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arxiv: 2605.20239 · v1 · pith:B6Y4HWGTnew · submitted 2026-05-17 · 🌀 gr-qc

Rotating Black Holes Surrounded by Massive Vector Fields in Kaluza Klein Gravity

Farokhnaz Hosseinifar , Shahin Mamedov , Kuantay Boshkayev , Soroush Zare , Filip Studnicka , Hassan Hassanabadi This is my paper

Pith reviewed 2026-05-21 07:58 UTC · model grok-4.3

classification 🌀 gr-qc
keywords rotating black holesKaluza-Klein gravitymassive vector fieldsthermodynamic topologyblack hole shadowphase transitionsextra dimensionsaccretion disk
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The pith

Extra dimensions in Kaluza-Klein gravity shift black hole phase transitions and shadow sizes while leaving the thermodynamic topological class unchanged.

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

The paper builds a rotating black hole solution in five-dimensional Kaluza-Klein gravity that includes both a massive vector field and a scalar field. It first locates the horizons and carves out the region of parameter space where black holes exist rather than naked singularities. Thermodynamic quantities such as Hawking temperature and heat capacity are then computed to locate phase transitions, while an off-shell generalized free energy is used to assign the system to a universal topological class. The geometry of the ergosphere, the size of the black hole shadow, and the structure of a thin accretion disk are examined next. The central result is that extra-dimensional parameters move the critical points and change the shadow radius, yet the topological classification of the thermodynamic potentials stays fixed.

Core claim

For rotating Kaluza-Klein black holes surrounded by massive vector fields and scalar fields, extra-dimensional contributions move the locations of thermodynamic phase transitions and alter the apparent size of the shadow, but the topological class assigned by the off-shell generalized free energy remains the same universal group.

What carries the argument

The rotating Kaluza-Klein metric ansatz that incorporates a massive vector field and a scalar field, which is solved for horizons, thermodynamic potentials, ergosphere, shadow, and accretion disk.

If this is right

  • Valid black hole solutions are separated from naked singularities by mapping the allowed ranges of the rotation, mass, and extra-dimensional parameters.
  • The Hawking temperature exhibits a conventional critical point whose location shifts with the extra-dimensional scale.
  • The black hole shadow radius changes measurably with the extra-dimensional parameters while the topological class of the free energy stays fixed.
  • The ergosphere boundary and the thin accretion disk properties depend on the black hole spin in the presence of the vector field.
  • The off-shell generalized free energy places the system in one specific universal topological group that is insensitive to the extra dimension.

Where Pith is reading between the lines

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

  • The reported stability of the topological class may serve as a diagnostic that distinguishes Kaluza-Klein models from other higher-dimensional theories even when direct shadow measurements are imprecise.
  • Similar topological invariance could be tested in non-rotating or charged versions of the same theory to see whether the result is tied to rotation.
  • If future shadow observations constrain the extra-dimensional parameter, the unchanged topological class would still allow the model to be placed in the same universal group as four-dimensional Kerr black holes.

Load-bearing premise

The chosen metric form for the rotating black hole with the added massive vector field and scalar field satisfies the coupled gravitational and matter field equations of Kaluza-Klein theory.

What would settle it

A numerical check showing that the proposed metric fails to satisfy the Kaluza-Klein field equations for any choice of parameters, or an observation that the measured shadow radius and critical temperature fail to move together while the topological class changes.

Figures

Figures reproduced from arXiv: 2605.20239 by Farokhnaz Hosseinifar, Filip Studnicka, Hassan Hassanabadi, Kuantay Boshkayev, Shahin Mamedov, Soroush Zare.

