REVIEW 3 major objections 4 minor 1 cited by
Analytically approximated scalarized black holes and their thermodynamic stability
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Scalarized black holes carry less horizon entropy than Schwarzschild black holes, so the paper argues they should radiate their scalar hair and decay.
desk verdict A clean continued-fraction fitting exercise for a non-standard scalarized EsGB family, but the entropy-based instability claim is undermined by the nonzero asymptotic scalar and a non-Wald entropy formula. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the continued-fraction parametrization of spherically symmetric black hole metrics in a compactified radial coordinate $x = 1 - r_0/r$. The metric functions are written as $A(x) = x J(x)$ and $\sqrt{A/B} = K(x)$, with $J,K$ and the scalar field represented as truncated continued fractions; the coefficients are fitted to the numerical solution. These analytical expressions $A_p(r), B_p(r), \varphi_p(r)$ reproduce the numerical metric within about two percent outside the horizon. The expressions are then inserted into the horizon first law $S = \int V'(r_+) C(r_+) dr_+$, where $C(r_+)$ is built from the radial stress-energy treated as thermodynamic pressure; the monotonic decrease of the integrand in $p$ is the fact that carries the entropy-ordering conclusion.
What would settle it
Compute the same scalarized solutions' entropy with the standard conserved-charge entropy for higher-curvature gravity instead of the horizon first law; if that entropy can exceed Schwarzschild's for any allowed parameter $p$, the paper's thermodynamic decay claim fails. Alternatively, evolve a scalarized black hole nonlinearly and check whether its scalar hair is actually radiated away: a persistent scalarized solution would contradict the decay prediction.
Extended reading notes
Core claim
On its own terms, the paper establishes that in the quadratic Einstein-scalar-Gauss-Bonnet theory there are no thermodynamically stable scalarized black holes. Labelling each solution by the dimensionless parameter $p = 384\alpha^2\varphi_0^2/r_0^4$, with $p=0$ the Schwarzschild case, it computes the horizon entropy from the horizon first law $S = \int V'(r_+) C(r_+) dr_+$ and finds the integrand decreases as $p$ increases. Hence the scalarized entropy is smaller than the Schwarzschild entropy for every allowed $p$. The paper concludes that the scalarized branch is thermodynamically unstable and should decay to Schwarzschild through scalar-wave emission, and that the existence of negative-energy states for scalar-charged test particles opens the possibility of extracting the energy of the scalar charge.
Load-bearing premise
The load-bearing premise is that the entropy obtained from the horizon first law, $S = \int V'(r_+) C(r_+) dr_+$, is the true entropy of Einstein-scalar-Gauss-Bonnet black holes; if the correct entropy for higher-curvature gravity follows a different law, the sign of the entropy gap, and with it the instability conclusion, could change.
Editorial extensions
If this is right
- Every scalarized solution in the quadratic family is thermodynamically less stable than the equal-mass Schwarzschild black hole.
- The scalarized black holes should decay to Schwarzschild by emitting scalar waves, making the bald solution the thermodynamic endpoint.
- The negative-energy region outside the horizon, an effective scalar ergosphere, permits extraction of energy from the scalar charge by a process analogous to energy extraction from a rotating black hole.
- Second-order continued-fraction expressions are accurate enough, with metric errors below about two percent, to compute thermodynamic quantities without full numerical integration.
Reading between the lines
- If the same horizon-first-law entropy is applied to other Einstein-scalar-Gauss-Bonnet couplings, such as exponential coupling, the sign of the entropy gap could differ; comparing those branches would show whether thermodynamic instability is generic to scalarization or special to the quadratic coupling.
- The analytic metric and scalar profile could be used to compute quasinormal modes and gravitational-wave signatures of scalarized holes, giving a dynamical test of whether the predicted decay to Schwarzschild is visible in a ringdown.
- A direct numerical experiment, starting from a scalarized configuration and evolving it in full general relativity, could measure the emitted scalar flux and check whether its total energy matches the entropy gap computed here.
