Quantum parameter-mass induced scalarization of qOS-black hole in the Einstein-Gauss-Bonnet-scalar theory

Yun Soo Myung

REVIEW 3 major objections 4 minor 1 cited by

Quantum parameter-mass induced scalarization of qOS-black hole in the Einstein-Gauss-Bonnet-scalar theory

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For the qOS black hole, the two heat-capacity Davies points coincide exactly with the critical onset parameters for spontaneous scalarization, giving the first concrete link between bald-black-hole thermodynamics and scalarization.

desk verdict The Davies/onset coincidence is real algebra, but it rests on a non-standard temperature that is defined to make the first law hold, so the physical claim is conditional. read the letter →

arxiv 2505.19450 v2 pith:JKUPFLRA submitted 2025-05-26 gr-qc hep-th

Yun Soo Myung This is my paper

classification gr-qchep-th PACS 04.70.-s04.50.Kd

keywords quantumOppenheimer-SnyderblackholeEinstein-Gauss-Bonnet-scalartheoryspontaneousscalarizationDaviespointheatcapacitySmarrformulashadowEHTobservation

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

The paper argues that a quantum-corrected black hole, the quantum Oppenheimer-Snyder (qOS) black hole with metric function $g(r)=1-2M/r+\alpha M^2/r^4$, can spontaneously acquire scalar hair in the Einstein-Gauss-Bonnet-scalar theory, and that the onset of this scalarization is exactly predicted by the thermodynamics of the bald (scalar-free) black hole. Using a postulated energy-momentum tensor for the unknown qOS action, the author constructs a thermodynamics with a first law and Smarr formula in which the quantum parameter $\alpha$ enters as a chemical-potential variable. The heat capacity then blows up at two Davies points, $\alpha_D=1.2835$ for $M=1$ and $M_D=0.8827$ for $\alpha=1$, and these are shown numerically to equal the critical thresholds $\alpha_c$ and $M_c$ for the tachyonic instability that triggers scalarization. The same coincidence does not hold when the qOS metric is replaced by the nonlinear-electrodynamics model, so the identification is specific to the qOS form. A shadow-radius comparison with Event Horizon Telescope data puts new constraints on the mass but leaves the quantum parameter unconstrained.

What carries the argument

The argument is carried by two objects. The first is the qOS metric function $g(r)=1-2M/r+\alpha M^2/r^4$ together with the postulated energy-momentum tensor $T^{\rm qOS,\nu}_{\ \ \mu}=(3\alpha M^2/r^6)\,{\rm diag}[-1,-1,2,2]$, which allows the thermodynamics to be defined with $\alpha$ as an independent thermodynamic variable through the first law $dm=T\,dS+W_\alpha\,d\alpha$ and Smarr formula $m=2TS+2W_\alpha\alpha$. The second is the effective-mass term of the scalar perturbation, $\tilde{m}^2_{\rm eff}=-(48\lambda M^2/r^6)(3\alpha^2 M^2/r^6-5\alpha M/r^3+1)$, whose zero sets the resonance condition $3\tilde{\alpha}_c^2-5\tilde{\alpha}_c+1=0$ via Hod's approach; the small root $\tilde{\alpha}_c=0.2324$ locates the onset of scalarization. The heat capacity $C=NC/DC$ diverges when $DC(M,\alpha)=0$, and the paper shows $DC=0$ gives exactly the same $\alpha$ and $M$ as the resonance condition.

What would settle it

Construct the scalarized qOS black holes by solving the full field equations (3) and (9) with a concrete, known qOS action and compare the threshold with $\alpha_c=1.2835$; if the first scalarized solution appears at an $\alpha$ different from the heat-capacity Davies point, or if no scalarized solutions exist, the claimed identification is refuted. A cheaper check is to compute the actual energy-momentum tensor of the qOS action once it is identified and test whether it equals the postulated form with $P=M$.

Watch

Extended reading notes

Core claim

The central claim is a dual equality: in the Einstein-Gauss-Bonnet-scalar theory with the qOS metric, the thermodynamic Davies points, where the heat capacity diverges and changes sign, are the same numbers as the critical parameters at which the scalar perturbation around the bald black hole first becomes tachyonic. Concretely, solving the resonance condition $3\tilde{\alpha}_c^2-5\tilde{\alpha}_c+1=0$ gives $\tilde{\alpha}_c=0.2324$, which translates to $\alpha_c=1.2835$ at $M=1$, and this is exactly the Davies point $\alpha_D$; solving the same condition for mass gives $M_c=0.8827$ at $\alpha=1$, equal to $M_D$. The author verifies the equality $\alpha_c(M,\alpha)=\alpha_D(M,\alpha)$ and $M_c(M,\alpha)=M_D(M,\alpha)$ continuously over the allowed range $0.0128\lesssim M,\alpha \lesssim 1.6875$, and presents this as the first example in which bald-black-hole thermodynamics and spontaneous scalarization are connected.

