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No smooth rotating bumblebee vacuum can extend through the rotation axis.

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2026-08-01 18:41 UTC pith:PIQ7FLFP

load-bearing objection Two clean no-go theorems for constant-norm vector vacua on rotating axes, but the Kerr-disformal application inherits an unproved exactness claim from earlier work. the 2 major comments →

arxiv 2607.17223 v1 pith:PIQ7FLFP submitted 2026-07-19 gr-qc

Axial Obstructions to Rotating Bumblebee Vacuum Solutions

Minyong Guo , Zhong-Ying Fan This is my paper
classification gr-qc MSC 83C5783C75
keywords bumblebee gravityLorentz symmetry breakingconstant-norm vector fieldrotating black holesKerr-disformal metricaxial singularityconicityKilling horizon
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes two general obstructions that prevent smooth rotating 'bumblebee' vacua — spacetimes where a vector field acquires a constant nonzero norm and breaks Lorentz symmetry. First, at any pole where a nondegenerate Killing horizon meets the rotation axis, symmetry inheritance forces the bumblebee one-form to vanish, contradicting its nonzero constant norm. Second, if the angular defect (conicity) of the axis varies along its length, the Kretschmann scalar diverges as one approaches the axis, so the metric cannot be smoothly extended. Applied to a three-parameter Kerr-disformal family, the axes become genuine curvature singularities rather than harmless conical defects, though a special nonextremal branch keeps a regular outer Killing horizon away from the poles. The upshot: these configurations are exact local rotating solutions only in the nonpolar region, not globally regular black holes.

Core claim

The paper's central claim is that two local, largely field-equation-independent mechanisms make a regular rotating constant-norm vector vacuum impossible on the axes. At a pole of a bifurcate horizon lying on the rotation axis, the horizon boost and the axial rotation are both fixed points, and their combined action on a smooth symmetry-inheriting one-form leaves no nonvanishing components, forcing Omega|_p = 0 and contradicting Omega_mu Omega^mu = b_0^2 != 0. Independently, a longitudinally varying conicity alpha(z) along the axis produces a universal curvature divergence R_rho^phi_z^phi ~ -d_z ln alpha / rho, hence a non-C^2 axis whenever d_z alpha != 0. For the Kerr-disformal family, both

What carries the argument

The two load-bearing tools are (i) the fixed-point isotropy argument: at p in B ∩ A, the boost generator ∇_a chi_b and the axial rotation generator ∇_a m_b are invertible on complementary subspaces, so Lie-invariance of Omega forces all four frame components to vanish; and (ii) the varying-conicity criterion: in Gaussian axial coordinates with proper radius f = alpha(z) rho, the mixed curvature component R_{\hat rho \hat phi \hat z \hat phi} = -∂_z ln alpha / rho + O(1) follows from the Hessian of f, proving that a nonuniform axial defect is a genuine curvature singularity. The application uses the disformal map g = g^(0) + ell Omega Omega with closed constant-norm Omega to generate the thre

Load-bearing premise

The application to the Kerr-disformal family leans on the assertion, made without derivation in Section III, that the disformal map (1) with a closed constant-norm one-form yields exact solutions of the closed vacuum sector (V = V' = 0) of action (7); if that mapping fails, the paper's concrete Kerr-disformal conclusions collapse, though the two generic obstructions would survive.

What would settle it

Directly substitute metric (9) and one-form (8) into the field equations from action (7) for generic nonzero ell, a, q; if the equations fail for any member of the family beyond the q^2 = r_+^2 branch, the claimed exact Kerr-disformal solutions are not established. Alternatively, exhibit a smooth symmetry-inheriting one-form with Omega_mu Omega^mu = b_0^2 != 0 on a regular bifurcate horizon containing a pole — that would refute the fixed-point theorem.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any smooth symmetry-inheriting constant-norm vector vacuum with a nondegenerate bifurcate horizon must fail at the horizon-axis poles; the obstruction is independent of the field equations.
  • A longitudinally varying conicity along a rotation axis is a sufficient condition for a curvature singularity, so nonuniform conical defects cannot be regularized by a global rescaling of the angular period.
  • The Kerr-disformal family with ell != 0 has curvature-singular polar axes for every real q, including the static sector with q != 0.
  • On the branch q^2 = r_+^2, the outer horizon is a regular Killing horizon away from the poles, defining an exact local rotating solution in the nonpolar exterior; the inner Kerr surface becomes a null curvature singularity.
  • A globally regular rotating bumblebee black hole must either relax symmetry inheritance, relax the strict nonzero constant norm, or modify the near-pole geometry.

