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

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 $\mu$ 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, ${\mathbb{E}}^3$. Such visualizations illustrate interesting and important information about horizon dynamics. Unfortunately, we could not visualize black holes with spin parameter $a_{*}>\sqrt{3}/2\approx 0.866$: such holes cannot be globally embedded into ${\mathbb{E}}^3$. 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, ${\mathbb{H}}^3$. 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 ${\mathbb{H}}^3$. As an application, we give a sequence of embeddings into ${\mathbb{H}}^3$ 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.

Figure 1

Embeddings of undistorted black holes with spins $a_{*}=0$ (left), $\sqrt{3}/2$ (center), and 1 (right) into the Poincaré ball. We set $M=\ell =1$. Horocycles (dashed red circles), geodesics (dotted gray curves), and the hyperbolic boundary at $r=1$ (black circles) are indicated. The $a_{*}=1$ black hole is negatively curved at the north and south poles and has no global isometric embedding into Euclidean three-space.

Figure 2

Embeddings of the $a_{*}=M=1$ black hole into Poincaré balls with $\ell =1/4$ (top left), $1/2$ (top right), $3/4$ (bottom left), and 1 (bottom right).

Figure 3

Embeddings of an unperturbed $a_{*}=0.85$ black hole into Euclidean three-space (solid black curve) and hyperbolic three-space (dotted curves) for $\ell =1$ (red), 2 (green), 3 (blue), and 5 (purple). In the top panel, the hyperbolic radial coordinate, $r$, has been rescaled and the hyperbolic boundary is at $r\approx 3.2$ (red), 4.8 (green), 6.6 (blue), and 10.4 (purple). In the bottom panel, the hyperbolic boundary is fixed at $r=1$.

Figure 4

Time series depicting the evolution of tidal bulges on an $a_{*}=0.9999$ black hole as its binary companion leaves periapsis. The mass ratio is $\epsilon =1/125$ (see text for details). The event horizon is embedded in the Poincaré ball ($\ell =1$). Colors indicate Gaussian curvature and the gray sphere is the hyperbolic boundary at $r=1$. The black hole is negatively curved at the poles and it can also be negatively curved at the equator during periods of strong tidal distortion.

Figure 5

Gaussian curvature of the $a_{*}=0.9999$ black hole at $\theta =\pi /2$ and $\varphi =0$ as a function of ingoing time $v$. The mass ratios are $\epsilon =1/50$ (solid curve) and $\epsilon =1/125$ (long-dashed curve). The curvature of the unperturbed horizon is indicated (short-dashed line). Red dots correspond to the embeddings shown in Fig. 6. The curvature oscillates despite the absence of significant tidal forces during this time interval, a phenomenon that was uncovered in Ref. [10], but whose origin remains somewhat mysterious.

Figure 6

Embeddings of the tidally distorted $a_{*}=0.9999$ horizon into ${\mathbb{H}}^3$ (we have set $\varphi =0$). The perturbed (solid curves) and unperturbed (dotted curves) horizons are indicated. Panels correspond to ingoing times $v/M=47.9$, 49.5, 51.1 and 52.7 (left to right). These times correspond to the red dots in Fig. 5. The horizon oscillates about its unperturbed position despite the absence of significant tidal forces over this time interval.