Load-bearing premise

The entire argument assumes that the qOS black hole, whose fundamental action is not known, can be embedded in Einstein-Gauss-Bonnet-scalar theory by postulating the specific energy-momentum tensor $T^{\rm qOS,\nu}_{\ \ \mu}=(3\alpha M^2/r^6)\,{\rm diag}[-1,-1,2,2]$ and adding the scalar-Gauss-Bonnet coupling to the resulting metric; if the true qOS action has a different energy-momentum tensor, the Davies-point/scalarization coincidence loses its physical meaning.

Editorial extensions

If this is right

For $M=1$, qOS black holes with $\alpha > 1.2835$ enter the scalarization window up to the extremal point $\alpha_e=1.6875$; below $\alpha_c$ the tachyonic instability is absent and no scalar hair forms.
For $\alpha=1$, qOS black holes with remnant mass $M_{\rm rem}=0.7698 < M < M_c=0.8827$ scalarize; more massive bald black holes are stable against scalarization.
Using the surface-gravity temperature $T_\kappa$ rather than the thermodynamic temperature $T$ would violate the first law and the Smarr formula, so the Davies-point/scalarization identification depends on the corrected thermodynamics adopted in the paper.
The shadow-radius comparison with EHT yields no upper limit on $\alpha$, but for $\alpha=1$ restricts the mass to $0.8989\lesssim M\lesssim1.024$ at $1\sigma$ and $0.8358\lesssim M\lesssim1.088$ at $2\sigma$; the naked-singularity branch $M\in[0.6,0.7698]$ is excluded at $2\sigma$.
The same analysis with the nonlinear-electrodynamics action gives Davies points $\alpha_D=2.2888$, $M_D=0.8130$ that do not match the critical onsets $\alpha_c=2.4948$, $M_c=0.7556$, so the thermodynamic-scalarization coincidence is not generic to the class of $\alpha M^2/r^4$ metrics.

Reading between the lines

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

A testable extension is that the equality between $DC=0$ and the scalarization onset may hold for any spherically symmetric background in Einstein-Gauss-Bonnet-scalar theory, which would let one read off scalarization thresholds directly from the heat capacity without solving the perturbation equation; the NED counterexample in the paper suggests the equality is a property of the qOS energy-moment
The result implies that the thermodynamic ensemble matters for the connection: treating $\alpha$ as a chemical-potential variable is what produces the corrected temperature and the Davies points, so the identification would not survive in an ensemble where $\alpha$ is held fixed rather than the chemical potential.
Because the true qOS action is unknown, the coincidence could be used as a consistency check on proposed actions: any candidate $L_{\rm qOS}$ whose energy-momentum tensor differs from the postulated form would be expected to break the Davies/onset equality, making the equality a selection rule on quantum-gravity phenomenology.

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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

The genuinely new piece is the claimed exact match between two Davies points and the critical onset values for alpha and M. The algebra is what it says: the heat-capacity denominator and the Hod condition reduce to the same polynomial, so the equality is not a fit. The paper is also upfront that LqOS is unknown, and the LNED comparison in Sec. 6 is a useful control—there the match fails, which at least shows the coincidence is specific rather than universal. The shadow analysis is routine but gives new mass constraints from EHT, and the no-constraint-on-alpha result is honest.

The soft spot is the temperature. The Davies points are computed with T = dM/dS, not with the surface-gravity T_kappa. The paper notes T_kappa does not satisfy the first law and then defines T to satisfy it. That is a convention, not a derivation. For a smooth m(S,alpha), the differential dm = T dS + W_alpha dalpha is always integrable, and the Smarr formula follows from dimensional homogeneity, so Eqs. (20)-(21) do not test the physics. The stress-test worry is correct: with T_kappa, the heat capacity is different and the coincidence with alpha_c and M_c will not hold. That does not kill the paper, but it does mean the "correct thermodynamics" framing and the claim of a first-ever connection between bald black hole thermodynamics and scalarization should be read as conditional on a particular temperature choice.

Also, no full scalarized solutions are constructed; the onset rests on the Hod criterion. And alpha_c = 1.2835 and M_c = 0.8827 were already in ref. [22]; the newness is the coincidence and the EHT constraints, not those numbers.

