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

The geometry of a two-dimensional surface in a curved space can be most easily visualized by using an isometric embedding in flat three-dimensional space. Here we present a new method for embedding surfaces with spherical topology in flat space when such an embedding exists. Our method is based on expanding the surface in spherical harmonics and minimizing the differences between the metric on the original surface and on the trial surface in the space of the expansion coefficients. We have applied this method to study the geometry of black-hole horizons in the presence of strong, non-axisymmetric, gravitational waves (Brill waves). We have noted that, in many cases, although the metric of the horizon seems to have large deviations from axisymmetry, the intrinsic geometry of the horizon is almost axisymmetric. The origin of the large apparent non-axisymmetry of the metric is the deformation of the coordinate system in which the metric was computed.