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arxiv: 2508.06725 · v2 · submitted 2025-08-08 · 🌀 gr-qc · hep-th

Thermal and Optical Signatures of Einstein-Dyonic ModMax Black Holes with GUP and Plasma Modifications

Erdem Sucu , Suat Dengiz , \.Izzet Sakall{\i} This is my paper

Pith reviewed 2026-05-18 23:28 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Einstein-Dyonic-ModMax black holesGeneralized Uncertainty PrincipleHawking radiationgravitational lensingplasma effectsthermodynamic phase transitionsnonlinear electrodynamicsquantum corrections
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The pith

GUP corrections to Einstein-Dyonic-ModMax black holes may produce stable remnants after Hawking evaporation

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

This paper examines the effects of the Generalized Uncertainty Principle and plasma on the properties of Einstein-Dyonic-ModMax black holes in nonlinear electrodynamics. It uses the Hamilton-Jacobi method to find that GUP alters the Hawking radiation spectrum in ways that could result in stable remnants rather than total evaporation. Light deflection angles are calculated via the Gauss-Bonnet theorem, showing clear dependence on the ModMax nonlinearity parameter and plasma density, with potential extensions to axion-plasmon media for dark matter detection. Thermodynamic analysis with corrected entropy reveals second-order phase transitions in heat capacity that vary with the nonlinearity parameter, while energy conditions hold classically but fail near the horizon once quantum effects are included.

Core claim

Applying the Hamilton-Jacobi tunneling formalism to the GUP-modified geometry of Einstein-Dyonic-ModMax black holes produces a Hawking radiation spectrum whose modifications by the Generalized Uncertainty Principle can halt evaporation at a stable remnant mass; simultaneously, the Gauss-Bonnet theorem applied to light rays in plasma yields deflection angles that depend sensitively on the ModMax parameter γ and plasma frequency, while quantum-corrected entropy leads to thermodynamic quantities including heat capacity that undergo second-order phase transitions at γ-dependent critical points.

What carries the argument

The Hamilton-Jacobi tunneling formalism, which computes the modified tunneling probability and Hawking temperature in the presence of GUP corrections to the EDM black hole metric.

If this is right

  • GUP-modified tunneling probabilities lead to a corrected Hawking temperature that permits stable black hole remnants.
  • Light deflection angles computed in vacuum and plasma media depend on the ModMax nonlinearity parameter γ and plasma density.
  • Frequency-dependent lensing modifications in axion-plasmon environments may provide observable signatures of dark matter.
  • Quantum-corrected heat capacity exhibits second-order phase transitions whose critical points vary with the value of γ.
  • Classical ModMax electrodynamics satisfies null and weak energy conditions, but near-horizon violations emerge after quantum entropy corrections.

Where Pith is reading between the lines

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

  • The existence of stable remnants could preserve quantum information that would otherwise be lost during complete evaporation.
  • The plasma-dependent lensing predictions might be probed through radio observations of black hole shadows or strong lensing events.
  • The shifting critical points in thermodynamic phase transitions indicate that quantum corrections can induce critical behavior similar to that seen in condensed matter systems.
  • Incorporating these modifications into models of astrophysical black holes could lead to revised predictions for their accretion disk emissions and stability.

Load-bearing premise

The Hamilton-Jacobi tunneling formalism remains valid when applied to the spacetime of Einstein-Dyonic-ModMax black holes after GUP corrections and plasma effects are included.

What would settle it

A direct observation of a black hole remnant whose mass is bounded below by the Planck scale set by the GUP parameter, or precise measurements of light deflection angles around a supermassive black hole that exhibit the predicted dependence on a nonlinearity parameter.

Figures

Figures reproduced from arXiv: 2508.06725 by Erdem Sucu, \.Izzet Sakall{\i}, Suat Dengiz.

