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REVIEW 4 major objections 6 minor 112 references

A power-law Yang–Mills charge relaxes the weak gravity bound below unity while cosmic censorship survives every tested charged-scalar absorption.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

T0 review · deepseek-v4-flash

2026-08-02 12:17 UTC pith:K2RN4EMH

load-bearing objection The paper's main WGC/WCCC results are invalidated by a bulk-vs-boundary charge dictionary error; only the thermodynamics and island sections hold up. the 4 major comments →

arxiv 2607.16216 v1 pith:K2RN4EMH submitted 2026-06-04 hep-th gr-qc

Holographic CFT Thermodynamics as a Bridge Between the Weak Gravity and Weak Cosmic Censorship Conjectures for EMPYM--AdS Black Holes

classification hep-th gr-qc MSC 83C5783C4783E05 PACS 04.70.-s04.70.Dy11.25.Tq
keywords weak cosmic censorship conjectureweak gravity conjectureEMPYM–AdS black holesholographic CFT thermodynamicssuperradiancecharged scalar perturbationsPage timeisland prescription
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to establish that the weak cosmic censorship conjecture survives charged-scalar absorption across the Einstein–Maxwell–power–Yang–Mills–AdS (EMPYM–AdS) black-hole family, and that the weak gravity conjecture enters only as a selector of the direction of evolution toward or away from extremality. The central quantitative claim is that the effective charge-to-mass threshold for a scalar obeying the weak gravity bound is lowered below the usual Reissner–Nordström value of unity once the non-Abelian Yang–Mills charge is switched on, with the threshold set by r_min/Q̃. A reader should care because the two conjectures constrain any consistent quantum gravity, and here they are shown to coexist rather than conflict: the horizon is protected to first order, and the relaxed threshold is tied to observable photon-sphere and shadow data. The paper also extends the same thermodynamic description to the island prescription, producing an explicit Page time written in terms of the full set of black-hole parameters.

Core claim

On the paper's own terms, the central discovery is a structural result: for an extremal EMPYM–AdS black hole, the leading change of the metric minimum under charged-scalar absorption is −2(ω−q̃_s φ̃_h)² r_h dt, non-positive for every frequency, so the minimum never lifts above zero and the horizon survives. The near-extremal case obeys the same non-positive first-order shift on top of a small negative deficit. The superradiance condition ω < q̃_s φ̃_h, combined with mass–energy equivalence μ_s = ω, becomes q̃_s/μ_s > r_min/Q̃. The Yang–Mills term pulls r_min inward, so this threshold falls below the Reissner–Nordström bound of unity when the non-Abelian charge is on. The weak gravity conject

What carries the argument

The load-bearing object is the metric function f(r) of the EMPYM–AdS solution, whose minimum r_min controls both tests: the censorship test tracks f_min through absorption, and the weak-gravity threshold is r_min/Q̃. The argument is carried by the horizon flux ratio of a charged massive scalar, dE/dt = ω(ω−q̃_s φ̃_h) r_h² and dQ/dt = q̃_s(ω−q̃_s φ̃_h) r_h², whose common factor (ω−q̃_s φ̃_h) squares in the shift of the metric minimum, making the extremal shift a perfect square. Around this sits the extended holographic first law dE = T̃ dS + φ̃ dQ̃ + ψ̃ dq̃ + μ dC − p dV, with a closed Euler relation that the paper verifies numerically, and the island-prescription Page time t_P expressed in t

Load-bearing premise

The bridge between the two conjectures rests on treating a scalar's frequency as its mass (μ_s = ω), so that superradiance becomes a charge-to-mass bound; if that identification is not granted, the relaxed weak-gravity threshold is only a statement about scalar scattering.

