A recent construction of binary black hole initial data prescribes the angular velocity Ω(θ)\Omega(\theta) on each horizon as Dirichlet boundary data for a coupled elliptic constraint system (Huang, Feng and Lau, 2025). When such data are evolved in the conformally flat case, non-uniform horizon profiles generate an oscillatory pattern superposed on the early inspiral gravitational waveform — reminiscent of ringdown, but occurring before merger — that is absent for constant-Ω\Omega and Bowen-York data and survives resolution studies. The physical origin of these pre-merger, quasinormal-mode-like features is unknown: it is not settled whether their frequencies and damping times match the near-horizon perturbation spectrum of the individual black holes, how they depend on the prescribed Ω(θ,ϕ)\Omega(\theta,\phi) profile and binary parameters, whether they persist beyond the conformally flat approximation, and whether their origin is the same as or distinct from the non-merging ringdown observed in close scattering of two black holes. This problem asks for the generating mechanism and a quantitative, validated account of the features.

Background

In the 3+1 split of general relativity, a numerical simulation of a binary black hole begins with initial data: a spatial metric γij\gamma_{ij} and extrinsic curvature KijK_{ij} (the embedding of the spatial slice in spacetime) satisfying the Hamiltonian and momentum constraint equations. In the standard conformal approach one writes γij=ψ4γˉij\gamma_{ij} = \psi^4 \bar{\gamma}_{ij} with a conformal factor ψ\psi and usually takes the conformally flat case γˉij=δij\bar{\gamma}_{ij} = \delta_{ij}; the classic Bowen-York solution (Bowen and York, 1980) supplies analytic extrinsic-curvature terms describing boosted and spinning holes and underlies the widely used puncture initial data. A drawback of any fixed analytic choice is that the slice is not in dynamical equilibrium, so the first phase of evolution sheds spurious transient radiation ("junk" radiation), and the rotational parameters of the holes are only loosely tied to the geometry of the resulting horizons.

Huang, Feng and Lau (2025), building on a geometric definition of angular velocity introduced by Feng, Yan, Gao, Lau and Yau (2025), proposed a different way to prescribe rotation. In that definition, the momentum constraint on a time slice of a rotating (Kerr) black hole can be rewritten as a Beltrami equation — the equation governing two-dimensional quasi-conformal maps, equivalent in this context to a steady two-dimensional compressible fluid flow — and the angular velocity Ω\Omega appears as the potential of a vector field built from the extrinsic curvature, without any reference to a stationarity (Killing vector) assumption. Huang, Feng and Lau extend this picture to binary black holes by decomposing the traceless extrinsic curvature as Kˉij=KˉijP+KˉijΩ\bar{K}_{ij} = \bar{K}^{P}_{ij} + \bar{K}^{\Omega}_{ij}, where KˉijP\bar{K}^{P}_{ij} is the Bowen-York linear-momentum part and KˉijΩ=−(ρ/2) ηˉ(i∂j)Ω\bar{K}^{\Omega}_{ij} = -(\rho/2)\,\bar{\eta}_{(i}\partial_{j)}\Omega is determined by the angular-velocity potential Ω\Omega (ρ\rho is the cylindrical radius and ηˉi\bar{\eta}^i the azimuthal unit vector). The constraints then become a coupled elliptic system,

Dˉ2ψ+18 KˉijKˉijψ−7=0,∂i ⁣(ρ3∂iΩ)=0,\bar{D}^2 \psi + \tfrac{1}{8}\,\bar{K}_{ij}\bar{K}^{ij}\psi^{-7} = 0, \qquad \partial_i\!\left(\rho^3 \partial^i \Omega\right) = 0,

solved with multi-domain spectral methods, where the essential novelty is the boundary treatment: values of Ω\Omega are prescribed (Dirichlet boundary conditions — fixed values rather than derivatives) on two inner coordinate spheres intended as the horizons of the two holes, together with a marginally-trapped-surface condition on ψ\psi there, while ψ→1\psi \to 1 and Ω=O(r−3)\Omega = O(r^{-3}) at infinity. The constructed data are evolved with a BSSN moving-puncture code (BSSN being a standard re-formulation of Einstein's equations for stable numerical evolution) in the conformally flat case, for equal-mass co-rotating and counter-rotating binaries. Single-hole tests show that as the prescribed horizon angular velocity grows, quasi-local spin measures approach but stay below extremality (χ≈0.984\chi \approx 0.984), consistent with earlier alternative spinning-hole data (Dain, Lousto and Takahashi, 2002).

With constant Ω\Omega on each horizon the waveforms are nearly identical to Bowen-York ones (the Bowen-York spin term corresponds to a particular Ω\Omega that is uniform on coordinate spheres), which calibrates the pipeline. The striking result appears for non-uniform profiles such as Ω∣r=rH=±82(3cos⁡2θ−1)\Omega|_{r=r_H} = \pm 82(3\cos^2\theta - 1) or 80(5cos⁡3θ−3cos⁡θ)80(5\cos^3\theta - 3\cos\theta) (proportional to the C20C^0_2 and C30C^0_3 spherical-harmonic components respectively): in quasi-circular inspirals — but not in head-on collisions — the early inspiral gravitational waveform develops an oscillatory pattern superposed on the inspiral signal, reminiscent of ringdown, the damped quasinormal-mode (QNM) ringing by which a perturbed black hole settles. Resolution studies and systematically decreasing constraint violations support the interpretation that these features are physical emissions of the initial configuration rather than numerical error or residual junk radiation. The authors note a resemblance to the "non-merging ringdown" that Bae, Hyun and Kang (2024) observed in close hyperbolic encounters of two black holes — waves at single-hole quasinormal frequencies emitted during the encounter, attributed to dynamical tidal deformation of the individual holes — while remarking that the physical origins in the two settings are likely different. The formulation has so far been implemented only in the conformally flat approximation, and the authors note one caveat in the construction itself: the trapped-surface boundary condition does not guarantee that the inner Dirichlet sphere coincides with the outermost apparent horizon (the outermost marginally trapped surface actually present in the data), although in practice the apparent horizon can be located as the outermost such surface once the data have been constructed. An extension with matter sources to neutron-star binaries is proposed but not carried out.