Where Pith is reading between the lines

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

  • The fixed-point obstruction likely extends beyond bumblebee gravity to any vector-tensor or Proca-type theory with a constant-norm vacuum, so the result may constrain a broader class of Lorentz-violating black holes.
  • The varying-conicity criterion could serve as a quick regularity test for other disformal or metric-generated rotating solutions with position-dependent conical defects.
  • The paper leaves open whether a one-sided future-horizon (collapse) spacetime with a nonseparable, non-symmetry-inheriting one-form could admit a smooth axis; a global boundary-value analysis would settle it.
  • If the disformal map's exactness were verified by direct substitution, the q^2 = r_+^2 branch would be a useful local rotating background for nonpolar strong-field phenomena, though global charges and polar trajectories remain undefined without an axis prescription.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves two general regularity obstructions for rotating constant-norm vector (bumblebee) vacua, then applies them to a three-parameter Kerr-disformal family in Einstein-bumblebee gravity. The fixed-point obstruction shows that a smooth one-form invariant under the stationary and axial Killing fields must vanish at a pole of a regular nondegenerate bifurcate Killing horizon, contradicting a constant nonzero norm. The varying-conicity criterion shows that a conicity α(z) depending on position along the rotation axis produces a 1/ρ curvature divergence and excludes a C^2 axial extension. The application claims that the Kerr-disformal metric (9) and one-form (8) are exact solutions in the closed vacuum sector, that both rotation axes are curvature singularities, and that the branch q^2=r_+^2 admits a regular outer Killing horizon away from the poles while r=r_- becomes a null curvature singularity.

Significance. The two Section II theorems are independent of the field equations and are valuable additions to the regularity literature for Lorentz-violating vector vacua. The fixed-point proof is concise and rigorous, using only the invertibility of the horizon boost and axial rotation linearizations. The varying-conicity criterion is also useful because it upgrades a nonuniform conical defect from a distributional to a genuine curvature singularity. The paper includes several good checks, such as the ℓ=0 limit recovering the Kerr axis and the a=q=0 limit recovering the regular static branch. The limitations (future horizons, extremal horizons, nonseparable fields) are stated honestly. However, the application's exactness claim is not derived in the manuscript, and the nonaxial horizon regularity on the selected branch rests partly on an announced 'direct evaluation' rather than a displayed computation.

major comments (2)
  1. [Section III, Eqs. (7)-(9)] The exactness of the Kerr-disformal family is the load-bearing step for the application. The sentence 'the field equations admit the disformal map (1)' is asserted with no derivation, and the cited checks [10,11] are for the q=0 sector. Eq. (7) contains the nonminimal coupling ξ B^μ B^ν R_{μν}; whether a disformal deformation of a Ricci-flat seed by the closed constant-norm Ω of Eq. (8) solves both the metric and vector equations is not automatic. The relation among ℓ, ξ, and b (if any) is also not stated. Since the abstract and Section VII describe the family as an 'exact local rotating solution', this missing verification must be supplied — either by a direct substitution of Eqs. (8)-(9) into the field equations, or by a precise theorem for map (1) and a check that its hypotheses hold for the three-parameter family, including the q^2=r_+^2 branch.
  2. [Section IV, Eq. (12)] Regularity of the outer horizon on the branch q^2=r_+^2 away from the axes is asserted through 'direct evaluation confirms' after Eq. (12). Eq. (10) displays only the leading singular term; the subleading O(sqrt(r^2-q^2)/sqrt(r-r+)) term is said to carry 'the same singular factors (up to a square root)', so the cancellation is not evident. Please provide the explicit nonaxial K_+(θ) or, more usefully, the horizon-penetrating coordinate chart in which the metric (9) and the one-form (8) are manifestly smooth on D_epsilon. This is required to substantiate the headline claim of a regular outer Killing horizon away from the poles.
minor comments (4)
  1. [Eq. (8)] The displayed one-form is difficult to parse because the square roots and fraction bars are not typeset unambiguously. For example, it should be written clearly as Ω = sqrt((r^2-q^2)/Δ) dr + sqrt(a^2 cos^2θ+q^2) dθ (or the intended equivalent).
  2. [Appendix C, Eqs. (C51)-(C52)] The expression for dz/dr appears dimensionally inconsistent as printed: the right-hand side has units of length. If the intended expression is sqrt(D0/Δ), please correct the formula and the subsequent ∂_z α expression.
  3. [Section III, V=V'=0] The role of setting V=V'=0 with a nonzero b0 should be clarified. If the potential is identically zero, what enforces the constant norm, or is b0 simply a free parameter in the vacuum ansatz? A short remark would remove ambiguity.
  4. [Section II.b] After Eq. (5), the statement that a regular axial segment requires α=1 for the standard 2π period could be made explicit, since some readers may conflate the conical-defect case α=const≠1 with the singular varying-α case.

