Thermodynamic and Dynamical Properties of Phantom Charged Black Holes in 4D Einstein-Gauss-Bonnet Gravity

Bilel Hamil

REVIEW 3 major objections 5 minor 76 references

Thermodynamic and Dynamical Properties of Phantom Charged Black Holes in 4D Einstein-Gauss-Bonnet Gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read An exact charged black hole solution combines four-dimensional Einstein-Gauss-Bonnet gravity with ModMax electrodynamics, screening the electric charge by e^−γ and predicting stable remnants.

desk verdict Same 4D-EGB charged black hole with q² screened by e^{-γ}; competently computed but neither new nor as advertised, and the stability claims sit on the unphysical T<0 branch. read the letter →

arxiv 2601.02717 v3 pith:JNA3KBIO submitted 2026-01-06 gr-qc hep-th

Bilel Hamil This is my paper

classification gr-qchep-th

keywords 4DEinstein-Gauss-BonnetgravityModMaxelectrodynamicschargedblackholesholethermodynamicsremnantsgeodesicmotionquasinormalmodesscalarperturbations

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 sets out to show that adding a conformally invariant nonlinear electrodynamics called ModMax to four-dimensional Einstein-Gauss-Bonnet gravity yields an exact, spherically symmetric charged black hole solution. The central result is a single metric function in which the ModMax parameter γ appears only as a screening factor e^−γ multiplying the electric charge squared, so every later quantity—horizon structure, mass, temperature, orbits, quasinormal modes—inherits a simple dependence on γ. The paper then reads off the physical consequences: black holes have a minimum mass and leave stable remnants, the Hawking temperature peaks at a second-order phase transition, and the innermost stable circular orbit moves outward as γ grows. It also computes scalar quasinormal modes and finds they all decay, which it interprets as linear stability, with the Gauss-Bonnet coupling α lengthening mode lifetimes and γ shortening them. A sympathetic reader would care because these are concrete, in-principle observable signatures of two otherwise hard-to-test ingredients—higher-curvature gravity and duality-invariant nonlinear electrodynamics.

What carries the argument

The load-bearing object is the negative-branch metric function F(r) of Eq. (22), obtained by the dimensional-regularization prescription α→α/(D−4) followed by D→4. It is the single input for every subsequent computation: horizon equation, mass, temperature, entropy, specific heat, free energy, effective potential for geodesics, and scalar-perturbation potential. The ModMax Lagrangian L=S cosh γ−√(S^2+P^2) sinh γ supplies the electromagnetic sector, a conformally invariant, electric-magnetic-duality-preserving nonlinear electrodynamics, but in the purely electric case its entire effect enters through the replacement q^2→q^2 e^{−γ}.

What would settle it

Take the full D-dimensional field equations of the action without the coupling rescaling, carry out the limit D→4 on the equations themselves, and check whether an extra scalar degree of freedom survives; if it does, the metric F(r) given in Eq. (22) is a solution of a different scalar-tensor theory, and the paper's identification with pure 4D Gauss-Bonnet gravity is falsified.

Watch

Extended reading notes

Core claim

The paper's core claim is that the negative-branch metric F(r)=1+r^2/(2α)[1−√(1+4α/r^2(2M/r−q^2 e^{−γ}/r^2))] is an exact static, spherically symmetric black hole solution of regularized 4D Einstein-Gauss-Bonnet gravity sourced by a purely electric ModMax field. The solution reduces to the known ModMax–Reissner-Nordström black hole as α→0 and to the usual 4D-EGB black hole as γ→0, and the parameter combination q^2 e^{−γ} controls the effective charge. From this metric the paper derives a minimum mass M_min=√(q^2 e^{−γ}+α), a Hawking temperature with a maximum and associated divergent specific heat, an ISCO whose radius shrinks with α and grows with γ, and a scalar quasinormal spectrum in whi

Load-bearing premise

Everything rests on the dimensional-regularization limit (rescaling the Gauss-Bonnet coupling by 1/(D−4) and taking D→4) being a valid route to pure-metric four-dimensional Gauss-Bonnet gravity; if that limit secretly requires an extra scalar field, the parameter α does not label a well-defined 4D higher-curvature theory.

