Spin-up and mass-gain in hyperbolic encounters of spinning black holes

TL;DR

This work uses numerical relativity to study spin-up and mass-gain in hyperbolic encounters of equal-mass spinning black holes across a broad parameter range. By tracking gravitational-wave emission and horizon properties, the authors quantify how orbital angular momentum and energy are re-absorbed, revealing a pronounced spin-up near the scattering-merger threshold and a concurrent mass gain up to about 15%. A linear relation emerges between spin-up at the threshold and the initial spin, with maximum spin-up around $0.3$ and a maximum spin-up efficiency near $5\%$, increasing with initial momentum and depending on spin alignment. The mass-gain behavior mirrors the spin dynamics, and the irreducible mass generally increases due to horizon-area growth, with larger gains for negative initial spins. The results illuminate tidal-torquing in strong-field gravity and offer insights applicable to GW source modeling, dense-cluster dynamics, and primordial-black-hole spin evolution, while pointing to future work on unequal masses, precession, and alternative theories.

Abstract

Scattering black holes spin up and gain mass through the re-absorption of orbital angular momentum and energy radiated in gravitational waves during their encounter. In this work, we perform a series of numerical relativity simulations to investigate the spin-up and mass-gain for equal-mass black holes with a wide range of equal initial spins, $χ_{\rm i}\in[-0.7,0.7]$, aligned (or anti-aligned) to the orbital angular momentum. We also consider a variety of initial momenta. Furthermore, we explore a range of incident angles and identify the threshold between scattering and merging configurations. The spin-up and mass-gain are typically largest in systems with incident angles close to the threshold value, large momenta, and negative (i.e. anti-aligned) initial spins. When evaluated at the threshold angle, we find that the spin-up decreases linearly with initial spin. Intriguingly, systems with initial spin $χ_{\rm i}=0.7$ sometimes experience a spin-down, in spite of an increase in the black-hole angular momentum, due to a corresponding gain in the black-hole mass. Across the simulation suite, we find a maximum spin-up of $0.3$ and a maximum increase in the black-hole mass of $15\%$.

Figures (23)

• Figure 1: Initial conditions of binary BHs with equal initial masses, $m_{\rm i}$, total mass $\mathrm{M}=2m_{\rm i}$, and equal initial spins, $\chi_{\rm i}$. The spins are aligned or anti-aligned with the orbital angular momentum that is pointing in the z direction. The setup has rotational symmetry such that the BHs have equal but opposite initial (linear) momenta, $|\boldsymbol{P}_{\rm i}|$, inclined at an incident angle, $\theta$, from the x-axis. Furthermore, the BHs have an initial separation $d=100\mathrm{M}$ along the x-axis.
• Figure 2: Trajectory and gravitational waveform of BH binaries with initial spin $\chi_{\rm i}=0.0$ and initial momentum $|\boldsymbol{P}_{\rm i}|=0.245\mathrm{M}$. From left to right, the panels depict systems with incident angles $\theta=0.0570$, $0.0580$, and $0.0590$ that result in a merger, zoom-whirl, and scattering, respectively. The threshold angle for this series of simulations is $\theta_{\rm th}=0.05825$. Top row: Trajectory of the BHs in the orbital (x-y) plane. Bottom row: Gravitational radiation as given by the real part of the quadrupole of the Weyl scalar, $\Psi_{4,22}$, rescaled by the extraction radius, $r_{\rm ex}=100\mathrm{M}$. The time is shifted by the extraction radius.
• Figure 3: Trajectory and gravitational waveform of BH binaries with initial spin $\chi_{\rm i}=0.7$ and initial momentum $|\boldsymbol{P}_{\rm i}|=0.245\mathrm{M}$. From left to right the panels depict systems with incident angles $\theta=0.0450$, $0.0470$, and $0.0490$ that result in a merger, zoom-whirl, and scattering, respectively. The threshold angle for this series of simulations is $\theta_{\rm th}=0.04825$. Top row: Trajectory of the BHs in the orbital (x-y) plane. Bottom row: Gravitational radiation as given by the real part of the quadrupole of the Weyl scalar, $\Psi_{4,22}$, rescaled by the extraction radius, $r_{\rm ex}=100\mathrm{M}$. The time is shifted by the extraction radius.
• Figure 4: Threshold angle, $\theta_{\rm th}$, as a function of the initial parameters. Left: Dependence on the initial spin for initial momenta $|\boldsymbol{P}_{\rm i}|=0.245 \mathrm{M}$ and $|\boldsymbol{P}_{\rm i}|=0.490\mathrm{M}$. Right: Dependence on the initial momentum for initial spin $\chi_{\rm i}=0.7$.
• Figure 5: Evolutions of the BH mass, $m$, irreducible mass, $m_{\rm irr}$, and (dimensionless) spin magnitude, $|\chi|$, for BHs scattering near the threshold angle, $\theta=\theta_{\rm th}$, with initial momentum $|\boldsymbol{P}_{\rm i}|=0.490 \mathrm{M}$. The dotted lines labeled $t_{\rm i}$ and $t_{\rm f}$ denote when initial and final quantities are measured. Top: This plot ($\chi_{\rm i}=-0.7$) is typical of anti-aligned spins, where the spin's magnitude decreases causing the BH mass and irreducible mass to approach in value. Middle: This plot ($\chi_{\rm i}=0.2$) is typical of small aligned spins, where the spin increases, but makes negligible contribution to the BH mass. Bottom: This plot ($\chi_{\rm i}=0.7$) is typical of large aligned spins, where the spin change is marginal.