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Spontaneous scalarization of charged black holes in the Scalar-Vector-Tensor theory

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Pith's one-line read In the f4 scalar-vector-tensor model, the double-dual Riemann coupling destabilizes the constant-scalar Reissner-Nordström solution, and the stable scalarized endpoint can carry charge-to-mass ratio greater than one.

desk verdict A solid extension of scalarization to the f4 L F F coupling with overcharged solutions; the main gap is that the end-state claim rests on scalar-only stability. read the letter →

arxiv 1908.09394 v2 pith:BTCXGGQC submitted 2019-08-25 gr-qc hep-th

Taishi Ikeda , Tomohiro Nakamura , Masato Minamitsuji This is my paper · ORCID
classification gr-qchep-th
keywords spontaneousscalarizationchargedblackholesscalar-vector-tensortheoryReissner-Nordströmdouble-dualRiemanntensortachyonicinstabilityoverchargedholescalarhair
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that a specific coupling in scalar-vector-tensor gravity, the product of the scalar function $H(\Phi)$ with the double-dual Riemann tensor contracted with the electromagnetic field strength, makes the constant-scalar Reissner-Nordström black hole unstable. For sufficiently large charge or coupling, the scalar field develops a tachyonic instability, so the black hole is expected to grow a scalar halo. The authors construct the scalarized black holes that would be the end state, and find solutions whose charge-to-mass ratio $Q/M$ can exceed 1, i.e. overcharged black holes. In the exponential model, where nonlinearity changes the sign of $H''(\Phi)$ at large field values, the nodeless scalarized solution has a positive effective potential for scalar perturbations and can be the endpoint of scalarization. The mechanism is special because the triggering invariant vanishes in flat spacetime, so curvature is essential.

What carries the argument

The central object is the invariant $L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}$, where $L^{\mu\nu\alpha\beta}=\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma\delta}R_{\rho\sigma\gamma\delta}$ is the double-dual Riemann tensor and $F_{\mu\nu}$ is the electromagnetic field strength. On the Reissner-Nordström background this combination is positive outside the horizon and proportional to $Q^2(1-f)/r^6$, giving the scalar field an effective negative mass squared near the horizon when $H''(0)>0$. In the nonlinear scalarized regime, the same coupling enters the equations of motion for the metric, vector, and scalar fields; the exponential form $H(\Phi)=\frac{\eta}{2}(1-e^{-\Phi^2})$ is crucial because $H''(\Phi)$ becomes negative for $|\Phi|>1/\sqrt{2}$, which flips the effective potential positive and stabilizes the nodeless solution.

What would settle it

Perform a linear perturbation analysis including metric and vector modes around the nodeless exponential scalarized solution constructed from Eqs. (28)-(30); a mode with $\omega^2<0$ would refute the endpoint claim. Alternatively, a full nonlinear evolution of a sufficiently charged Reissner-Nordström black hole in this theory that fails to settle on the scalarized solution would settle the question.

Watch

Extended reading notes

Core claim

Within the f4 model defined by $S=\int d^4x\sqrt{-g}\left(\frac{1}{16\pi G}R-\frac{1}{2}\nabla_\mu\Phi\nabla^\mu\Phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+H(\Phi)L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\right)$, with $H(0)=H'(0)=0$, the Reissner-Nordström solution with $\Phi=0$ is not the whole story. Around RN, $L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}=16Q^2(1-f)/r^6$ is positive outside the horizon, so the scalar perturbation has effective mass squared $m_{\rm eff}^2=-H''(0)L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}$; when $H''(0)>0$ this is negative and the $l=0$ mode becomes tachyonic for $Q/M$ above a threshold. The paper constructs static, spherically symmetric scalarized solutions by solving Eqs. (28)-(30) with the vector field determined by Eq. (31). These solutions carry mass $M$, electric charge $Q$, and scalar charge $Q_s$; in both the quadratic and exponential models the nodeless branch reaches $Q/M>1$, meaning the scalarized black hole is overcharged. Stability is checked only for scalar-field perturbations; within that check, the exponential model has an everywhere positive effective potential for the nodeless solution, so the paper concludes this scalarized black hole can be the end state of scalarization.

Load-bearing premise

The endpoint claim assumes the full coupled metric-vector-scalar perturbations around the scalarized solution are stable, while the paper checks only scalar-field perturbations.