Editorial extensions

If this is right

If the solution is exact, the black hole cannot evaporate below the mass M_min=√(q^2 e^{−γ}+α); Hawking radiation halts, leaving a stable remnant.
The Hawking temperature has a maximum and the specific heat diverges at two radii, so the evaporation history and thermodynamic-stability regions are qualitatively different from the Maxwell case.
The ISCO radius decreases with the Gauss-Bonnet coupling α and increases with the ModMax parameter γ, so accretion-disk observations could in principle distinguish these two parameters.
All computed scalar quasinormal modes have negative imaginary parts; if this extends to the full perturbation spectrum, the black hole is linearly stable.
The quasiclassical spectrum shifts systematically with α and γ (higher α → longer-lived oscillations; higher γ → shorter-lived), giving a target pattern for black-hole spectroscopy.

Reading between the lines

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

I note that the document's outer abstract and title describe a 'phantom' electromagnetic field, while the full text consistently works with ModMax electrodynamics; these are physically different theories, and the body's identification should be taken as canonical for the results presented.
Even granting the metric, the claimed linear stability is computed only for a test scalar field; extrapolating to gravitational perturbations would require a separate analysis, and Gauss-Bonnet theories in other dimensions are known to exhibit tensor instabilities.
Since the entropy S=πr_+^2+4πα ln r_+ is independent of γ while mass and temperature depend on γ only through q^2 e^{−γ}, a measurement of any two of these could isolate the ModMax screening from the bare charge.
The eikonal QNM damping tends to a constant independent of ℓ, consistent with the geometric-optics relation to the photon sphere; comparing the photon-sphere radius inferred from QNMs with the ISCO inferred from accretion could provide a consistency test of the e^{−γ} screening.

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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

Quick take: this is a competent parameter scan of a known solution, not a new one. The metric (22) is the standard 4D-EGB-Reissner-Nordström solution with q² replaced by q²e^{-γ}; the paper's own introduction admits the ModMax parameter screens the charge. Every subsequent formula—mass, temperature, entropy, specific heat, ISCO, QNMs—depends on γ only through that factor. So the "combined effects" are just the old effects at a rescaled charge.

Credit where due: the derivation of F(r) is clean, the mass and temperature are internally consistent, entropy from dM/T integrates to the expected πr+² + 4πα ln r+, and the WKB-Padé vs Pöschl-Teller comparison in Table 4 agrees to ~0.1% at high l. That cross-check is genuine evidence the numerics are not garbage.

Now the soft spots.

First, the title and abstract say "phantom electromagnetic field" and promise accretion efficiency, no critical behavior, and time-domain verification. The manuscript is about ModMax electrodynamics, contains second-order phase transitions, has no accretion analysis, and no time-domain code. That is a serious packaging problem.

Second, the foundational premise: the Glavan-Lin D→4 limit is adopted without engaging the substantial literature arguing the limit is not a well-defined purely metric theory. If that critique holds, α in Eq. (22) is not uniquely defined. The paper cites equivalence to Horndeski but never shows Eq. (22) solves that theory. This is the load-bearing assumption; it needs at least a paragraph.

Third, the claimed global stability of small black holes (Sec. 6, Fig. 5) is for r+ < rF ≲ 0.2, which is entirely inside r+ < r_root where T < 0 from Eq. (29). Those are unphysical configurations. The "stable remnant" story is tied to the same negative-temperature branch.

Fourth, "linear stability confirmed" is an overclaim: positivity of a scalar-field effective potential plus a few WKB modes does not establish linear stability of the spacetime.

Minor items: Table 3 has an l=0, μ=0.2 mode with Re ω < μ, which is not a standard QNM; numerical tables have no convergence checks or error bars; the WKB reliability condition is cited to [107], which does not exist in the reference list (which ends at [76]).

Who this is for: someone compiling 4D-EGB black hole catalog entries or needing quick parameter dependencies for ISCO/QNM shifts. A serious referee could fix the overclaims, remove the phantom mislabeling, and pin down the Glavan-Lin caveat. I'd send it to review rather than desk reject, mainly because the algebra and numerics are checkable and the cross-check is real.