Figure 1
Figure 1. Figure 1: FIG. 1. The behavior of the function [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Roots of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variations of event horizon radius in terms of spin and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The variation of the ergosphere in the X–Y plane for different selections of spin parameter (upper panels), [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The behavior of Hawking temperature as function of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Variation of the black hole remnant in terms of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Heat capacity of the rotating KK black hole as function of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Variations of generalized free energy in terms of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: displays the vector space for the potential Φ considering γ = 0.5, λ = 10, and a = 0.6. In this vector space, there exists a zero point at rc1 = 1.2945. By examining the rotation of ϕr − ϕθ curve, we find that the topological number of rc1 is −1. This topological number indicates the presence of a conventional critical point within the vector space of potential Φ [72]. To achieve a universal topological cl… view at source ↗
Figure 10
Figure 10. Figure 10: illustrates the behavior of r+ as a function of τ with the choice of γ = 0.5, λ = 10, and a = 0.6. This curve changes 20 40 60 80 100 0 2 4 6 8 τ r+ FIG. 10. r+ − τ curve considering γ = 0.5, λ = 10, and a = 0.6. The curve has two branches for τ > 21.2. direction at τc = 21.2 and there are two possible solutions for τ > τc, one corresponding to a smaller black hole and one to a larger black hole [PITH_FU… view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The black hole shadow silhouettes under variations of the black hole parameters for an observer on equatorial plane. [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: displays Rs as a function of spin and the parameter γ. Our results are consistent with previous study [52], which FIG. 13. The variation of Rsh/M as a function of a/M and γ. Dashed lines represent the lower bounds of 1σ and 2σ regions for Sgr A∗ based on the data of EHT. investigated the shadow of a KK black hole and noted that the effect of the λ parameter on the shadow’s size is minimal compared to that… view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The shadow of a rotating KK black hole and its associated observables, considering [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The distortion parameter as a function of the spin parameter. The plot shows results for several values of the KK parameters, while [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: As shown, for a = 0, the oblateness is D = 1. Also, in the regime of small a/M, varying KK parameters does not significantly affect Dsh but, by increasing a/M, enhancing KK parameters causes the black hole becomes more oblate. To analyze the overall size of the black hole shadow, its area can also be calculated by integrating over the boundary of the shadow as Ash = 2 Z r + ph r − ph drph Y (rph)∂rphX(rph… view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The variations of oblateness in terms of spin for some cases of KK parameters. [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The shadow area as a function of [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Variation of the ISCO radius and the radiation efficiency for [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. The behavior of the angular velocity, specific energy, and angular momentum as a function of the radial coordinate for [PITH_FULL_IMAGE:figures/full_fig_p015_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The energy flux and the radiation temperature of the thin accretion disk as a function of radius for [PITH_FULL_IMAGE:figures/full_fig_p016_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Density profile of the radiation temperature considering [PITH_FULL_IMAGE:figures/full_fig_p016_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The differential luminosity in terms of [PITH_FULL_IMAGE:figures/full_fig_p017_22.png] view at source ↗
read the original abstract

In this paper, we introduce a rotating Kaluza-Klein black hole characterized by a massive vector field and a scalar field. We begin by identifying the horizons and mapping the allowed parameter space to differentiate black hole solutions from naked singularities. The thermodynamic analysis shows a phase transition by examining Hawking temperature and heat capacity. We also conduct a topological study of the thermodynamic potentials. The Hawking temperature indicates a conventional critical point, while the off-shell generalized free energy classifies the system into a specific universal group. We further investigate the geometry of the ergosphere and how it relates to the black holes spin. Additionally, we look at astrophysical signs, such as the black hole shadow and the features of the thin accretion disk. Our results indicate that while the extra-dimensional changes significantly shift phase transition points and modify the shadow size, the essential topological class remains stable. This study provides a solid framework for distinguishing higher-dimensional gravity models through both thermodynamic and observational signs.

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 paper introduces a rotating Kaluza-Klein black hole with a massive vector field and scalar field. It identifies horizons and maps the parameter space separating black hole solutions from naked singularities, performs thermodynamic analysis via Hawking temperature and heat capacity to identify phase transitions, classifies the topology of the off-shell generalized free energy, examines ergosphere geometry in relation to spin, and studies astrophysical observables including the black hole shadow and thin accretion disk. The central claim is that extra-dimensional effects shift phase transition points and shadow size while the essential topological class remains stable.

