Embeddings of dynamical event horizons

Event horizons are distorted by tidal stresses that arise from a binary companion. Studying and quantifying these distortions was the main Ph.D. project of Stephen O’Sullivan. On this page, we show videos illustrating situations in which a black hole’s horizon distortions vary dynamically, following the dynamics of the applied tide which itself follows the dynamics of the binary’s orbit. The calculations which underlie these videos are described here and here.

What’s an embedding?

The following visualizations show embeddings of the geometry of event horizons in large mass-ratio binary systems. An embedding is a way to show a surface whose curvature depends on the structure of the geometry in which this surface exists. In these visualizations, we show ellipsoids whose intrinsic geometry in Euclidean space is identical to that of the event horizons in large mass-ratio black hole binary spacetimes.

In the case of the event horizons shown below, it should be emphasized that every one of these surfaces is actually a sphere in the coordinate system that we use for the underlying calculations. We choose a gauge so that the horizon lies at the coordinate $$r_H = M + \sqrt{M^2 - a^2}$$ (using geometrized units with G = 1 = c). Because of the way that spacetime near the black hole is distored by the orbiting body, the geometry of that surface can be highly nonspherical, despite its spherical coordinate representation. It is useful to think of these embeddings as being like a soap bubble whose shape is chosen so that its intrinsic geometry, particularly its Gaussian curvature, matches that of the tidally distorted black hole.

In the absence of a binary companion, the embedding of a black hole is a spheroid that is flattened near the poles defined by its axis of rotation. It is perfectly spherical for non-spinning (Schwarzschild) black holes, and has a “squashedness” that varies with spin for rotating (Kerr) black holes. Larry Smarr first described how to make embedding diagrams of the horizons of spinning black holes in a 1973 paper. Interestingly, it turns out that one cannot embed even undistorted Kerr black holes in Euclidean space if their spin exceeds about 87% of the Kerr maximum (although one can make such embeddings in other geometries). For the visualizations shown below, we choose the spin parameter in the Kerr cases to be slightly below the maximum allowed value that allows a Euclidean embedding.

Simplest case: Spherical orbits of Schwarzschild black holes

We begin with a simple case: A spherical (i.e., constant radius) orbit of a Schwarzschild black hole. The orbit is at r = 6M (the smallest radius stable Schwarzschild orbit), and inclined with respect to our chosen equatorial plane by 60°. For Schwarzschild, orbital inclination doesn’t mean much: The hole is spherically symmetric, and any orbital plane is equivalent to any other. We see this here by the fact that the horizon’s distortion moves in perfect lockstep with the orbit. If we rode along with the orbiting body, the event horizon would look static; all of its dynamics are perfectly locked to those of the orbit.

Note, in the following video and those which follow, it should be emphasized that the magnitude of the distortion due to the binary companion has been magnified for the purpose of the visualization. The size of the effect scales with the mass ratio of the system, and for the perturbative tools that we use to be valid, the mass of the perturber should be much smaller than the mass of the black hole.

Some dynamics: Spherical orbits of Kerr black holes

We next examine a more interesting case: A spherical orbit of radius r = 6M inclined at 60° around a Kerr black hole with spin a = 0.6M. Orbital inclination is very important here; the black hole’s spin axis picks out a preferred direction, and the black hole is not spherically symmetric. Even though this orbit is of constant radius, the tidal field that acts on the horizon varies over the course of the orbit. We see “bending” dynamics for the black hole as the orbit proceeds in this case; the bulge’s behavior is not “locked” to the orbit in the same way that we saw in the Schwarzschild case. The rotating axes seen in this case (and the next two as well) are tied to the black hole’s rotation, and rotate at the horizon spin frequency

$$ \Omega_H = \frac{a}{2Mr_H}. $$

where $$r_H = M + \sqrt{M^2 - a^2}$$ is the coordinate radius of the event horizon.

More dynamics: Eccentric but equatorial orbits of Kerr black holes

We next move away from circular orbits. Our third example is a black hole with spin a = 0.85M. The orbit is confined to the equatorial plane; it has eccentricity e = 0.7, and semi-latus rectum p = 4M. This means the orbital radius varies from r = 2.35M to r = 13.33M over an orbit. The tidal field (which varies roughly as \(1/r^3\)) thus varies by a factor of about 180 over an orbit. We see this here in that the horizon’s distortion varies from essentially undistorted at large radius to highly distorted as the orbit passes near its closest approach.

Lots of dynamics: Generic orbits of Kerr black holes

The final case we show combines all of these features: As in the previous case, the black hole spin is again a = 0.85M, the orbit has eccentricity e = 0.7, and semi-latus rectum p = 4M. Now, we incline orbit 30° above the equatorial plane. We again see large-scale variation in horizon geometry as the orbit varies from minimum to maximum radius and back; in addition to this, we see the interesting “bending” dynamics associated with the smaller body’s orbital motion above and below the equatorial plane.