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

This paper derives an exact electrically charged black hole solution in higher-order curvature gravity coupled with ModMax electrodynamics and quintessence dark energy, demonstrating that while thermodynamic geometry confirms phase transition consistency, quintessence and curvature corrections significantly enlarge the black hole shadow radius, whereas electric charge reduces it.

The Gist

Imagine the universe as a giant, complex machine. For decades, scientists have been trying to figure out why this machine is speeding up its expansion. They call the invisible force pushing it apart "Dark Energy." At the same time, they are trying to understand the most extreme objects in the universe: Black Holes.

This paper is like a team of physicists building a new, more detailed simulation of a black hole. They aren't just looking at a simple black hole; they are building a "super-black hole" that combines three different, very complex ingredients to see how they interact.

Here is a breakdown of what they did, using simple analogies:

1. The Three Ingredients

To build their new black hole model, the authors mixed three specific "flavors" of physics:

• The Gravity Sauce (Higher-Order Curvature): Standard gravity (Einstein's General Relativity) is like a smooth, flat sheet. But the authors added "higher-order curvature," which is like adding wrinkles, bumps, and extra texture to that sheet. It's a more complex version of gravity that might explain things standard gravity can't.
• The Electric Spark (ModMax Electrodynamics): Black holes often have electric charges. Usually, we think of electricity like water flowing in a pipe (linear). But this paper uses "ModMax" theory, which is like electricity that behaves like a stretchy rubber band. It can snap and change shape in extreme conditions, but it still follows specific rules.
• The Invisible Fog (Quintessence Dark Energy): This is the "Dark Energy" ingredient. Imagine the space around the black hole isn't empty, but filled with a thin, invisible fog that pushes things apart. This fog has a specific "personality" (called the state parameter, $\omega$) that determines how strongly it pushes.

2. Building the Black Hole

The authors took these three ingredients and mixed them together in a mathematical recipe. They found a perfect, exact solution for what this black hole looks like.

• The Result: They created a map (a mathematical formula) that describes the shape of space and time around this black hole.
• The Check: They tested this map against known black holes. When they turned off the "wrinkles" in gravity or the "rubber band" electricity, their new map turned back into the old, familiar maps of standard black holes. This proved their new recipe works correctly.

3. Testing the Stability (The Thermodynamics)

Once they built the black hole, they asked: "Is it stable? Will it fall apart?"

• Heat Capacity: They checked how much energy it takes to change the black hole's temperature. Think of this like checking if a pot of water is about to boil over. They found that for some sizes, the black hole is unstable (like a pot about to boil), but for others, it is stable.
• The "Geometric" Check: They used a special mathematical tool called "thermodynamic geometry." Imagine the black hole's energy state as a landscape with hills and valleys. They looked for "cliffs" (divergences) in this landscape. They found that whenever the black hole was unstable (the heat capacity hit zero), there was a cliff in this geometric landscape. This confirmed their findings were consistent.
• Global vs. Local: They found that while the black hole might have some local "twitches" or instability, the whole system remains globally stable, like a wobbly tower that doesn't actually fall down.

4. The Hawking Radiation (The Leak)

Black holes aren't truly black; they slowly leak energy (radiation) and shrink over time. This is called Hawking radiation.

• Sparsity: The authors looked at how "sparse" or "clumpy" this leak is. Imagine a steady stream of water versus a dripping faucet. They found that because of their complex ingredients (the wrinkles in gravity and the dark energy fog), the radiation from this black hole is much "sparser" (more like a slow drip) than a standard black hole.
• The Effect: The "fog" of dark energy and the "wrinkles" in gravity actually slow down the evaporation process, making the black hole last longer than it would in a simpler universe.

5. The Shadow (What We Would See)

Finally, they asked: "If we took a picture of this black hole, what would it look like?" This is the "shadow" (like the dark circle seen in the famous EHT images of M87*).

• The Photon Sphere: Light orbits the black hole in a specific ring before either falling in or escaping. This ring is the edge of the shadow.
• The Findings:
  • More Wrinkles = Bigger Shadow: The more complex the gravity (the "wrinkles"), the bigger the shadow gets.
  • More Fog = Bigger Shadow: The more dark energy (the "fog") there is, the bigger the shadow becomes.
  • More Charge = Smaller Shadow: Interestingly, if the black hole has more electric charge, the shadow actually gets smaller.
  • The Winner: The "fog" (dark energy) has a much stronger effect on the size of the shadow than the electric charge does.

The Bottom Line

This paper doesn't claim to have found a new black hole in the sky yet. Instead, it provides a new, highly detailed mathematical blueprint for a black hole that includes complex gravity, weird electricity, and dark energy.

The main takeaway is that if we ever look closely enough at a black hole's shadow (like with future telescopes), the size of that shadow could tell us if the universe is filled with this "wrinkled" gravity and "foggy" dark energy. The authors suggest that dark energy leaves a much bigger fingerprint on a black hole's shadow than electric charge does, offering a potential way to test these theories in the future.

Technical Summary

Technical Summary: Thermodynamics and Optical Aspects of ModMax Black Holes in Higher Order Curvature Gravity with Quintessence Dark Energy

Problem Statement
The paper addresses the need to understand the interplay between nonlinear electrodynamics, higher-order curvature gravity, and dark energy within the context of black hole (BH) physics. While Modified Maxwell (ModMax) electrodynamics, $F(R)$-gravity, and quintessence dark energy (DE) have been studied independently or in pairwise combinations, their combined impact on the thermodynamic stability, microscopic phase structure, and optical properties of black holes remains unexplored. Specifically, the authors aim to derive an exact black hole solution that incorporates ModMax nonlinear electrodynamics, $F(R)$-gravity (as a representative of higher-order curvature corrections), and a quintessence field, and to analyze how these components collectively influence the BH's horizon structure, thermodynamic behavior, and shadow observables.