Circularity Check

0 steps flagged

No significant circularity: the two axial obstructions are parameter-free local derivations, and the disformal exactness claim rests on external prior work, not on a self-referential chain.

full rationale

The paper's central derivations are self-contained and do not reduce to their inputs. The fixed-point obstruction (Section II.a and Appendix A) uses only smoothness, symmetry inheritance, and the invertible linearized boost/rotation actions (A1)-(A2) to force Ω|_p = 0; it does not assume the conclusion or fit any parameter. The varying-conicity criterion (Section II.b and Appendix C) derives the universal 1/ρ divergence (C35) from the Hessian of f with f = α(z)ρ + O(ρ²), and the proof shows the divergence cannot be removed by bounded frame changes (C36)-(C37). Neither theorem invokes the bumblebee field equations or the disformal ansatz. The applied Kerr-disformal family does rely on the statement in Section III that 'the field equations admit the disformal map (1)', but this is attributed to external references [10,11], not to the present authors, so it is not a self-citation load-bearing step. The absence of an independent re-derivation is a correctness or verification concern, not circularity. Similarly, the branch choice q² = r_+² is explicitly derived from the requirement that the leading nonaxial divergence in (10) cancel, giving (11); the paper then checks regularity rather than presenting a fitted quantity as a prediction. The paper also transparently states its scope limitations (Section VI and Appendix F), including future-horizon, nonseparable, and extremal cases. Against external benchmarks, the ℓ = 0 limit recovers the regular Kerr axis and a = q = 0 recovers the regular static branch (D41), providing independent checks. No equation in the paper is equivalent by construction to a claimed output, and no self-citation chain forces the conclusions.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The two general theorems rest on standard bifurcate-horizon geometry (Racz-Wald structure) plus the explicitly stated regularity hypotheses of the conicity criterion, which the application verifies against the explicit metric. The application additionally assumes the disformal-map exactness inherited from [10,11]. The led parameter count is modest: one separation constant (q, selected to r_+^2 on the distinguished branch), the theory coupling ell, and the vector norm b_0. No new fields, forces, or particles are introduced.