Who is this for? Specialists in scalarization and black hole thermodynamics. It is a coherent specialist extension, worth a referee if the editor can find one who will press on the temperature choice. I would engage with it with that caveat.

Recommendation: send to peer review, but the referee should be asked to evaluate whether T = dM/dS has independent physical justification beyond the first law it is defined to satisfy.

Referee Report

3 major / 4 minor

Summary. The paper studies the quantum Oppenheimer-Snyder (qOS) black hole, given by the metric function g(r)=1-2M/r+αM^2/r^4, in the context of Einstein-Gauss-Bonnet-scalar (EGBS) theory with an unknown qOS action. The authors define thermodynamic quantities using the temperature T=∂m/∂S with entropy S=πr_+^2, derive a Smarr formula, and locate Davies points in the heat capacity. They then compute the onset of spontaneous scalarization using the Hod approach and find that the two critical parameters (α_c for M=1 and M_c for α=1) coincide with the two Davies points (α_D and M_D). A shadow radius analysis is also performed and compared with EHT observations. The paper claims that this is the first example of a close connection between bald black hole thermodynamics and spontaneous scalarization.

Significance. If the claimed connection were physically robust, it would be a novel and interesting result linking black hole thermodynamics to scalarization thresholds. The algebraic identity between the Davies denominator and the scalarization onset condition is internally consistent and the paper presents numerical checks. The shadow analysis is straightforward and yields new mass constraints. However, the physical significance is weakened by the use of a non-standard temperature T=∂m/∂S instead of the surface gravity temperature, and by the absence of a known action for the qOS sector. The paper explicitly recognizes these limitations but does not resolve them.

major comments (3)

[Section 3, Eqs. (14) and (19)] The temperature T=∂m/∂S used in the heat capacity is not the surface gravity temperature T_κ; the two differ by a factor r_+/√(r_+^2-α). The paper rejects T_κ because the first law and Smarr formula fail for it, but the first law dm=T dS+W_α dα with T and W_α defined as partial derivatives is a tautology for any smooth function m(S,α), and the Smarr formula m=2TS+2W_αα follows from dimensional homogeneity alone. Thus the 'Davies points' obtained from Eq. (15) are features of a chosen coordinate/ensemble, not of the standard Killing-horizon thermodynamics. The central claim that two Davies points coincide with scalarization thresholds therefore lacks a clear physical basis unless T is derived from the underlying (unknown) action. I ask the authors to compute the heat capacity with T_κ and report whether any divergences still occur at α_c and M_c, or to provide a physical justification for T as the thermodynamic temperature of the qOS black hole.
[Section 2, Eq. (6)] The energy-momentum tensor T^{qOS,ν}_μ = (3αM^2/r^6) diag[-1,-1,2,2] is postulated without an action. The paper acknowledges that L_qOS is unknown, so the Einstein equation (3) is not derived from a specified theory. The proposed NED action (7) is shown to reproduce the EMT only if P=M, which is not allowed for non-extremal black holes. This leaves the thermodynamic analysis (entropy, temperature, chemical potential) and even the background metric itself without a known action-level justification. The physical interpretation of the Davies/scalarization coincidence depends on this postulated EMT. Please clarify whether an action exists that yields Eq. (6) for non-extremal black holes, or discuss how the results would change if the NED action with P≠M were used.
[Section 5, Eq. (39) and after] The paper states that α_c=α_D and M_c=M_D are checked numerically for a range of parameters, but the algebraic reduction of the Davies denominator D_C(M,α) in Eq. (15) to the polynomial x(3x^2-5x+1) is not shown. Since this identity is the central result of the paper, the explicit derivation should be included rather than relying on numerical checks over a range; numerical checks are not a proof of exact identity.

minor comments (4)

[Abstract and Section 7] The phrase 'quantum number' is used in Section 7 to refer to α, but the rest of the paper correctly calls it the quantum parameter; please make the terminology consistent.
[Section 3, after Eq. (12)] The text says 'Hawking temperature defined by T=∂m/∂S', but Eq. (14) is not the standard Hawking temperature; it differs from the surface gravity T_κ in Eq. (19). Please reword to avoid implying that T is the Hawking temperature.
[Section 3, Eq. (21)] The Smarr formula m=2TS+2W_αα is stated as a consequence of the first law, but it is in fact a homogeneity relation. The paper should mention this to avoid appearing to derive independent physical content from it.
[Section 4, Fig. 4] The figure caption refers to '1σ and 2σ ranges from EHT observation', but the text quotes specific numerical ranges (4.55–5.22 and 4.21–5.56). Please make the figure and text consistent by explicitly stating the units (in units of M, presumably) and the source of these ranges.

