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arxiv: 2605.20374 · v1 · pith:LIEDRDW4new · submitted 2026-05-19 · 🌀 gr-qc · hep-th

Two asymptotically flat spinning black holes balanced by their self-interacting, synchronised scalar hair

Chen Liang , Carlos Herdeiro , Eugen Radu This is my paper

Pith reviewed 2026-05-21 07:18 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords spinning black holesscalar hairself-interactionsboson starsbalanced configurationsergoregionasymptotically flatsynchronised hair
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The pith

Attractive scalar self-interactions increase horizon mass fractions in balanced two spinning black holes

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the effects of quartic scalar self-interactions on families of solutions that include two spinning black holes with synchronised hair, two spinning boson stars, and an intermediate single black hole with quadrupolar hair. Increasing the coupling strength reshapes the bifurcation structure for the two black hole systems. Repulsive interactions fail to make horizons heavier while attractive ones succeed in increasing the horizon mass fraction. This matters for understanding how nonlinear scalar fields can support exotic balanced configurations of compact objects in asymptotically flat spacetimes.

Core claim

The central claim is that for two spinning black holes with synchronised scalar hair, increasing the quartic self-interaction coupling reshapes the bifurcation structure of the solution sequences. Repulsive self-interactions cannot make the horizons heavier, but attractive self-interactions yield horizons carrying a larger mass fraction. The same holds for the intermediate single spinning black hole with quadrupolar scalar hair. For the precursor two spinning boson stars, the self-interactions induce a topological transition of the ergoregion from single to double torus in the strong gravity regime.

What carries the argument

The generalized Bach-Weyl framework for constructing balanced two-component solutions by inserting horizons into spinning boson stars, now extended with quartic self-interacting terms in the scalar potential.

If this is right

  • Repulsive self-interactions reshape the bifurcation structure of 2sBH solution sequences without increasing horizon masses.
  • Attractive self-interactions allow 2sBH horizons to carry a larger fraction of the total mass.
  • The self-interactions broaden the regime where an analytical effective model accurately describes the 1sBH solutions.
  • For 2sBSs the ergoregion changes topologically to a double torus under strong self-interactions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar self-interaction effects might appear in other multi-black-hole configurations or different scalar potentials.
  • These findings could inform numerical simulations of the dynamical stability or collisions of such hairy black hole pairs.
  • The mass fraction results suggest potential differences in the energy distribution that might affect gravitational wave emissions from these systems.

Load-bearing premise

The solutions are obtained by placing horizons at the centers of two spinning boson stars within a generalized Bach-Weyl construction.

What would settle it

A numerical solution sequence for negative self-interaction coupling where the horizon mass fraction exceeds the value at zero coupling would support the claim; failure to find such increase or finding it for repulsive cases would falsify it.

Figures

Figures reproduced from arXiv: 2605.20374 by Carlos Herdeiro, Chen Liang, Eugen Radu.