Referee Report

3 major / 5 minor

Summary. The paper derives a static, spherically symmetric charged black hole solution in four-dimensional Einstein-Gauss-Bonnet gravity coupled to ModMax nonlinear electrodynamics, Eq. (22), using the Glavan-Lin regularization α→α/(D−4). From this metric function the authors compute the horizon structure, mass, Hawking temperature, entropy, specific heat, Helmholtz free energy, geodesic motion and ISCO parameters, and the quasinormal spectrum of a massive scalar field using sixth-order WKB with Padé approximants and the Pöschl–Teller approximation. The paper reports a minimum-mass remnant, a temperature maximum, modified ISCO radii as α and γ vary, and QNM frequencies that depend on α, γ, and μ, and concludes linear stability from the negativity of Im ω.

Significance. If the underlying 4D EGB framework is accepted, the paper provides a complete and internally consistent thermodynamic and dynamical analysis of a new exact solution, with a useful parameter study. The algebraic derivation of the metric function is standard and correct conditional on the regularization, and the entropy follows self-consistently from dM/T. A genuine strength is the high-ℓ agreement between the WKB-Padé and Pöschl–Teller frequencies in Table 4 (agreement to ~0.1%), which gives confidence in the QNM numerics. The work is less significant if the Glavan-Lin limit is not a well-defined pure-metric theory, because the same expressions can be reinterpreted in a well-defined scalar-tensor theory only with extra work. The paper does not engage the substantial literature questioning the D→4 limit, and it contains an error in the local thermodynamic stability analysis. These issues currently prevent the results from being fully established.

major comments (3)

[Sec. 2, Eq. (18)] The D→4 limit of the Gauss-Bonnet term is not unique for a purely metric theory; a large body of work (e.g., Gurses–Sisman–Tekin, Arrechea–del Rio–Senovilla, Fernandes–Mulhem–et al.) shows that the limit introduces a scalar degree of freedom and depends on the regularization scheme. The paper cites Glavan–Lin [17] and the Horndeski equivalence [21] but does not demonstrate that Eq. (22) is a solution of that scalar-tensor theory, nor does it address the counterarguments. Since every subsequent result is computed from Eq. (22), the claim that α labels a well-defined 4D-EGB theory is load-bearing and currently unsupported.
[Sec. 3, Eqs. (37)-(38) and Fig. 4] The paper claims two divergence points for the specific heat, r1 and r2, and two locally stable physical regions (0<r+<r1 and rroot<r+<r2). However, for Q=e^{-γ}q²>0 and α>0, the expression (9Q+11α)(Q+3α) is always greater than (3Q+5α)², so the quantity inside the square root in Eq. (37) yields r1²<0. Thus r1 is always imaginary; the specific heat has only one positive divergence, r2. Consequently, the stated stable regions are incorrect, and the claim that small black holes with 0<r+<r1 are physical contradicts the earlier statement that r+<sqrt(α+e^{-γ}q²) have negative temperature. This is a substantive error in the thermodynamic analysis.
[Sec. 3, Eq. (40) and Fig. 5] The Helmholtz free energy is used to conclude that small black holes are globally stable and large black holes are not. However, the spacetime is asymptotically flat (F→1 as r→∞), and the sign of F=M−TS relative to zero is not a valid global-stability criterion in the canonical ensemble without a reference background; e.g., Schwarzschild black holes have negative F but are thermodynamically unstable. The global-stability conclusion is therefore not established and needs either a proper ensemble comparison or a demonstrated reference state.

minor comments (5)

[Title and Abstract] The title refers to 'phantom charged black holes', while the full text consistently uses 'ModMax' electrodynamics, which is not a phantom field. The abstract in the header also mentions 'absence of critical thermodynamic behavior', a claim not developed in the body. These inconsistencies should be resolved.
[Sec. 3, Eq. (28)] The surface gravity is defined as κ=(1/2π)F'(r)|_{r+}, which would give T=F'/(4π²), inconsistent with the standard T=F'/(4π) used in Eq. (29). The prefactor should be 1/2, not 1/2π.
[Sec. 5, Eqs. (55)-(56)] The symbol G(r) is reused for the metric function F(r), whereas G was earlier defined as F−1 in Eq. (20). This causes confusion in the radial wave equation and effective potential.
[Sec. 5.2] The conclusion of 'linear stability' is based on WKB results for a finite set of low-lying modes and on the positivity of the effective potential. This supports stability for the modes considered but is not a proof; the wording should be softened.
[General] There are numerous typographical and reference-quality issues (e.g., 'gravty', 'tthe', Ref. [8] dated 2025 instead of 2005, Ref. [12] incomplete) that should be corrected in a revised version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the metric is derived algebraically from the stated action and all subsequent results are direct computations from it.

