Dynamics of marginally trapped surfaces in a binary black hole merger: Growth and approach to equilibrium

The behavior of quasilocal black hole horizons in a binary black hole merger is studied numerically. We compute the horizon multipole moments, fluxes, and other quantities on black hole horizons throughout the merger. These lead to a better qualitative and quantitative understanding of the coalescence of two black holes: how the final black hole is formed, initially grows, and then settles down to a Kerr black hole. We calculate the rate at which the final black hole approaches equilibrium in a fully nonperturbative situation and identify a time at which the linear ringdown phase begins. Finally, we provide additional support for the conjecture that fields at the horizon are correlated with fields in the wave zone by comparing the in-falling gravitational wave flux at the horizon to the outgoing flux as estimated from the gravitational waveform.

Figure 1

This figure shows, as a function of time, the areas of the common outer and inner horizons and the sum of the areas of the two individual horizons. The common horizon first appears at the intersection of the green and blue curves, which then splits into an inner (blue) and outer (green) horizon. The area of the inner horizon decreases in area, while the outer horizon increases and eventually reaches an equilibrium value. The purple curve refers to the sum of the areas of the two separate individual horizons. We have plotted only up to $t=50M$ since the area of the common outer horizon after that point is essentially constant. The small dip in the area of the individual horizons at the end indicates a problem with numerical accuracy for locating the inner horizon and is not to be trusted at that point.

Figure 2

A speculative scenario for the fate of the two individual and the common inner horizons. After the common outer and -inner horizons form, the initial horizons continue to orbit, eventually intersecting with one another and then merging with the common inner horizon. Current horizon tracking algorithms are unable to track the individual horizons this far into the evolution because of the high level of horizon distortion, and this scenario remains speculation. The plot is not drawn to scale.

Figure 3

The shape of the inner and outer common horizons and the two individual horizons on the equatorial plane at four selected times: $t/M=18.75$, 20, 21, 25. The first ($t=18.75M$) is shortly after the common horizon is formed, and the third ($t=21M$) is shortly before we lose track of the common inner horizon. In the last panel ($t=25M$), we are unable to locate the common inner horizon, and thus only the outer and individual horizons are shown; we lose track of the individual horizons soon after this time. In particular, note that at $t=21M$ portions of the common inner horizon are almost tangential to the $y$ axis, indicating that the inner horizon is close to violating the property of being star shaped. Note also that the inner horizons are rapidly decreasing in size in the coordinate system used in the simulation, which causes the horizon finder to lose track of them. This is a gauge effect, and the area of these horizons shows no such effect. Different gauge conditions can be used, which would make it easier to locate the individual horizons.

Figure 4

The shape of the outer common horizon on the equatorial plane as a function of time. See the text for explanation.

Figure 5

The shear at the horizon at $t=19$, 20, 25.

Figure 6

The angular momentum and mass of the final black hole. The first figure shows the mass $M(t)$ of the common outer horizon as a function of time $t$. The second shows the dimensionless spin, i.e. $J(t)/M(t{)}^2$ with $J$ being the angular momentum.

Figure 7

The mass and spin moments for the common outer horizon. The inset for both plots is a zoomed version of the bigger plot, showing better the time variation of the moments.

Figure 8

The behavior of the ratios of the mass and spin moments to the corresponding Kerr moments at each instant of the simulation.

Figure 9

Logarithmic plot of the mass and spin multipoles. A change in the slope is clearly identifiable around the time $27M$. This time corresponds to a time approximately $10M$ after the merger and delineates a substantial change in the behavior of the horizon. The change is most clearly visible in the higher moments but also occurs in $M_2$ and $J_3$. The figure also shows the best fit straight lines to the portion of the plots before and after $t=27M$ (gray lines).

Figure 10

A spacetime diagram demonstrating cross-correlations between the horizon and null infinity. Future null infinity is ${\mathcal{I}}^{+}$ where the gravitational waveform is extracted. The event horizon is $\mathcal{E}$, and the dynamical horizon is the spacelike surface $\mathcal{H}$. Future timelike infinity is $i^0$ where $\mathcal{E}$, $\mathcal{H}$, ${\mathcal{I}}^{+}$, and the singularity (the bold dashed horizontal line) all meet. Cauchy surfaces used in the numerical simulation are represented by dashed lines. The common outer horizon is formed at time $t_0$ when the Cauchy surface just touches $\mathcal{H}$ in this figure. At later times, the intersection of $\mathcal{H}$ with the Cauchy surfaces yield the outer and inner marginally trapped surfaces ${\mathcal{S}}_t^{\text{outer}}$ and ${\mathcal{S}}_t^{\text{inner}}$ respectively. The common source is the shaded region which is conjectured to lead to correlations between the horizon (either the event horizon or preferably the dynamical horizon). In this picture, if the formation of the common horizon is to be correlated with the maximum of the outgoing energy flux at ${\mathcal{I}}^{+}$, the common source for this must be at some earlier time, which can causally affect both fields at $\mathcal{H}$ and ${\mathcal{I}}^{+}$.

Figure 11

The incoming and outgoing fluxes plotted as functions of time. The time axis for the ingoing flux $|\overline{\sigma }{|}^2$ has been shifted so that it is aligned with the peak outgoing luminosity.