Figure 1
Figure 1. Figure 1: Embedding diagrams for Schwarzschild and selected dyonic ModMax BH configurations with varying [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hawking temperature TH as a function of horizon radius rh for various values of the ModMax parameter γ. Parameters: M = 1.0, Q˜ = 0.5, P = 0.3. The plot demonstrates how the nonlinearity parameter γ modifies the thermal emission compared to the classical Reissner-Nordström case. This leads to a divergent integral for the radial action as follows: W(r) = ± Z ω f ′(rh) dr r − rh . (18) The integral contains … view at source ↗
Figure 3
Figure 3. Figure 3: Deflection angle α of light in plasma media for parameters M = 1.0, Q˜ = 0.5, P˜ = 0.5 and γ = 1, in terms of the impact parameter b and the plasma parameter δ. The color scale shows the deflection angle values, with yellow regions representing large positive deflections and purple regions representing negative deflections. The graph demonstrates that plasma density and small impact parameters significantl… view at source ↗
Figure 4
Figure 4. Figure 4: Deflection angle α˜ of light in terms of the impact parameter b and the ModMax parameter γ for the case without plasma effects. Parameters: M = 1.0, Q˜ = 0.5, P˜ = 0.5. The color scale represents the deflection angle values; yellow regions indicate large positive deflections, and purple regions indicate negative deflections. The graph reveals that the deflection angle increases strongly in the regime of sm… view at source ↗
Figure 5
Figure 5. Figure 5: Three-dimensional visualization of the deflection angle [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Density plot of deflection angle Θ as a function of impact parameter b and ModMax parameter γ in axion-plasmon environments. Fixed parameters: M = 1.0, Q˜ = P˜ = 0.5, ω0 = 1.0, ωφ = 0.5, ωp = 0.5, and B0 = 1.0. The color scale reveals how axion-photon coupling modifies the lensing signature compared to pure plasma effects. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Density plot of the BH energy E as a function of the event horizon radius rh and the ModMax parameter γ, for fixed parameters M = 1.0, Q˜ = 0.5, and P˜ = 0.5. The plot illustrates that the energy increases predominantly with rh, while the influence of γ remains subdominant yet monotonic. with total particle number N = Psini and energy E = Psiniϵi , where ni is the occupancy of state i with energy ϵi . Usin… view at source ↗
Figure 8
Figure 8. Figure 8: Density plot of the Helmholtz free energy [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Density plot of the thermodynamic pressure [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Heat capacity profile for EDM BH with parameters [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Heat capacity C in terms of the event horizon radius rh and the modification parameter γ by fixing the parameters M = 1.0, Q˜ = 0.5, P˜ = 0.5 within the scope of EDM theory. The color scale shows the C values that determine the thermodynamic stability of the system; red regions represent positive heat capacity (stable phase), blue regions represent negative heat capacity (unstable phase). The phase transi… view at source ↗
read the original abstract

We explore the thermodynamic and optical properties of Einstein-Dyonic-ModMax (EDM) black holes (BHs), incorporating quantum gravity corrections and plasma effects. The ModMax theory promotes the classical Maxwell theory to a non-linear electrodynamics with a larger symmetry structure (electromagnetic duality plus conformal invariance), and provides dyonic BH solutions characterized by both electric and magnetic charges modulated by the nonlinearity parameter $\gamma$. Using the Hamilton-Jacobi tunneling formalism, we derive the Hawking radiation spectrum and demonstrate how the Generalized Uncertainty Principle (GUP) modifies the thermal emission, potentially leading to stable remnants. Our analysis of gravitational lensing employs the Gauss-Bonnet theorem to compute light deflection angles in both vacuum and plasma environments, revealing strong dependencies on the ModMax parameter and plasma density. We extend this to axion-plasmon environments, uncovering frequency-dependent modifications that could serve as dark matter signatures. The photon motion analysis in plasma media shows how the exponential damping term $e^{-\gamma}$ affects electromagnetic backreaction on spacetime geometry. We compute quantum-corrected thermodynamic quantities, including internal energy, Helmholtz free energy, pressure, and heat capacity, using exponentially modified entropy models. The heat capacity exhibits second-order phase transitions with critical points shifting as functions of $\gamma$, indicating rich thermodynamic phase structures. The energy condition analysis shows that classical ModMax electrodynamics satisfies the null and weak energy conditions, while the observed near-horizon violations arise only after incorporating quantum-corrected entropy effects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper explores thermodynamic and optical properties of Einstein-Dyonic-ModMax black holes incorporating GUP quantum corrections and plasma effects. It uses the Hamilton-Jacobi tunneling formalism to derive a GUP-modified Hawking radiation spectrum claimed to produce stable remnants, applies the Gauss-Bonnet theorem to compute light deflection angles in vacuum and plasma (including axion-plasmon media), computes quantum-corrected thermodynamic quantities (internal energy, free energy, pressure, heat capacity) via exponentially modified entropy models, identifies second-order phase transitions whose critical points depend on the ModMax parameter γ, and analyzes energy conditions with near-horizon violations attributed to quantum entropy corrections.

Significance. If the core derivations hold, the results would offer a concrete parameter space (γ, GUP deformation, plasma density) for studying how nonlinear electrodynamics and quantum gravity corrections jointly affect black-hole remnants, phase structure, and lensing observables, potentially linking to dark-matter signatures via frequency-dependent axion-plasmon effects. The explicit inclusion of both electric and magnetic charges modulated by ModMax nonlinearity is a strength relative to purely electric cases.

