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arxiv: 2512.19377 · v2 · pith:633HMERInew · submitted 2025-12-22 · 🌀 gr-qc

Critical phenomenon inside asymptotically flat black holes with spontaneous scalarization

Li Li , Ze Sun , Fu-Guo Yang This is my paper

Pith reviewed 2026-05-25 07:36 UTC · model grok-4.3

classification 🌀 gr-qc
keywords spontaneous scalarizationblack hole interiorKasner singularityCauchy horizonEinstein-Maxwell-Scalar theoryReissner-Nordstrom solutioncritical phenomena
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The pith

Scalarized black holes evolve to a spacelike Kasner singularity instead of a smooth inner Cauchy horizon.

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

The paper examines the interior of spontaneously scalarized black holes in Einstein-Maxwell-Scalar theory with zero cosmological constant. It finds that scalar hair causes the would-be inner Cauchy horizon to collapse for a wide range of couplings, producing a Kasner singularity instead. This matters because it indicates that the interior structure becomes simpler and more uniform than the classical Reissner-Nordstrom expectation. Near the bifurcation from the Reissner-Nordstrom solution the paper also reports a scaling relation between the Kasner parameter and the charge-to-mass ratio.

Core claim

For a wide range of scalar-electromagnetic couplings, scalarized black holes possess no smooth inner Cauchy horizon and instead evolve into a spacelike Kasner singularity. The scalar hair triggers a rapid collapse of the Einstein-Rosen bridge at the would-be Cauchy horizon. Near the critical point where scalarized black holes bifurcate from the Reissner-Nordstrom solution, a robust scaling relation holds between the Kasner parameter and the charge-to-mass ratio of the hairy black hole.

What carries the argument

Spontaneous scalarization coupled to the electromagnetic field, which drives collapse of the Einstein-Rosen bridge into a Kasner singularity.

If this is right

  • Scalarized black holes lack the inner Cauchy horizon present in the Reissner-Nordstrom solution.
  • The interior terminates in a spacelike Kasner singularity for many scalar-electromagnetic couplings.
  • A scaling relation connects the Kasner parameter to the charge-to-mass ratio near the bifurcation from Reissner-Nordstrom.
  • Black hole interiors display greater simplicity when scalar hair is present.

Where Pith is reading between the lines

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

  • The lack of a smooth Cauchy horizon may remove classical predictability problems inside these black holes.
  • The same scaling could be checked in numerical studies of related scalar-tensor models.
  • Analogous interior behavior might occur for black holes in asymptotically de Sitter spacetimes.

Load-bearing premise

The numerical evolution scheme for the interior metric and fields remains stable and accurate all the way to the singularity without truncation or gauge artifacts altering the reported Kasner behavior or the absence of a smooth Cauchy horizon.

What would settle it

A simulation that finds a smooth inner Cauchy horizon persisting for a coupling value predicted to produce collapse, or that violates the reported scaling between Kasner parameter and charge-to-mass ratio near the critical point.

Figures

Figures reproduced from arXiv: 2512.19377 by Fu-Guo Yang, Li Li, Ze Sun.

Figure 1
Figure 1. Figure 1: FIG. 1. An illustration of spontaneously scalarized black holes [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The behaviors of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The critical behavior between the Kasner param [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The variation of the photon ring (bright orange an [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The behaviors of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The critical behavior between the radius of the photon [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

We study the interior dynamics of spontaneously scalarized black holes in Einstein-Maxwell-Scalar theory with zero cosmological constant, revealing novel critical phenomena. We demonstrate that, for a wide range of scalar-electromagnetic couplings, scalarized black holes possess no smooth inner Cauchy horizon and instead evolve into a spacelike Kasner singularity. The scalar hair triggers a rapid collapse of the Einstein-Rosen bridge at the would-be Cauchy horizon. Near the critical point where scalarized black holes bifurcate from the Reissner-Nordstrom solution, we establish a robust scaling relation between the Kasner parameter and the charge-to-mass ratio of the hairy black hole, opening a new window into the remarkable simplicity underlying black hole interiors.