free parameters (3)
  • q (separation constant) = q^2 = r_+^2 on the distinguished nonextremal branch; otherwise free
    Enters Omega and the metric via Eq. (8); Eq. (11) selects q^2 = r_+^2 to cancel the leading nonaxial Kretschmann divergence near r_+ (Eq. 10). The paper presents this as explicit branch selection, not prediction.
  • ell (disformal/bumblebee coupling) = real, with 1+ell > 0; axial singularities require ell != 0
    Coupling in construction (1) and action (7); a theory input rather than a fitted number. The paper's conclusions require ell(a^2+q^2) != 0 for the varying-conicity mechanism to activate; the ell=0 limit correctly reduces to Kerr.
  • b_0 (constant norm of the vector vacuum) = b_0^2 > 0, normalized to 1 in the main construction
    Strict nonzero constant norm is the defining premise of the bumblebee vacuum; it drives the fixed-point contradiction and appears explicitly in the separable obstruction (Appendix F, Eqs. F1-F2). The normalization is a choice of units.
axioms (4)
  • standard math Standard bifurcate Killing horizon structure: at the bifurcation surface, grad_a chi^b|_p = kappa(e0_a e1_b - e1_a e0_b) with kappa != 0, and at the axis pole grad_a m^b|_p = omega(e2_a e3_b - e3_a e2_b) with omega != 0.
    Used in Section II.a and Appendix A to conclude that the two linearized actions are invertible on their respective planes, forcing all four components of Omega to vanish at p. Standard results from [14-16] (Wald; Racz-Wald).
  • domain assumption The disformal map g = g0 + ell Omega(x)Omega, B = b Omega with closed constant-norm Omega and Ricci-flat seed yields exact solutions of the closed-sector field equations of action (7) with V = V' = 0.
    Stated in Section III ('the field equations admit the disformal map (1)'); the entire application inherits this from [10,11] without re-derivation. The bumblebee equation in the closed sector requires R_mu nu Omega^nu = 0, trivially satisfied by a Ricci-flat seed, but the modified Einstein equations for the disformal metric are not re-checked here.
  • domain assumption Regularity hypotheses of the conicity criterion: quotient metric has a regular C^2 limit in Gaussian coordinates with rho proper distance from the axis, g_i phi = O(rho^2), A_i bounded with bounded derivatives, and f = alpha(z) rho + O(rho^2).
    These are the stated hypotheses of the Appendix C theorem (Eqs. C2-C4); verified for the Kerr-disformal metric in Appendix C.6. A proposed application that violated them would not be covered by the 1/rho criterion.
  • domain assumption The one-form Omega of Eq. (8) has constant positive norm on the Kerr seed.
    The construction is a constant-norm vector vacuum; norm constancy is part of the ansatz inherited from [10]. Used implicitly whenever the constant norm b_0^2 enters, including the axis-smoothness evaluation Omega_theta^2|_theta=0,pi = a^2 + q^2 (Eq. 15).

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0 comments
read the original abstract

We identify two general obstructions to a regular rotating constant-norm vector vacua. Firstly, at the poles of a nondegenerate bifurcation surface, horizon and axial invariance force every smooth symmetry-inheriting one-form to vanish, contradicting a strictly nonzero constant norm. Secondly, we show that a conicity varying along a rotation axis generates a singular curvature that diverges as the inverse proper distance from the axis. We apply these results to a general three parameter family Kerr-disformal solution in Einstein--bumblebee gravity. Consequently, its vector vacuum is not smooth on the axes because of a nonzero constant norm, and the position-dependent conicity leads to genuine curvature singularities on the polar axes. A distinguished nonextremal branch nevertheless admits a regular outer Killing horizon away from the axes and defines an exact local rotating solution in the nonpolar region.

Figures

Figures reproduced from arXiv: 2607.17223 by Minyong Guo, Zhong-Ying Fan.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic radial causal structure of the nonextremal [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

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Reference graph

Works this paper leans on

37 extracted references · 10 linked inside Pith

  1. [1]

    V. A. Kosteleck´ y and S. Samuel, Spontaneous breaking of lorentz symmetry in string theory, Phys. Rev. D39, 683 (1989)

    1989

  2. [2]

    S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane, and T. Okamoto, Noncommutative field theory and Lorentz violation, Phys. Rev. Lett.87, 141601 (2001), arXiv:hep-th/0105082

    Pith/arXiv arXiv 2001

  3. [3]

    V. A. Kostelecky and S. Samuel, Phenomenological Grav- itational Constraints on Strings and Higher Dimensional Theories, Phys. Rev. Lett.63, 224 (1989)

    1989

  4. [4]

    Gambini and J

    R. Gambini and J. Pullin, Nonstandard optics from quan- tum space-time, Phys. Rev. D59, 124021 (1999), arXiv:gr- qc/9809038

    arXiv 1999

  5. [5]

    V. A. Kostelecky and S. Samuel, Gravitational Phe- nomenology in Higher Dimensional Theories and Strings, Phys. Rev. D40, 1886 (1989)

    1989

  6. [6]

