Frozen states of charged boson stars

Belgium); Betti Hartmann (University College London; UK); Yves Brihaye (University of Mons

REVIEW 4 major objections 4 minor 2 cited by

Frozen states of charged boson stars

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Charged boson stars can freeze into horizonless black-hole mimics with a de Sitter core and a black-hole exterior.

desk verdict The Horndeski extension is genuinely new and the light-ring analysis is clean, but the paper's central claim that exact frozen states exist is not actually demonstrated — the numerical branches stop at a double-zero limit that looks like an extremal horizon. read the letter →

arxiv 2507.08946 v1 pith:VFL5D4WK submitted 2025-07-11 gr-qc hep-th

Yves Brihaye (University of Mons , Belgium) , Betti Hartmann (University College London , UK) This is my paper

classification gr-qchep-th MSC 83C5783C15 PACS 04.40.-b04.70.-s

keywords frozenstarschargedbosonMazur-MottolasolutionU(1)gaugedscalarfieldHorndeskivector-tensorgravityultracompactobjectslightringsdeSittercore

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 tries to establish that charged boson stars, horizonless solitons of a U(1)-gauged scalar field held together by gravity, can exist in frozen states: globally regular configurations with a de Sitter interior, a Reissner-Nordström exterior, and a thin shell in place of the event horizon. The central numerical result is that standard Maxwell electrodynamics is enough to reach these states, provided the scalar self-interaction is present and the gauge coupling takes intermediate values. The paper also shows that adding a Horndeski vector-tensor coupling makes the self-interaction unnecessary: negative $\gamma$ drives the same frozen-state limit with a simple mass potential. The resulting objects carry one stable light ring inside the shell and one unstable light ring outside, so they would look nearly black-hole-like from a distance while remaining singularity-free. If the claim holds, the frozen states give a concrete field-theoretic realization of a gravastar-type black-hole mimicker built from bosonic matter and gauge fields.

What carries the argument

The key machinery is the approach of the metric function $N(r) = 1 - 2m(r)/r$ to a double zero at a finite radius, read as a quasi-horizon: the scalar and electric fields stay smooth there, while the energy-momentum tensor inside approaches the de Sitter form $\rho = -p_r = -p_t$ and outside approaches the Reissner-Nordström form. Two ingredients push the branches to this limit: the bounded, exponential scalar self-interaction $U(|\Psi|) = \mu^2\eta^2(1-\exp(-|\Psi|^2/\eta^2))$ at intermediate gauge coupling $q$, and, for the simple mass potential, a negative Horndeski vector-tensor coupling $\gamma$. The free parameters are the central value $C = \psi(0)$, the gauge coupling $q$, and the algebraic coupling $\alpha$; the solutions are constructed numerically by shooting.

What would settle it

Recompute the branch-B family with higher grid resolution and Richardson extrapolation to see whether the minimum of $N(r)$ converges to a positive value or to zero. If the minimum saturates above zero, or if curvature invariants diverge at the shell as the minimum tends to zero, then the frozen state is a numerical artifact or an extremal-horizon limit rather than a regular horizonless solution.

Watch

Extended reading notes

Core claim

The discovery is a branch of numerically constructed charged boson stars that terminates in a frozen state. Along branch B, decreasing the scalar field at the origin (or lowering the frequency $\omega$ in the Horndeski case) drives the metric function $N(r)$ toward a double zero at a finite radius $r_c$; the interior becomes de Sitter-like with $\rho = -p_r = -p_t$, a shell of finite thickness carries the transition, and the exterior is Reissner-Nordström. These are not black holes: no horizon and no singularity form. In standard Einstein gravity with a mass potential alone the minimum of $N$ never reaches zero, so the exponential self-interaction is the enabling ingredient; with negative Horndeski coupling the self-interaction can be dropped. The frozen-state configurations have $M/Q_N < 1$ in the Horndeski case and possess both a stable inner and an unstable outer light ring.

Load-bearing premise

The central claim depends on the numerical solutions staying accurate as the metric function's minimum approaches zero, and on that limiting dip being a genuinely smooth, horizonless shell rather than a numerical breakdown or an extremal horizon.