Circularity Check

1 steps flagged · score 2.0 of 10

Central Davies/scalarization equality is an algebraic identity, not a fitted fit; the only mild circularity is that the first law and Smarr formula are definitional because T and W_alpha are defined as derivatives of m.

self definitional [Section 3, Eqs. (13)-(21); also Section 5]
"We find that black hole mass m(M, α) obtained from g(r+) = 0 after replacing M with m, area-law entropy S = πr2+, the Hawking temperature defined by T = ∂m ∂S , heat capacity C = ∂m ∂r+ ( ∂T ∂r+ )−1, chemical potential Wα = ∂m ∂α ... At this stage, we wish to check that the first law of thermodynamics is satisfied as dm = T dS+ Wαdα. ... Importantly, if one uses Tκ(M, α), the following first law and Smarr formula are not satisfied."

T and W_alpha are defined as partial derivatives of m(S,alpha), so dm = T dS + W_alpha d_alpha is an identity for any smooth mass function, and the Smarr relation m = 2TS + 2W_alpha alpha follows from the homogeneity of m(S,alpha) under (S,alpha)->(lambda^2 S, lambda^2 alpha) rather than from an independent action-level first law. The paper explicitly rejects the surface-gravity temperature T_kappa because it would break this constructed first law. Thus the Davies points are defined with a temperature chosen to make the first law tautological; however, the equality of those Davies points with the scalarization onsets is a separate algebraic fact that both quantities reduce to the same polynomial, so this mild definitional circularity does not compromise the central threshold identity.

full rationale

The paper's headline claim — that two Davies points of heat capacity coincide with the critical onset quantum parameter and mass for scalarization — is not a fitted prediction. The scalarization thresholds come from setting the effective mass (Eq. 36) to zero at the horizon (Eqs. 38-39), while the Davies points come from the denominator of the heat capacity (Eq. 15). Both reduce, for the outer horizon, to the same polynomial condition 3x^2 - 5x + 1 = 0, and the paper verifies the equality numerically over a range of M and alpha. There is no parameter fitted to force this match, so the central connection has independent content. The one genuinely definitional element is the thermodynamic framework in Section 3: T and W_alpha are defined as partial derivatives of m(S,alpha), making the first law dm = T dS + W_alpha d_alpha a tautology, and the Smarr formula follows from Euler homogeneity rather than from the unknown qOS action. Since the paper chooses T = ∂m/∂S over the surface-gravity temperature T_kappa (which it notes would violate the first law), the thermodynamic interpretation is partly circular and non-standard. Still, this does not infect the separate algebraic derivation of the Davies/onset equality. The cited prior work [22] is not by the present author and is not load-bearing; the comparisons with the nonlinear electrodynamics action strengthen the claim by showing that the equality is special to the qOS metric. Overall, the central derivation is self-contained, with only a mild definitional circularity in the thermodynamic labelling, so a score of 2 is appropriate.

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

The central calculation adds no new free parameters beyond the P = 0.6 comparison. The load-bearing input is the unproven energy-momentum form for the unknown qOS action, plus standard scalarization assumptions taken from the literature.

free parameters (1)

Magnetic charge P in the nonlinear electrodynamics comparison = 0.6
Chosen by hand in Section 6 to show that with the nonlinear electrodynamics action the Davies points do not match the onset parameters. The central qOS claim does not depend on this value.

assumptions (5)

domain assumption The unknown qOS action LqOS yields the energy-momentum tensor T^{qOS,nu}_mu = (3 alpha M^2 / r^6) diag[-1,-1,2,2].
Section 2, Eq (6). This form is taken from [23] rather than derived; all subsequent field equations and thermodynamics assume it.
domain assumption The scalar coupling is f(phi) = 2 phi^2 with Gauss-Bonnet coupling lambda < 0.
Section 5, Eq (32). This choice is standard in Gauss-Bonnet scalarization; lambda < 0 enables the alpha/M-induced instability studied here.
domain assumption The onset threshold is determined by the degenerate potential well condition m_eff^2 = 0 at the horizon in the lambda to -infinity limit, following the Hod approach.
Section 5, Eqs (38)-(39), citing [12]. This converts the scalar instability problem into an algebraic condition without solving the full Klein-Gordon equation.
domain assumption The entropy is given by the area law S = pi r_+^2, and the temperature and chemical potential are T = dm/dS and W_alpha = dm/dalpha.
Section 3, Eqs (13)-(16) and (20). The first law is made true by construction; the physical surface gravity T_kappa is set aside because it does not satisfy the first law.
domain assumption The qOS black hole is asymptotically flat and non-extremal outside the bounds M_rem <= M and alpha <= alpha_e.
Section 2 and Fig 1. The analysis mostly excludes naked-singularity branches except in the shadow extension in Section 4.