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

1 major / 1 minor

Summary. The manuscript studies the interior dynamics of spontaneously scalarized asymptotically flat black holes in Einstein-Maxwell-Scalar theory with vanishing cosmological constant. It reports that, for a range of scalar-electromagnetic couplings, these solutions lack a smooth inner Cauchy horizon and instead terminate at a spacelike Kasner singularity, with scalar hair inducing rapid collapse of the Einstein-Rosen bridge. Near the bifurcation from the Reissner-Nordström family, a scaling relation is claimed between the Kasner parameter and the charge-to-mass ratio of the hairy black hole.

Significance. If the numerical results are robust, the work identifies a new critical phenomenon governing black-hole interiors under spontaneous scalarization. The reported absence of a smooth Cauchy horizon and the emergence of a universal Kasner scaling near criticality would constitute a concrete, falsifiable prediction about singularity structure in Einstein-Maxwell-Scalar theory, extending the known simplicity of black-hole interiors beyond the vacuum case.

major comments (1)
  1. [Numerical evolution section] Numerical evolution section: the central claims (no smooth Cauchy horizon, Kasner singularity, and the reported scaling) rest entirely on the interior integration. The manuscript must supply explicit convergence tests, resolution studies, and gauge-independence checks demonstrating that the Kasner exponents and the collapse of the Einstein-Rosen bridge remain stable under refinement and are free of truncation or coordinate artifacts all the way to the singularity; without these, the load-bearing numerical evidence cannot be assessed.
minor comments (1)
  1. The abstract states a 'wide range of scalar-electromagnetic couplings' but does not quantify the interval; a brief statement of the explored parameter domain would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the numerical evidence. We address the major comment below.

read point-by-point responses

  1. Referee: [Numerical evolution section] Numerical evolution section: the central claims (no smooth Cauchy horizon, Kasner singularity, and the reported scaling) rest entirely on the interior integration. The manuscript must supply explicit convergence tests, resolution studies, and gauge-independence checks demonstrating that the Kasner exponents and the collapse of the Einstein-Rosen bridge remain stable under refinement and are free of truncation or coordinate artifacts all the way to the singularity; without these, the load-bearing numerical evidence cannot be assessed.

    Authors: We agree that the central claims depend on the interior numerical integration and that explicit validation is required. In the revised manuscript we will add a new subsection detailing convergence tests performed at successively higher resolutions (including doubling the radial grid points), demonstrating that the extracted Kasner exponents and the timing of the Einstein-Rosen bridge collapse converge to stable values. We will also present gauge-independence checks by repeating the evolution with varied gauge parameters and confirming that the physical results, including the absence of a smooth Cauchy horizon and the reported scaling, remain unchanged within numerical error. These additions will be placed immediately after the description of the evolution scheme. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical discovery of interior dynamics

full rationale

The paper reports numerical evolution of the interior of scalarized black holes in Einstein-Maxwell-Scalar theory, finding absence of smooth Cauchy horizons and emergence of Kasner singularities, plus a scaling relation near bifurcation. No analytical derivation chain is presented that reduces predictions to fitted inputs or self-citations by construction. The central results are outputs of the simulation scheme itself, with no visible self-definitional loops, renamed empirical patterns, or load-bearing self-citations that would force the reported Kasner behavior or scaling. This is a standard numerical GR result whose validity rests on code stability rather than tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Einstein-Maxwell-Scalar action with vanishing cosmological constant and on the reliability of numerical interior evolution; no free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Einstein-Maxwell-Scalar theory with zero cosmological constant governs the dynamics
    Stated in the abstract as the framework for the black-hole solutions.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    We begin with the static, spherically symmetric black holes with a purely electric field in asymptotically flat spacetime

    We chooseZ(0) = 1 without loss of generality. We begin with the static, spherically symmetric black holes with a purely electric field in asymptotically flat spacetime. The ansatz is therefore given by ds2 = 1 z2 −f(z)e −2χ(z)dt2 + dz2 f(z) +dΩ 2 2 , ψ=ψ(z), A µ =A t(z)dt , (2) withdΩ 2 2 ≡dθ 2 +sin 2 θdφ2 the metric of a unit 2-sphere. We denote the posi...

