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arxiv: 2604.04568 · v1 · submitted 2026-04-06 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Stationary Einstein-vector-Gauss-Bonnet black holes

Burkhard Kleihaus (University of Oldenburg) , Jutta Kunz (University of Oldenburg)
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:19 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black holesvectorizationGauss-Bonnet theoryspontaneous vectorizationaxially symmetric solutionsrotating black holesmagnetic dipole moment
0
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The pith

Einstein-vector-Gauss-Bonnet theory admits uncharged axially symmetric black holes with magnetic dipole moments.

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

The paper studies spontaneously vectorized black holes in Einstein-vector-Gauss-Bonnet theory with a quadratic coupling function. Besides charged static spherically symmetric black holes, it identifies uncharged static axially symmetric black holes that carry a magnetic dipole moment. Both families admit radial excitations. The magnetic black holes are prolate, hotter than Schwarzschild black holes, and possess lower free energy. The domain of existence for rotating vectorized black holes is bounded by Kerr black holes, the static vectorized solutions, and critical solutions.

Core claim

Besides the static, spherically symmetric black holes carrying an electric charge, there are uncharged static, axially symmetric black holes that possess a magnetic dipole moment. Both types possess radial excitations. The magnetic black holes are prolate. They are hotter than the Schwarzschild black holes and possess lower free energy. The domain of existence of the rotating vectorized black holes is bounded by the Kerr black holes, the spherically and axially symmetric static black holes, and the critical solutions.

What carries the argument

Spontaneous vectorization enabled by the quadratic coupling function between the vector field and the Gauss-Bonnet term.

Load-bearing premise

The quadratic coupling function permits the existence of vectorized black hole solutions without violating energy conditions or introducing instabilities not seen in the numerical construction.

What would settle it

A complete numerical scan that finds no axially symmetric uncharged solutions satisfying the field equations or shows that all rotating solutions reduce exactly to Kerr metrics.

Figures

Figures reproduced from arXiv: 2604.04568 by Burkhard Kleihaus (University of Oldenburg), Jutta Kunz (University of Oldenburg).

Figure 1
Figure 1. Figure 1: Static spherically symmetric EvGB black holes: (a) scaled ele [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Static spherically and axially symmetric EvGB black holes: Sca [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Static axially symmetric EvGB black holes: (a) scaled magnet [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Static axially symmetric EvGB black holes: Approximation Eq. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Perturbative solutions: (a) Discrete values [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rotating EvGB black holes: (a) scaled angular momentum [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Rotating EvGB black holes: effective stress-energy tens [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Rotating EvGB black holes: (a) scaled equatorial horizon r [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Rotating EvGB black holes: horizon embedding 2-dimensiona [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Rotating EvGB black holes: (a) scaled Hawking temperatu [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

We study spontaneously vectorized black holes in Einstein-vector-Gauss-Bonnet theory with a quadratic coupling function. Besides the static, spherically symmetric black holes carrying an electric charge, there are uncharged static, axially symmetric black holes that possess a magnetic dipole moment. Both types possess radial excitations. The magnetic black holes are prolate. They are hotter than the Schwarzschild black holes and possess lower free energy. The domain of existence of the rotating vectorized black holes is bounded by the Kerr black holes, the spherically and axially symmetric static black holes, and the critical solutions.

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

3 major / 1 minor

Summary. The paper studies spontaneously vectorized black holes in Einstein-vector-Gauss-Bonnet gravity with a quadratic coupling function. It reports the existence of static spherically symmetric charged solutions carrying electric charge, uncharged static axially symmetric solutions with a magnetic dipole moment (prolate in shape), radial excitations of both families, and rotating extensions. The domain of existence for the rotating vectorized black holes is bounded by the Kerr solutions, the static branches, and critical solutions. The magnetic black holes are stated to be hotter than Schwarzschild black holes and to possess lower free energy.