What would settle it

Run a full numerical evolution, rather than the first-order expansion, of a charged massive scalar impinging on a near-extremal EMPYM–AdS black hole with γ and C held fixed; if f(r_min) ever becomes positive at order dt while dγ = dC = 0, the non-positive perfect square in Eq. (4.11) is violated and the censorship claim collapses. A second check is to evaluate r_min/Q̃ directly from Eq. (5.3) for any reported parameter pair: if a q>0 configuration yields r_min/Q̃ ≥ 1 while the paper claims <1, the relaxed-threshold statement fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For every extremal and near-extremal configuration examined, charged-scalar absorption leaves f_min ≤ 0 to first order, so the horizon survives and weak cosmic censorship is respected in this family.
  • A scalar satisfying the superradiance condition has q̃_s/μ_s > r_min/Q̃, and switching on the Yang–Mills charge makes r_min/Q̃ < 1, so the model tolerates charge-to-mass ratios below the Reissner–Nordström value.
  • The weak gravity conjecture plays a directional role: emission of a high-charge particle drives the hole away from extremality, toward the protected side, rather than toward naked singularity.
  • The relaxed threshold moves the photon-sphere radius and shadow size in a γ- and q-dependent way, giving a potential observational handle on the bound.
  • The Page time for these black holes is set by the same thermodynamic data (charges, exponent γ, pressure), so information recovery is governed by black-hole chemistry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the relaxed threshold holds beyond first order, one would expect second-order gedanken-experiment corrections to still respect censorship for small fluxes; testing that would tell whether the perfect-square protection is an artifact of the linearized flux bookkeeping.
  • The identification μ_s = ω is a kinematic choice; a natural extension is to substitute a full quantum-field-theoretic mass-shell condition, which could shift the threshold and test how robust the 'relaxed weak gravity bound' really is.
  • The same machinery could be applied to rotating or higher-dimensional EMPYM–AdS black holes, where frame dragging and different sphere topologies may sharpen or erase the relaxation below unity.
  • The predicted shadow shift is measurable in principle: a horizon-scale image of an EMPYM–AdS black hole could constrain γ and q, and thereby indirectly test whether the charge-to-mass bound is truly below one.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies EMPYM-AdS black holes in holographic CFT thermodynamics. It builds an extended first law with the central charge and CFT volume as independent variables, derives equations of state and an Euler relation, and checks them numerically. A charged massive scalar perturbation gives horizon fluxes, from which the superradiance threshold is converted into the charge-to-mass bound q̃_s/μ_s > r_min/Q̃. Expanding the metric function, the paper claims that absorption keeps f_min ≤ 0, so WCCC holds, and that the Yang-Mills charge lowers the threshold below the Reissner-Nordström value. The same setup is used to derive an island-prescription Page curve and an analytic Page time.

Significance. The paper is clearly organized, and the algebraic part of the thermodynamics is self-consistent; the finite-difference checks of the equations of state and the limiting cases q→0 and γ→1 are useful. If the central claims were correct, the paper would provide an explicit family in which a nonlinear gauge sector relaxes the WGC bound while protecting the horizon, together with a Page time expressed in extended thermodynamic variables. However, the headline WCCC and WGC results currently rest on a charge-dictionary inconsistency (Secs. 4–5) and on identifying superradiance with the WGC via μ_s=ω (Sec. 5.1), so the significance is conditional pending a correct derivation.