    Bluhm and V

    R. Bluhm and V. A. Kostelecky, Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity, Phys. Rev. D71, 065008 (2005), arXiv:hep-th/0412320

    Pith/arXiv arXiv 2005

  7. [7]

    Casana, A

    R. Casana, A. Cavalcante, F. P. Poulis, and E. B. Santos, Exact schwarzschild-like solution in a bumblebee gravity model, Phys. Rev. D97, 104001 (2018), arXiv:1711.02273 [gr-qc]

    Pith/arXiv arXiv 2018

  8. [8]

    Chen and H.-S

    Y.-Q. Chen and H.-S. Liu, Taub-NUT-like black holes in Einstein-bumblebee gravity, Phys. Rev. D112, 084040 (2025), arXiv:2505.23104 [gr-qc]

    Pith/arXiv arXiv 2025

  9. [9]

    C. Ding, C. Liu, R. Casana, and A. Cavalcante, Exact kerr-like solution and its shadow in a gravity model with spontaneous lorentz symmetry breaking, Eur. Phys. J. C 80, 178 (2020), arXiv:1910.02674 [gr-qc]

    Pith/arXiv arXiv 2020

  10. [10]

    F. P. Poulis and M. A. C. Soares, Exact modifications on a vacuum spacetime due to a gradient bumblebee field at its vacuum expectation value, Eur. Phys. J. C82, 613 (2022), arXiv:2112.04040 [gr-qc]

    Pith/arXiv arXiv 2022

  11. [11]

    Ovcharenko, Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique, (2026), arXiv:2601.16037 [gr-qc]

    H. Ovcharenko, Exact Kerr-Newman-(A)dS and other spacetimes in bumblebee gravity: employing a simple generating technique, (2026), arXiv:2601.16037 [gr-qc]

    Pith/arXiv arXiv 2026

  12. [12]

    J. D. Bekenstein, The Relation between physical and gravitational geometry, Phys. Rev. D48, 3641 (1993), arXiv:gr-qc/9211017

    Pith/arXiv arXiv 1993

  13. [13]

    Bettoni and S

    D. Bettoni and S. Liberati, Disformal invariance of sec- ond order scalar-tensor theories: Framing the Horndeski action, Phys. Rev. D88, 084020 (2013), arXiv:1306.6724 [gr-qc]

    Pith/arXiv arXiv 2013

  14. [14]

    R. M. Wald,General Relativity(Chicago Univ. Pr., Chicago, USA, 1984)

    1984

  15. [15]

    Racz and R

    I. Racz and R. M. Wald, Extension of space-times with Killing horizon, Class. Quant. Grav.9, 2643 (1992)

    1992

  16. [16]

    Racz and R

    I. Racz and R. M. Wald, Global extensions of space-times describing asymptotic final states of black holes, Class. Quant. Grav.13, 539 (1996), arXiv:gr-qc/9507055

    Pith/arXiv arXiv 1996

  17. [17]

    R. P. Geroch, A Method for generating solutions of Ein- stein’s equations, J. Math. Phys.12, 918 (1971)

    1971

  18. [18]

    Vilenkin, Gravitational Field of Vacuum Domain Walls and Strings, Phys

    A. Vilenkin, Gravitational Field of Vacuum Domain Walls and Strings, Phys. Rev. D23, 852 (1981)

    1981

  19. [19]

    Vilenkin, Cosmic Strings and Domain Walls, Phys

    A. Vilenkin, Cosmic Strings and Domain Walls, Phys. Rept.121, 263 (1985)

    1985

  20. [20]

    Kinnersley and M

    W. Kinnersley and M. Walker, Uniformly accelerating charged mass in general relativity, Phys. Rev. D2, 1359 (1970)

    1970

  21. [21]

    J. F. Plebanski and M. Demianski, Rotating, charged, and uniformly accelerating mass in general relativity, Annals Phys.98, 98 (1976)

    1976

  22. [22]

    B. S. Kay and R. M. Wald, Theorems on the Unique- ness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon, Phys. Rept.207, 49 (1991)

    1991

  23. [23]

    Carter, Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations, Commun

    B. Carter, Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10, 280 (1968). 6 Appendix A: Adapted-frame fixed-point proof At p∈ B ∩ A, choose an orthonormal frame {eˆ0, eˆ1, eˆ2, eˆ3}. The first two vectors span the normal plane toB, while the last two spanT pB. For a nondegenerate bifurcate horizon, ∇aχb p ...

