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arxiv: 2605.13908 · v1 · submitted 2026-05-13 · 🌀 gr-qc · hep-th

Recognition: 2 theorem links

· Lean Theorem

Thermodynamics and optical aspects of ModMax black holes in higher order curvature gravity with quintessence dark energy

Ahmad Al-Badawi , Usman Zafar , Abdul Jawad , Kazuharu Bamba
Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:04 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords black hole shadowquintessenceModMax electrodynamicshigher-order curvature gravitythermodynamic geometryphoton spheredark energy
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The pith

Exact solutions show quintessence and higher curvature corrections enlarge ModMax black hole shadows while charge reduces them.

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

The paper derives an exact black hole solution by coupling higher-order curvature gravity to ModMax nonlinear electrodynamics and a quintessence dark energy fluid. It analyzes thermodynamic stability through heat capacity and thermodynamic geometry, confirming that phase transition points match divergences in the curvature scalar. Optically, it computes the photon sphere radius in closed form and shows that curvature corrections plus quintessence increase the resulting shadow size, with quintessence exerting a stronger influence than the opposing effect from electric charge.

Core claim

An exact black hole metric exists in higher-order curvature gravity with ModMax electrodynamics and quintessence, for which the photon-sphere radius admits an analytic expression; this radius grows with the curvature coupling and quintessence strength but shrinks with electric charge, and the quintessence contribution dominates the charge contribution in setting the shadow radius.

What carries the argument

The exact analytic black hole metric obtained from the coupled action, which reduces ModMax to a rescaled Maxwell theory and permits closed-form null geodesic integration for the photon sphere.

If this is right

  • The shadow radius grows with higher-order curvature corrections and with quintessence strength.
  • Electric charge reduces the shadow radius, but this reduction is weaker than the enhancement from quintessence.
  • Divergences in the thermodynamic curvature scalar coincide exactly with the vanishing points of the heat capacity.
  • The ModMax sector reduces to rescaled Maxwell electrodynamics for the purely electric configurations studied.

Where Pith is reading between the lines

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

  • Precision shadow measurements could place limits on the strength of higher-order curvature corrections and on the quintessence equation-of-state parameter.
  • The model predicts that dark-energy effects on photon spheres may be detectable even when nonlinear electromagnetic corrections remain small.
  • The same parameter dependencies may persist in slowly rotating generalizations, offering a route to test the solution against spin-resolved shadow data.

Load-bearing premise

The higher-order curvature term and its linear coupling to the quintessence fluid take the specific forms that permit an exact closed-form metric solution.

What would settle it

An observed black hole shadow radius that shows no dependence on the quintessence state parameter or that increases with electric charge would contradict the derived analytic expressions.

Figures

Figures reproduced from arXiv: 2605.13908 by Abdul Jawad, Ahmad Al-Badawi, Kazuharu Bamba, Usman Zafar.

Figure 1
Figure 1. Figure 1: FIG. 1: A comparison of the metric function [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Behavior of Ricci squared invariant [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Plot of mass in terms of entropy [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Plot of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Plot of pressure in terms of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The behavior of HFE [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The behavior of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Plots of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: displays how η behaves in different ModMax BH models in the form of S by employing q = 0.5, f0 = 0.9, c = 0.1, γ = 0.6, R0 = −0.01 and w = −2/3. For different models of ModMax BH, we obtain different curves, including red, black, blue, and green curves for ModMax, F(R)−ModMax BH, ModMax BH with QF, and F(R)−ModMax BH with QF, respectively. We notice that QF and F(R)-gravity significantly impact the sparsi… view at source ↗
Figure 11
Figure 11. Figure 11: displays how εtΩ varies in different ModMax BH models in terms of entropy S by plugging q = 0.5, f0 = 0.9, c = 0.1, γ = 0.6, R0 = −0.01 and w = −2/3. For various models of ModMax BH, we get different trajec￾tories, such as red, black, blue, and green curves for ModMax, F(R)−ModMax BH, ModMax BH with QF, and F(R)−ModMax BH with QF, respectively. It is observed that in all the models of ModMax BHs, εtΩ is i… view at source ↗
Figure 12
Figure 12. Figure 12: presents a detailed analysis of the impact of BH parameters on rph for the special choice w = −2/3. The left panel displays the dependence of the photon sphere radius rph on the parameters γ and c, with fR0 and q held fixed. It demonstrates that rph increases monotonically with both γ and c. In contrast, the right panel illustrates the variation of rph with respect to fR0 and q, while keeping γ and c cons… view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: BH shadows for numerous values of [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
read the original abstract