major comments (4)
  1. [§4.2, Eq. (4.3); dictionary (2.7)] The derivative ∂f/∂Q̃ is computed as if f contained Q̃²/r². But Eq. (2.7) gives Q=Q̃/(2√C), so the Maxwell term is Q̃²/(4C r²). The correct derivatives are ∂f/∂Q̃ = Q̃/(2C r²) and ∂f/∂C = −r²/(4C²) − Q̃²/(4C² r²). Consequently Eq. (4.10) is not the perfect square claimed; the scalar contribution reads (ω−q̃_s φ̃_h)[−2ω r_h + Q̃ q̃_s/(2C)]dt and is not sign-definite. The WCCC conclusion in §4.2 and its near-extremal counterpart in §4.3 rest on this artifact. The same variable swap invalidates the extremal computation and Table 8 in §5.1: for q=0, ℓ=50, C=625, the RN-AdS extremal radius is r_ext≈Q̃/50, not 0.9994Q̃.
  2. [§5.1, Eqs. (5.1)–(5.8)] The advertised WGC bound is obtained by substituting μ_s=ω into the superradiance inequality. This is a kinematic rewriting of test-field fluxes, not a derivation that a WGC-satisfying state exists in the UV completion; without that identification, the bound is only a statement about scalar scattering. In addition, r_min/Q̃ in Eq. (5.8), Fig. 7, and Table 8 is computed with Q̃ treated as the bulk charge (see previous comment), so the numerical 'relaxed threshold' and its Yang-Mills dependence are not trustworthy. The authors should either provide an independent derivation of the WGC statement or explicitly reframe the result as the charge-to-mass form of the superradiance condition.
  3. [§2.5, Eqs. (2.8), (2.19), Tables 3–4] Equation (2.19) is just Eq. (2.8) rearranged, so it is an identity by construction; Table 4 consequently verifies a tautology. The finite-difference checks in Table 3 are legitimate consistency checks of the closed-form derivatives, but the text's claim that a 'closed Euler relation follows' and is 'confirmed to machine precision' overstates the content. Please state explicitly that μ is defined so that the Euler relation is exact, or derive μ independently before checking Eq. (2.19).
  4. [§3, Eq. (3.13), Table 6] The dictionary (2.7) gives φ̃=φ/R=Q/(R r_h), whereas the text uses φ̃_h=Q̃/r_h. This is the same bulk/boundary charge conflation as in Sec. 4, and it affects the superradiance threshold and the flux equations. The paper should specify which charge variable enters f(r), φ_h, and the probe fluxes, and keep that convention consistent throughout.
minor comments (6)
  1. [§2.1, Eqs. (2.2)–(2.3)] The symbol Q denotes both the Maxwell charge and the Yang-Mills amplitude, which invites the dictionary error noted above. Distinct symbols (e.g., Q_M and Q_YM) would remove the ambiguity.
  2. [Table 6] The entries use φ̃_h=Q̃/r_h without comment; please align the table with the corrected dictionary used in the text.
  3. [Captions] There are typos in table captions ('T able 1', 'T able 5'); please unify table and figure references.
  4. [References] Several references are duplicated ([13]–[15], [18]–[20]); please consolidate the bibliography.
  5. [§7.3, Eq. (7.11)] The island-boundary expression χ = c e^{−κr∗(b)}/(12πr_h²) is introduced without showing the extremization steps; a short derivation would improve readability.
  6. [Abstract/§5.2] The abstract says the WGC fixes only the direction of evolution, while §5.1 also claims a quantitative threshold; please clarify the logical status of these two statements.

Circularity Check

3 steps flagged

Euler relation and WGC bound are identities by construction; the WCCC/WGC numbers rest on treating the boundary charge Q̃ as the bulk charge Q, contradicting the paper's own dictionary.

specific steps
  1. self definitional [Sec. 2.3 and 2.5, Eqs. (2.8), (2.13), (2.19)]
    "The chemical potential conjugate to C is µ= 1/C (E− ˜T ˜S− ˜ϕ ˜Q− (3γ−1)/(2γ) ˜ψ˜q). ... Using the closed forms (2.14)–(2.17) together with the definition (2.8), one finds the Euler relation E= ˜T S+ ˜ϕ ˜Q+ (3γ−1)/(2γ) ˜ψ˜q+µC."

    Equation (2.19) is exactly Eq. (2.8) multiplied by C: µ is defined as the residual combination (E−T̃S−φ̃Q̃−...)/C, so the 'derived' Euler relation is a tautology. The independent, non-circular content would be proving that this µ coincides with ∂E/∂C in Eq. (2.17). The numerical checks in Tables 3–4 verify that consistency, but they do not turn the definitional identity into a derivation. The paper nevertheless presents Eq. (2.19) as a closed Euler relation and as the integrated form of the first law.

  2. self definitional [Sec. 5, Eqs. (5.1), (5.5), (5.8)]
    "Mass–energy equivalence then gives µs=ω ... µs<˜qs ˜ϕh ⇐⇒ ˜qs/µs>1/˜ϕh. ... the effective bound becomes ˜qs/µs>rmin/˜Q (extremal), ˜qs/µs>rmin/˜Q(1+ε/rmin) (near-extremal)."

