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Spin-induced Scalarized Black Holes in Einstein-Maxwell-scalar Models

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Pith's one-line read The paper constructs spin-induced scalarized Kerr-Newman black holes in Einstein-Maxwell-scalar models with negative coupling, showing that scalar hair appears only above a spin threshold and that coexisting hair-free Kerr-Newman black…

desk verdict A solid numerical construction of spin-induced scalarized Kerr-Newman black holes with negative coupling; the outer existence boundary is a numerical limit, but the core results hold up. read the letter →

arxiv 2506.01773 v2 pith:2ZL77KSI submitted 2025-06-02 gr-qc

Lang Cheng , Guangzhou Guo , Peng Wang , Haitang Yang This is my paper
classification gr-qc
keywords spontaneousscalarizationEinstein-Maxwell-scalarmodelKerr-Newmanblackholespin-inducedscalarhairtachyonicinstabilitynegativecouplingspectralmethods
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 numerically constructs rotating charged black holes with scalar hair in Einstein-Maxwell-scalar models when the scalar-electromagnetic coupling constant $\alpha$ is negative. It claims that, in this regime, scalar hair appears only for rapidly spinning Kerr-Newman black holes, while slowly spinning ones remain described by the hair-free Kerr-Newman metric. The scalar field carries only a small fraction of the total mass ($E_\phi/M\sim10^{-3}$ for $\alpha=-100$ and $10^{-4}$ for $\alpha=-1000$), so its backreaction is weak, and the paper attributes the narrow existence domain and the similarity between different coupling functions to this suppression. Inside the existence domain, scalarized black holes coexist with linearly stable Kerr-Newman black holes that have larger horizon area, so the hairy solutions are entropically disfavored and may be metastable.

What carries the argument

The mechanism is spin-induced tachyonic scalarization driven by the non-minimal coupling $f(\phi)F_{\mu\nu}F^{\mu\nu}$. On a Kerr-Newman background with $\phi=0$, a linearized scalar perturbation obeys $(\Box-\mu_{\rm eff}^2)\delta\phi=0$ with $\mu_{\rm eff}^2=\alpha F_{\mu\nu}F^{\mu\nu}$, equal to $-\alpha Q^2(r^4-6a^2r^2\cos^2\theta+a^4\cos^4\theta)/(r^2+a^2\cos^2\theta)^4$; for $\alpha<0$ the negative-mass-squared region appears only at high spin, and the scalar-cloud thresholds found in the linear problem mark the bifurcation line. To construct the full solutions, the paper uses a seven-function stationary axisymmetric ansatz, expands each function in Chebyshev polynomials in a compactified radial coordinate and $\cos(2j\theta)$ in the angular coordinate, and solves the coupled Einstein-Maxwell-scalar equations by Newton-Raphson iteration with boundary conditions at the horizon, infinity, the symmetry axis, and the equatorial plane. The Smarr relation $M=2T_H S+2\Omega_H J+\Phi Q$ and the absence of conical singularities are used as independent accuracy checks, and the convergence tests reported in the appendices justify the chosen resolution.

What would settle it

Recompute the scalarized branch with an independent numerical scheme (finite differences or a null evolution code) for a point just outside the claimed domain, e.g. $\alpha=-100$, $q=0.5$, and $\chi$ slightly below the red critical line of Fig. 1; a regular, asymptotically flat solution with residuals below $10^{-5}$ would show the boundary is numerical rather than physical. Conversely, a fully nonlinear evolution starting just above the bifurcation line that settles to a hair-free Kerr-Newman solution rather than the scalarized branch would undermine the dynamical-coexistence interpretation.

