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arxiv: 2605.17870 · v1 · pith:H5KBY7KMnew · submitted 2026-05-18 · 🌀 gr-qc · hep-th

Rotating black holes with primary hair in five-dimensional generalized Proca theory

Mokhtar Hassaine , Ulises Hernandez-Vera This is my paper

Pith reviewed 2026-05-20 10:08 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords rotating black holesprimary hairgeneralized Proca theoryfive-dimensional gravityKerr-Schild ansatzMyers-Perry solutionsvector-tensor theories
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The pith

Five-dimensional generalized Proca theories admit rotating black holes with primary hair given by an arbitrary function of one angular coordinate.

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

This paper constructs exact analytic rotating black hole solutions in five-dimensional generalized Proca theories that carry primary hair. The hair appears as an arbitrary function of the non-Killing angular coordinate, while the solutions also include a cosmological constant and two independent angular momenta. The construction proceeds by adopting a Kerr-Schild metric ansatz in which the Proca field lies along a null geodesic congruence, reducing the nonlinear equations to three consistent master equations. A sympathetic reader would care because these examples show how vector-tensor theories can support black holes that encode more information than mass and spin alone, while the metric keeps the same structural form as the Myers-Perry solution plus one extra term quadratic in the Proca field.

Core claim

By employing a Kerr-Schild ansatz in which the Proca field is aligned along a null geodesic congruence in five-dimensional spacetime, the nonlinear field equations of generalized Proca theory reduce to a set of three master equations. The solutions obtained are rotating black holes with two independent angular momenta and a cosmological constant, featuring primary hair expressed as an arbitrary function of the non-Killing angular coordinate. The metric preserves the Kerr-Schild structure of the Myers-Perry solution augmented by a term quadratic in the Proca one-form, and different branches arise from specific relations among the theory's coupling constants.

What carries the argument

The Kerr-Schild ansatz with the Proca vector field aligned to a null geodesic congruence, which keeps the field light-like on shell and reduces the nonlinear equations to three master equations that determine the metric and the hair function.

If this is right

  • The solutions support a cosmological constant together with two independent angular momenta.
  • Different branches of solutions exist according to algebraic relations satisfied by the Proca coupling constants.
  • The metric retains exactly the Kerr-Schild form of the Myers-Perry solution plus an added term constructed from the outer product of the Proca one-form with itself.
  • The primary hair remains an arbitrary function of the non-Killing angular coordinate without preventing analytic solvability.

Where Pith is reading between the lines

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

  • The same alignment technique could be tested in other dimensions or in related vector-tensor theories to look for additional analytic hairy solutions.
  • Numerical evolution of these spacetimes could reveal whether the primary hair influences late-time behavior or stability.
  • Specific choices of the arbitrary hair function might produce black holes whose shadows or quasinormal modes differ measurably from the Myers-Perry case.

Load-bearing premise

The Proca field must remain light-like on the solution by alignment with a null geodesic congruence, which restricts the allowed couplings in the theory to discrete constant values.

What would settle it

An explicit substitution of a non-constant hair function into the full field equations that produces a nonzero residual, or a demonstration that the three master equations admit no solutions unless the couplings satisfy the specific algebraic relations required by the ansatz.

Figures

Figures reproduced from arXiv: 2605.17870 by Mokhtar Hassaine, Ulises Hernandez-Vera.

Figure 1
Figure 1. Figure 1: FIG. 1. Horizon function [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

This work presents a new class of exact analytic rotating black hole solutions within five-dimensional generalized Proca theories. Through a Kerr-Schild ansatz where the Proca field is set along a null geodesic congruence, the non-linear field equations reduce to a consistent set of three master equations. This geometric configuration ensures that the vector field remains light-like on-shell, effectively restricting the theory's functional couplings to discrete constants and allowing for a fully analytic treatment. The resulting solutions, incorporating a cosmological constant and two independent angular momenta, exhibit primary hair given by an arbitrary function of the non-Killing angular coordinate. We identify several solution branches defined by specific algebraic relations between the Proca coupling constants, providing a significant generalization of the Myers-Perry family. Notably, the metric retains a Kerr-Schild form identical to the Myers-Perry representation, with an additional contribution constructed from the tensor product of the Proca one-form with itself.

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

0 major / 2 minor

Summary. The manuscript presents a new class of exact analytic rotating black hole solutions in five-dimensional generalized Proca theories. By using a Kerr-Schild ansatz with the Proca field aligned along a null geodesic congruence, the nonlinear field equations are reduced to three master equations. The solutions include a cosmological constant and two independent angular momenta, and exhibit primary hair in the form of an arbitrary function of the non-Killing angular coordinate. The metric retains the Myers-Perry Kerr-Schild form with an additional contribution from the tensor product of the Proca one-form with itself. Several solution branches are identified based on algebraic relations among the Proca coupling constants.

Significance. If the derivations hold, this work is significant as it provides exact solutions with primary hair in a vector-tensor theory, generalizing the Myers-Perry family in higher dimensions. Exact analytic solutions in modified gravity are valuable for testing the theory and understanding black hole properties. The reduction to master equations and the consistency for specific coupling branches demonstrate a careful analytic treatment. The presence of primary hair allows for more general black hole configurations than in standard GR.

minor comments (2)
  1. The abstract states that the Proca field remains light-like on-shell and restricts couplings to discrete constants; a brief explicit listing of the allowed algebraic branches on the couplings would improve accessibility without requiring the reader to reach the main text.
  2. The Kerr-Schild form is stated to be identical to the Myers-Perry representation plus an A⊗A term; a short comparison table or equation highlighting the precise difference from the vacuum Myers-Perry metric would aid clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the accurate summary of the Kerr-Schild ansatz, reduction to master equations, and the presence of primary hair. The recommendation for minor revision is noted.

read point-by-point responses

  1. Referee: No specific major comments listed; recommendation is minor_revision.

    Authors: We observe that the major comments section contains no specific points or criticisms. The referee summary correctly describes the construction via the null Proca field, the retention of the Myers-Perry Kerr-Schild form, and the identification of solution branches from algebraic relations among couplings. We are happy to implement any minor editorial or clarification changes in the revised version. revision: partial

Circularity Check

0 steps flagged

Derivation self-contained via standard ansatz reduction

full rationale

The paper selects a Kerr-Schild ansatz with the Proca one-form aligned to a null geodesic congruence, which reduces the nonlinear generalized Proca equations to three master equations whose solutions are then reported for specific algebraic branches on the couplings. This is a conventional exact-solution technique in higher-dimensional gravity; the arbitrary function of the non-Killing coordinate enters directly as primary hair without being fitted to data or redefined from the output. No self-citation chains, uniqueness theorems imported from prior work, or ansatz smuggling are described, and the metric retains the Myers-Perry Kerr-Schild form plus an A⊗A term as a direct consequence of the ansatz. The construction is therefore independent of its own results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the geometric ansatz that forces the Proca field to be light-like and on the restriction of the theory's couplings to discrete constants that permit analytic solutions.

free parameters (1)
  • Proca coupling constants
    Restricted to discrete values by the requirement that the vector field remain light-like on-shell.
axioms (1)
  • domain assumption Kerr-Schild ansatz with Proca field aligned to a null geodesic congruence
    This configuration reduces the nonlinear equations to three master equations and keeps the vector field light-like.

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Reference graph

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