full rationale

The paper's central claim is the metric function Eq. (22), obtained by substituting a static, spherically symmetric ansatz and a purely electric ModMax field into the action, applying the stated D→4 regularization, and solving the resulting ordinary differential equation Eq. (19). No input quantity is defined in terms of an output: M, q, α, and γ are fixed inputs, and the reported thermodynamic quantities, geodesic relations, ISCO values, and scalar quasinormal frequencies are all computed directly from that metric function. The entropy, for example, follows from integrating dM/T rather than being imposed beforehand, so there is no fitted-input-called-prediction or self-definitional loop. The self-citations (Refs. [38], [61], [72-75]) appear only in literature surveys or as methodological references for WKB methods; none carries the central derivation. The Glavan-Lin D→4 limit is indeed contested in the literature, but a contested premise is a correctness risk, not circularity. Likewise, the abstract's claim of verification by time-domain evolution is not substantiated in the text, but that is an evidentiary gap, not a circular reduction. Thus no circular step is present and the derivation chain is self-contained in the relevant sense.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on the contested Glavan-Lin regularization, the choice of the ModMax model, and the standard first-law/WKB machinery. No new particles or fields are introduced; alpha and gamma are borrowed from prior literature. The only genuinely ad hoc element is the unexamined status of the D -> 4 limit, which the paper treats as settled.

free parameters (5)

GB coupling alpha = scanned: 0.1-1.0 (tables); 0.1-0.8 (ISCO)
Higher-curvature coupling of 4D-EGB; chosen by hand for each figure and table, not measured or derived. It controls horizon structure, remnant mass, the logarithmic entropy term, and QNM shifts.
ModMax parameter gamma = scanned: 0-1.0 (tables); 0.1-0.8 (ISCO)
Nonlinear electrodynamics parameter taken from prior literature; tuned by hand. In every equation it appears only as e^{-gamma} multiplying q^2.
electric charge q = q = 1 in all numerical work
Set to unity in Tables 1-4 and all figures; results are scans at fixed charge.
scalar field mass mu = 0 and 0.2 (Table 3); 0-0.5 (Fig. 13)
Chosen by hand for the QNM study; affects the potential asymptotics and produces the omega_R < mu row for l=0, mu=0.2.
black hole mass M = M = 1 (normalization)
Unit normalization for the scale-free solution; listed for completeness, not a fitted quantity.

assumptions (6)

domain assumption Glavan-Lin D -> 4 regularization (alpha -> alpha/(D-4), then D -> 4) yields a consistent 4D theory of Gauss-Bonnet gravity.
Adopted in Sec. 1 and Eq. (18). Contested in the literature: the naive limit is not well-defined as a pure metric theory and carries a hidden scalar degree of freedom. The paper cites [17] only and does not address the critique. If false, the alpha-dependence of all results is not a prediction of an actual 4D GB theory.
domain assumption Purely electric ModMax configuration (P = 0) with gauge potential A_t = epsilon(r) captures the physically relevant sector.
Sec. 2, after Eqs. (8)-(12); restricts to electric charge and excludes magnetic charge or P-dependent effects.
domain assumption ModMax Lagrangian (Eq. 3) with gamma >= 0 is the correct conformal and duality-invariant extension of Maxwell theory.
Taken from [47-49]. All novel results are consequences of the specific combination q^2 e^{-gamma}, so the content depends on this model choice.
standard math Entropy follows from dM = T dS at fixed charge, giving S = pi r+^2 + 4 pi alpha ln r+.
Sec. 3, Eqs. (31)-(33). The logarithmic term is the known 4D-EGB entropy from the prior literature [17, 38]; gamma drops out. No independent derivation is given in this paper.
domain assumption Sixth-order WKB with Pade approximants reliably computes the scalar QNM spectrum when l >= n, and the series has converged at sixth order.
Sec. 5.2. Known WKB caveats for small l are acknowledged only partially; no error bars or convergence checks are provided. The Pade-Poschl-Teller agreement in Table 4 is partial support, mainly at high l.
standard math The negative branch of the metric is physical; the positive branch is discarded as graviton-unstable.
Sec. 2, Eqs. (22)-(24), citing [63]; standard practice in the 4D-EGB literature.