In this work, we derive an exact black hole solution in higher-order curvature gravity by coupling an electromagnetic sector formulated within the ModMax framework to a quintessence dark energy component. Focusing on purely electrically charged configurations, we analyze the thermodynamic and geothermodynamic properties of the solution to investigate its stability and phase structure. Within this sector, the ModMax theory effectively reduces Maxwell electrodynamics up to a rescaling of the electric charge, and thus the obtained solution corresponds to a consistent subset of the broader nonlinear theory. Using thermodynamic geometry, we examine microscopic interactions and phase transitions, showing that divergences in the thermodynamic curvature coincide with the vanishing of the heat capacity, confirming the consistency of the phase structure. We further explore the optical properties of the black hole by studying null geodesics and determining the photon sphere and the corresponding shadow radius for different values of the quintessence state parameter $\omega$. Exact analytical expressions for the photon-sphere radius are derived, revealing that higher-order curvature corrections and quintessence significantly enhance the shadow size, whereas the electric charge has the opposite effect. Notably, quintessence is found to have a more pronounced impact on the shadow than the charge. These results highlight that dark energy and higher-order curvature corrections can yield potentially observable signatures in black hole shadows.

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 manuscript derives an exact black hole solution in higher-order curvature gravity by coupling ModMax nonlinear electrodynamics (which reduces to rescaled Maxwell) to a quintessence dark energy fluid. It analyzes thermodynamic stability and phase structure via heat capacity and thermodynamic geometry, showing that divergences in the thermodynamic curvature scalar coincide with heat-capacity zeros. Exact analytic expressions for the photon-sphere radius are obtained from the null geodesic condition, and the shadow radius is studied as a function of the higher-order curvature coupling, quintessence state parameter ω, and charge, with the conclusion that curvature corrections and quintessence increase the shadow while charge decreases it and quintessence exerts the stronger effect.

Significance. If the exact solution and analytic photon-sphere expressions hold under the stated assumptions, the work supplies a rare closed-form example in modified gravity with dark energy, enabling precise analytic study of thermodynamic phase transitions and potential observational signatures in black-hole shadows. The explicit matching between thermodynamic curvature divergences and heat-capacity zeros, together with the parameter dependence of the shadow radius, constitutes a concrete strength that could support falsifiable predictions once the coupling assumptions are clarified.

major comments (2)
  1. [§3] §3 (Black Hole Solution): The exact metric function f(r) is obtained only after imposing a linear coupling between the quintessence fluid and the higher-order curvature sector that permits closed-form integration. This choice must be shown to follow directly from varying the full action without additional integrability assumptions; otherwise the claimed analytic r_ph and all subsequent shadow results rest on a tuned ansatz rather than the unmodified theory.
  2. [§5] §5 (Optical Properties): The algebraic equation for the photon-sphere radius r_ph is solved in closed form using the specific f(r) derived under the linear quintessence coupling. Any modification to that coupling alters f(r) and therefore invalidates the explicit r_ph expression and the comparative statements about the relative impact of quintessence versus charge.
minor comments (2)
  1. [Abstract and §1] The abstract and §1 should explicitly state the precise form of the higher-order curvature Lagrangian (e.g., the coefficient of the R^2 or Gauss-Bonnet term) rather than referring only to “higher-order curvature gravity.”
  2. [Figure captions] Figure captions for the shadow-radius plots should include the fixed values of all parameters not being varied (e.g., the curvature coupling constant and mass) so that the curves are reproducible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points about the assumptions needed for our exact solution. We address each major comment below and will revise the manuscript accordingly to improve clarity on the coupling choice and the scope of the analytic results.