    1968

  24. [24]

    Azimuthal decomposition Away from the axis, the metric admits the exact local decomposition ds2 =q ij(y) dyi dyj +f 2 dϕ+A i dyi 2 ,(C5) where yi = (ρ, t, z),(C6) and qij =g ij − giϕgjϕ gϕϕ .(C7) The tensor qij is the Lorentzian metric on the local quo- tient by the axialU(1) action. Because ρ is a Gaussian normal coordinate for the quotient geometry,qcan...

  25. [25]

    (C11) gives de ˆϕ = df∧(dϕ+A) +fdA = d lnf∧e ˆϕ +f F ,(C13) where F = dA

    Connection one-forms Taking the exterior derivative of Eq. (C11) gives de ˆϕ = df∧(dϕ+A) +fdA = d lnf∧e ˆϕ +f F ,(C13) where F = dA. In the quotient orthonormal frame, write d lnf=u ˆie ˆi , u ˆi =e ˆi(lnf),(C14) and F= 1 2 Fˆiˆje ˆi ∧e ˆj .(C15) It follows that de ˆϕ =u ˆie ˆi ∧e ˆϕ + f 2 Fˆiˆje ˆi ∧e ˆj .(C16) Let ωˆiˆj denote the Levi-Civita connection...

  26. [26]

    Curvature involving the shrinking orbit The curvature two-forms are defined by the second Cartan equation, Rˆaˆb = dωˆaˆb +ω ˆa ˆc∧ω ˆcˆb ,(C20) with Rˆaˆb = 1 2 Rˆaˆbˆcˆdeˆc∧e ˆd .(C21) Substituting the connection one-forms above into Eq. (C20), the curvature components with two azimuthal indices can be organized as Rˆi ˆϕˆj ˆϕ =− Hij f +T ˆiˆj ,(C22) wh...

  27. [27]

    Near-axis behavior We now evaluate the mixed component with ˆi = ˆρand ˆj= ˆz. Sinceρis a Gaussian normal coordinate, ∇e ˆρe ˆρ= 0.(C27) Choose eˆzon the axis and parallel transport it along the radial Gaussian geodesics: ∇e ˆρeˆz= 0.(C28) This is a regular choice of orthonormal frame and places no additional restriction on the quotient geometry. For a sc...

  28. [28]

    (C35) implies R ˆρˆϕˆzˆϕ ∼ |∂z lnα(z 0)| ρ − → ∞(ρ→0).(C36) The component in Eq

    Consequence for axial regularity At an axial point z = z0 satisfying ∂zα(z0) ̸= 0, Eq. (C35) implies R ˆρˆϕˆzˆϕ ∼ |∂z lnα(z 0)| ρ − → ∞(ρ→0).(C36) The component in Eq. (C36) is measured in a unit orthonormal frame. Along any fixed radial approach to the axis, this adapted frame differs from a regular local orthonormal frame only by a bounded Lorentz trans...

  29. [29]

    Applicability to the Kerr-disformal axis We verify that the Kerr-disformal metric satisfies the hypotheses of the Cartan-frame varying-conicity criterion on every open exterior axial segment. Let R=r 2 +a 2, P=r 2 −q 2, Q 0 =a 2 +q 2,(C38) and define D0 =R+ℓP.(C39) Atθ= 0, the (r, θ) block has grr = D0 ∆ +O(θ 2),(C40) grθ = ℓ√P Q0√ ∆ +O(θ 2),(C41) gθθ =R+...

  30. [30]

    Near the axis, gtϕ = O(sin2 θ) and gϕϕ = O(sin2 θ), so At and its required derivatives remain bounded

    Quotient geometry and azimuthal radius Recall that away from the axis, the metric admits the exact decomposition ds2 =q ij dyi dyj +f 2 dϕ+A i dyi 2 , y i = (t, r, θ), (D3) where f = √gϕϕ is the proper radius of an azimuthal orbit and Ai = giϕ gϕϕ , q ij =g ij − giϕgjϕ gϕϕ .(D4) For the Kerr-disformal metric, grϕ = gθϕ = 0, and only At is nonzero. Near th...