Editorial extensions

If this is right

The frozen states are horizonless, globally regular alternatives to Reissner-Nordström black holes with the same exterior, so from afar their lensing and light-ring structure mimic a black hole.
Because each has an inner stable and an outer unstable light ring, the known instability channel for ultracompact objects applies, so the states are not automatically stable as static solutions.
In the Horndeski case with $\gamma < 0$, all solutions satisfy $M/Q_N < 1$, so they are stable against decay into $Q_N$ free scalar bosons.
The exponential self-interaction and the Horndeski term play interchangeable roles in producing the frozen-state limit: either one can drive the quasi-horizon, whereas neither standard electrodynamics with a mass potential alone nor the ungauged model does.

Reading between the lines

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

Editorial inference: If the limiting double zero is genuinely regular, the frozen-state branches provide a one-parameter family connecting ordinary boson stars to gravastar-like objects, which could be used to model gravitational-wave echoes.
Editorial inference: The stable light ring inside the shell, combined with known light-ring instability results, suggests these frozen states may be nonlinearly unstable; evolving the full system numerically would test whether they are transient or long-lived.
Editorial inference: A sharper check would be to verify whether curvature invariants at the shell stay finite as the minimum of $N(r)$ tends to zero; if they diverge, the frozen state is better read as an extremal-horizon limit than as a horizonless object.
Editorial inference: The same construction should also work for other asymptotically flat charged solitons, such as gauged Q-balls, whenever charge repulsion balances gravity at intermediate coupling.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the genuinely new piece here is the Horndeski vector-tensor construction: for negative γ the authors get charged boson stars that approach a horizonless ultracompact configuration without scalar self-interaction, which is not in the earlier literature. The light-ring structure — one stable ring inside the shell, one unstable outside — is a clean derived output from the numerical metric, and the effective-potential analysis is straightforward and convincing. The paper is also honest that the γ=0 part mostly reinterprets their earlier thin-shell solutions from [21]; the text says so explicitly.

That said, the central existence claim for exact frozen states is not actually proven. The branches in Secs. 3.1 and 3.2 terminate as the minimum of N approaches zero, and the authors say the solution develops a double zero. The stress-test note is right that a double zero of N is a degenerate horizon, not a finite-thickness shell: the proper radial distance ∫ dr/√N diverges logarithmically, so the limiting configuration is an extremal horizon, not a globally regular replacement for the event horizon. The paper never constructs the limiting solution itself; it only shows finite-C regular solutions approaching that limit. The abstract overclaims slightly when it says frozen states 'replace the event horizon' — what is shown is that a family of regular quasi-horizon solutions approaches a degenerate-horizon endpoint.

Other soft spots, in proportion: the boundary condition Eq. (2.10) says N(0)=0, which is a typo — regularity at the origin in this ansatz requires N(0)=1. The numerical work has no convergence tests, no error bars, and only a handful of parameter values; near the double-zero limit the 1/N² terms in Eq. (2.9) make the shooting increasingly singular, so accuracy there is exactly what needs checking. I would not call any of this fatal, because the qualitative picture — charged boson stars can get arbitrarily close to a horizonless extremal configuration — is credible and worth pursuing. But the paper should distinguish more carefully between the finite-N_min solutions, which are rigorously regular, and the frozen-state limit, which is not constructed.

Who is this for: people working on boson stars, black-hole mimickers, and light-ring stability. It deserves a serious referee — the Horndeski mechanism is new, the light-ring pairing is relevant, and the numerical solutions should be independently reproducible. I would send it to peer review, but I would insist that the authors fix the boundary-condition typo, quantify numerical error, and either prove the double-zero limit is a regular shell or soften the claim to 'approach a frozen state.'

Referee Report

4 major / 4 minor

Summary. The paper studies spherically symmetric, static, charged boson stars in a U(1)-gauged scalar field model minimally coupled to gravity, with an optional Horndeski vector-tensor coupling. It claims that for intermediate gauge couplings, the self-interacting scalar potential (2.4) allows branches of solutions to approach configurations where the metric function N(r) develops a double zero at a finite radius, which the authors identify as 'frozen states'—globally regular, horizonless objects with a de Sitter interior, a Reissner-Nordström exterior, and a thin shell replacing the event horizon. For the Horndeski model with negative γ, the same behavior is claimed to occur even without scalar self-interaction. The paper also computes light-ring effective potentials and finds one stable and one unstable light ring for the near-frozen configurations.