Watch

Extended reading notes

Core claim

The central claim is that Einstein-Maxwell-scalar models with $f(\phi)=1+\alpha\phi^2+\cdots$ and $\alpha<0$ admit spin-induced scalarized Kerr-Newman black holes: stationary, axisymmetric, asymptotically flat solutions with a nodeless scalar field that branches off the hair-free Kerr-Newman family at a bifurcation line set by scalar clouds. In the reduced parameter space of charge $q=Q/M$ and spin $\chi=J/M^2$, the scalarized solutions occupy a narrow region bounded by the bifurcation line and a critical line at which the numerical solutions stop converging; the two lines merge at points on the extremal line $q^2+\chi^2=1$, so no hair appears at very low or very high spin. The scalar wave function is concentrated near the poles, decays outward, and carries only a tiny fraction of the mass, $E_\phi/M\sim10^{-3}$ for $\alpha=-100$ and $10^{-4}$ for $\alpha=-1000$. Within the coexistence region, the hair-free Kerr-Newman solution has larger reduced horizon area $a_H=A_H/(16\pi M^2)$ and is linearly stable, so it is entropically favored; the paper argues that angular momentum loss during scalarization can stabilize the coexisting Kerr-Newman solution and may make the scalarized black holes metastable.

Load-bearing premise

The load-bearing premise is that the red critical line, defined by where the numerical solver stops producing solutions with error below $10^{-5}$, marks a real physical boundary of the scalarized-solution family rather than a limit of the numerical method; the lower boundary is likewise imported from an earlier linear scalar-cloud calculation instead of being re-derived in the full nonlinear problem.

Editorial extensions

If this is right

  • For $\alpha<0$, slowly rotating charged black holes in these models remain effectively indistinguishable from Kerr-Newman, so any scalar hair must be triggered by spin above a threshold.
  • The scalarized solutions occupy a narrow band in the $(q,\chi)$ plane that widens as $|\alpha|$ grows, and no scalarized solutions are found at very low or very high spin.
  • In the coexistence region, the hair-free Kerr-Newman solution has larger horizon area and is linearly stable, so by entropy arguments the scalarized black hole is not the preferred equilibrium.
  • Because the scalar field's energy fraction is tiny, the detailed form of the coupling function has little effect: exponential and quadratic couplings produce very similar existence domains.
  • Whether scalarized solutions are persistent remnants or transient stages is left to future linear stability analysis and nonlinear evolution calculations.

Reading between the lines

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

  • A testable extension: run a fully nonlinear evolution of a Kerr-Newman black hole just above the bifurcation line with $\alpha<0$; the paper's angular-momentum-loss argument predicts the end state is a lower-spin scalarized black hole (or a spun-down hair-free one), a prediction that nonlinear simulations can check directly.
  • If the critical line is physical rather than a numerical artifact, the polar concentration of scalar hair should give observational signatures distinct from both $\alpha>0$ scalarized black holes and bald Kerr-Newman, such as polar features in very-long-baseline interferometric images or ringdown echoes.
  • The coexistence with a linearly stable hair-free solution suggests a metastable branch whose decay could be triggered by finite perturbations; searching for an analogous entropy ordering in other scalarization theories (e.g., scalar-Gauss-Bonnet or scalar-tensor) would test how generic this phase structure is.
  • Because the bifurcation line is imported from a linear scalar-cloud calculation, one could close the loop by deriving the small-amplitude nonlinear branch directly and verifying that it emanates exactly at the predicted bifurcation, which would also sharpen the claimed boundary of the existence domain.
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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

3 major / 4 minor

Summary. The authors construct stationary, axisymmetric, asymptotically flat black hole solutions with scalar hair in Einstein-Maxwell-scalar (EMS) models with negative coupling, where scalarization is spin-induced. Using a spectral solver with compactified coordinates, they obtain families of scalarized Kerr-Newman black holes for exponential and quadratic coupling functions and map their existence domain in the (χ, q) plane. They report that the scalar field carries only a small fraction of the total energy, that the scalarized branch coexists with linearly stable KN black holes, and that KN black holes have larger horizon area in the coexistence region. Accuracy is monitored via the Smarr relation and the conical-singularity condition, with exponential convergence away from the numerical critical line.

