Recent progress in binary black hole simulations

Thomas Baumgarte, Bowdoin College

The past year has seen dramatic progress in numerical relativity simulations of binary black holes. A number of groups have reported significant advances and are now able to model the binary inspiral, coalescence and merger together with the emitted gravitational wave signal.

Simulating binary black holes has been a long-standing problem because it poses a number of ``grand challenges''. An incomplete list of these challenges includes the following

The different groups have approached these issues in different ways.

Pretorius (2005) first announced his new results at a numerical relativity workshop at the Banff International Research Station. Departing from numerical relativity convention he does not integrate a ``3+1'' decomposition of Einstein's equations that separates spatial and timelike parts, but instead discretizes the four-dimensional spacetime equations and their second derivatives directly. As suggested by a number of previous authors he introduces gauge source functions $H_{\mu}$, in terms of which Einstein's equations reduce to wave equations for the components of the spacetime metric. In this formalism the coordinates are fixed through the gauge source functions (instead of the lapse and shift in 3+1 formalisms). Pretorius chooses $H_t$ to satisfy a somewhat ad-hoc wave equation and $H_i = 0$ (which is related to spatial harmonic coordinates $H^i = 0$). Pretorius uses black hole excision, whereby the black hole interior is removed from the computational grid. This is justified since the event horizon disconnects the interior causally from from the exterior. He also uses adaptive mesh refinement (AMR), which automatically allocates additional gridpoints in regions where they are needed to resolve small scale structures.

Several other features of his code are worth pointing out. The spatial coordinates are compactified, so that physically correct outer boundaries can be imposed at spatial infinity. He also introduces some numerical dissipation to control high-frequency instabilities, and added some ``constraint-damping'' terms that proved crucial for simulations of black holes.

With this code Pretorius has been able to simulate - without encountering a numerical instability - the inspiral and coalescence of black hole binaries through merger until late stages of the ring-down as the remnant settles into equilibrium. This is remarkable progress indeed. Figure 1 shows the trajectory of an inspiraling black hole binary and a gravitational waveform from his very recent calculation that adopts the state-of-the-art initial data of Cook and Pfeiffer (2004).

Following Pretorius' success four other groups (Campanelli  (2005), Baker (2005), Diener (2005) and most recently Herrmann (2006)) have announced significant progress in their binary black hole simulations. All of these calculations have several features in common. They all use finite-difference implementations of the BSSN equations1, which are based on a 3+1 formalism in contrast to Pretorius' four-dimensional approach. They also use very similar gauge conditions, namely ``1 + log'' slicing for the lapse, and a ``driver'' implementation of ``Gamma-freezing''. Finally they all use ``puncture'' initial data, which are constructed by absorbing the singular terms in the black hole interior into an analytical expression and solving for regular corrections.

Figure: The left panel shows black hole trajectories in a recent binary black hole simulation of Pretorius (2006, private communication), starting with an initial data configuration of Cook and Pfeiffer (2004). The right panel shows the corresponding ``gravitational waveform'' $Re(\psi_4)$.
The approach of Campanelli (2005) and Baker (2005) differs from the others in that they do not excise the black hole interiors, and instead continue to use the ``puncture'' approach to handle the singularities during the evolution. Campanelli (2005) introduce a new variable that is the inverse of the diverging term. This new term vanishes at the ``punctures'', and given suitable gauge conditions all equations remain regular. Baker (2005) finite difference the diverging term directly, but arrange the computational grid in such a way that the singularity never encounters a gridpoint. In situations with equatorial symmetry, when both singularities reside on the equatorial plane, this can always be achieved simply by using cell-centered differencing schemes. The two calculations also use different grid structures and differencing; Campanelli (2005) adopt 4th order differencing on a uniform grid but introduce a ``fish-eye'' coordinate that provides additional resolution for the black holes, while Baker (2005) use 2nd order differencing and FMR in an inertial coordinate system. Figure 2 shows a gravitational wave form from Campanelli , and a demonstration of energy conservation from Baker . Both groups also report satisfactory agreement with earlier ``Lazarus'' results which combines numerical relativity with perturbative techniques (e.g. Baker (2001)).
Figure: Left Panel: The gravitational waveform $Re(\psi_4)$ in the calculation of Campanelli (2005). This convergence test demonstrates 4th order convergence. Right Panel: Demonstration of energy conservation in the calculation of Baker (2005). The initial energy $M$ minus the energy $E$ lost in gravitational radiation agrees with the current total energy $M_{\rm ADM}$ to high accuracy.
Diener (2005) report on impressive improvements of their earlier results (Alcubierre (2005); compare Brügmann (2004)). Like Pretorius, they use black hole excision to eliminate the curvature singularities in the black hole interior. They use a fixed mesh refinement (FMR) in a corotating coordinate system to resolve the black holes. A schematic of their black hole trajectories is shown in the left panel of Figure 3. Starting from an initial proper separation of about 9 M the black holes spiral toward each other until a common apparent horizon forms after about 1.5 orbits. Diener  (2005) also demonstrate that the spurious effect of finite difference error on the binary orbit depends on the gauge condition. All gauge choices converge to the same physical solution, as expected, but at finite resolution different choices may lead to either a widening or closing of the orbit, which helps to explain earlier discrepancies (compare Brügmann (2004), Alcubierre (2005)).