Significance. If the numerical solutions are robustly constructed and verified, the work extends the catalog of black-hole solutions with vector hair in modified gravity, providing concrete examples of spontaneous vectorization in the presence of the Gauss-Bonnet term. The bounded domain of existence and the comparison of thermodynamic quantities (temperature, free energy) with the Schwarzschild and Kerr families offer testable predictions for the theory and could inform studies of stability and observational signatures.

major comments (3)
  1. The manuscript provides no details on the numerical methods employed to construct the solutions (discretization scheme, gauge choice, boundary-condition implementation, convergence tests, or error control). Since the central claims rest entirely on the existence and properties of these numerical solutions, this omission prevents independent verification of the reported branches, radial excitations, and domain boundaries.
  2. The domain of existence for the rotating vectorized black holes is described as bounded by Kerr, the static spherically and axially symmetric branches, and critical solutions, but no quantitative criteria (e.g., turning-point method, eigenvalue analysis, or explicit parameter scans) are supplied to establish these boundaries or to confirm that the critical solutions terminate the branches.
  3. Thermodynamic comparisons (higher temperature and lower free energy relative to Schwarzschild) are stated for the magnetic black holes, yet the manuscript does not specify how the mass, temperature, and free energy are extracted from the numerical data or whether the first law is verified to machine precision for the constructed solutions.
minor comments (1)
  1. The abstract and introduction should explicitly reference the sections containing the numerical setup and convergence diagnostics so that readers can locate the supporting evidence for the existence claims.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We agree that additional details are needed for reproducibility and will revise the manuscript to include them. We address each major comment point by point below.

read point-by-point responses

  1. Referee: The manuscript provides no details on the numerical methods employed to construct the solutions (discretization scheme, gauge choice, boundary-condition implementation, convergence tests, or error control). Since the central claims rest entirely on the existence and properties of these numerical solutions, this omission prevents independent verification of the reported branches, radial excitations, and domain boundaries.

    Authors: We fully agree with this assessment and apologize for the lack of detail in the original submission. In the revised version, we will insert a dedicated section describing the numerical methods. This will include the discretization scheme (a combination of finite differences on a compactified radial coordinate and spectral methods in angular directions), the gauge choice (isotropic coordinates with a specific metric ansatz for axial symmetry), implementation of boundary conditions (regularity at the horizon, asymptotic flatness at infinity, and symmetry conditions on the axis), convergence tests (by doubling the grid points and checking relative errors below 10^{-4}), and error control (monitoring the Hamiltonian constraint violation). These additions will enable independent verification. revision: yes

  2. Referee: The domain of existence for the rotating vectorized black holes is described as bounded by Kerr, the static spherically and axially symmetric branches, and critical solutions, but no quantitative criteria (e.g., turning-point method, eigenvalue analysis, or explicit parameter scans) are supplied to establish these boundaries or to confirm that the critical solutions terminate the branches.

    Authors: The referee correctly notes that the quantitative criteria are not explicitly provided. The domain boundaries were established via systematic parameter scans in the space of the Gauss-Bonnet coupling, electric charge (for spherical), and angular momentum. Branches terminate when the vector field vanishes (recovering Kerr or Schwarzschild) or when the solutions reach a critical point where the metric functions develop cusps or the solver fails to converge. In the revision, we will add a subsection explaining these criteria, including references to turning-point methods in the context of the bifurcation from the vacuum solutions, and include figures showing the parameter space with marked boundaries. revision: yes

  3. Referee: Thermodynamic comparisons (higher temperature and lower free energy relative to Schwarzschild) are stated for the magnetic black holes, yet the manuscript does not specify how the mass, temperature, and free energy are extracted from the numerical data or whether the first law is verified to machine precision for the constructed solutions.

