Black-hole binaries, gravitational waves, and numerical relativity

Understanding the predictions of general relativity for the dynamical interactions of two black holes has been a long-standing unsolved problem in theoretical physics. Black-hole mergers are monumental astrophysical events, releasing tremendous amounts of energy in the form of gravitational radiation, and are key sources for both ground- and space-based gravitational-wave detectors. The black-hole merger dynamics and the resulting gravitational wave forms can only be calculated through numerical simulations of Einstein’s equations of general relativity. For many years, numerical relativists attempting to model these mergers encountered a host of problems, causing their codes to crash after just a fraction of a binary orbit could be simulated. Recently, however, a series of dramatic advances in numerical relativity has allowed stable, robust black-hole merger simulations. This remarkable progress in the rapidly maturing field of numerical relativity and the new understanding of black-hole binary dynamics that is emerging is chronicled. Important applications of these fundamental physics results to astrophysics, to gravitational-wave astronomy, and in other areas are also discussed.

Figure 1

Lines of force for plane gravitational waves propagating along the $z$ axis. The wave on the left is purely in the + polarization state and the one on the right is purely in the × polarization state. The gravitational waves produce tidal forces in planes transverse to the propagation direction. From 4.

Figure 2

The $3+1$ split into space and time. Two spatial slices at $t=0$ and $t=dt$ are depicted. $\alpha$ is the lapse and $\alpha dt$ represents the proper time lapse between slices. ${\beta }^a$ is the shift and ${\beta }^{a}dt$ represents the amount by which the spatial coordinates shift between slices. $n^a$ is normal to the slice at $t=0$. If a ray parallel to $n^a$ intersects the $t=0$ slice at a point $x^a$, then it will intersect the $t=dt$ slice at $x^a-{\beta }^{a}dt$. $t^a\equiv \alpha n^a+{\beta }^a$ is a coordinate time vector. If a ray parallel to $t^a$ intersects the $t=0$ slice at point $x^a$, then it will also intersect the $t=dt$ slice at $x^a$.

Figure 3

In the wormhole representation of a black hole, the initial slice typically just touches the horizon. The upper “sheet” represents the exterior space, while the lower sheet is a duplicate, joined to the upper sheet by a “throat.”

Figure 4

(Color online) Lazarus wave forms for equal-mass nonspinning black-hole mergers. The real $\mathcal{l}=2$, $m=2$ part of $r{\Psi }_4$, corresponding to the + polarization (measured on the system’s axis), is shown for ten simulations: five different initial black-hole separations (designated QC-0, etc.), each with two transition times to perturbative evolution. From 35.

Figure 5

The first gravitational wave form for black holes evolving through a single plunge orbit, merger, and ringdown achieved by 262. The waves were extracted at four radii and shifted in time to overlap for comparison. Note that the amplitudes decrease for larger $r$ due to lower resolution in the outer regions.

Figure 6

(Color online) The trajectories of the black-hole punctures. The individual apparent horizons are shown at times $t=0$, $10M$, and $18.8M$. The solid circles denote the centroids of the apparent horizons every $2.5M$ during the evolution. The first common horizon, marking the time of merger, forms at $18.8M$, just before the punctures complete $1∕2$ orbit. From 103.

Figure 7

(Color online) Gravitational wave forms from the black-hole merger simulations by 39. The real part of the $\mathcal{l}=2$, $m=2$ mode of $r{\Psi }_4$ was extracted from the numerical simulation on spheres of radii $r=20M$ and $40M$ for the medium- and high-resolution runs. The wave forms extracted at different radii have been rescaled by $r$ and shifted in time to account for the wave propagation time between the two extraction spheres. Curves are shown for medium ($M∕24$ in the finest grid) and high $(M∕32)$ resolutions. A wave form from a Lazarus calculation starting with the same initial data is shown for comparison (35). The Lazarus wave form shown in this figure is scaled differently from those in Fig. 4.

Figure 8

(Color online) Puncture tracks from equal-mass nonspinning black-hole merger simulations starting at increasingly larger separations by 38. For clarity, the trajectory of only one black hole in each run is shown. The tracks lock on to a universal trajectory $\sim 1$ orbit before the merger (denoted by ∗).

Figure 9

(Color online) The universal wave form ($\mathcal{l}=2$, $m=2$ mode) produced by the four simulations whose puncture tracks are shown in Fig. 8. The signals have been shifted in time so that the peak radiation amplitude occurs at $t=0$. The agreement among the wave forms is excellent, with differences $\sim 1\mathrm{\%}$ for the final burst of radiation starting at $t\sim -50M_f$. From 38.

Figure 10

(Color online) Comparison of three wave forms from simulations of equal-mass black-hole mergers computed by Pretorius and the UTB and Goddard groups. The UTB and Goddard simulations used nonspinning black holes, whereas Pretorius’ initial conditions produced corotating black holes, each with a small spin ${(a∕M)}_{1,2}=0.08$. The differences between Pretorius’ wave form and the others during the ringdown, $t∕M_f\gtrsim 25$, are consistent with his simulation producing a slightly faster rotating final black hole due to these initial spins. From 37.

