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arxiv: 2606.24041 · v1 · pith:R3KGRBHWnew · submitted 2026-06-23 · 🌀 gr-qc

Nonrotating and rotating black holes with secondary disformal hair in a ghost-free metric-affine parity-violating scalar-torsion theory

E. Brito , G. Macedo , J. R. Nascimento , A. Yu. Petrov , P. J. Porf\'irio This is my paper

Pith reviewed 2026-06-25 23:24 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black holesscalar hairdisformal transformationsmetric-affine gravitytorsionNieh-Yan densityKerr metricghost-free theories
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The pith

Kerr black holes in the h-metric frame acquire generically axisymmetric scalar profiles that manifest as secondary disformal hair in the physical g-metric frame.

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

The paper constructs a projectively invariant metric-affine scalar-torsion theory with a pseudoscalar coupled to the Nieh-Yan density and the Einstein-Palatini tensor. After gauge fixing, the connection decomposes into the Levi-Civita connection of an auxiliary metric h disformally related to the physical metric g, plus axial torsion sourced by the scalar gradient. In the degenerate branch where the parameter α equals 6β²κ², Ricci-flat h-geometries satisfy the metric equations identically and the Noether current vanishes, allowing standard Schwarzschild and Kerr solutions in the h-frame to induce nontrivial scalar profiles and disformal corrections in the g-frame. For the Kerr case the fixed-norm scalar is axisymmetric and the configuration is stealth only in the h-frame, yielding secondary disformal hair rather than primary Noether-charge hair. A sympathetic reader would care because the construction supplies explicit rotating black-hole solutions with scalar hair that arise purely from the disformal relation between two metric frames.

Core claim

In the degenerate ghost-free branch α=6β²κ² the theory admits Ricci-flat geometries of the auxiliary metric h as exact solutions with vanishing Noether current; Schwarzschild and Kerr metrics in the h-frame therefore generate disformal deformations together with nontrivial scalar profiles when viewed in the physical g-frame, and the Kerr solution with fixed-norm scalar is stealth solely in the h-frame while representing secondary disformal scalar hair in the g-frame.

What carries the argument

The degenerate branch α=6β²κ², which renders the metric equations satisfied identically by any Ricci-flat h-geometry while forcing the Noether current to vanish, thereby permitting disformal scalar hair on black-hole backgrounds.

If this is right

  • Schwarzschild and Kerr geometries placed in the h-metric frame produce disformal deformations of the metric together with nontrivial scalar profiles when expressed in the g-metric frame.
  • The fixed-norm scalar profile on the Kerr background is generically axisymmetric, depending on both radial and angular coordinates.
  • A stationary future-regular scalar profile can be constructed in ingoing Kerr coordinates.
  • The resulting configuration is stealth only in the h-frame and carries secondary disformal hair, not primary Noether-charge hair, in the g-frame.

Where Pith is reading between the lines

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

  • The same disformal mechanism could be applied to other vacuum solutions, such as charged or higher-dimensional black holes, to generate families of hairy geometries without introducing new primary charges.
  • Axisymmetric scalar profiles on rotating backgrounds may imprint distinctive signatures on quasinormal-mode spectra or shadow images that differ from those of primary-hair solutions.
  • Because the torsion is purely axial and sourced only by the scalar gradient, the construction might extend to other parity-violating couplings while preserving the ghost-free degenerate branch.

Load-bearing premise

The assumption that the specific parameter choice α=6β²κ² simultaneously eliminates ghosts and makes every Ricci-flat h-geometry solve the metric field equations with zero Noether current.

What would settle it

An explicit evaluation of the Noether current for the Kerr solution in the degenerate branch that yields a nonzero result, or a direct integration showing the fixed-norm scalar profile cannot be made axisymmetric while remaining regular at the future horizon.

read the original abstract

In this paper, we formulate a ghost-free metric-affine scalar-torsion model in which a pseudoscalar field is coupled to the Nieh--Yan density and derivatively coupled to the Einstein-Palatini tensor. The theory is projectively invariant and, in the shift-symmetric sector, depends on the scalar only through its gradient. After fixing the projective gauge, the independent connection is solved exactly: its symmetric part is the Levi-Civita connection associated with the metric \(h_{\mu\nu}\), disformally related to the metric $g_{\mu\nu}$, while its antisymmetric part is a purely axial torsion sourced by $v_\mu=\partial_\mu\vartheta$. We identify a degenerate ghost-free branch, $\alpha=6\beta^2\kappa^2$, for which Ricci-flat geometries in the $h$-metric frame solve the metric equations and the Noether current vanishes identically. For example, we consider Schwarzschild and Kerr geometries in the $h$-metric frame, which generate disformal deformations on these geometries from the perspective of the $g$-metric frame. For the Schwarzschild black hole in the $h$-metric frame, we construct static and Babichev--Charmousis-like time-dependent scalar profiles, and discuss horizon regularity in Eddington--Finkelstein coordinates. For the Kerr black hole in the $h$-metric frame, we show that the fixed-norm scalar profile is generically axisymmetric, depending on both the radial and angular coordinates, and we construct a stationary future-regular profile in ingoing Kerr coordinates. The Kerr solution with the nontrivial scalar field is stealth only in the $h$-metric frame. In the $g$-metric frame, it produces disformal corrections stemming from the nontrivial scalar field profile and it represents secondary disformal scalar hair rather than primary Noether-charge hair.