Methodology
The authors construct a theoretical framework in four-dimensional spacetime by defining an action that couples the Einstein-Hilbert term modified by an $F(R)$ function to a ModMax nonlinear electromagnetic sector and a quintessence scalar field.

1. Field Equations and Solution: Assuming a static, spherically symmetric metric and a constant scalar curvature $R_0$, the authors derive the field equations. By focusing on purely electrically charged configurations (setting the pseudoscalar invariant $P=0$), they obtain an exact analytical metric function $g(r)$. This solution generalizes known models, reducing to Reissner-Nordström-AdS, ModMax, or $F(R)$-ModMax solutions under specific limiting cases of the parameters (mass $m_0$, charge $q$, ModMax parameter $\gamma$, curvature parameter $f_{R0}$, and quintessence parameters $c$ and $\omega$).
2. Thermodynamics: The authors employ Wald's entropy formalism to derive the entropy, accounting for higher-order curvature corrections. They reconstruct the mass function in terms of entropy and derive expressions for temperature, pressure, and volume. They analyze local and global stability using heat capacity ($C$), Helmholtz free energy ($F$), and Gibbs free energy ($G$).
3. Geothermodynamics: To investigate microscopic interactions and phase transitions, the authors utilize thermodynamic geometry, specifically the Weinhold and Ruppeiner metrics. They examine the Ricci curvature scalars ($R_{Wein}$ and $R_{Rup}$) to identify singularities that correspond to phase transitions.
4. Hawking Radiation: The sparsity of Hawking radiation and the energy emission rate are calculated using the derived temperature and entropy relations.
5. Optical Properties: The authors analyze null geodesics to determine the photon sphere radius ($r_{ph}$) and the shadow radius ($R_{sh}$). They derive exact analytical expressions for $r_{ph}$ for specific values of the quintessence state parameter $\omega$ ($-1/3, -2/3, -1$) and numerically evaluate the shadow size for a static observer at a finite distance, acknowledging the lack of asymptotic flatness.

Key Contributions and Results

• Exact Solution: The paper presents a novel exact black hole solution (Eq. 17) that unifies $F(R)$-gravity, ModMax electrodynamics, and quintessence. The metric function reveals that higher-order curvature terms rescale the electromagnetic coupling, while quintessence introduces a radial modification affecting the asymptotic regime and potentially generating cosmological horizons.
• Thermodynamic Stability:
  • Local Stability: Analysis of the heat capacity ($C$) reveals regions of negative values, indicating local thermodynamic instability. The points where $C$ vanishes or diverges correspond to phase transitions.
  • Global Stability: The Helmholtz and Gibbs free energies remain positive and exhibit smooth behavior without "swallow-tail" features, suggesting the system is globally stable across the parameter space.
  • Geothermodynamic Consistency: The study confirms that the divergences in the Ruppeiner curvature scalar ($R_{Rup}$) coincide precisely with the zero points of the heat capacity, validating the consistency of the phase structure. The Weinhold metric, however, does not show this coincidence, highlighting the superiority of the Ruppeiner metric for this analysis.
  • Microscopic Interactions: The sign of the Ruppeiner curvature indicates that microscopic interactions within the BH are both attractive and repulsive, depending on the entropy and parameter values.
• Hawking Radiation: The sparsity parameter ($\eta$) and energy emission rate ($\epsilon_{\omega t}$) are found to be significantly influenced by the quintessence field and higher-order curvature corrections. The results indicate that these factors can suppress the evaporation process, making the radiation sparser than in standard Schwarzschild cases, particularly for smaller black holes.
• Optical Signatures:
  • Photon Sphere and Shadow: Exact analytical expressions for the photon sphere radius are derived for specific $\omega$ values.
  • Parameter Dependence: The shadow radius ($R_{sh}$) increases monotonically with the ModMax parameter ($\gamma$) and the curvature correction parameter ($f_{R0}$). Conversely, it decreases with increasing electric charge ($q$) and quintessence normalization factor ($c$).
  • Dominant Effects: The study finds that quintessence has a more pronounced impact on the shadow size than the electric charge. The combined effect of higher-order curvature corrections and dark energy yields unique deviations in the shadow size, potentially distinguishable from standard General Relativity predictions.

Significance and Claims
The paper claims that its primary contribution is the systematic exploration of the combined effects of dark energy and higher-order curvature corrections on black hole thermodynamics and optical properties within the ModMax framework. The authors assert that their solution provides a consistent generalization of previously known models, recovering standard limits when appropriate parameters are set to zero.

The significance of the work lies in demonstrating that:

1. Dark energy (quintessence) and higher-order curvature corrections can yield potentially observable signatures in black hole shadows, distinct from those predicted by standard GR or linear electrodynamics.
2. The thermodynamic phase structure of these black holes is consistent, with geothermodynamic curvature singularities aligning with heat capacity divergences.
3. The optical properties, specifically the shadow size, are sensitive to the state parameter of quintessence and the nonlinear electromagnetic parameter, suggesting that future observational research could use black hole shadows as probes for these specific higher-order gravity and dark energy theories.

The authors conclude that while their results highlight potential observational signatures, further work is required to extend stability analysis to include perturbations (scalar, gravitational, electromagnetic) and to compute quasinormal modes for gravitational wave signatures. They do not claim immediate experimental verification but position their analytical results as a foundation for future observational tests.