Problem Statement

Determine the physical origin of the pre-merger, quasinormal-mode-like oscillatory features that appear in the inspiral-phase gravitational waveform when binary black hole initial data are constructed with non-uniform angular-velocity profiles Ω\Omega prescribed on the two horizons — axisymmetric profiles Ω(θ)\Omega(\theta) in the simulations of Huang, Feng and Lau (2025), general profiles Ω(θ,ϕ)\Omega(\theta,\phi) in the underlying formulation — and establish whether the phenomenon is genuine and generic. The common objective is to decide whether and how the rotational structure prescribed on the horizons at the initial slice imprints a predictable, quantitatively understood signature on the pre-merger gravitational waveform. The investigation should: (i) identify the generating mechanism and give a quantitative account of the observed oscillation frequencies and damping times — in particular whether they match the quasinormal-mode (near-horizon perturbation) spectrum of the individual black holes as characterized by the constructed initial data, or arise from another mechanism; (ii) characterize how the features depend on the prescribed horizon profile Ω(θ,ϕ)\Omega(\theta,\phi) (amplitude, angular structure, co- versus counter-rotation) and on binary parameters such as mass ratio and initial separation; (iii) establish whether the effect survives relaxing the conformally flat approximation used in the original simulations — that is, whether it persists for non-conformally-flat data within the Ω\Omega-based formulation — or whether it is contingent on the conformally flat restriction or on residual initial-data transients; and (iv) settle whether the physical origin is the same as or distinct from the non-merging ringdown observed in close scattering of two black holes (Bae, Hyun and Kang, 2024), where the signal is attributed to dynamical tidal deformation of the individual holes during the encounter. The mandatory core is (i), together with the determination that the features are genuine emissions of the initial configuration rather than numerical error or residual junk radiation: a solution that identifies the generating mechanism and gives the quantitative account of the observed frequencies and damping times resolves the problem. Parts (ii)–(iv) — dependence on the prescribed profile Ω(θ,ϕ)\Omega(\theta,\phi) and on binary parameters, robustness beyond the conformally flat approximation, and relation to the non-merging ringdown of close scattering — are the further programme that establishes the genericity, robustness and novelty of the phenomenon; they strengthen a resolution and are not prerequisites, so a partial answer that settles the core without them is a valid resolution rather than an unresolved task.

References

  1. Shuanglin Huang, Xuefeng Feng, Yun-Kau Lau, 'Angular velocity of rotating black holes — a new way to construct initial data for binary black holes', Classical and Quantum Gravity 42 (2025) 235021, DOI 10.1088/1361-6382/ae2414, arXiv:2511.09050, https://arxiv.org/abs/2511.09050
  2. Xuefeng Feng, Ruodi Yan, Sijie Gao, Yun-Kau Lau, Shing-Tung Yau, 'Geometric inequality for axisymmetric black holes with angular momentum', Classical and Quantum Gravity 42 (2025) 065022, DOI 10.1088/1361-6382/adb82a, arXiv:2312.10590, https://arxiv.org/abs/2312.10590
  3. Jeffrey M. Bowen, James W. York Jr., 'Time-asymmetric initial data for black holes and black-hole collisions', Physical Review D 21 (1980) 2047, DOI 10.1103/PhysRevD.21.2047, https://doi.org/10.1103/PhysRevD.21.2047
  4. Sergio Dain, Carlos O. Lousto, Ryoji Takahashi, 'New conformally flat initial data for spinning black holes', Physical Review D 65 (2002) 104038, DOI 10.1103/PhysRevD.65.104038, https://doi.org/10.1103/PhysRevD.65.104038
  5. Yeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang, 'Ringdown gravitational waves from close scattering of two black holes', Physical Review Letters 132 (2024) 261401, DOI 10.1103/PhysRevLett.132.261401, arXiv:2310.18686, https://arxiv.org/abs/2310.18686
  6. Yeong-Bok Bae, Young-Hwan Hyun, Gungwon Kang, 'Conformal-Mapping Method for Horizon Multipoles in Numerical Relativity: Implementation, Kerr Validation, and Applications Beyond Axisymmetry', arXiv:2608.07985 (2026), https://arxiv.org/abs/2608.07985
  7. Iago B. Mendes, Nils L. Vu, Oliver Long, Harald P. Pfeiffer, Robert Owen, 'Parameter control for binary black hole initial data', Physical Review D 112 (2025) 124049, DOI 10.1103/zh31-bbtm, arXiv:2509.07291, https://arxiv.org/abs/2509.07291