Event horizon deformations in extreme mass-ratio black hole mergers

We study the geometry of the event horizon of a spacetime in which a small compact object plunges into a large Schwarzschild black hole. We first use the Regge-Wheeler and Zerilli formalisms to calculate the metric perturbations induced by this small compact object, then find the new event horizon by propagating null geodesics near the unperturbed horizon. A caustic is shown to exist before the merger. Focusing on the geometry near the caustic, we show that it is determined predominantly by large-$l$ perturbations, which in turn have simple asymptotic forms near the point at which the particle plunges into the horizon. It is therefore possible to obtain an analytic characterization of the geometry that is independent of the details of the plunge. We compute the invariant length of the caustic. We further show that among the leading-order horizon area increase, half arises from generators that enter the horizon through the caustic, and the rest arises from area increase near the caustic, induced by the gravitational field of the compact object.

Figure 1

Spacetime diagram of the merger of two black holes. The horizontal cross sections give the 3-dimensional geometry of the event horizons as time progresses. The null rays which trace out the horizon are given by the black and red directed lines. The black lines originate from the horizon at past infinity, while the red lines enter the horizon through the caustic. A caustic point $\mathcal{Q}$ is shown with its two null generators entering the horizon.

Figure 2

Plot of $f_{lm}^{(e)}/\mu l^{-1}$ versus $v/l^{-1}$ for a radial plunge, values $2<l<24$, $m=0$ shown. Larger values of $l$ are denoted by darker lines. The red dashed line is the empirical limit curve $8\pi e^{-x/2}$; the significance of this curve is touched on in Sec. 3c.

Figure 3

Event horizon and caustic of a black-hole merger with $m=0.15$, plotted in an ingoing coordinate system at times $v=-1$, $-0.75$, $-0.5$, and $-0.25$, respectively. Black dots indicate null generators on the horizon; red diamonds indicate rays which have yet to enter the horizon via the caustic. The large black dot is the infalling black hole.

Figure 4

Null event-horizon generators, traced back in time. The rays inside the caustic horizon cross each other and exit the event horizon, forming caustics. The rays outside the caustic horizon always remain on the event horizon.

Figure 5

Viewed as a 2-surface in 3-space, the horizon near the caustic looks like a cone of angle $\pi -\alpha$. This is shown for a small cone angle (top) and a larger cone angle (bottom), both cones propagating to the right as time progresses (three time slices of the horizon are shown in the figures). Like a Cherenkov cone, the speed $V_c>c$ increases as the cone angle decreases.

Figure 6

The event horizon of the large black hole as a surface in spacetime. The small black hole has a mass $m=0.15$. Null generators are shown as black lines in the solid region, while the region spanned by the generators entering through the caustic is patterned and shaded red.

Figure 7

As black holes merge, the surface area of the large hole’s horizon increases. This is due to rays entering the horizon through the caustic ($\Delta A_c$, shown in red), expansion of rays near the caustic ($\Delta A_n$, blue), and expansion in the bulk of the horizon ($\Delta A_b$, gray).

Figure 8

A diagram of two strings falling into a black hole, one parallel to the horizon and the other orthogonal. In reality, most objects would fall in at oblique angles.

Figure 9

A Gaussian surface chosen around a string falling into the event horizon.

Figure 10

Plot of $|Z(\omega )|$ versus $\omega$. Larger values of $l$ are denoted by darker lines.

Figure 11

Plot of $|\mathrm{Re}(Z(\omega ))|$ versus $\omega$. Larger values of $l$ are denoted by darker lines.

Figure 12

Plot of $Z(v)/l^{-3}{Y}_{lm}^{*(0)}$ versus $v/l^{-1}$. Larger values of $l$ are denoted by darker lines. The limit curve is shown by a red dotted line.

Figure 13

Plot of $⟨Q(\omega )⟩$ versus $\omega$. Larger values of $l$ are denoted by darker lines.

Figure 14

Plot of the horizon generators as lensed by Eq. (d9) for three masses: $\mu =0.1$ (light gray), 0.01 (dark gray), and 0.001 (black), and the delta function (red, dashed).