Significance. If the existence domain is as claimed, this is a new family of charged rapidly rotating hairy black holes with negative scalar-electromagnetic coupling, and it provides a concrete contrast with EsGB scalarization, where scalarized solutions are entropically favored. The paper's numerical hygiene is a clear strength: solutions are checked with two independent global relations, the convergence study in Appendix A is explicit, and the error levels away from the critical line are small (≲10^-8). The main caveat is that the boundary of the existence region is partly defined by numerical failure, so the precise extent of the domain is not established to the same standard as the individual solutions.

major comments (3)
  1. [Sec. III (Fig. 1) and Appendix A] The critical line (red solid in Fig. 1) is defined operationally: "Beyond this line, numerical solutions cannot be reliably obtained with an error below 10^-5" (Sec. III). This is not a physical existence criterion. Near that line the reported Smarr and conical-singularity errors are already at the 10^-5–10^-6 level (right columns of Figs. 4 and 5), so the boundary sits at the solver's accuracy limit. The claim that scalarized black holes exist only in the light blue region, and in particular that they do not exist below a threshold spin, therefore requires an independent demonstration that the branch actually terminates: for example, a check that the Newton-Raphson Jacobian becomes singular, a study of spectral coefficients at fixed high resolution across the line, or continuation with a different radial compactification. As written, the existence domain is a statement about the solver, not about the field equations.
  2. [Sec. II.A and Sec. III] The bifurcation line is taken from the scalar-cloud analysis of Ref. [55] rather than re-derived in this work. Because the headline claim that scalar hair appears only above a threshold spin and the statement that coexisting KN black holes are linearly stable both rely on this boundary, the authors should either include an independent computation of the scalar-cloud threshold in the same numerical setup or provide a quantitative consistency check between the bifurcation points obtained here and those of [55].
  3. [Sec. III and Fig. 1] The sentence "KN black holes below the bifurcation line are stable ... while those above exhibit a tachyonic instability. Therefore, Fig. 1 demonstrates that spin-induced scalarized KN black holes coexist with stable KN black holes" requires the coexistence region to lie on the stable side of the bifurcation line. With the light-blue region bounded by the bifurcation and critical lines, the orientation is not immediately obvious from the figure or the text. Please state explicitly which side of the bifurcation line the coexistence region lies on and verify the stability status of the KN branch with the same (χ, q) as the computed scalarized solutions; this is a central claim and should not rest only on a cited classification.
minor comments (4)
  1. [Sec. II.A] There is a typo: "Boyer-Linquist coordinates" should be "Boyer-Lindquist coordinates".
  2. [Eq. (18)] In the definition of E_φ, the volume element and the unit normal vector n^μ are not explicitly defined; please specify the integration measure and the foliation used.
  3. [Appendix A] Please state explicitly which (χ, q) points correspond to the left, middle, and right columns of Figs. 4 and 5; this would make it easier to assess how the reported convergence behavior depends on proximity to the critical line.
  4. [Fig. 1] The labels C1, C2, B1, B2, and B3 are difficult to read in the text version; enlarging the figure or listing the coordinates of these points in the caption would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No construction-level circularity: nonlinear solutions are obtained from the full field equations and verified with independent Smarr and conical-singularity checks.

full rationale

The paper's central derivation is self-contained. Scalarized KN black holes are constructed by numerically solving the coupled Einstein-Maxwell-scalar equations (Eq. (2)) under the axisymmetric ansatz (Eq. (7)), using spectral decomposition and Newton-Raphson iteration; no parameter is fitted to the headline outputs, and the existence domain, scalar energy ratio E_phi/M, and horizon-area ordering are computed from the solutions rather than imposed. The convergence tests in Appendix A use two independent relations, the Smarr relation (Eq. (11)) and the conical-singularity condition F1 = F2, and demonstrate exponential convergence, so the numerical branch is not an artifact built into the ansatz. The bifurcation line and the statement that KN black holes below it are scalar-stable are imported from the authors' earlier scalar-cloud analysis [55]; this is a self-citation, but it is a parameter-free linear result on the KN background that does not assume the nonlinear target, and the same bifurcation endpoint is independently visible in the nonlinear branch as the scalar amplitude tends to zero. The main caveat is the critical line, which is operationally defined by numerical non-convergence ('Beyond this line, numerical solutions cannot be reliably obtained with an error below 10^{-5}'), so the claim that scalarized solutions do not exist beyond it is a numerical robustness limitation rather than a proven physical boundary; this is a correctness concern, not a circular reduction. Overall, no claim in the paper reduces by construction to its inputs.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard field equations, a cited linear-instability analysis from the same group (scalar clouds in [55]), and a numerical ansatz. No new particles or parameters are introduced beyond the model's coupling alpha, which is chosen by hand.