Significance. If the metric ansatz is confirmed as an exact solution, the work supplies a concrete example of how Kaluza-Klein compactification and massive vector fields alter quantitative thermodynamic and lensing features without changing the topological classification. This stability could serve as a robust discriminator among higher-dimensional gravity models when combined with shadow and accretion-disk observables.

major comments (2)
  1. [Metric ansatz section] Section introducing the metric ansatz: The rotating black hole metric together with the massive vector and scalar profiles is presented as a solution, yet the manuscript does not substitute the ansatz into the full set of coupled Einstein-Proca-scalar equations derived from the Kaluza-Klein action and demonstrate that all components are satisfied identically. This verification is load-bearing for the thermodynamic potentials, topological classification, ergosphere analysis, and shadow calculations that follow.
  2. [Topological study section] Topological classification section: The statement that the off-shell generalized free energy places the system in a specific universal group is given, but the explicit computation of the topological charge (winding number or equivalent) from the thermodynamic potential is not shown in sufficient detail to confirm that the class remains unchanged under variation of the extra-dimensional parameters.
minor comments (2)
  1. [Abstract] The abstract refers to a 'conventional critical point' without quoting the critical temperature or the order of the transition; adding these values would improve context for the phase-transition claim.
  2. [Figure captions] Figure captions for the shadow and parameter-space plots should explicitly list the fixed values of the vector-field mass and coupling used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make to improve the clarity and rigor of the paper.

read point-by-point responses

  1. Referee: [Metric ansatz section] Section introducing the metric ansatz: The rotating black hole metric together with the massive vector and scalar profiles is presented as a solution, yet the manuscript does not substitute the ansatz into the full set of coupled Einstein-Proca-scalar equations derived from the Kaluza-Klein action and demonstrate that all components are satisfied identically. This verification is load-bearing for the thermodynamic potentials, topological classification, ergosphere analysis, and shadow calculations that follow.

    Authors: We appreciate the referee's emphasis on this point. The manuscript indeed presents the metric as a solution without showing the explicit substitution into the field equations. This omission could raise questions about the validity of the ansatz. To address this, we will revise the manuscript by adding the detailed verification in the metric ansatz section, substituting the proposed forms for the metric, vector field, and scalar field into the Einstein-Proca-scalar equations and confirming that they are satisfied. This addition will provide the necessary foundation for all subsequent analyses. revision: yes

  2. Referee: [Topological study section] Topological classification section: The statement that the off-shell generalized free energy places the system in a specific universal group is given, but the explicit computation of the topological charge (winding number or equivalent) from the thermodynamic potential is not shown in sufficient detail to confirm that the class remains unchanged under variation of the extra-dimensional parameters.

    Authors: We agree that providing the explicit computation would enhance the transparency of our topological analysis. In the revised manuscript, we will include a more detailed derivation of the topological charge, specifically computing the winding number from the off-shell generalized free energy. We will show the steps involved and demonstrate that the topological class remains stable despite variations in the extra-dimensional parameters, as claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a metric ansatz for the rotating Kaluza-Klein black hole with massive vector and scalar fields, then derives horizons, thermodynamic quantities (Hawking temperature, heat capacity, phase transitions), topological classification via off-shell generalized free energy, ergosphere geometry, and shadow/accretion disk features. These steps apply standard black-hole thermodynamics and topological methods to the assumed solution without any result reducing to its inputs by construction, self-definition, or fitted-parameter renaming. The central claim of stable topological class under extra-dimensional shifts follows from the free-energy analysis and is independent of the metric construction itself. No load-bearing self-citations, uniqueness theorems from prior author work, or smuggled ansatze are present in the provided text. The derivation remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the metric solution in the specified gravity theory and the application of standard thermodynamic and topological methods to it.

free parameters (1)
  • massive vector field mass and coupling
    Parameters defining the massive vector field are introduced to construct the solution and likely chosen or fitted to satisfy equations.
axioms (1)
  • domain assumption Existence of a stationary axisymmetric metric ansatz in Kaluza-Klein theory coupled to massive vector and scalar fields that solves the field equations.
    Invoked at the start to introduce the black hole solution.

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