Simulation of Binary Black Hole Spacetimes with a Harmonic Evolution Scheme

TL;DR

This paper advances binary black hole simulations by detailing a generalized harmonic (GH) evolution scheme augmented with constraint damping and dynamic source-function gauge control. It describes numerical strategies—excision, dissipation, and multigrid constraint solving—that enable stable evolutions in both axisymmetric head-on and full 3D mergers, with gravitational waves extracted from $\Psi_4$. It also reports scalar-field collapse–driven BBH spacetimes that reveal potential zoom-whirl–like behavior and emphasizes convergence and robustness tests across resolutions. The findings suggest GH with constraint damping is a viable path toward accurate, long-time BBH waveforms, while highlighting current limitations and avenues for higher-order methods and broader parameter exploration for gravitational-wave modeling.

Abstract

A numerical solution scheme for the Einstein field equations based on generalized harmonic coordinates is described, focusing on details not provided before in the literature and that are of particular relevance to the binary black hole problem. This includes demonstrations of the effectiveness of constraint damping, and how the time slicing can be controlled through the use of a source function evolution equation. In addition, some results from an ongoing study of binary black hole coalescence, where the black holes are formed via scalar field collapse, are shown. Scalar fields offer a convenient route to exploring certain aspects of black hole interactions, and one interesting, though tentative suggestion from this early study is that behavior reminiscent of "zoom-whirl" orbits in particle trajectories is also present in the merger of equal mass, non-spinning binaries, with appropriately fine-tuned initial conditions.

Figures (14)

• Figure 1: The normalized sum of total AH mass (\ref{['smarr']}) as a function of time, for the head-on collision simulations described in Sec. \ref{['sec_cdg']}with constraint damping (compare Fig.\ref{['fig_M_thc_wo_cd']}). This plot demonstrates convergence to a conserved mass before and after the merger. The "spiky" behavior about the merger point (around $t=37M_0$) is due to AH finder problems as discussed in Sec.\ref{['sec_eah']}.
• Figure 2: A norm of the residual of the Einstein equations (\ref{['efe']}), calculated as discussed in Sec. \ref{['sec_cnv']}, for the head-on collision simulations described in Sec. \ref{['sec_cdg']}with constraint damping (compare Fig.\ref{['fig_efe_thc_wo_cd']}, though note the different vertical and horizontal scales). This demonstrates the convergence of the solution---see also Fig. \ref{['fig_cf_efe_thc_w_cd']}.
• Figure 3: The order of convergence (\ref{['n_h1h2']}) calculated from the head-on collision simulations described in Sec. \ref{['sec_cdg']} (with constraint damping), and using the residual data shown in Fig. \ref{['fig_efe_thc_w_cd']}. The discretization scheme is second order, and so one would expect $n(h_1,h_2)$ to asymptote to 2 as the resolution is increased. This is evident in the figure, particularly after most of the scalar field has left the domain (around $t\approx 40-50M_0$).
• Figure 4: The normalized sum of total AH mass (\ref{['smarr']}) as a function of time, for the head-on collision simulations described in Sec. \ref{['sec_cdg']} and without constraint damping; compare Fig. \ref{['fig_M_thc_w_cd']}. The curves end when the simulations crashed.
• Figure 5: A norm of the residual of the Einstein equations (\ref{['efe']}), calculated as discussed in Sec. \ref{['sec_cnv']}, for the head-on collision simulations described in Sec. \ref{['sec_cdg']}without constraint damping; compare Fig.\ref{['fig_efe_thc_w_cd']}, though note the different vertical and horizontal scales. Again, convergence is evident, though the rapid growth of the residual with time prevents useful results from being obtained at modest resolution.