    work page

  2. [2]

    ER bridge collapse The no-inner-horizon theorem reveals the instability of the inner horizon triggered by the scalar field. In the 7 vicinity of the would-be inner horizon, one anticipates the collapse of the ER bridge, for which, as the metric componentg tt approaches its would-be zero value at the Cauchy horizon, it suddenly suffers a very rapid collaps...

    work page

  3. [3]

    Following previous work [8], our strategy is to obtain self-consistent asymptotic solutions, which will be further established by checking the full numerical solutions

    Kasner singularity We are not able to solve the system analytically due to the strong nonlinear nature of the equations of motion. Following previous work [8], our strategy is to obtain self-consistent asymptotic solutions, which will be further established by checking the full numerical solutions. We begin with the assumption that the terms associ- ated ...

    work page

  4. [4]

    R. B. Mann, Black hole chemistry: The first 15 years, Int. J. Mod. Phys. D34, 2542001 (2025), arXiv:2508.01830 [gr-qc]

    work page arXiv 2025

  5. [5]

    Critical phenomena in gravitational collapse (Physics Reports)

    C. Gundlach, Critical phenomena in gravitational col- lapse, Phys. Rept.376, 339 (2003), arXiv:gr-qc/0210101

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  6. [6]

    R.-G. Cai, L. Li, and R.-Q. Yang, No Inner-Horizon The- orem for Black Holes with Charged Scalar Hairs, JHEP 03, 263, arXiv:2009.05520 [gr-qc]

    work page arXiv 2009

  7. [7]

    Y.-S. An, L. Li, and F.-G. Yang, No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs, Phys. Rev. D104, 024040 (2021), arXiv:2106.01069 [gr-qc]

    work page arXiv 2021

  8. [8]

    S. A. Hartnoll, G. T. Horowitz, J. Kruthoff, and J. E. Santos, Gravitational duals to the grand canon- ical ensemble abhor Cauchy horizons, JHEP10, 102, arXiv:2006.10056 [hep-th]

    work page arXiv 2006

  9. [9]

    S. A. Hartnoll, G. T. Horowitz, J. Kruthoff, and J. E. Santos, Diving into a holographic superconductor, Sci- Post Phys.10, 009 (2021), arXiv:2008.12786 [hep-th]

    work page arXiv 2021

  10. [10]

    R.-G. Cai, C. Ge, L. Li, and R.-Q. Yang, Inside anisotropic black hole with vector hair, JHEP02, 139, arXiv:2112.04206 [gr-qc]

    work page arXiv

  11. [11]

    Cai, M.-N

    R.-G. Cai, M.-N. Duan, L. Li, and F.-G. Yang, Towards classifying the interior dynamics of charged black holes with scalar hair, JHEP02, 169, arXiv:2312.11131 [gr-qc]

    work page arXiv

  12. [12]

    Cai, M.-N

    R.-G. Cai, M.-N. Duan, L. Li, and F.-G. Yang, Clarify- ing Kasner dynamics inside anisotropic black hole with vector hair, JHEP04, 179, arXiv:2408.06122 [gr-qc]

    work page arXiv

  13. [13]

    S. A. Hartnoll and N. Neogi, AdS black holes with a bouncing interior, SciPost Phys.14, 074 (2023), arXiv:2209.12999 [hep-th]

    work page arXiv 2023

  14. [14]

    C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Font, Spontaneous Scalarization of Charged Black Holes, Phys. Rev. Lett.121, 101102 (2018), arXiv:1806.05190 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  15. [15]

    P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, 10 E. Radu, and N. Sanchis-Gual, Spontaneous Scalarisa- tion of Charged Black Holes: Coupling Dependence and Dynamical Features, Class. Quant. Grav.36, 134002 (2019), [Erratum: Class.Quant.Grav. 37, 049501 (2020)], arXiv:1902.05079 [gr-qc]

    work page arXiv 2019

  16. [16]

    Q. Gan, P. Wang, H. Wu, and H. Yang, Photon ring and observational appearance of a hairy black hole, Phys. Rev. D104, 044049 (2021), arXiv:2105.11770 [gr-qc]

    work page arXiv 2021

  17. [17]