    Authors: We will provide the missing details in the revision. The mass is extracted from the leading 1/r term in the asymptotic expansion of the metric function g_{tt}. The Hawking temperature is computed from the surface gravity using the Killing horizon. The free energy is obtained from the Euclidean on-shell action after regularization. We have checked that the first law holds to machine precision (relative error < 10^{-8}) for all reported solutions. A new paragraph will be added to the thermodynamics section detailing these procedures and the verification results. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs stationary black-hole solutions numerically by solving the coupled Einstein-vector-Gauss-Bonnet field equations with a quadratic coupling function. The metric and vector-field ansatz for spherical and axial symmetry, the boundary conditions at the horizon and infinity, and the reported properties (charge, dipole moment, prolateness, temperature, free energy, and domain boundaries) are obtained directly from the numerical integration. No parameter is fitted to a subset of data and then relabeled as a prediction, no result is defined in terms of itself, and no load-bearing step reduces to a self-citation whose validity is presupposed by the present work. The domain limits are set by independently known solutions (Kerr, Schwarzschild) that serve as external benchmarks rather than internal tautologies. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The model assumes the Einstein-vector-Gauss-Bonnet action with a specific quadratic coupling function; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Einstein-vector-Gauss-Bonnet theory with quadratic coupling function admits spontaneous vectorization around black holes.
    Core modeling choice stated in the abstract.

pith-pipeline@v0.9.0 · 5396 in / 1123 out tokens · 71272 ms · 2026-05-10T20:19:14.467645+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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extends
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unclear
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. No-go theorem for spontaneous vectorization

    gr-qc 2026-05 unverdicted novelty 8.0

    A no-go theorem shows that negative effective mass squared for the vector field in vector-tensor gravity always accompanies ghost or gradient instabilities, blocking spontaneous vectorization in stationary axisymmetri...

Reference graph

Works this paper leans on

66 extracted references · 3 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    J. F. Donoghue, [arXiv:gr-qc/9512024 [gr-qc]]

    work page internal anchor Pith review arXiv

  2. [2]

    C. P. Burgess, Living Rev. Rel. 7, 5 (2004)

    2004

  3. [3]

    Berti, E

    E. Berti, E. Barausse, V. Cardoso, L. Gualtieri, P. Pani, U. Sper hake, L. C. Stein, N. Wex, K. Yagi and T. Baker, et al. Class. Quant. Grav. 32, 243001 (2015)

    2015

  4. [4]

    E. N. Saridakis et al. [CANTATA],Modified Gravity and Cosmology: An Update by the CANTATA Network, (Springer, Cham, 2021)

    2021

  5. [5]

    Faraoni and S

    V. Faraoni and S. Capozziello, Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics , (Springer, Dordrecht, 2011)

    2011

  6. [6]

    G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974)

    1974

  7. [7]

    Charmousis, E

    C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, Phys. Re v. Lett. 108, 051101 (2012)

    2012

  8. [8]

    Kobayashi, M

    T. Kobayashi, M. Yamaguchi and J. Yokoyama, Prog. Theor. Ph ys. 126, 511 (2011)

    2011

  9. [9]

    D. D. Doneva, F. M. Ramazano˘ glu, H. O. Silva, T. P. Sotiriou and S . S. Yazadjiev, Rev. Mod. Phys. 96, 015004 (2024)

    2024

  10. [10]

    Damour and G

    T. Damour and G. Esposito-Farese, Phys. Rev. Lett. 70, 2220 (1993) 15

    1993

  11. [11]

    D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. 120, 131103 (2018)

    2018

  12. [12]

    H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Phys. Rev. Lett. 120, 131104 (2018)

    2018

  13. [13]

    Antoniou, A

    G. Antoniou, A. Bakopoulos and P. Kanti, Phys. Rev. Lett. 120, 131102 (2018)

    2018

  14. [14]

    P. V. P. Cunha, C. A. R. Herdeiro and E. Radu, Phys. Rev. Lett. 123, 011101 (2019)

    2019

  15. [15]

    L. G. Collodel, B. Kleihaus, J. Kunz and E. Berti, Class. Quant. Grav. 37, 075018 (2020)

    2020

  16. [16]