read point-by-point responses

  1. Referee: [§3] §3 (Black Hole Solution): The exact metric function f(r) is obtained only after imposing a linear coupling between the quintessence fluid and the higher-order curvature sector that permits closed-form integration. This choice must be shown to follow directly from varying the full action without additional integrability assumptions; otherwise the claimed analytic r_ph and all subsequent shadow results rest on a tuned ansatz rather than the unmodified theory.

    Authors: We acknowledge that the exact analytic form of f(r) relies on adopting a linear coupling between the quintessence fluid and the higher-order curvature terms that allows the field equations to integrate in closed form. This is a standard and physically motivated choice in the literature on exact solutions in modified gravity with dark energy fluids, corresponding to an effective stress-energy contribution that preserves the structure permitting analytic integration. The full action is varied to obtain the field equations; the linear coupling is then imposed as an ansatz to achieve exact solvability rather than an arbitrary tuning. We will revise §3 to explicitly derive the metric under this coupling, state the integrability assumption clearly, and discuss its motivation as an effective model within the broader theory. This does not alter the validity of the subsequent analytic expressions but specifies their regime of applicability. revision: yes

  2. Referee: [§5] §5 (Optical Properties): The algebraic equation for the photon-sphere radius r_ph is solved in closed form using the specific f(r) derived under the linear quintessence coupling. Any modification to that coupling alters f(r) and therefore invalidates the explicit r_ph expression and the comparative statements about the relative impact of quintessence versus charge.

    Authors: We agree that the closed-form expression for r_ph and the quantitative comparisons (e.g., quintessence exerting a stronger effect on shadow size than charge) are tied to the specific f(r) obtained under the linear coupling. These results are exact and analytic only within the solvable model we consider. We will revise §5 to emphasize that the photon-sphere and shadow expressions, as well as the relative impact statements, hold specifically for this exact solution. We will also note that more general couplings would generally require numerical solution of the null geodesic equation, thereby clarifying the scope without weakening the analytic insights provided by the solvable case. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation proceeds from explicit action to exact metric to geodesic observables without reduction to inputs.

full rationale

The paper specifies a higher-order curvature action, ModMax electromagnetic sector, and linear quintessence coupling chosen to permit exact integration of the field equations, yielding a closed-form metric function f(r). The photon-sphere radius is obtained by imposing the standard condition for unstable circular null geodesics on this metric (effective potential extremum), producing explicit algebraic expressions in the model parameters. The ModMax reduction to rescaled Maxwell is stated as a property of the theory rather than a derived or fitted result here. No load-bearing step equates any claimed prediction to a prior fit, self-definition, or self-citation chain; the analytic results follow directly from the stated assumptions and standard GR calculations on the solved metric. The model choice is explicit, so the derivation remains self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The solution rests on the existence of an exact integrable ansatz for the metric in the chosen higher-order action plus quintessence stress-energy; no new particles or forces are postulated beyond the standard higher-curvature and quintessence terms already present in the literature.

free parameters (2)
  • higher-order curvature coupling constant
    Dimensionful parameter controlling the strength of curvature corrections; its value is left free and varied in the analysis.
  • quintessence state parameter ω
    Constant equation-of-state parameter for the dark-energy fluid; scanned over a range of values.
axioms (2)
  • domain assumption The spacetime is static and spherically symmetric with a purely electric vector potential.
    Standard ansatz invoked to reduce the field equations to ordinary differential equations.
  • domain assumption Quintessence is modeled as a perfect fluid with constant equation-of-state parameter.
    Common phenomenological choice that allows analytic integration.

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