    Equation (5.8) is Eq. (5.1) rewritten with ω replaced by µs and with φ̃h=Q̃/rh, rh≈rmin. The advertised 'effective WGC threshold' is therefore a re-arrangement of the superradiance inequality under the imposed identification µs=ω; no independent WGC input or derivation is added. The paper calls the result a renormalized WGC bound, but by its own equations it is the superradiance condition itself, so the 'relaxed WGC threshold' has no content beyond the kinematic identification.

  3. other [Sec. 2.3 Eq. (2.7); Sec. 4 Eq. (4.3); Sec. 5 Table 8]
    "˜Q= Qℓ/√G = 2Q√C ... ∂f/∂ ˜Q |_{rmin}=2 ˜Q/r²_min ... The q= 0 column reproduces the Reissner–Nordström bound of unity."

    Under the dictionary (2.7), the Maxwell term in f(r) is Q̃²/(4C r²), so ∂f/∂Q̃ = Q̃/(2C r²), not 2Q̃/r², and ∂f/∂C contains an additional −Q̃²/(4C²r²) contribution. The sign-definite cancellation in Eq. (4.11) and the RN-unity entries in Table 8 are obtained by using Q̃ directly as the bulk charge Q. At ℓ=50, C=625, the correct extremal ratio is r_ext/Q̃ = Q/Q̃ = 1/(2√C)=0.02, not 0.9994. Thus the WCCC and WGC quantitative conclusions reduce to an input identification (Q̃=Q) that contradicts the holographic dictionary the paper explicitly adopts.

full rationale

The paper's central WCCC computation in Sec. 4 is, in structure, an independent first-order calculation: it expands f about its minimum and follows the shift under charged-scalar fluxes. However, the printed sign-definite perfect square in Eq. (4.11) depends on the derivative ∂f/∂Q̃=2Q̃/r², which is incompatible with the paper's own dictionary Q̃=2Q√C. With the dictionary-correct derivative, the scalar contribution is not sign-definite, so the advertised censorship conclusion is not established by the equations presented. Similarly, the WGC threshold in Eq. (5.8) is not a separate derivation: it is the superradiance inequality (5.1) rewritten via the assumed mass–energy equivalence µs=ω. The Euler relation (2.19) is also definitional, since µ is defined in Eq. (2.8) as the precise leftover that makes the relation true; the numerical verification confirms consistency of two definitions of µ but does not add independent content. I checked for self-citation circularity: the paper cites earlier work by the same groups (e.g., Refs. [104]–[107]) for context and comparison, but the main argument does not rely on a self-citation chain or an imported uniqueness theorem, so this is not the dominant issue. The dominant issue is that the two headline quantitative results reduce by construction to their inputs: the Euler relation reduces to the definition of µ, and the WGC bound reduces to the superradiance condition. Combined with the charge-dictionary conflation that forces the WCCC and RN-bound numbers, the burden is substantial but not total: the flux framework and the first-order expansion are legitimate techniques, and the paper is transparent about the µs=ω step. A score of 7 reflects partial circularity and a load-bearing internal inconsistency, rather than a fully tautological derivation.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claims rest on the cited EMPYM-AdS solution, the standard holographic dictionary, the identification of the superradiance condition with the WGC, first-order backreaction with dγ=dC=0, and the island formula. No new entities are introduced. The model parameters γ, q, Q, M, ℓ are inputs; γ and q are surveyed by hand, not fitted.

free parameters (3)
  • γ (power-Yang-Mills exponent)
    Chosen by hand in the survey (0.9–2.0); controls the Yang-Mills amplitude and the r_min/Q̃ threshold; not fitted to data.
  • q (Yang-Mills charge)
    Chosen by hand (0–1.1); controls the lowering of r_min/Q̃; not determined by the theory.
  • q_s, μ_s (probe scalar charge and mass)
    Parameters of the test field; the WGC-type bound is a relation between them and the background, not a prediction of their values.
axioms (5)
  • domain assumption The EMPYM-AdS metric function (2.2)-(2.3) is the correct solution of the chosen theory.
    Taken from refs. [109] and [118]; no action variation or independent derivation is given in this paper.
  • domain assumption Holographic dictionary: C=ℓ²/4G, V=4πR², and rescalings (2.7) are the correct bulk-boundary map.
    Standard in CFT thermodynamics; used to derive the first law and equations of state.
  • ad hoc to paper The superradiance condition for a probe scalar is equivalent to the WGC when μ_s=ω.
    This identification is not derived from quantum gravity and is the load-bearing step for the WGC claim; see Eqs. (5.1)-(5.8).
  • domain assumption First-order backreaction suffices; dγ=dC=0 during absorption.
    The WCCC conclusion in Sec. 4 truncates the expansion at first order and treats γ,C as fixed; second-order terms are dropped.
  • domain assumption Island formula (7.7) and its extremization are valid for this spacetime.
    Borrowed from [110,111]; applied without independent derivation.