  31. [31]

    Quotient-space Hessian Since only∂ θf= √ Ris nonzero on the axis, Hij =∂ i∂jf− Γ θ ij √ R .(D16) We now calculate its independent nonzero components. a. Time component.—On the axis, qtt =− ∆ R , ∂ rqtt =− 2M(r 2 −a 2) R2 .(D17) Stationarity gives Γ θ tt =− 1 2 qθr∂rqtt .(D18) It follows that Htt = ℓM(r 2 −a 2)√P Q0∆ (1 +ℓ)R 7/2 .(D19) b. Radial component....

  32. [32]

    Hessian contraction The leading curvature components containing the shrinking azimuthal direction are Rˆi ˆϕˆj ˆϕ =− Hˆiˆj f +O(1).(D27) The bounded rotational terms do not modify the coeffi- cient of the 1/fdivergence. The required quotient contraction is H2 ≡ HijHij =q ikqjlHijHkl .(D28) The time contribution is (qtt)2H2 tt = ℓ2M 2(r2 −a 2)2P Q0 (1 +ℓ) ...

  33. [33]

    The complete 1 /f 2 contribution to the Kretschmann scalar therefore comes from Eq

    Leading Kretschmann divergence The quotient curvature and all twist-dependent terms are bounded under the assumptions stated above. The complete 1 /f 2 contribution to the Kretschmann scalar therefore comes from Eq. (D27). Accounting for the algebraic symmetries of the Riemann tensor gives K= 4H2 f 2 +O(f −1).(D35) Since f 2 =Rsin 2 θ+O(sin 4 θ),(D36) we ...

  34. [34]

    (D38) have finite outer-horizon limits: ∆ r2 −r 2 + − →r+ −r − 2r+ , r2 −r 2 + ∆ − → 2r+ r+ −r −

    Horizon-selected branch For the nonextremal horizon-selected branch q2 =r 2 + ,∆ = (r−r +)(r−r −),(D42) the two apparently singular ratios in Eq. (D38) have finite outer-horizon limits: ∆ r2 −r 2 + − →r+ −r − 2r+ , r2 −r 2 + ∆ − → 2r+ r+ −r − . (D43) It follows that CH ≡lim r→r+ Cax(r) = 4ℓ2r+(r+ −r −) (1 +ℓ) 2(r2 + +a 2)3 .(D44) We used a2 = r+r− and r2 ...

  35. [35]

    Axis smoothness requires L = 0

    Separable expansioin Locally, Ω = dS, with S=−Et+Lϕ+S r(r) +S θ(θ). Axis smoothness requires L = 0. The constant positive Kerr norm then gives ∆(S′ r)2 + (S′ θ)2 − E2(r2 +a 2)2 ∆ +a 2E2 sin2 θ =b 2 0 r2 +a 2 cos2 θ .(F1) Separating the radial and angular terms gives (S′ θ)2 +a 2E2 sin2 θ−b 2 0a2 cos2 θ=λ .(F2) Smoothness requires S′ θ = O(sinθ ). Evaluati...

  36. [36]

    F uture-horizon nonseparable expansion For a closed stationary and axisymmetric one-form, axis regularity allows Ω =−Edt+∂ rS(r, θ) dr+∂ θS(r, θ) dθ . The constant-norm equation is ∆(∂rS)2 + (∂θS)2 − E2(r2 +a 2)2 ∆ +a 2E2 sin2 θ =b 2 0 r2 +a 2 cos2 θ .(F3) In ingoing Kerr coordinates, d v = dt + (r2 + a2) dr/∆, future-horizon regularity requires ∂rS=− E(r...

  37. [37]

    Thus no local contradiction arises for a genuinely nonseparable field regular only on a future horizon

    = 0.(F4) Unlike the additively separable equation, the derivative term can compensate the positive algebraic terms. Thus no local contradiction arises for a genuinely nonseparable field regular only on a future horizon. Existence of a global solution remains an open boundary-value problem