Editorial extensions

If this is right

  • In this theory the usual no-hair expectation fails: a charged black hole with large enough $Q/M$ or coupling spontaneously acquires a scalar profile.
  • The scalarized solutions can be overcharged, with $Q/M>1$, so the f4 model predicts a class of black holes outside the extremal Reissner-Nordström bound.
  • In the exponential model the nodeless scalarized solution is stable against radial scalar perturbations, so it is a viable final state of the instability; the quadratic model is not.
  • Because the triggering invariant vanishes in flat spacetime, scalarization here requires a curved background, unlike scalarization through $F^{\mu\nu}F_{\mu\nu}$.
  • The scalar charge $Q_s$ can be of order one, making the asymptotic metric differ measurably from the Reissner-Nordström metric.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step is to include metric and vector perturbations around the scalarized solutions; if any coupled mode is unstable, the endpoint claim would need revision.
  • Overcharged scalarized black holes, if stable in the full theory, would sit above the extremal charge bound and could serve as testbeds for weak cosmic censorship in modified gravity.
  • The same coupling should act around a rotating black hole immersed in an external magnetic field, an environment the paper mentions as the likely astrophysical setting for this scalarization.
  • The critical line of maximum $Q/M$ could be mapped numerically and compared with the extremal condition (39), which would tell whether overcharged solutions approach extremality or stay bounded away from it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper investigates spontaneous scalarization of charged black holes in a scalar-vector-tensor theory with the nonminimal coupling H(Φ)L^{μναβ}F_{μν}F_{αβ}. It first derives a no-hair condition for the constant-scalar Reissner-Nordström solution and shows that for H''(0)>0 the scalar field can develop a tachyonic instability, with the threshold computed via the zero-mode method. It then constructs test-field bound states for quadratic and exponential forms of H(Φ), finding that the quadratic bound state is radially unstable while the exponential nodeless bound state can have a positive effective potential. Finally, it constructs fully backreacted scalarized black hole solutions and reports that they can have Q/M > 1 (overcharged), and that in the exponential model the nodeless solution has a positive scalar effective potential, leading the authors to conclude that it can be the endpoint of scalarization.

Significance. The paper presents a new mechanism for black hole scalarization in an SVT theory, and the derived instability threshold follows directly from the equations of motion without fitted parameters. The overcharging result, if confirmed, is a novel feature of the f4 model. The zero-mode stability method is applied consistently and the numerical solutions appear to satisfy the stated equations and boundary conditions. However, the central final-state claim is currently supported only by a scalar-perturbation analysis; the full coupled stability remains an open issue, which tempers the significance of the conclusion.

major comments (1)
  1. [Sec. IV B, Eq. (42)] The stability analysis of the scalarized solutions considers only scalar-field perturbations around the fixed background, as the text states: 'Here, we only consider the stability against the perturbation of the scalar field.' In the action (1), the scalar field is nonminimally coupled to both the metric and the electromagnetic field strength through H(Φ)L^{μναβ}F_{μν}F_{αβ}; at first order, a scalar perturbation sources metric and vector perturbations. Positivity of U_eff in Eq. (42) therefore does not rule out coupled instabilities, and Appendix C's justification of the test-field limit only covers small G, not the strongly backreacted solutions of Sec. IV. The conclusion that 'the scalarized BH solution in the exponential model can be the end state of scalarization' is accordingly not established by the analysis presented. I recommend either performing a full linear stability analysis including metric and vector perturbations, or revising the conclusion to state that these are candidate end states.
minor comments (5)
  1. [Sec. II B] The phrase 'the scalar field must be constant under the RH BH' should read 'RN BH'.
  2. [Sec. III A] The statement 'for each electric charge Q, there is a solution' is imprecise because the solution is also characterized by the number of nodes, and in the quadratic model the overall normalization is arbitrary.
  3. [Sec. IV A, Tables III and IV] The numerical solutions leading to the overcharging claim would be more convincing with a brief description of the numerical method, grid resolution, and estimated errors.
  4. [Fig. 3] The vertical axis label uses 'H''(Φ0)' while the text discusses 'H''(0)'; please unify the notation.
  5. [Sec. III A, Figs. 5 and 7] The statement that the critical line 'almost coincides' with the line of Fig. 3 is qualitative; a quantitative comparison would strengthen the claim.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; instability, bound states, and scalarized solutions are computed from the action, with the only self-citation non-load-bearing.