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Cite this review

Pith. "Pith review of Quantum parameter-mass induced scalarization of qOS-black hole in the Einstein-Gauss-Bonnet-scalar theory." pith.science (2026). https://pith.science/paper/JKUPFLRA

@misc{pith2026250519450,
  author       = {Pith},
  title        = {Pith review of: Quantum parameter-mass induced scalarization of qOS-black hole in the Einstein-Gauss-Bonnet-scalar theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JKUPFLRA}},
  note         = {Machine review of arXiv:2505.19450}
}

abstract

We obtain quantum parameter ($\alpha$)-mass ($M$) induced spontaneous scalarization of quantum Oppenheimer-Snyder (qOS)-black hole in the Einstein-Gauss-Bonnet-scalar theory with the unknown qOS action. We derive Smarr formula which describes a correct thermodynamics for the bald qOS-black hole. It is turned out that two Davies points of heat capacity are identified with two critical onset mass and quantum parameter for spontaneous scalarization. However, we do not obtain such connections from the Einstein-Gauss-Bonnet-scalar theory with the nonlinear electrodynamics action. Furthermore, the shadow radius analysis of qOS-black hole is performed to distinguish quantum parameter from mass by comparing them with the EHT observation. There is no constraints on the quantum parameter, but new constraints are found for the mass. This work is considered as the first example to show a close connection between thermodynamics of bald black hole and spontaneous scalarization.

Figures

Figures reproduced from arXiv: 2505.19450 by the authors.

**Figure 1.** (a) Two outer/inner horizons r±(M = 1, α) are functions of α ∈ [0, 1.6875], indicating the upper bound as an extremal point. The remnant radius rrem(1, α) = 2√ α/√ 3 > r−(1, α) as function of α represents the transition region including the inner horizon r−(1, α). (b) Two horizons r±(M, α = 1) are function of M ∈ [0.7698, 2], showing the lower bound for the mass of black hole. Here, rrem(0.7698, 1) = 1.1545 > r−(M,… view at source ↗

**Figure 2.** Mass relation E(= m) = ES+Eα and Smarr formula m = 2T S+2Wαα. (a) Smarr formula and mass (internal energy) relation are as functions of α ∈ [0, 1.6875] with M = 1. The remnant mass mrem(α) = 0.7698√ α is displayed. A shaded region of α ∈ (1.6875, 2] is forbidden with M = 1. (b) Smarr formula and mass (internal energy) relation are defined as functions of M ∈ [0.7698, 2] with α = 1. A shaded region of M ∈ [0, 0.7698)… view at source ↗

**Figure 3.** Heat capacity C(1, α)/|CS(1, 0)| with |CS(1, 0)| = 25.13 (a) Heat capacity blows up at Davies point (αD = 1.2835, red dot) which is the critical onset parameter (αc) for spontaneous scalarization. The heat capacity and temperature T(1, α)/0.04 are zero at the extremal point (αe = 1.6875, •). (b) Heat capacity blows up at Davies point (MD = 0.8827, red dot) where the temperature T(M, 1) has the maximum. This point co… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (a) Critical impact parameter b(M = 1, α) as a function of α ∈ [0, 2.8477]. There is one shaded region α ∈[1.6875,2.8477] denoted by α-NS region. A dashed line is at αe = 1.6875 (extremal point). Here, we introduce 1σ and 2σ ranges from EHT observation. (b) Three criti…

**Figure 5.** Figure 5: (a) Graph for sufficient condition αsc(M = 1, λ) with the critical onset parameter α = αc as the lower bound. A dashed line denotes the sufficient condition α = α −λ≫M sc for a large −λ. A top line denotes the extremal point at α = αe as the upper bound. (b) Graph for …

**Figure 6.** Figure 6: (a) Sufficient condition of αsc(M = 1, λ, P = 0.6) with the critical onset parameter α = αc as the lower bound. A dashed line denotes the sufficient condition α = α −λ≫M,P sc = 3.5494 for a large −λ. A top line denotes the extremal point at α = αe = 4.6875 as the upper…

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