Spinning-black-hole binaries: The orbital hang up

Abstract

We present the first fully-nonlinear numerical study of the dynamics of highly spinning black-hole binaries. We evolve binaries from quasicircular orbits (as inferred from Post-Newtonian theory), and find that the last stages of the orbital motion of black-hole binaries are profoundly affected by their individual spins. In order to cleanly display its effects, we consider two equal mass holes with individual spin parameters S/m^2=0.757, both aligned and anti-aligned with the orbital angular momentum (and compare with the spinless case), and with an initial orbital period of 125M. We find that the aligned case completes three orbits and merges significantly after the anti-aligned case, which completes less than one orbit. The total energy radiated for the former case is ~7% while for the latter it is only ~2%. The final Kerr hole remnants have rotation parameters a/M=0.89 and a/M=0.44 respectively, showing the unlikeliness of creating a maximally rotating black hole out of the merger of two spinning holes.

Figures (5)

• Figure 1: The real $(\ell=2, m=2)$ component of $r\psi_4$ in the Quasi-Kinnersley tetrad at $r=10M$ for the - - 0.757 case. The lower inset shows the differences $r\psi_4(M/25) - r\psi_4(M/30)$ (solid line) and $r\psi_4(M/30) - r\psi_4(M/35)$ (dotted line), the latter rescaled by $2.33$ to demonstrate fourth-order convergence. The lack of convergence for $t<10M$ is due to roundoff effects in the initial data solver. The upper inset shows the real part of the phase-corrected $(\ell=2,m=2)$ mode of $\psi_4$ at the same radius. Note the near-perfect agreement after $t=45M$.
• Figure 2: The puncture trajectories on the $xy$ plane for the '- -' case with resolution $M/35$. The spirals are the puncture trajectories with ticks every 10M of evolution. The dot-dash 'peanut shaped' figure is the first detected common horizon at $105.5M$. The (extrapolated) period of the last orbit is around $120M$.
• Figure 3: The real part of the $(\ell=2,m=2)$ mode of $r\,\psi_4$ in the Quasi-Kinnersley frame at $r=10M$ from the '$++0.757$' configuration. (The small circles are the early-time waveform from conformal Kerr data.) The top inset shows a magnified view of the early orbital motion. Note that the '++0.757' waveform has 6 wavelengths of orbital motion prior to the plunge waveform (at $t\sim 232.5M$), indicating that the binary orbited approximately three times before merging. The bottom inset shows the real (solid) and imaginary (dotted) components of the (2,2) component of the strain $h$ calculated at $r=10M$.
• Figure 4: The puncture trajectories on the $xy$ plane for '++' configuration with resolution $M/30$. The spirals are the puncture trajectories with ticks every 10M of evolution. The dot-dash 'peanut shaped' figure is the first detected common horizon at $t=232.5M$. The period of the last orbit is around $36M$. The last orbit begins when the punctures are located at $1.4M$ from the origin (in these coordinates).
• Figure 5: The Hamiltonian constraint violation at $t=45M$ along the x-axis (top plot) and at $t=80M$ along the $y$-axis (bottom plot) for the $M/30$ and $M/35$ runs (the latter rescaled by $(35/30)^4$) for the '- -' configuration. The punctures crossed the x-axis at $t=45M$ and crossed the y-axis for the second time at $t=75M$. Note the reasonable fourth-order convergence (except at the puncture). Points contaminated by boundary errors have been excluded from the plot. The high frequency violations near the numerical coordinate $y/M=\pm9$ are due to the extreme fisheye deresolution near the boundary, and converge with resolution.