- If scalarized black holes are indeed transient, they would appear in observations as a short-lived hairy phase that rapidly sheds its hair rather than as stationary hairy remnants, which could affect binary-merger waveform templates.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs approximate analytic solutions for static, spherically symmetric scalarized black holes in Einstein-scalar-Gauss-Bonnet theory with a quadratic coupling φ²R²_GB. Using the Rezzolla-Zhidenko continued fraction parametrization, the authors fix α=0.1 and r0=1, obtain numerical solutions, fit the parametrization coefficients as functions of p=384α²φ0²/r0⁴ (Eqs. 37-45), and provide closed-form expressions for A(r), B(r), and φ(r) in Eqs. (46)-(52). They then compute the Hawking temperature and, via a horizon first law, obtain the horizon entropy S=∫V'(r+)C(r+)dr+ (Eq. 64). The paper's central claim, stated in Sec. V.B and the abstract, is that the horizon entropy of scalarized black holes is always smaller than that of Schwarzschild, implying that scalarized black holes should decay to Schwarzschild by emitting scalar waves and that energy extraction from scalar charge may be possible.
Significance. If the central claim were fully established, the paper would provide a useful analytic demonstration of thermodynamic instability for scalarized black holes in quadratic EsGB theory, consistent with the numerical radial-instability result of Ref. [32], and it would showcase the continued fraction method as a practical tool for hairy black hole thermodynamics. The reported fitting accuracy (relative metric errors below about 2% over most of the exterior, and below 0.5% for the temperature) is a concrete positive achievement. However, the thermodynamic conclusion currently rests on two load-bearing assumptions that are not adequately justified: the asymptotic scalar sector of the constructed solutions, and the validity of the horizon first-law entropy formula for this higher-curvature theory. These issues must be resolved before the stability claim can be accepted.
major comments (3)
- [Secs. III and V.B, Eqs. (24), (45)] The solutions studied have a nonzero asymptotic scalar field φ∞ throughout the fitted range: the text reports φ∞≈−0.047 for α=0.1, r0=1, φ0=0.5, and Eq. (45) gives φ∞≈−0.0354√p+O(p). For the action (1), a Schwarzschild metric with constant nonzero scalar is not a solution of the scalar equation (4), because 2αφ∞R²_GB=96αφ∞M²/r⁶≠0. Hence the p=0 Schwarzschild solution and the p>0 scalarized solutions belong to different asymptotic scalar-field sectors, and the comparison in Sec. V.B of entropies does not have the stated decay interpretation. The standard scalarized branch in Refs. [11-13] satisfies φ(r→∞)=0; the present shooting calculation did not impose this condition, so the approximated family appears to be a different, asymptotically nonvanishing-scalar family. This directly undermines the central claim that scalarized black holes should decay to Schwarzschild through scalar-wave emission.
- [Sec. V.B, Eq. (64)] The horizon entropy is computed from S=∫V'(r+)C(r+)dr+, where C(r+) is defined in Eq. (62), following the horizon first-law approach of Refs. [29-31]. No derivation is provided showing that this first-law entropy coincides with the Iyer-Wald Noether charge entropy for the higher-curvature EsGB action (1). Because the sign of ΔS=S_scalarized−S_Schwarzschild is the entire basis for the thermodynamic stability conclusion, the use of an unvalidated entropy definition is load-bearing. If the correct Wald entropy differs from Eq. (64) by a positive or sign-changing term, the instability conclusion could fail. The authors should either justify Eq. (64) for this theory from first principles or recompute the entropy using the Noether charge formalism.
- [Secs. IV.B and V.B, Eqs. (37)-(45), Fig. 7] The analytic approximants (46)-(52) are built from high-order polynomial fits in p (up to degree 14) with no reported fit uncertainties, and Fig. 3 shows relative metric errors increasing near p→0 and p→1. The entropy integrand V'(r+)C(r+) plotted in Fig. 7 is a functional of these fitted coefficients, and the conclusion 'always smaller' is inferred from the monotonic decrease of that integrand. No error propagation is provided from the fit residuals to the entropy difference, so it is not demonstrated that the sign of ΔS is robust. Given that the claim is universal over 0<p<1, the authors should quantify how fit uncertainty affects the monotonicity and the sign of the entropy difference.
minor comments (4)
- [Abstract and Sec. I] There are typographical errors, including 'Schwarzshcild' in the abstract and 'scalarzied' in Sec. I; the paper would benefit from a careful proofreading pass.
- [Sec. V.C, Eq. (69)] The notation Eeff=−p0 with p0 defined in Eq. (65) is confusing because p0 is already a momentum component; the authors should clarify the relationship between E, p0, and the effective potential.