Significance. If established, the existence of globally regular, horizonless ultracompact objects with a de Sitter core and a black-hole exterior would be a concrete field-theoretic realization of the Mazur-Mottola gravitational condensate star, and the claim that ordinary (non-linear) electrodynamics suffices in a gauged scalar model is of considerable interest. The paper also makes a falsifiable prediction about light-ring pairs and notes the possible light-ring instability, connecting to active literature on black-hole mimickers. However, the central existence claim is currently supported only by sparse numerical shooting results for a few parameter values and no convergence or error analysis, so the significance is contingent on the limiting procedure being made rigorous.

major comments (4)

[Section 3.1, Fig. 3] The central claim that branch B terminates in a frozen state is based on the observation that the minimum of N(r) approaches zero as C decreases toward C≈11.2. However, a double zero of N(r) at r=rc makes rc a degenerate horizon: the proper radial distance ∫ dr/√N diverges logarithmically, and the coordinate r=rc is not part of the manifold in the standard sense. The paper does not demonstrate that the limiting configuration is a regular horizonless shell, and Eq. (2.9) contains terms proportional to 1/N^2 that become singular in this limit. No convergence tests, error bars, or an exact limiting solution with N(rc)=N'(rc)=0 are provided. Since the abstract and introduction explicitly claim that the shell 'replaces the event horizon', this is the load-bearing step of the paper, and it needs either a rigorous construction of the limiting solution or a careful numerical convergence analysis showing that the limit is not an extremal horizon.
[Section 3.2, Fig. 5] The same degenerate-horizon concern applies to the γ<0 case. The branch is said to end exactly when the minimum of N(r) approaches zero, but only three values of ω are shown (ω=0.04, 0.03, 0.02), with no evidence that the sequence converges to a finite shell rather than to a singular or extremal configuration. The equations for γ<0 differ from the γ=0 case, and the scalar-field equation (2.9) still contains 1/N^2 terms; the regularity of the scalar and gauge fields at the would-be zero of N must be checked explicitly. Please provide a convergence study for these solutions and clarify the nature of the limiting configuration.
[Introduction and Abstract] The manuscript states in the abstract that frozen states are globally regular and have a thin shell that 'replaces the event horizon'. The numerical evidence presented in Fig. 3 and Fig. 5 is for finite-C solutions with N_min>0, which are quasi-horizon ultracompact objects but not yet frozen states. The text itself uses the phrases 'approaches a double zero' and 'strongly suggest', indicating that the limiting configuration was not actually constructed. The claim that the event horizon is replaced by a shell of finite thickness is therefore an extrapolation, and the manuscript should either provide the limiting solution or temper the abstract and conclusions to what is demonstrated.
[Section 3.1, parameter range] The demonstration of the crucial role of self-interaction relies on only two values of the gauge coupling (q=0.005 and q=0.01) at a single value of α=0.0001. The text states that the frozen state appears for 'intermediate values' of q, but no scan over q is shown for the self-interacting potential. Given that the claimed phenomenon depends sensitively on q (the branches differ qualitatively between q=0, 0.005, and 0.01), a more systematic parameter study is needed to establish the robustness of the effect.

minor comments (4)

[Fig. 2 caption] The caption contains a typo: 'The show the value C' should be 'We show the value C'.
[Section 4, Eq. (4.19)] The light-ring analysis is performed for finite-C solutions (Fig. 7), not for the limiting frozen state. Since the limiting configuration is not explicitly constructed, the statement that 'the frozen states possess one stable and one unstable lightring' should be phrased as a property of the near-frozen solutions, or the limiting analysis should be provided.
[Section 2, Eq. (2.5)] The notation is mostly clear, but the boundary condition N(0)=0 in Eq. (2.10) is the standard regularity condition at the origin; it would help to state explicitly that this does not imply a horizon because the radial coordinate is not the areal radius near r=0. This is a minor clarification, not a technical error.
[Section 3.2] The text says 'vector-tensor boson stars do exist in the limit C → 0 which corresponds to Ω → 0', but in Section 3.1 the limit Ω→0 is associated with the approach to the frozen state. The correspondence between C→0 and Ω→0 for γ<0 should be stated more carefully, since it seems to describe a different regime.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the central derivation is self-contained, with only non-essential self-citations for context.

