Angular velocity of rotating black holes -- a new way to construct initial data for binary black holes

TL;DR

The authors introduce a geometric framework that defines the angular velocity of a rotating black hole through a Beltrami-flow-inspired representation of the momentum constraint, and extend it to binary black holes by solving a coupled elliptic system for the conformal factor $\psi$ and horizon angular velocity $\Omega$ under horizon and infinity boundary conditions. In the conformally flat setting, they augment the Bowen–York extrinsic curvature with a rotational part derived from $\Omega$, enabling initial data that encode horizon rotation and inspiral without relying solely on traditional Bowen–York punctures. Numerical experiments show that constant $\Omega$ reproduces Bowen–York-like waveforms, while nonuniform horizon rotation yields new, ringdown-like oscillatory features early in the inspiral, with convergence tests confirming numerical reliability. This approach establishes a direct link between near-horizon dynamics and gravitational wave signals, offering a path to probe horizon physics and potentially reveal novel pre-merger waveform characteristics, and it can be extended to include matter such as neutron stars in future work.

Abstract

Motivated by a geometric understanding of the angular velocity of a Kerr black hole in terms of a quasi-conformal map that describes a 2d Beltrami fluid flow, a new way to construct initial data sets for binary rotating black holes by prescribing the angular velocities of the two black holes at their horizons is discussed. A set of elliptic equations with prescribed Dirichlet boundary conditions at the horizons and at spatial infinity is established for constructing the initial data. To explore the dynamics encoded in these initial data, we consider the conformally flat three-metric case and numerically evolve it using the BSSN code for two co-rotating and counter-rotating black holes with angular velocities prescribed at the horizons. When the angular velocities are non-uniform and deviate from a constant value at the horizons, new gravitational waveforms are generated which display certain oscillatory pattern reminiscent of that of quasi-normal ringing in the inspiral phase before merger takes place.

Figures (8)

• Figure 1: A Beltrami flow defined near the horizon by $\Omega$ and its conjugate stream function $\Psi$.
• Figure 2: The three spin parameters $\chi$, $\epsilon_{J}$, and $\zeta$ for a single rotating black hole are plotted, where a value of zero denotes a non-spinning black hole and a value of one corresponds to an extremal case. As the value of $\Omega_{H}/r_{H}^2$ increases, the parameters approach the following asymptotic values: $\chi_{\rm max}=0.984$, $\zeta_{\rm max}=0.823$, and $\epsilon_{J,\rm{max}}(S/M_{\rm ADM}^2)=0.928$.
• Figure 3: Angular velocity function $\Omega(\theta)$ evaluated at the horizon. The profile in the left panel corresponds to the models in Fig. \ref{['fig_headon']}, \ref{['Inspiral_aligned_20']}, and \ref{['Inspiral_anti_20']}, while the one in the right panel is employed in Fig. \ref{['Inspiral_30']}.
• Figure 4: The gravitational radiation $r\Psi_{4}^{20}$ from the head-on collision of equal-mass black holes produced by our initial data with the inner boundary condition for $\Omega$ prescribed as $\Omega|_{r=r_H}=\pm 82(3\cos^2{\theta}-1)$. Black holes have mass $M_1=M_2=0.5493$, and are initially placed at $z = \pm 6$. This is compared with Bowen-York black holes of the same masses and spins. Left: Both black holes have spins aligned along the $+z$ axis. Right: The black holes have opposite spin directions, aligned along the $+z$ and $–z$ axes, respectively.
• Figure 5: The gravitational radiation $r\Psi_{4}^{22}$ from the black hole inpiral of equal-mass, with aligned spin direction. The inner boundary conditions for $\Omega$ is $\Omega|_{r=r_H}=82(3\cos^2{\theta}-1)$. The black holes have masses $M_1=M_2=0.5511$, spins $S_1=S_2=(0.1312,0,0)$, and are initially placed at $x = \pm 6$. In the left panel, the enclosed box highlights an oscillatory pattern during the early phase of the merger. The right panel provides a zoomed-in view of this pattern.