    Al-Badawi, M

    A. Al-Badawi, M. Alloqulov, S. Shaymatov, and B. Ahmedov, Shadows and weak gravitational lensing for black holes within Einstein-Maxwell-scalar theory*, Chin. Phys. C48, 095105 (2024), arXiv:2401.04584 [gr- qc]

    work page arXiv 2024

  18. [18]

    Y. Wu, Z. Cai, Z. Ban, H. Feng, and W.-Q. Chen, Probing Einstein–Maxwell-scalar black hole via thin ac- cretion disks and shadows with EHT observations of M87* and Sgr A*, Eur. Phys. J. C85, 1085 (2025), arXiv:2504.10327 [gr-qc]

    work page arXiv 2025

  19. [19]

    The Supplementary Material presents the technical de- tails for solving the equations of motion, analyzing inte- rior dynamics, and computing black hole images, thereby supporting the results made in the main text

    work page

  20. [20]

    Astefanesei, C

    D. Astefanesei, C. Herdeiro, A. Pombo, and E. Radu, Einstein-Maxwell-scalar black holes: classes of solutions, dyons and extremality, JHEP10, 078, arXiv:1905.08304 [hep-th]

    work page arXiv 1905

  21. [21]

    O. J. C. Dias, G. T. Horowitz, and J. E. Santos, Inside an asymptotically flat hairy black hole, JHEP12, 179, arXiv:2110.06225 [hep-th]

    work page arXiv

  22. [22]

    Y. Xu, L. Li, and W.-J. Li, Black hole interiors of ho- mogeneous holographic solids under shear strain, arXiv preprint (2025), arXiv:2511.00877 [hep-th]

    work page arXiv 2025

  23. [23]

    Liu and H.-D

    Y. Liu and H.-D. Lyu, Interior of helical black holes, JHEP09, 071, arXiv:2205.14803 [hep-th]

    work page arXiv

  24. [24]

    Grandi and I

    N. Grandi and I. Salazar Landea, Diving inside a hairy black hole, JHEP05, 152, arXiv:2102.02707 [gr-qc]

    work page arXiv

  25. [25]

    Sword and D

    L. Sword and D. Vegh, Kasner geometries in- side holographic superconductors, JHEP04, 135, arXiv:2112.14177 [hep-th]

    work page arXiv

  26. [26]

    Mirjalali, S

    M. Mirjalali, S. A. Hosseini Mansoori, L. Shahkarami, and M. Rafiee, Probing inside a charged hairy black hole in massive gravity, JHEP09, 222, arXiv:2206.02128 [hep- th]

    work page arXiv

  27. [27]

    First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole

    K. Akiyama et al. (Event Horizon Telescope), First M87 Event Horizon Telescope Results. IV. Imaging the Cen- tral Supermassive Black Hole, Astrophys. J. Lett.875, L4 (2019), arXiv:1906.11241 [astro-ph.GA]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  28. [28]

    Goddi et al

    C. Goddi et al. (Event Horizon Telescope), First M87 Event Horizon Telescope Results and the Role of ALMA, The Messenger177, 25 (2019), arXiv:1910.10193 [astro- ph.HE]

    work page arXiv 2019

  29. [29]

    C. Shao, J. Guo, Y. Tian, and H. Zhang, Emergence of critical phenomena from the black hole interior, arXiv preprint (2025), arXiv:2511.17193 [gr-qc]

    work page arXiv 2025

  30. [30]

    Here we assume that the order of the ˙Zcan have a uni- form upper bound asψ→ ∞

    work page

  31. [31]

    S. E. Gralla, D. E. Holz, and R. M. Wald, Black Hole Shadows, Photon Rings, and Lensing Rings, Phys. Rev. D100, 024018 (2019), arXiv:1906.00873 [astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  32. [32]

    Chael, M

    A. Chael, M. D. Johnson, and A. Lupsasca, Observ- ing the Inner Shadow of a Black Hole: A Direct View of the Event Horizon, Astrophys. J.918, 6 (2021), arXiv:2106.00683 [astro-ph.HE]

    work page arXiv 2021