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

TL;DR

This paper introduces a new spherical harmonic expansion method for isometrically embedding black hole horizons in flat space, revealing that apparent non-axisymmetry often results from coordinate deformation rather than intrinsic geometry.

Contribution

The authors develop a novel embedding technique using spherical harmonics to visualize black hole horizons in flat space, addressing non-axisymmetric geometries caused by gravitational waves.

Findings

Horizon geometries often appear non-axisymmetric due to coordinate effects.

Intrinsic horizon geometry remains nearly axisymmetric despite metric deviations.

The method effectively visualizes complex horizon shapes in flat space.

Abstract

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 a embedding exists. Our method is based on expanding the surface in spherical harmonics and minimizing for the differences between the metric on the original surface and the metric on the trial surface in the space of the expansion coefficients. We have applied this method to study the geometry of back hole horizons in the presence of strong, non-axisymmetric, gravitational waves (Brill waves). We have noticed 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.