major comments (2)
  1. [Hawking radiation / tunneling section] The Hamilton-Jacobi tunneling calculation applies the standard eikonal equation directly to the classical EDM metric; no re-derivation of the WKB ansatz or dispersion relation under the GUP-deformed algebra [x,p]=iħ(1+βp²) is shown. Consequently the claimed modification to the Hawking spectrum and the existence of stable remnants rest on an unverified hybrid construction rather than a consistent semi-classical derivation.
  2. [Thermodynamic quantities and heat capacity] Exponentially modified entropy models are introduced by hand to encode GUP effects; the resulting heat-capacity critical points and second-order phase-transition locations therefore shift parametrically with the GUP deformation parameter rather than emerging from a first-principles modification of the action or metric. This renders the thermodynamic phase-structure claims dependent on the ad-hoc functional form chosen for the entropy correction.
minor comments (2)
  1. [Energy conditions] The abstract and energy-condition discussion state that near-horizon violations appear only after quantum-corrected entropy is included, yet no explicit limiting-case check (e.g., γ→0 or β→0) is referenced to confirm recovery of the standard ModMax or Reissner-Nordström results.
  2. [Optical properties / lensing] Notation for the plasma frequency and axion-plasmon coupling is introduced without a dedicated table or equation reference; readers must infer the precise functional dependence of the deflection angle on these quantities from the surrounding text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below with clarifications on our methodology and indicate the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Hawking radiation / tunneling section] The Hamilton-Jacobi tunneling calculation applies the standard eikonal equation directly to the classical EDM metric; no re-derivation of the WKB ansatz or dispersion relation under the GUP-deformed algebra [x,p]=iħ(1+βp²) is shown. Consequently the claimed modification to the Hawking spectrum and the existence of stable remnants rest on an unverified hybrid construction rather than a consistent semi-classical derivation.

    Authors: We appreciate the referee highlighting this aspect of the presentation. The GUP correction is incorporated via the standard modified dispersion relation arising from the deformed commutator, which is substituted into the Hamilton-Jacobi equation before applying the eikonal approximation to the EDM metric; this follows the procedure used in numerous prior works on GUP-corrected tunneling. While the manuscript emphasizes the resulting spectrum for the specific black-hole solution, we agree that an explicit intermediate derivation of the WKB ansatz from the GUP algebra would remove any ambiguity. We will add this derivation, together with the explicit form of the modified dispersion, to the tunneling section in the revised manuscript. revision: yes

  2. Referee: [Thermodynamic quantities and heat capacity] Exponentially modified entropy models are introduced by hand to encode GUP effects; the resulting heat-capacity critical points and second-order phase-transition locations therefore shift parametrically with the GUP deformation parameter rather than emerging from a first-principles modification of the action or metric. This renders the thermodynamic phase-structure claims dependent on the ad-hoc functional form chosen for the entropy correction.

    Authors: The referee correctly identifies that the exponential entropy correction is a phenomenological ansatz chosen to encode GUP effects while preserving thermodynamic stability and the existence of remnants. Such forms appear in the effective quantum-gravity literature as alternatives to purely logarithmic corrections. We acknowledge that the model is not obtained by varying a modified action. In the revision we will expand the motivation for this specific functional choice by citing the relevant GUP entropy derivations, explicitly state its effective character, and discuss how the parametric shifts in the critical points are a direct consequence of the chosen correction rather than a universal feature. revision: partial

Circularity Check

1 steps flagged

GUP corrections enter via chosen exponentially modified entropy ansatz; thermodynamic critical points follow by construction

specific steps
  1. self definitional [Quantum-corrected thermodynamics (abstract and implied thermodynamics section)]
    "We compute quantum-corrected thermodynamic quantities, including internal energy, Helmholtz free energy, pressure, and heat capacity, using exponentially modified entropy models. The heat capacity exhibits second-order phase transitions with critical points shifting as functions of γ"

    The functional form of the entropy is chosen to incorporate GUP; the resulting heat-capacity critical points and phase structure are therefore fixed by the parameters introduced to encode the quantum correction, rather than being derived from the spacetime geometry or a re-computed tunneling probability independent of that choice.

full rationale

The paper states it derives Hawking spectrum via standard Hamilton-Jacobi tunneling then demonstrates GUP modifications, but provides no re-derivation of the WKB step under the deformed [x,p] algebra. Separately, thermodynamic quantities are obtained from an exponentially modified entropy whose functional form is selected to encode GUP; heat-capacity critical points and phase-transition locations are therefore controlled by the parameters of that chosen form. This matches the self-definitional pattern: the 'quantum-corrected' outputs are fixed once the input modification ansatz is written down. No equations are shown reducing the final expressions to the unmodified case, and no external benchmark or independent derivation is invoked. The central claim of stable remnants therefore rests on the modeling choice rather than an independent first-principles step. Score remains moderate because the spacetime metric itself is derived classically and the optical/lensing sections appear independent of the entropy ansatz.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents exhaustive identification of all free parameters and axioms; the ModMax nonlinearity parameter gamma and the specific GUP deformation function are treated as inputs whose values are not derived within the paper.

free parameters (2)
  • ModMax nonlinearity parameter gamma
    Introduced to modulate electric and magnetic charges; its value is not fixed by any internal consistency condition shown in the abstract.
  • GUP deformation parameter
    Controls the strength of quantum correction to the tunneling rate and remnant mass; chosen rather than derived.
axioms (1)
  • domain assumption Validity of the Hamilton-Jacobi tunneling method for the modified metric
    Invoked to obtain the Hawking spectrum without additional justification in the provided abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On gravitating dyonic configurations in nonlinear electrodynamics

    gr-qc 2026-04 unverdicted novelty 6.0

    For dyonic nonlinear electrodynamics with equal charges, the electromagnetic invariant f vanishes identically, enabling simple gravitating solutions in GR and extended gravity theories.

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