Figure 11

Puncture tracks for equal-mass black-hole mergers with initial orbital eccentricity $e=0.1$ (left) and $e=0.3$ (right). In these cases, the initial eccentricity is radiated away and the binaries circularize before the merger. From 182.

Figure 12

Real part of (2,2) strain mode (labeled $h_{+}$ here) for black-hole mergers shown in Fig. 11. These runs have initial orbital eccentricity $e=0.1$ (top) and $e=0.3$ (bottom) and enter a universal plunge by $t\sim 50M_f$ before the gravitational radiation reaches its peak value. From 182.

Figure 13

(Color online) Trajectories for the merger of equal-mass nonspinning black holes computed by the Caltech-Cornell group using their spectral evolution code (285). The circles (ellipses) are the initial (final) coordinate locations of the apparent-horizon surfaces and the peanut-shaped contour is the common apparent horizon just after it appeared. The black holes complete 16 orbits before merging. From H. Pfeiffer.

Figure 14

(Color online) Gravitational wave forms from the Caltech-Cornell merger simulation seen in Fig. 13 showing the $\mathcal{l}=2$, $m=2$ component of $\mathrm{Re}\left(r{\Psi }_4\right)$. The left panel shows a zoom of the inspiral wave form and the right panel shows a zoom of the merger and ringdown. From 285.

Figure 15

(Color online) The gravitational-wave phase $\varphi$ as a function of the dimensionless frequency $M\omega$ for the five codes from the Samurai comparison (177). The scale along the top of the panel labels the frequency in kHz scaled with respect to the total binary mass in solar units.

Figure 16

(Color online) Amplitudes of the $\mathcal{l}=2$, $m=2$ component of the gravitational waves produced by the five codes in the Samurai comparison (177) are shown as a function of frequency.

Figure 17

(Color online) Puncture tracks for an unequal-mass nonspinning binary with $q=2$ produced by the Caltech-Cornell spectral code. The solid curve gives the trajectory of the smaller black hole and the dotted curve gives the path of the larger one. The contours are the final coordinate locations of the apparent-horizon surfaces and the peanut-shaped contour is the common apparent horizon just after it appeared. From H. Pfeiffer.

Figure 18

(Color online) The amplitudes of several $(\mathcal{l},m)$ modes of $\mathit{Mr}{\Psi }_4$ for mass ratio $q=2$ from simulations by 57. The initial burst of radiation is an artifact of the initial data, and the wiggles at late times result from numerical noise.

Figure 19

(Color online) Strain wave forms scaled by $\eta$ for different-mass ratios. $h_{+}$ is calculated for different-mass ratios using the $\mathcal{l}=2$ and 3 modes. The observer is located at distance $r$ along the axis of the system. From 31.

Figure 20

The $\mathcal{l}=m=2$ and $\mathcal{l}=m=3$ component of $\mathrm{Re}\left(rM{\Psi }_4\right)$ from the 10:1 mass ratio simulation. From 159.

Figure 21

Schematic of the physical basis of the recoil or kick produced by a merging black-hole binary with unequal masses. A net flux of momentum $P_{\text{ejected}}$ is emitted parallel to the smaller hole’s velocity. Momentum conservation requires that the system center-of-mass recoil in the opposite direction $P_{\text{recoil}}$. From 185.

Figure 22

(Color online) The magnitude of the kick velocity for a nonspinning unequal-mass merger with $q=1.5$. Results are shown from three simulations, with increasingly wider initial black-hole coordinate separations $d_{\mathrm{init}}$. The small-dotted curve shows a 2 PN calculation starting with initial conditions commensurate with the $d_{\mathrm{init}}=7.0M$ case (dotted line). From 40.

Figure 23

(Color online) Gravitational wave forms for mergers of equal-mass spinning binaries as in 107. The dominant $\mathcal{l}=2$, $m=2$ mode is shown for three cases: aligned spins (solid), antialigned spins (dotted), and no spins (dashed). All three simulations start with the same initial orbital period at $t=0$. From M. Campanelli.

Figure 24

(Color online) The difference in the black-hole trajectories ${\vec{x}}_1-{\vec{x}}_2$ for the generic binary evolution. The precession of the system spin induces precession of the orbital plane. From 104.

Figure 25

(Color online) Precession-induced amplitude modulations in the wave forms from the generic black-hole binary merger. The $\mathcal{l}=2$, $m=1$ mode of the strain $\left(h_{+}\right)$ is shown for the numerical-relativity simulation (solid) and a 3.5 PN calculation (dotted). The amplitude oscillations are induced by precession. From 104.

