it was shown that embeddings are very useful tools to aid in the understanding of the dynamics of black holes. For example, distorted rotating black-hole horizons were found to oscillate, about their oblate equilibrium shape, at their quasi-normal frequency.…”