- [Sec. V.C] The term 'effective scalar ergosphere' is introduced for the region where Eeff<0, but no rigorous definition is given and no relation to an actual ergoregion is established; the Penrose-like energy extraction claim should be labeled as speculative.
- [Eq. (24)] The asymptotic expansion for B(r) contains a D²/(4r²) term, which is unusual for a massless scalar with nonzero φ∞; a brief comment on the consistency of this expansion with the field equations would be helpful.
Circularity Check
No significant circularity: the entropy conclusion is an approximate computation from a numerically fitted metric, not a quantity forced by definition or by a self-citation chain.
full rationale
The paper's derivation chain is: solve the EsGB field equations numerically for scalarized black holes, fit the Rezzolla-Zhidenko continued-fraction coefficients (37)-(45) to those numerical solutions, evaluate the horizon first-law entropy (64) using the externally prescribed horizon-entropy formula of Refs. [29-31], and compare the result with the p=0 Schwarzschild case. No load-bearing step reduces to its own input: the fitted coefficients are fits to solutions of the field equations, not fits to the entropy; the entropy integrand V'(r+)C(r+) is evaluated after the fitting, and its monotonic decrease with p is a derived property rather than an imposed one. The agreement with the independent radial-instability analysis of Ref. [32] is an external consistency check, not a self-citation. The only self-citation, Ref. [33], concerns the rotating scalarization regime and does not justify the present entropy claim. The paper's own observation that phi(infty) is approximately -0.047 rather than 0 is a boundary-condition concern that may affect the physical interpretation of the comparison to Schwarzschild, but it is a correctness issue, not a circular reduction: the entropy statement is not true by construction of the ansatz.
Assumptions & free parameters
free parameters (2)
- Fit coefficients ϵ(p), j1(p), j2(p), k1(p), k2(p), l0(p), l1(p), l2(p) =
Polynomials in p, Eqs. (37)-(44)
- Asymptotic scalar field φ∞(p) =
Polynomial in p, Eq. (45)
assumptions (5)
- domain assumption The Einstein-scalar-Gauss-Bonnet action (1) with quadratic coupling is the correct effective theory for the scalarized black holes.
- ad hoc to paper The continued fraction parametrization (29)-(33) with truncation at second order can represent the numerical solutions to sufficient accuracy.
- domain assumption The horizon first law, S = integral V'(r+) C(r+) dr+ (Eq. 64), gives the horizon entropy for EsGB black holes.
- ad hoc to paper Comparison to Schwarzschild is meaningful even when the scalar field at infinity is nonzero (φ∞ ≈ -0.047).
- domain assumption The numerical solutions obtained by the shooting method for α = 0.1, r0 = 1 are accurate and representative.
invented entities (1)
-
Effective scalar ergosphere
Cite this review
Pith. "Pith review of Analytically approximated scalarized black holes and their thermodynamic stability." pith.science (2026). https://pith.science/paper/64LPUDVU
@misc{pith2026190801346,
author = {Pith},
title = {Pith review of: Analytically approximated scalarized black holes and their thermodynamic stability},
year = {2026},
howpublished = {\url{https://pith.science/paper/64LPUDVU}},
note = {Machine review of arXiv:1908.01346}
}
read the original abstract
It is recently shown that, besides the Schwarzshcild black hole solution, there exist also scalarized black hole solutions in some Einstein-scalar-Gauss-Bonnet theories. In this paper, we construct analytical expressions for the metric functions and scalar field configurations for these scalarized black hole solutions approximately by employing the continued fraction parametrization method and investigate their thermodynamic stability. It is found that the horizon entropy of a scalarized black hole is always smaller than that of a Schwarzschild black hole, which indicates that these scalarized black holes may decay to Schwarzschild black holes by emission of scalar waves. This fact also implies the possibility to extract the energy of scalar charges.
Figures
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Forward citations
Cited by 1 Pith paper
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Compact Objects in Einstein-scalar-Gauss-Bonnet Theory and beyond
A review of compact-object solutions in Einstein-scalar-Gauss-Bonnet and Horndeski theories, emphasizing scalarized black holes, traversable wormholes, and bubble-like particle solutions.
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Reviewed August 14, 2026 · model on record in the stance chip above.
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