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The weak cosmic censorship conjecture (\WCCC) and the weak gravity conjecture (\WGC) sit at the heart of how singularities and charged matter are constrained in any consistent theory of quantum gravity. We study both conjectures for Einstein--Maxwell--power--Yang--Mills--AdS (\EMPYM) black holes, a family in which a non-Abelian power-law Yang--Mills field, a Maxwell field, and a negative cosmological constant act together. Working in holographic conformal field theory (CFT) thermodynamics, we treat the central charge $C$ and the CFT volume $\Vol$ as independent variables and obtain an extended first law that carries variations of the Yang--Mills charge and of the nonlinearity exponent $\gamma$. A closed Euler relation follows, and we check it together with the full set of equations of state to machine precision. The perturbation is a charged massive scalar minimally coupled to the Maxwell sector. Its horizon energy and charge fluxes fix the superradiance threshold $\omega<\tilde{q}_s\tilde{\phi}_h$, and the mass--energy relation $\mu_s=\omega$ turns this into the bound $\tilde{q}_s/\mu_s>r_{\min}/\tilde{Q}$. Expanding the metric function $f(r)$ about its minimum for extremal and near-extremal configurations, we find that absorption keeps $f_{\min}$ non-positive, so the horizon survives. The Yang--Mills sector lowers the effective charge-to-mass threshold below the Reissner--Nordstr\"om value of unity, while a local stability analysis locates a Davies point in the heat capacity. Across the parameter range examined the \EMPYM\ family respects cosmic censorship under scalar perturbations, with the \WGC\ fixing only the direction of evolution toward or away from extremality.

Figures

Figures reproduced from arXiv: 2607.16216 by Behnam Pourhassan, Izzet Sakalli, Saeed Noori Gashti, Saheb Soroushfar.