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

2 major / 2 minor

Summary. The manuscript formulates a ghost-free metric-affine scalar-torsion theory with a pseudoscalar coupled to the Nieh-Yan density and derivatively to the Einstein-Palatini tensor. After projective gauge fixing the independent connection is solved exactly (symmetric part the Levi-Civita of a disformally related metric h, antisymmetric part axial torsion from v=∂ϑ). A degenerate branch α=6β²κ² is identified in which any Ricci-flat h-geometry solves the metric equations identically while the Noether current vanishes; explicit scalar profiles are constructed on Schwarzschild and Kerr backgrounds in the h-frame, yielding disformal deformations in the g-frame that are classified as secondary rather than primary Noether-charge hair.

Significance. If the central identification of the branch holds, the construction supplies explicit, exactly solvable examples of stealth black-hole solutions carrying secondary disformal scalar hair in a parity-violating scalar-torsion theory. The exact solvability of the connection after gauge fixing and the parameter relation that decouples the scalar from the metric equations constitute concrete technical strengths.

major comments (2)
  1. [section on degenerate branch and Noether current] Degenerate branch (statement following projective gauge fixing and connection solution): the claim that the single algebraic relation α=6β²κ² renders the metric equations identically satisfied by any Ricci-flat h-geometry and forces the Noether current J^μ to vanish identically is load-bearing for the secondary-hair classification; an explicit substitution into the metric field equations and the Noether current expression is required to confirm both statements.
  2. [section identifying the ghost-free branch] Ghost-freedom assertion for the branch α=6β²κ²: the statement that the theory remains ghost-free on this slice must be supported by an explicit quadratic-action expansion or dispersion-relation analysis around the background solutions, because absence of ghosts is required for the physical viability of the constructed Kerr and Schwarzschild configurations.
minor comments (2)
  1. [connection solution paragraph] Notation for the disformal relation between g and h should be stated once with all coefficients explicit before being used in the solution constructions.
  2. [Kerr solution subsection] The axisymmetric scalar profile on the Kerr background is stated to depend on both r and θ; a brief explicit functional form or plot would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: Degenerate branch (statement following projective gauge fixing and connection solution): the claim that the single algebraic relation α=6β²κ² renders the metric equations identically satisfied by any Ricci-flat h-geometry and forces the Noether current J^μ to vanish identically is load-bearing for the secondary-hair classification; an explicit substitution into the metric field equations and the Noether current expression is required to confirm both statements.

    Authors: We agree that an explicit substitution strengthens the presentation. The derivation in the manuscript proceeds from the solved connection (symmetric part Levi-Civita of h, axial torsion from v=∂ϑ) into the metric equations, which reduce to a multiple of the Einstein tensor of h plus scalar-gradient terms that cancel precisely when α=6β²κ²; the same cancellation sets the Noether current to zero. To make this transparent, the revised manuscript will insert the intermediate steps of the substitution for both the metric equations and J^μ. revision: yes

  2. Referee: Ghost-freedom assertion for the branch α=6β²κ²: the statement that the theory remains ghost-free on this slice must be supported by an explicit quadratic-action expansion or dispersion-relation analysis around the background solutions, because absence of ghosts is required for the physical viability of the constructed Kerr and Schwarzschild configurations.

    Authors: The branch is identified as ghost-free because the condition α=6β²κ² eliminates the higher-order derivative contributions that would otherwise produce ghosts in the quadratic action. We will add an explicit quadratic expansion (or equivalent dispersion analysis) around the Schwarzschild and Kerr backgrounds in the revised manuscript to confirm the absence of ghost modes. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from action, gauge fixing, and explicit parameter choice without self-referential reduction

full rationale

The paper begins from a stated action with projective invariance and shift symmetry, fixes the projective gauge, solves the connection explicitly (symmetric part Levi-Civita of h, antisymmetric axial torsion from v=∂ϑ), then identifies the specific branch α=6β²κ² under which Ricci-flat h solves the metric equations identically and the Noether current vanishes. The Kerr profile is then constructed in the h-frame and reinterpreted in the g-frame as secondary disformal hair precisely because the current vanishes by that choice. No equation equates the final hair profile to a quantity fitted from the same data, no self-citation supplies a uniqueness theorem or ansatz, and the classification follows from the stated consequences of the branch rather than presupposing the result. The construction is therefore self-contained against its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on projective invariance of the action, the existence of a ghost-free degenerate branch defined by a specific algebraic relation among couplings, and the exact solvability of the independent connection once the projective gauge is fixed. No new particles or forces are postulated beyond the pseudoscalar and the axial torsion already present in the metric-affine setup.

free parameters (1)
  • α, β, κ relation
    The branch α=6β²κ² is imposed by hand to eliminate ghosts and make the Noether current vanish; it is a discrete choice among possible coupling values.
axioms (2)
  • domain assumption The theory is projectively invariant
    Invoked to reduce the connection degrees of freedom and allow exact solution after gauge fixing.
  • domain assumption The shift-symmetric sector depends on the scalar only through its gradient
    Stated to ensure the scalar enters the equations only via v_μ = ∂_μ ϑ.

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discussion (0)

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