Spinning-black-hole binaries: The orbital hang-up

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 antialigned 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 antialigned case, which completes less than one orbit. The total energy radiated for the former case is $\approx 7\%$ while for the latter it is only $\approx 2\%$. 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 two highly spinning holes.

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 dashed- dotted “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 3 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 dashed-dotted “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 $y/M=\pm 9$ are due to the extreme fisheye deresolution near the boundary, and converge with resolution.