free parameters (1)
  • alpha (scalar-EM coupling constant) = -100, -1000
    The coupling constant in f(phi)=e^{alpha phi^2} is fixed to two representative negative values. The existence of spin-induced scalarized solutions requires alpha<0 with sufficiently large magnitude; the paper does not scan alpha continuously or fit it to data.
assumptions (5)
  • standard math KN black hole with f'(0)=0 and f(0)=1 is a solution of the EMS model
    Derived in Sec. II.A: substituting the KN metric with phi=0 into Eq. (2) gives zero scalar equation because the F^{mu nu}F_{mu nu} term is multiplied by f'(0)=0.
  • domain assumption Existence domains of stationary scalar clouds around KN black holes from [55] mark the bifurcation points where scalarized solutions branch off
    Used in Sec. III to define the blue dashed bifurcation line. This is a cited result from the same group, not re-derived in this paper.
  • domain assumption The ansatz (7) with equatorial symmetry captures all relevant stationary, axisymmetric scalarized solutions
    Sec. II.B: the seven functions F0,F1,F2,W,At,Aphi,phi depend only on r and theta; the paper restricts to equatorial-plane symmetric solutions. This excludes potentially different symmetry classes.
  • domain assumption KN black holes below the bifurcation line are linearly stable against scalar perturbations
    Sec. III states this following [55]; the paper relies on it to claim coexistence with linearly stable KN black holes.
  • standard math Smarr relation (11) and absence of conical singularities are valid consistency checks
    Sec. II.B and Appendix A use these to assess numerical accuracy; standard in the literature.

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Cite this review

Pith. "Pith review of Spin-induced Scalarized Black Holes in Einstein-Maxwell-scalar Models." pith.science (2026). https://pith.science/paper/2ZL77KSI

@misc{pith2026250601773,
  author       = {Pith},
  title        = {Pith review of: Spin-induced Scalarized Black Holes in Einstein-Maxwell-scalar Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ZL77KSI}},
  note         = {Machine review of arXiv:2506.01773}
}
read the original abstract

We construct spin-induced scalarized black hole solutions in a class of Einstein-Maxwell-scalar models, where a scalar field is non-minimally coupled to the electromagnetic field. Our results show that scalar hair develops only for rapidly rotating black holes, while slowly spinning ones remain well described by the Kerr-Newman (KN) metric. The scalar field contributes only a small fraction of the total mass, indicating suppressed nonlinear effects. This suppression may account for the narrow existence domains of scalarized black holes and the similarities observed in their existence domains across different coupling functions. Moreover, scalarized black holes are found to coexist with linearly stable, entropically favored KN black holes. These results motivate further investigations into the nonlinear dynamics and stability of scalarized black holes in these models.

Figures

Figures reproduced from arXiv: 2506.01773 by the authors.

Figure 1
Figure 1. FIG. 1. Domain of existence for spin-induced scalarized KN black holes in the ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. presents representative scalar wave functions ϕ (x, θ) for α = −100 and −1000. Three scalarized black hole solutions along the constant-q line connecting B1 and B2, are selected. All scalar wave functions exhibit maxima near the black hole poles and decay outward from the event horizon, consistent with the fundamental scalar cloud wave functions reported in [55]. Notably, rapidly rotating KN black holes with α > 0 d… view at source ↗
Figure 3
Figure 3. FIG. 3. Ratio of scalar field energy to black hole mass, [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Logarithmic plots of the conical singularity error [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Logarithmic plots of the conical singularity error [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Domain of existence for spin-induced scalarized KN black holes in the ( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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