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Cite this review

Pith. "Pith review of Thermodynamic and Dynamical Properties of Phantom Charged Black Holes in 4D Einstein-Gauss-Bonnet Gravity." pith.science (2026). https://pith.science/paper/JNA3KBIO

@misc{pith2026260102717,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic and Dynamical Properties of Phantom Charged Black Holes in 4D Einstein-Gauss-Bonnet Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNA3KBIO}},
  note         = {Machine review of arXiv:2601.02717}
}

read the original abstract

We present charged black hole solutions of regularized four dimensional Einstein Gauss Bonnet gravity coupled to a phantom electromagnetic field. We investigate the combined effects of higher curvature corrections and phantom charge on the horizon structure, thermodynamics, geodesic motion, and scalar perturbations. The phantom sector exhibits properties that differ qualitatively from those of the ordinary Maxwell case, including the absence of critical thermodynamic behavior and persistent thermal instability. Circular geodesics and accretion efficiency are also significantly modified. Quasinormal modes are computed using the sixth-order WKB approximation and verified through time-domain evolution. The results reveal characteristic signatures of both the Gauss Bonnet coupling and phantom electrodynamics, while confirming linear stability against scalar perturbations.

Figures

Figures reproduced from arXiv: 2601.02717 by the authors.

**Figure 1.** Metric function F (r) as a function of the radial coordinate r. Parameters: M = 1, q = 1. 3 Thermodynamics To investigate the thermodynamic properties of black holes in ModMax–EGB gravity, we first express the black hole mass M in terms of the event horizon radius r+, the electric charge q, the ModMax parameter γ, and the GB coupling constant α . This can be achieved by solving the horizon equation F (r+) = 0, which… view at source ↗

**Figure 2.** Black hole mass M versus event horizon radius r+ for q = 1. The Hawking temperature is defined as T = κ 2π , (28) where κ is the surface gravity defined as κ = 1 2π ∂ ∂rF (r) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** The Hawking temperature as a function of the event horizon [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: The specific heat as a function of the event horizon [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: The Helmholtz free energy as a function of the event horizon [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: The effective potential as a function of radial distance for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: The effective potential of massive particle for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Trajectories of massive particle for different initial velocities. Parameters: [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: The radial profiles of E of the circular orbits.Parameters: M = 1, q = 1. Determining the innermost stable circular orbit (ISCO), where the maximum and minimum of the effective potential coincide, is of critical importance. This requires an additional condition for the…

**Figure 10.** Figure 10: The Variation of the scalar effective potential as a function of radial distance. Parameters: [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: QNM dependence on GB coupling α. Parameters: ℓ = 5, M = 1, q = 1, γ = 0.8 and µ = 0.2. Next, we compute the quasinormal modes as functions of the ModMax parameter γ for ℓ = 5, n = 0, α = 0.1 , q = 1 and m = 1 as shown in [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: QNM dependence on ModMax Parameter γ. Parameters: ℓ = 5, M = 1, q = 1, α = 0.1 and µ = 0.2. As shown in Fig.10c, increasing the field mas µ, causes the peak of the potential to grow. Beyond a certain value, however, the height of this peak becomes lower than the asymp…

**Figure 13.** Figure 13: QNM dependence on the scalar field mass µ. Parameters: ℓ = 5, M = 1, q = 1, α = 0.1 and γ = 0.5. 5.3 Posch-Teller method In the Pöschl-Teller method, the effective potential is approximated by the Pöschl-Teller function [76], which allows Eq. (58) to be solved analyti…

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