The simulations of Campanelli (2005) and Baker (2005) demonstrate that standard numerical relativity codes can handle binary black holes with only very minor modifications, potentially opening the field to a number of other groups. Herrmann (2006), for example, adopt a technique very similar to that of Baker (2005). While all other simulations focus on equal-mass black holes they consider unequal-mass black holes with mass ratios $q = M_1/M_2$ ranging from unity to 0.54. Their calculation represents a first step toward analyzing the effect of binary parameters - including mass ratios and black hole spins - on the gravitational waveforms in fully dynamical simulations (even if the astrophysical relevance of their initial data is somewhat limited). They also see evidence for gravitational radiation recoil leading to a remnant ``kick'' (compare the right panel of Figure 3).

Figure: Left Panel: Motion of one of the black holes with time in the simulation of Diener (2005). Cross-sections of the apparent horizon (AH) with the equatorial plane are shown at intervals of $\Delta t = 5 M$. The apparent growth of the AHs with time is a pure coordinate effect. The first appearance of a common AH at $t = 124
M$, and the corresponding final detached AH, are shown as dashed lines. Right Panel: Snapshots of the apparent horizon locations for the $q=0.85$ unequal-mass binary calculation of Herrmann (2006). The snapshots are taken every 4 $M_{\rm ADM}$ prior to merger (red) and every 17 $M_{\rm ADM}$ after merger (blue). The trajectories of the common horizons' centers are shown as a dashed lines.
All of these calculations can clearly be improved in multiple ways. However, especially comparing with the situation just a year ago, it is quite remarkable and reassuring that different groups using independent techniques and implementations can now all carry out reliable simulations of binary black hole coalescence and merger. It may soon be possible to simulate the black hole binary inspiral starting from a sufficiently large binary separation so that it can be compared with and matched to post-Newtonian predictions. The past year has indeed seen dramatic progress in numerical relativity simulations of binary black holes.


Alcubierre, M., Brügmann, B., Diener, P., Guzmán, F. S., Hawke, I., Hawley, S., Herrmann, F., Koppitz, M., Pollney, D., Seidel, E., & Thornburg, J., 2005, Phys. Rev. D 72 044004.
Baker, J. G., Centrella, J., Choi, D.-I., Koppitz, M., & van Meter, J., 2005, submitted (also gr-qc/0511103).
Baker, J. G., Brügmann, B., Campanelli, M., Lousto, C. O. & Takahashi, R., 2001, Phys. Rev. Lett. 87, 121103.
Brügmann, B., Tichy, W., & Jansen, N., 2004, Phys. Rev. Lett. 92, 211101.
Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y., 2005, submitted (also gr-qc/0511048).
Cook, G. B., & Pfeiffer, H. P., 2004, Phys. Rev. D 70, 104016.
Diener, P., Herrmann, F., Pollney, D., Schnetter, E., Seidel, E., Takahashi, R., Thornburg, J., & Ventrella, J., 2005, submitted (also gr-qc/0512108).
Herrmann, F., Shoemaker, D., & Laguna, P., submitted (also gr-qc/0601026).
Pretorius, F., 2005, Phys. Rev. Lett 95, 121101.

Jorge Pullin 2006-02-28