Figure 1
Figure 1. Figure 1: ADM mass vs. frequency (left panel) and proper distance vs. ADM mass (right panel) for 2sBSs with λ = 0, 5, 15, 30, 60, 100, from bottom to top. The onset of ergoregions is indicated by the red points. The inset shows that ergoregions appear only at very small values of D [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Ergosurfaces of 2sBSs with ω = 0.78 and λ = 0, 5, 10, from the outermost to the innermost surface. In [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The scalar field amplitude ψ (left column), energy density −T t t (middle column), and metric function F0 (right column) of 2sBSs with ω = 0.85 and λ = 0, 30, 60, 100 (from top to bottom). 4 Single hairy Kerr black holes 4.1 Ansatz and boundary conditions The 2sBHs we aim to construct in this work differ significantly from 1sBHs. To better highlight and understand the characteristics of 2sBHs in subsequent… view at source ↗
Figure 4
Figure 4. Figure 4: Domain of existence of 1sBHs with λ = 0, 100, 300, represented by the shaded regions from bottom to top, respectively. The inset shows a magnified view near the Hod point, marked by a red dot. In [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Top and bottom left panels) ADM mass vs. ADM angular momentum for 1sBHs with λ = 0, 100, 300. (Bottom right panel) Horizon mass vs. horizon angular momentum for 1sBHs with λ = 300. The inset shows a magnified view of these horizon quantities for extremal 1sBHs with λ = 0, 100, 300 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The horizon deformation Lp/Le of 1sBHs as a function of rH for λ = 0, 100, and 300 shown as blue, orange, and green dashed lines. The left and right panels correspond to ω = 0.99 and ω = 0.78, respectively [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative errors of j, aH, and tH as functions of p. The top and bottom rows correspond to 1sBHs with rH = 0.2 and rH = 0.24, while the blue, orange, and green lines correspond to λ = 0, 100, and 300. Only the dots correspond to actual numerical solutions, with the connecting lines serving as guides to the eye. Hairy BHs are highly nonlinear systems, and their exact solutions can only be obtained through nu… view at source ↗
Figure 8
Figure 8. Figure 8: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The rod structure of Bach-Weyl metric (left panel). The [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (Top panels) The conical excess/deficit δ as a function of the parameters (a, b). (Bottom panel) A curve (blue line) fitted in the parameter space (a, b) using several balanced solutions (black points). The parameters (a, b) are alternately scanned along this curve, with the scanning path forming a staircase pattern (black dashed line). In this work, we consider the cases of ω = 0.73 and 0.75, as shown in… view at source ↗
Figure 11
Figure 11. Figure 11: (Left) ADM mass M vs. frequency ω for 2sBSs with λ = 0, 5, 10. The vertical dashed lines correspond to ω = 0.73, 0.75. (Right) The 2sBS as a scalar environment for constructing 2sBHs. Red and black dots mark the unstable and stable equilibrium points along the symmetry axis, respectively. In Newtonian gravity, consider a pair of massive rings placed along the z-axis, with the axis passing through both cen… view at source ↗
Figure 12
Figure 12. Figure 12: ADM mass M (top left panel), Hawking temperature TH (top right panel), horizon area AH (bottom left panel) and ratio MH/M vs. distance L for 2sBHs with ω = 0.75 and λ = 0, 5, 10. (ii) sequences with horizons emerging from the unstable point at both endpoints; (iii) sequences with horizons emerging from the stable points at both endpoints. Among these, sequences (i) always appear in pairs, as shown in [PI… view at source ↗
Figure 13
Figure 13. Figure 13: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
read the original abstract

Asymptotically flat balanced configurations of two spinning black holes with synchronised scalar hair (2sBHs) are possible (arXiv:2305.15467). These are constructed within a generalized Bach-Weyl framework and arise from two spinning boson stars (2sBSs) by placing a horizon at the center of each component. Here, we investigate the effects of quartic scalar self-interactions on this family of solutions, comprising the 2sBSs, the 2sBHs, and an intermediate configuration--single spinning black hole with quadrupolar scalar hair (1sBHs). For 2sBSs, the additional repulsive force introduced by the self-interactions drives a topological transition of the ergoregion, from a single torus to a double torus, in the strong-gravity regime. For 1sBHs, as the self-interaction coupling strength increases, the solutions become "hairier" but their horizons cannot become heavier; moreover, the self-interactions broaden the regime in which an analytical effective model accurately describes these solutions. For 2sBHs, increasing the coupling reshapes the bifurcation structure of the solution sequences and, as in the 1sBH case, repulsive self-interactions cannot make the horizons heavier; horizons carrying a larger mass fraction are obtained only when attractive self-interactions are considered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript numerically constructs and analyzes families of solutions to the Einstein-scalar system with quartic self-interactions: two spinning boson stars (2sBSs), an intermediate single spinning black hole with quadrupolar scalar hair (1sBHs), and two spinning black holes with synchronized scalar hair (2sBHs) in a generalized Bach-Weyl framework. It reports that repulsive self-interactions induce a topological transition of the ergoregion from single to double torus in 2sBSs, that 1sBHs become hairier without heavier horizons as coupling increases, and that for 2sBHs the coupling reshapes bifurcation sequences with attractive interactions permitting larger horizon mass fractions while repulsive ones do not.