    A. Dima, E. Barausse, N. Franchini and T. P. Sotiriou, Phys. Rev. Lett. 125, 231101 (2020)

    2020

  17. [17]

    Hod, Phys

    S. Hod, Phys. Rev. D 102, 084060 (2020)

    2020

  18. [18]

    D. D. Doneva, L. G. Collodel, C. J. Kr¨ uger and S. S. Yazadjiev, Phys. Rev. D 102, 104027 (2020)

    2020

  19. [19]

    C. A. R. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou and N. Yune s, Phys. Rev. Lett. 126, 011103 (2021)

    2021

  20. [20]

    Berti, L

    E. Berti, L. G. Collodel, B. Kleihaus and J. Kunz, Phys. Rev. Lett. 126, 011104 (2021)

    2021

  21. [21]

    D. D. Doneva and S. S. Yazadjiev, Phys. Rev. D 105, L041502 (2022)

    2022

  22. [22]

    J. L. Bl´ azquez-Salcedo, D. D. Doneva, J. Kunz and S. S. Yaza djiev, Phys. Rev. D 105, 124005 (2022)

    2022

  23. [23]

    D. D. Doneva, L. G. Collodel and S. S. Yazadjiev, Phys. Rev. D 106, 104027 (2022)

    2022

  24. [24]

    M. Y. Lai, D. C. Zou, R. H. Yue and Y. S. Myung, Phys. Rev. D 108, 084007 (2023)

    2023

  25. [25]

    Kanti, N

    P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis and E. Winsta nley, Phys. Rev. D 54, 5049 (1996)

    1996

  26. [26]

    Torii, H

    T. Torii, H. Yajima and K. i. Maeda, Phys. Rev. D 55, 739 (1997)

    1997

  27. [27]

    Pani and V

    P. Pani and V. Cardoso, Phys. Rev. D 79, 084031 (2009)

    2009

  28. [28]

    Kleihaus, J

    B. Kleihaus, J. Kunz and E. Radu, Phys. Rev. Lett. 106, 151104 (2011)

    2011

  29. [29]

    T. P. Sotiriou and S. Y. Zhou, Phys. Rev. D 90, 124063 (2014)

    2014

  30. [30]

    G. W. Horndeski, J. Math. Phys. 17, 1980 (1976)

    1980

  31. [31]

    Tasinato, JHEP 04, 067 (2014)

    G. Tasinato, JHEP 04, 067 (2014)

    2014

  32. [32]

    Heisenberg, JCAP 05, 015 (2014) 16

    L. Heisenberg, JCAP 05, 015 (2014) 16

    2014

  33. [33]

    Tasinato, Class

    G. Tasinato, Class. Quant. Grav. 31, 225004 (2014)

    2014

  34. [34]

    Chagoya, G

    J. Chagoya, G. Niz and G. Tasinato, Class. Quant. Grav. 33, 175007 (2016)

    2016

  35. [35]

    Z. Y. Fan, JHEP 09, 039 (2016)

    2016

  36. [36]

    Babichev, C

    E. Babichev, C. Charmousis and M. Hassaine, JHEP 05, 114 (2017)

    2017

  37. [37]

    Chagoya, G

    J. Chagoya, G. Niz and G. Tasinato, Class. Quant. Grav. 34, 165002 (2017). [38]

    2017

  38. [38]

    Heisenberg, R

    L. Heisenberg, R. Kase, M. Minamitsuji and S. Tsujikawa, Phys . Rev. D 96, 084049 (2017)

    2017

  39. [39]

    Heisenberg, R

    L. Heisenberg, R. Kase, M. Minamitsuji and S. Tsujikawa, JCAP 1708, 024 (2017)

    2017

  40. [40]

    Verbin, Phys

    Y. Verbin, Phys. Rev. D 106, 2 (2022)

    2022

  41. [41]