Figure 26

(Color online) Black-hole trajectories for the spin-flip case. The black holes have equal masses and equal spins initially pointing 45° above the initial orbital plane. The spin directions are shown as arrows along each trajectory every $4M$ until merger. From 108.

Figure 27

(Color online) The spin flip produced by the evolution in Fig. 26. The spin direction of the component holes, shown by the arrows plotted every $4M$ until merger, varies continuously due to precession. The dark arrow shows the final black-hole spin, with direction flipped discontinuously from that of the component holes just prior to merger. From 108.

Figure 28

(Color online) Puncture trajectories from a superkick simulation carried out by 158. The black holes have equal masses and spins ${(a∕M)}_{1,2}\sim 0.8$, initially oppositely directed in the orbital plane. During the evolution, the black holes move out of the original plane, and the final black-hole recoils with velocity $v_{\text{kick}}\sim 2500\mathrm{km}{\mathrm{s}}^{-1}$ in the negative $z$ direction.

Figure 29

(Color online) Comparison of NR and PN wave forms provided mutual affirmation of the results from each approach and showed that in combination NR and PN results could treat the complete signal. From 42.

Figure 30

(Color online) Phase differences between a numerical simulation and various PN models. The left panel shows phase differences from an initial (wave-form) angular frequency of $M\omega =0.04$ up to $M\omega =0.063$, while the comparison in right panel extends this range to $M\omega =0.10$. From 72.

Figure 31

(Color online) Comparison of EOB-model and numerical-relativity wave forms for the $h_{+}$ component of the gravitational-wave strain from the merger of binary with mass ratio of 4:1. The wave form is for an observer located an at inclination angle $\theta =\pi ∕3$ from the axis of rotation. From 95.

Figure 32

(Color online) Agreement of refined EOB-model $\mathcal{l}=m=2$ wave form (130) with numerical result from 285 for equal-mass nonspinning merger. Slight differences near the merger time are difficult to perceive without color.

Figure 33

(Color online) Design sensitivity curves for the LIGO and Virgo detectors (dashed lines), as well as the approximate curves used for the NINJA project (solid lines). LIGO sensitivity is effectively zero below $\sim 30\mathrm{Hz}$, while Virgo does a little better (note that achieved sensitivities may not perfectly mirror the design curves). From 27.

Figure 34

(Color online) Relationship between exact wave form $h_e$, model wave form with the same physical parameters $h_m$, actual “best-fit” model wave form $h_{\overline{m}}$, and template bank wave forms $h_b$, $h_{b^{\prime }}$. From 216.

Figure 35

The importance of including NR merger wave forms in data analysis can be seen here for Advanced LIGO (top) and LISA (bottom). In both panels, the dashed curve is the achieved SNR (for a given redshifted source mass) when only the early inspiral signal (up to $\sim 1000M$ before the merger) is used; the dotted curve uses the signal between this time and when final plunge commences ($1000M$ to $50M$ before the merger); the thin solid curve uses just the merger wave form (starting $50M$ before the merger). The thick solid curve is the combined result of using the entire wave form. The SNR and hence distance reach of the detector is clearly greatly enhanced by the final merger portion. From 41.

Figure 36

The importance of binary mass for the LISA observation window demonstrated by the characteristic amplitudes (41) of three different sources relative to the rms noise amplitude of the LISA detector. On each $h_{\mathrm{char}}$ curve, we mark times before the peak amplitude (circle)—$1\mathrm{h}$ (square), one day (diamond), and one month (triangle).

Figure 37

(Color online) The power distribution in 10% segments of the wave form before the merger for a range of LIGO-appropriate equal-mass binaries. Note that as the total mass $M$ increases, the final merger and ringdown account for a greater fraction of the total power. Adapted from 248.

Figure 38

(Color online) Maximum signal-to-noise ratio (SNR) $\rho$ below which different “long” numerical wave forms are indistinguishable for the Advanced LIGO detector. Even the earliest, lowest-accuracy published long wave forms are indistinguishable for SNR levels below 14, as demonstrated in this figure from the Samurai paper (177).

Figure 39

(Color online) Accuracy in parameter estimation for EOBNR templates in the NINJA project. The top panel shows the fractional error in estimated total mass $(M_{\text{injected}}-M_{\text{detected}})∕M_{\text{injected}}$ for all detected signals, while the lower shows the error in merger time. Adapted from 27.

Figure 40

Magnetic and electric-field lines around inspiralling equal-mass nonspinning black holes at $\approx 40M$ before the merger. The electric-field lines are twisted around the black holes, while the magnetic-field lines are mainly aligned with the $z$ axis. From 247.

Figure 41

Same as Fig. 40, except at a later time, $\approx 20M$ before the merger.

Figure 42

(Color online) Numerical simulations are applied to study the critical transition separating mergers from scattering events in high-velocity black-hole encounters. The curves show the path of one black hole in each of three simulations begun with different initial conditions near the critical impact parameter. The trajectories track closely together coming in from the right until the black holes encounter each other near the origin. From 311.