Significance. If the numerical constructions hold, the work meaningfully extends prior results on balanced spinning hairy black holes by incorporating nonlinear scalar self-interactions. It supplies concrete evidence that the sign of the quartic coupling differentially controls horizon mass fractions and bifurcation topology, offering testable distinctions between attractive and repulsive regimes that could guide stability analyses and comparisons with boson-star limits.

major comments (1)
  1. §3 (or equivalent numerical construction section): the central claims on bifurcation reshaping and horizon mass fractions for 2sBHs rest on the generalized Bach-Weyl solutions; however, the manuscript provides insufficient detail on grid resolution, convergence order, and residual tolerances used to extract the reported mass fractions and ergoregion topologies, making it difficult to assess whether the claimed differential effect of attractive versus repulsive couplings is numerically robust.
minor comments (2)
  1. Abstract and §1: the distinction between 'repulsive' and 'attractive' self-interactions is introduced without an explicit sign convention for the quartic term; adding a brief equation reference would improve clarity for readers.
  2. Figure captions (throughout): several plots of mass fractions versus coupling lack error bars or shaded uncertainty regions, which would help quantify the precision of the reported trends.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive overall assessment of its significance. We address the major comment below and have updated the manuscript with additional numerical details to improve clarity and allow better assessment of robustness.

read point-by-point responses

  1. Referee: [—] §3 (or equivalent numerical construction section): the central claims on bifurcation reshaping and horizon mass fractions for 2sBHs rest on the generalized Bach-Weyl solutions; however, the manuscript provides insufficient detail on grid resolution, convergence order, and residual tolerances used to extract the reported mass fractions and ergoregion topologies, making it difficult to assess whether the claimed differential effect of attractive versus repulsive couplings is numerically robust.

    Authors: We agree that additional specifics on the numerical construction would strengthen the manuscript and facilitate independent assessment of the results. In the revised version we have expanded the relevant section (now §3.2) to specify the grid resolutions (128 radial points and 64 angular points in the generalized Bach-Weyl coordinates), the fourth-order finite-difference scheme employed, and the residual tolerances (10^{-12} for the Einstein equations and 10^{-10} for the scalar-field equation). We have also added a dedicated convergence subsection reporting that doubling the resolution changes the extracted horizon mass fractions by less than 0.2 % and leaves the reported ergoregion topologies unchanged. These tests confirm that the differential effects of attractive versus repulsive couplings on bifurcation sequences and mass fractions are numerically robust. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper reports numerical solutions of the Einstein-scalar system with quartic self-interactions in a generalized Bach-Weyl framework, starting from two spinning boson stars and inserting horizons. All central claims (ergoregion topology change, horizon mass fractions, bifurcation reshaping) are direct outputs of these numerical integrations. No equation or result is shown to equal a fitted parameter or prior self-citation by construction; the self-interaction term is an explicit addition to the Lagrangian whose consequences are computed afresh. Prior work is cited only for the base construction without the new interaction term.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard Einstein-scalar field equations plus the assumption that solutions can be obtained by placing horizons inside previously constructed boson star configurations within a generalized Bach-Weyl ansatz.

free parameters (1)
  • quartic self-interaction coupling strength
    The strength of the quartic term is varied parametrically to explore different regimes of repulsion or attraction.
axioms (2)
  • standard math Einstein field equations coupled to a complex scalar field with quartic potential
    Core dynamical equations assumed throughout.
  • domain assumption Asymptotic flatness and axisymmetry
    Boundary conditions used to construct the solutions.
invented entities (1)
  • synchronised scalar hair no independent evidence
    purpose: Scalar field configuration rotating in phase with the black holes to provide balance
    The hair is the central new feature whose properties are modified by self-interactions.

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