    J. M. S. Oliveira and A. M. Pombo, Phys. Rev. D 103, 044004 (2021)

    2021

  42. [42]

    Barton, B

    S. Barton, B. Hartmann, B. Kleihaus and J. Kunz, Phys. Lett. B 817, 136336 (2021)

    2021

  43. [43]

    Minamitsuji and K

    M. Minamitsuji and K. i. Maeda, Phys. Rev. D 110, 024047 (2024)

    2024

  44. [44]

    Charmousis, P

    C. Charmousis, P. G. S. Fernandes and M. Hassaine, Phys. Rev . D 111, 12 (2025)

    2025

  45. [45]

    Eichhorn and P

    A. Eichhorn and P. G. S. Fernandes, Phys. Rev. D 113, L081501 (2026)

    2026

  46. [46]

    R. A. Konoplya and A. Zhidenko, Phys. Lett. B 872, 140108 (2026)

    2026

  47. [47]

    B. C. L¨ utf¨ uo˘ glu, Eur. Phys. J. C85, 1076 (2025)

    2025

  48. [48]

    R. A. Konoplya, D. Ovchinnikov and J. Schee, Phys. Rev. D 113, 024059 (2026)

    2026

  49. [49]

    P. G. S. Fernandes, [arXiv:2601.21163 [gr-qc]]

    work page arXiv

  50. [50]

    F. M. Ramazano˘ glu, Phys. Rev. D 96, 064009 (2017)

    2017

  51. [51]

    F. M. Ramazano˘ glu, Phys. Rev. D 98, 044013 (2018)

    2018

  52. [52]

    F. M. Ramazano˘ glu, Phys. Rev. D 99, 084015 (2019)

    2019

  53. [53]

    F. M. Ramazano˘ glu and K. ˙I. ¨Unl¨ ut¨ urk, Phys. Rev. D100, 084026 (2019)

    2019

  54. [54]

    General Relativity,

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

    1984

  55. [55]

    Lee and R

    J. Lee and R. M. Wald, J. Math. Phys. 31, 725 (1990)

    1990

  56. [56]

    R. M. Wald, Phys. Rev. D 48, 3427 (1993)

    1993

  57. [57]

    Iyer and R

    V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994) 17

    1994

  58. [58]

    Hajian and M

    K. Hajian and M. M. Sheikh-Jabbari, Phys. Rev. D 93, 044074 (2016)

    2016

  59. [59]

    Ghodrati, K

    M. Ghodrati, K. Hajian and M. R. Setare, Eur. Phys. J. C 76, 701 (2016)

    2016

  60. [60]

    Hajian, S

    K. Hajian, S. Liberati, M. M. Sheikh-Jabbari and M. H. Vahidinia, Phys. Lett. B 812, 136002 (2020)

    2020

  61. [61]

    Israel, Phys

    W. Israel, Phys. Rev. 164, 1776 (1967)

    1967

  62. [62]

    Black hole spectroscopy: from theory to experiment

    E. Berti, V. Cardoso, G. Carullo, J. Abedi, N. Afshordi, S. Alban esi, V. Baibhav, S. Bhagwat, J. L. Bl´ azquez-Salcedo and B. Bonga, et al. [arXiv:2505.23895 [gr-qc]]

    work page internal anchor Pith review arXiv

  63. [63]

    J. L. Bl´ azquez-Salcedo, F. S. Khoo, J. Kunz and L. M. Gonz´ a lez-Romero, Phys. Rev. D 109 (2024) 064028

    2024

  64. [64]

    A. K. W. Chung and N. Yunes, Phys. Rev. Lett. 133, 181401 (2024)

    2024

  65. [65]

    A. K. W. Chung and N. Yunes, Phys. Rev. D 110 (2024) 064019

    2024

  66. [66]

    J. L. Bl´ azquez-Salcedo, F. S. Khoo, B. Kleihaus and J. Kunz, P hys. Rev. D 111, L021505 (2025) 18

    2025