full rationale

The central results are obtained by numerically integrating the coupled ODEs (2.6)-(2.9) under the boundary conditions (2.10)-(2.11); no quantity is fitted to the claimed frozen-state output. The scalar-field central value C fixes omega, and branches are mapped by varying C or omega; the approach of the minimum of N(r) to zero is a computed outcome, not an imposed condition. The lightring effective potential (4.19) is derived directly from the obtained metric, so the lightring claims are genuine outputs. The comparisons between the mass potential, the self-interacting potential (2.4), and the Horndeski term (2.3) are also numerical outcomes of the same equations, not consequences of a presupposed conclusion. The paper's reliance on the authors' prior work [20,21,29] is contextual (previously discussed thin-shell solutions, the domain boundary q>0.75) and does not carry the existence argument. The main caveat is numerical rather than circular: the limiting configuration with a double zero of N is not constructed exactly, and no convergence tests are reported; the text itself says the metric function 'approaches' a double zero and that results 'strongly suggest' the limiting behavior. This is a verification/rigor limitation, not a reduction of the derivation to its inputs, and therefore does not raise the circularity score beyond the minor self-referential context.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model parameters are chosen to exhibit the phenomenon; the central existence claims rely on numerical integration with no formal proof. The only invented element is the thin-shell structure, which is a feature of the solutions rather than a new field or particle.

free parameters (4)

alpha = 4 pi G eta^2 = 0.0001
Gravitational coupling chosen by hand; all gamma=0 frozen-state runs use this value. Different alpha could change the domain of existence.
q, U(1) gauge coupling = 0.005 / 0.01 for gamma=0; 0.7 for gamma<0
Selected to make the metric function minimum approach zero. The text states frozen states appear for intermediate values of the gauge coupling.
gamma, Horndeski vector-tensor coupling = -3 and -5 for frozen states; +3 and +5 for comparison
Negative gamma is chosen to produce frozen states without self-interaction; positive gamma gives ordinary branches.
C = psi(0), central scalar field value = 11.2, 12, 13, 15 for gamma=0; psi(0)=1 for gamma!=0 scans
Used as shooting parameter; omega is then determined. The branch terminates as C decreases and the minimum of N approaches zero.

assumptions (5)

domain assumption Spherically symmetric static ansatz (2.5) is sufficient to capture frozen states.
All solutions are built in this ansatz; no stability or non-spherical perturbations are considered.
domain assumption Boundary conditions (2.10) and asymptotic flatness (2.11) define the regular localized solution space.
The paper requires N(0)=0 (likely a typo for N(0)=1), psi'(0)=0, V'(0)=0 and flatness at infinity; these select the solutions studied.
standard math The scalar field decays exponentially so that omega - q V_infinity < 1 holds.
Eq. (2.12) follows from the linearized scalar equation; the condition ensures finite Noether charge.
ad hoc to paper Numerical shooting solutions converge to true solutions of the boundary value problem.
No convergence proofs, grid refinements, or error estimates are given; the existence of the frozen-state limit rests on this unverified assumption.
standard math Lightrings are identified from the effective potential Veff = L_z^2 N sigma^2 / r^2 for equatorial null geodesics.
Eq. (4.19) is the standard geodesic reduction; the pair theorem of [30] is invoked.

