Binary Black Hole Coalescence

TL;DR

This paper surveys the non-linear GR two-body problem, emphasizing the final coalescence of binary black holes and the advent of stable numerical relativity methods to simulate merger, energy emission, and gravitational-wave signals. It contrasts two leading numerical frameworks— generalized harmonic coordinates with constraint damping and Baumgarte-Shapiro-Shibata-Nakamura (BSSN) with moving punctures—describing the historical challenges, key ingredients for stable evolutions, and practical computational strategies. The results demonstrate that equal-mass mergers radiate a few percent of the system's rest mass energy, produce quadrupole-dominated waveforms in good agreement with post-Newtonian and effective-one-body models, and can impart large recoil kicks depending on spins and mass ratio. These findings have direct implications for gravitational-wave detection, the astrophysical evolution of black holes in galaxies, and even speculative high-energy collisions, highlighting the central role of numerical relativity in connecting theory to observable signals.

Abstract

The two-body problem in general relativity is reviewed, focusing on the final stages of the coalescence of the black holes as uncovered by recent successes in numerical solution of the field equations.

Figures (7)

• Figure 1: A depiction of the trajectories of the black holes from a merger simulation (the "d=16" Cook-Pfeiffer case, from Buonanno:2006ui). The green lines are the centers of the apparent horizons of each black hole. The trajectories end once a common horizon is found. Also shown are the coordinate shapes of the apparent horizons at several key moments.
• Figure 2: A depiction of the gravitational waves emitted during the merger of two equal mass black holes (specifically "d=19" Cook-Pfeiffer initial data Buonanno:2006ui). Shown is a color-map of the real component of the Newman-Penrose scalar $\Psi_4$ multiplied by $r$ along a slice through the orbital plane, which far from the blackholes is proportional to the second time derivative of the plus polarization (green is $0$, toward violet (red) positive (negative)). The time sequence is from top to bottom, and left to right within each row. Each imagine is $25M$ apart, and a common apparent horizon is first detected at $t=529M$ (i.e., the "merger"), which is a little after the frame in row 5, column 3. In the first several frames the spurious radiation associated with the initial data, and how quickly it leaves the domain, is clearly evident. The width/height of each box is around $100M$.
• Figure 3: The plus (left) and cross (right) polarizations of the waveform (multiplied by coordinate distance $r$ from the source, and by the total mass $M$ of the spacetime to non-dimensionalize) from the simulation shown in Fig.\ref{['orbits']}, though here measured along the axis normal to the orbital plane. $t_{CAH}$ is the time when a common apparent horizon is first detected.
• Figure 4: Several phases of the merger as a function of time (horizontal axis) and orbital/wave angular frequency (vertical axis), from Buonanno:2006ui. $\omega_c$ is the orbital angular frequency of the apparent horizons in coordinate space (multiplied by the total mass $M$ to non-dimensionalize); this curve ends once a common apparent horizon forms. $\omega_\lambda$ is the instantaneous frequency of the emitted gravitational wave divided by 2 (and normalized by $M$ again). $dE/dt$ is the luminosity of the wave integrated over the wave extraction 2-sphere. $J_z$ is the component of the angular momentum of the gravitational waves normal to the orbital plane. The "light ring" here is defined as the coordinate location of the unstable equatorial photon orbit of the final Kerr black hole. One cannot define this precisely or unambigously in the binary spacetime, though it is interesting that the orbital and gravitational wave frequencies decouple roughly at this separation. This is also the time when the EOB approach advocates stitching together the inspiral waveform from resummed PN calculations to a ringdown signal.
• Figure 5: A depiction of the orbital configuration resulting in the largest kick velocities. The orbital angular momentum points out of the paper in this case, and the spin vectors for each black hole is in the orbital plane within the paper as shown by the solid blue vectors. The grey curved lines illustrate the dragging of spacetime about the black hole caused by its spin.