full rationale

The derivation is self-contained. The action (1) fixes the model through H(Φ) with H(0)=H'(0)=0; H''(0)>0 then gives the tachyonic effective mass in Eq. (12) directly from the linearized scalar equation, so the instability region in Fig. 3 is a computed consequence, not a fitted output. The bound-state solutions in Sec. III solve the test-field equation (19) with the same H(Φ), and the threshold coinciding with the instability line is a consistency check (for the quadratic model the zero-node bound state and the zero mode of Eq. (18) are the same linear problem), not an independent prediction. The scalarized solutions in Sec. IV are obtained by solving the full ODEs (28)-(30) with boundary conditions; the overcharging Q/M>1 is read off from the asymptotic coefficients, so it is emergent. The stability conclusion for the exponential model rests on the explicit positivity of the effective potential (42) for the nodeless solution, which is checked numerically. A limitation is that only scalar-field perturbations are treated (Sec. IV B: 'Here, we only consider the stability against the perturbation of the scalar field'), so the final-state claim is not fully proven, but this is an incompleteness, not a circular reduction. The only self-citation, footnote 3's use of [26] as an analogy for test-field behavior, is not load-bearing: the paper gives its own small-G justification in Appendix C and its own stability computation for the constructed solutions.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central construction rests on symmetry restrictions, on the chosen form of H(Φ), and on neglecting metric/vector perturbations in the stability analysis. There are no invented particles or forces; the scalar and vector fields are part of the theory under study.

free parameters (3)
  • η (scalarization coupling constant)
    Defines H(Φ) in the quadratic and exponential models, Eqs. (21) and (22); scanned over ranges such as η/M²=1,10 rather than fitted to data.
  • Q/M or Q/r_H (electric charge parameter)
    Input charge-to-mass ratio of the Reissner-Nordström background and of the scalarized solutions; scanned to map existence regions.
  • Φ_H (horizon value of scalar field in exponential bound states)
    Used as a parameter when describing exponential-model bound states; for the full backreacted solutions it is fixed by shooting with asymptotic flatness.
assumptions (3)
  • domain assumption Spacetime is static, spherically symmetric, and asymptotically flat, with scalar and vector fields sharing the spacetime symmetries.
    Enter at metric ansatz Eq. (27) and in the no-hair theorem Appendix B; the analysis does not treat non-symmetric or time-dependent end states.
  • domain assumption Stability of scalarized solutions can be assessed by scalar-field perturbations on a fixed background.
    Sec. IV B explicitly ignores metric and vector perturbations; this is not proven for the backreacted solutions.
  • ad hoc to paper The quadratic and exponential forms of H(Φ) are representative scalarization couplings.
    Chosen for simplicity in Secs. III and IV; results (e.g., stability of nodeless solutions) may depend on this choice.

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Cite this review

Pith. "Pith review of Spontaneous scalarization of charged black holes in the Scalar-Vector-Tensor theory." pith.science (2026). https://pith.science/paper/BTCXGGQC

@misc{pith2026190809394,
  author       = {Pith},
  title        = {Pith review of: Spontaneous scalarization of charged black holes in the Scalar-Vector-Tensor theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BTCXGGQC}},
  note         = {Machine review of arXiv:1908.09394}
}
abstract

We present spontaneous scalarization of charged black holes (BHs) which is induced by the coupling of the scalar field to the electromagnetic field strength and the double-dual Riemann tensor $L^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}$ in a scalar-vector-tensor theory. In our model, the scalarization can be realized under the curved background with a non-trivial electromagnetic field, such as Reissner-Nordstr$\ddot{\rm o}$m Black Holes (RN BHs). Firstly, we investigate the stability of the constant scalar field around RN BHs in the model, and show that the scalar field can suffer a tachyonic instability. Secondly, the bound state solution of the test scalar field around a RN BH and its stability are discussed. Finally, we construct scalarized BH solutions, and investigate their stability.

Figures

Figures reproduced from arXiv: 1908.09394 by the authors.

Figure 1
Figure 1. FIG. 1. Definiton of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The parameter region in which the constant scalar [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The effective potential for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The typical behavior of the scalar field for the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The relation between the coupling [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The typical behavior of the non-trivial profile of the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The effective potential for [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The typical profile of the scalarized BH solutions for [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The sequence of the scalarized BH solutions for [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The typical behavior of the scalarized BH solutions [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The effective potential for the scalarized BH solu [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The sequence of the scalarized BH solutions for the [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The effective potential for the scalarized BH solu [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]

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Reference graph

Works this paper leans on

54 extracted references · 17 canonical work pages · cited by 1 Pith paper

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    Here,Gi andfi are functions of the SVT theory, andF = FµνFµν, ˜F = ˜FµνFµν (see AppendixA).Fµν and ˜Fµν are the field strength of the vector field, and its dual field (i.e

    f3(X) vanishes at X = 0. Here,Gi andfi are functions of the SVT theory, andF = FµνFµν, ˜F = ˜FµνFµν (see AppendixA).Fµν and ˜Fµν are the field strength of the vector field, and its dual field (i.e. Fµν =∇µAν−∇νAµ, ˜Fµν = 1 2ϵµνρσFρσ). X is a kinetic arXiv:1908.09394v2 [gr-qc] 9 Nov 2019 2 term of the scalar field Φ (i.e. X =− 1 2∇µΦ∇µΦ). Un- der these assumpt...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.