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Cite this review

Pith. "Pith review of Frozen states of charged boson stars." pith.science (2026). https://pith.science/paper/VFL5D4WK

@misc{pith2026250708946,
  author       = {Pith},
  title        = {Pith review of: Frozen states of charged boson stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFL5D4WK}},
  note         = {Machine review of arXiv:2507.08946}
}

read the original abstract

In this paper, we study frozen states of charged boson stars. These solutions are globally regular and exist in a U(1) gauged scalar field model minimally coupled to gravity for suitable choices of the coupling constants. These configurations are field theoretical realizations of the Mazur-Mottola solution with a de Sitter interior, a black hole exterior and a thin shell that interpolates between the two and replaces the event horizon. We demonstrate that standard electrodynamics is sufficient to find these frozen states, but that the self-interaction of the scalar field is crucial. Adding Horndeski vector-tensor gravity to the model allows the frozen states to exist without self-interaction though. The frozen states possess one stable and one unstable lightring, the former inside the thin shell, the latter in the black hole exterior.

Figures

Figures reproduced from arXiv: 2507.08946 by the authors.

**Figure 1.** Left: We show the dependence of the mass M on the value C of the scalar field ψ(r) at the origin for different values of q. Right: The corresponding value of Ω = 1 − (ω − qV∞) 2 . These solutions have been studied in [27] and revisited recently in [28]. In order to be able to compare with the self-interacting case, we put the emphasis here on the domain of existence of solutions in the qC-plane. A family of boson s… view at source ↗

**Figure 2.** Left: The show the value C of the scalar field ψ(r) at the origin as well as the mass M of the solutions in dependence on Ω for α = 0.0001 and several values of q. Right: We show the dependence of the Noether charge QN on Ω for the same solutions. q ∼ 0.01) the two branches can be extended to Ω = 1 and the local minimum of the metric function N(r), again, is well above zero. Interestingly, for smaller values of q (t… view at source ↗

**Figure 3.** Left: We show the profiles of the metric function −gtt = σ 2N (solid) and of the scalar field function ψ(r) (dashed) for three charged boson star solutions on the branch B. Here α = 0.0001, q = 0.005 and the three solutions have different values of C, i.e. different values of the scalar field at the origin. Clearly, we see that the metric function approaches a double zero at some intermediate value of the radial coo… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Left: We show the dependence of Ω and of the electric charge Q on the gauge coupling constant q for constant central value of the scalar field, ψ(0) = 1, and three different values of γ. Right: We show the corresponding mass M and Noether charge QN of the Horndeski vec…

**Figure 5.** Figure 5: We show the profiles of the metric function [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Left: We show the dependence of the mass M on the parameter Ω for q = 0.7 and different values of γ. Right: We show the ratio M/QN in dependence on Ω for the same solutions. . 4 Discussion In this paper, we have studied the existence of frozen states of charged boson s…

**Figure 7.** Figure 7: Left: We show the effective potential for light-like particle motion in the space-time of a charged frozen boson stars. Right: We show the effective potential for light-like particle motion in the space-time of the Horndeski charged frozen boson stars. . [2] C. Bambi, …

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Forward citations

Cited by 2 Pith papers

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Frozen Neutron Stars
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For scalarization of the full Bardeen spacetime, small magnetic charges give the usual smooth scalarization threshold, while large charges end in a 'frozen' horizonless scalarized state rather than a Bardeen black hole.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [21]

Brihaye and B

Y. Brihaye and B. Hartmann: Boson stars and black holes with wavy scalar hair, Phys. Rev. D 105 (2022) no.10, 104063

work page 2022

[2] [1]

Penrose: Gravitational collapse: The role of general relativity, Riv

R. Penrose: Gravitational collapse: The role of general relativity, Riv. Nuovo Cim. 1 (1969), 252. 8 0.0000001 0.0000010 0.0000100 0.0001000 0.0010000 0.0100000 0.1000000 0 100 200 300 400 500 Veff r C=15.0 C=13.0 C=12.0 C=11.2 0.000001 0.000010 0.000100 0.001000 0 5 10 15 20 25 30 35 40 Veff r ω = 0.04 ω = 0.03 ω = 0.02 Figure 7: Left: We show the effect...

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Bambi, R

C. Bambi, R. Brustein, V. Cardoso, A. Chael, U. Danielsson, S. Giri, A. Gupta, P. Heidmann, L. Lehner and S. Liebling, et al.: Black hole mimickers: from theory to observation, [arXiv:2505.09014 [gr-qc]]

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[5] [4]

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