Figure 1
Figure 1. Figure 1: Metric function f(r) versus r/M for the EMPYM–AdS black hole. γ Q rmin fmin rh (outer) horizons 0.90 0.81832 0.36991 −2.50260 2.00860 2 1.00 0.49000 0.31984 −2.12340 1.73400 2 1.30 0.22138 – – 1.49770 1 1.60 0.14238 – – 1.43010 1 2.00 0.09604 – – 1.39520 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: it climbs out of a small-entropy regime, where the Yang–Mills and charge terms of Eq. (2.14) dominate, and settles onto the large-entropy branch fixed by the √ C and S 1/2 terms. Past S ≃ 3 the four curves crowd into a single line width, which is why the inset magnifies the minimum near S ∈ [8, 20]. The ordering there is not arbitrary: the leading Yang–Mills piece scales as S 1/2−2γ , so a larger γ depress… view at source ↗
Figure 3
Figure 3. Figure 3: Yang–Mills potential ψ˜ versus ˜q for γ = 0.90, 1.00, 1.30, 1.60 at S = 6, C = 4, Q˜ = 1, R = 2, from Eq. (2.16). 5 10 15 20 25 S 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 ¹ ° = 0:90 ° = 1:00 ° = 1:30 ° = 1:60 12.5 15.0 17.5 20.0 22.5 0.182 0.184 0.186 0.188 0.190 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Chemical potential µ conjugate to the central charge versus entropy S for γ = 0.90, 1.00, 1.30, 1.60 at C = 4, Q˜ = 1, ˜q = 0.8, R = 2, from Eq. (2.17). The inset magnifies the peak region S ∈ [11, 23], where the curves order by γ. S C Q˜ q˜ γ T˜ ψ˜ µ 4 4 1.0 0.8 1.0 0.163365 0.088623 0.118468 6 4 1.2 0.9 1.3 0.149936 0.016413 0.142850 9 6 1.5 1.0 1.6 0.122024 0.001446 0.117096 12 8 1.8 1.2 0.9 0.100730 0.… view at source ↗
Figure 5
Figure 5. Figure 5: Heat capacity CQ˜q˜ from Eq. (2.21) versus entropy S for γ = 0.90, 1.00, 1.30, 1.60 at C = 4, Q˜ = 1, ˜q = 0.8, R = 2. The divergence near S ≃ 1 is a Davies point separating the small-S stable branch from the large-S unstable branch. The inset magnifies the sign-change region S ∈ [0.6, 2.6]. In [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Extremal censorship test (left) and the near-extremal horizon structure (right). That tangency in Fig. 6b is the geometric content of the extremal condition f(rext) = f ′ (rext) = 0. Since the curve meets the axis with zero slope, a small absorbed mass moves it down rather than up, keeping fmin ≤ 0 and protecting the horizon. The heavier curve crosses zero twice, which confirms the two-horizon count of [P… view at source ↗
Figure 7
Figure 7. Figure 7: Effective WGC threshold (left) and the censorship phase diagram (right). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Normalised entanglement entropy S(R)/(2SBH) as a function of time t for EMPYM–AdS black holes at rh = 1.5, Q = 0.2, ℓ = 3, c = 1. (a) Maxwell charge Q = 0.1, 0.3, 0.5 at γ = 1. (b) Yang–Mills exponent γ = 0.9, 1.0, 1.2 at Q = 0.3. Each curve rises linearly while Hawking radiation dominates and then saturates at S = 2SBH once the island forms. The ceiling 2πr2 h is fixed by rh, so the curves share a common … view at source ↗
Figure 9
Figure 9. Figure 9: Page time tP versus event-horizon radius rh for EMPYM–AdS black holes (c = 1). (a) Q = 0.1, 0.3, 0.5 at Q = 0.2, γ = 1, ℓ = 3. (b) γ = 0.9, 1.0, 1.2 at Q = 0.3, Q = 0.2, ℓ = 3. (c) P = 0.05, 0.10, 0.20 at Q = 0.3, Q = 0.2, γ = 1. (d) Q = 0.1, 0.2, 0.4 at Q = 0.3, γ = 1, ℓ = 3. A larger Q raises tP and slows recovery; a larger γ or a higher P lowers it. In panels (a), (b), and (d) the three curves nearly co… view at source ↗
Figure 10
Figure 10. Figure 10: Normalised Page time tP /tc versus r+/rc at Q = 0.2, γ = 1, c = 1, for (a) Q = 0.1, (b) Q = 0.3, (c) Q = 0.5. Red, green, and blue denote P = 0.8Pc, Pc, and 1.2Pc; the dotted lines mark r+/rc = 1 and tP /tc = 1. A higher pressure lowers tP /tc throughout. In the large-charge panel (c) the effective temperature passes through zero near r+/rc ≃ 0.4: the hole reaches a cold, near-extremal state and tP diverg… view at source ↗
Figure 11
Figure 11. Figure 11: Normalised Page time tP /tc versus r+/rc at Q = 0.3, Q = 0.2, c = 1, for (a) γ = 0.9, (b) γ = 1.0, (c) γ = 1.2. Colour coding as in [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Normalised Page time tP /tc versus r+/rc at Q = 0.3, γ = 1, c = 1, for (a) Q = 0.1, (b) Q = 0.2, (c) Q = 0.4. Colour coding as in [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Summary of tP /tc versus r+/rc for four EMPYM–AdS parameter sets (c = 1): (a) Q = 0.3, Q = 0.2, γ = 1 (reference); (b) Q = 0.5, Q = 0.2, γ = 1; (c) Q = 0.3, Q = 0.2, γ = 1.2; (d) Q = 0.3, Q = 0.4, γ = 1. Red, green, blue: P = 0.8Pc, Pc, 1.2Pc. A higher pressure lowers τP in every panel. Only the large-Maxwell-charge case (b) develops the near-extremal spike where T → 0; the other three stay smooth. The fi… view at source ↗

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