Strong-field tidal distortions of rotating black holes. III. Embeddings in hyperbolic three-space

Abstract

In previous work, we developed tools for quantifying the tidal distortion of a black hole’s event horizon due to an orbiting companion. These tools use techniques which require large mass ratios (companion mass μ much smaller than black hole mass M), but can be used for arbitrary bound orbits and for any black hole spin. We also showed how to visualize these distorted black holes by embedding their horizons in a global Euclidean three-space, E^{3}. Such visualizations illustrate interesting and important information about horizon dynamics. Unfortunately, we could not visualize black holes with spin parameter a_{*}>sqrt[3]/2≈0.866: such holes cannot be globally embedded into E^{3}. In this paper, we overcome this difficulty by showing how to embed the horizons of tidally distorted Kerr black holes in a hyperbolic three-space, H^{3}. We use black hole perturbation theory to compute the Gaussian curvatures of tidally distorted event horizons, from which we build a two-dimensional metric of their distorted horizons. We develop a numerical method for embedding the tidally distorted horizons in H^{3}. As an application, we give a sequence of embeddings into H^{3} of a tidally interacting black hole with spin a_{*}=0.9999. A small-amplitude, high-frequency oscillation seen in previous work shows up particularly clearly in these embeddings.