⚛️ general relativity

Extended Thermodynamics and Throttling Process of Charged AdS Black Holes in ModMax-dRGT Massive Gravity with Sharma-Mittal Entropy

This paper investigates the extended thermodynamics and Joule-Thomson expansion of four-dimensional charged AdS black holes in ModMax-dRGT massive gravity using Sharma-Mittal entropy, revealing how ModMax nonlinearities, non-extensive statistical parameters, and massive gravity distinctively govern the cooling domain, local stability, and global phase landscape respectively.

Original authors: Naba Jyoti Gogoi, Hassan Hassanabadi, Himasri Pinapothu, Dhruba Jyoti Gogoi

Published 2026-06-26

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CC BY 4.0

Original authors: Naba Jyoti Gogoi, Hassan Hassanabadi, Himasri Pinapothu, Dhruba Jyoti Gogoi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a black hole not as a terrifying cosmic vacuum cleaner, but as a very strange, super-dense "balloon" floating in a universe with a negative pressure (like a stretched rubber sheet pulling inward). This paper treats that black hole like a gas in a piston, asking: What happens if we let this cosmic balloon expand without adding or removing energy?

This process is called the Joule-Thomson expansion (or "throttling"). In everyday life, think of a spray can of air. When you spray it, the gas rushes out, expands, and the can gets cold. That's cooling. But if you release a gas that's already too hot or under specific conditions, it might actually get hotter. Scientists want to know exactly when a black hole switches from getting cold to getting hot.

Here is the story of the paper, broken down into simple concepts:

1. The Ingredients: A Cosmic Smoothie

The researchers mixed three very complex ingredients to create a new type of black hole model:

The Gravity (The Container): They used a theory called dRGT Massive Gravity. Imagine gravity usually as a ghost that has no weight. In this theory, the "ghost" (the graviton) has a tiny bit of mass. This changes how the black hole's "container" (spacetime) behaves, making it stiffer or more rigid depending on the settings.
The Electricity (The Fuel): They used ModMax Electrodynamics. Standard electricity is like a straight line. ModMax is like a rubber band that stretches and snaps back in a non-linear way. It has a special "knob" (called $\gamma$ ) that can turn down the strength of the electric repulsion inside the black hole.
The Entropy (The Counting System): They used Sharma-Mittal Entropy. Usually, when we count the tiny particles inside a system, we use standard math. But at the edge of a black hole, things get weird and "non-extensive" (meaning the whole isn't just the sum of its parts). This new math is like using a different ruler that accounts for these weird, long-range connections between particles.

2. The Experiment: The Cosmic Throttle

The team asked: "If we let this black hole expand (throttle) while keeping its total energy (mass) the same, does it cool down or heat up?"

To answer this, they calculated the Inversion Curve. Think of this as a "weather map" for the black hole:

Above the line: The black hole cools down (like the spray can).
Below the line: The black hole heats up.

3. The Big Discoveries

The paper found some fascinating things about how these three ingredients interact:

A. The "ModMax" Knob Expands the Cooling Zone
When they turned up the ModMax knob (increasing $\gamma$ ), the electric force inside the black hole got weaker.

The Result: The "cooling zone" got bigger. The black hole could expand and cool down at much higher pressures than before.
The Analogy: Imagine a crowded room (the black hole). If the people (electric charges) stop pushing against each other as hard, the room can expand more easily and cool down faster. The ModMax field essentially "calms down" the electric repulsion, allowing the black hole to cool over a wider range of conditions.

B. The "Sharma-Mittal" Ruler Controls the Local Stability
The new entropy math (Sharma-Mittal) acted like a fine-tuner for the black hole's internal stability.

The Result: Changing these parameters made the black hole's "heat capacity" (how hard it is to change its temperature) go up and down wildly. It determined whether the black hole was stable or prone to chaotic fluctuations locally.
The Analogy: Think of this as adjusting the tension on a drum skin. Tightening or loosening it changes how the drum vibrates (stability), but it doesn't change the shape of the room the drum is in.

C. The "Massive Gravity" is the Architect
Even though the other two ingredients changed the details, the Massive Gravity (the weight of the graviton) was the boss of the big picture.

The Result: The overall shape of the "cooling vs. heating" map was dictated almost entirely by the massive gravity. The entropy math didn't change the global boundary; it just changed the local vibrations.
The Analogy: If the black hole is a house, Massive Gravity built the foundation and the walls. ModMax and Sharma-Mittal just painted the walls and arranged the furniture. The house's shape is determined by the foundation, not the paint.

4. The Phase Transition: The "Swallow-Tail"

The paper also looked at what happens when the black hole changes from a "small" state to a "large" state.

They found a Swallow-Tail shape in their graphs. This is a classic sign of a Phase Transition (like water turning into ice or steam).
It means the black hole can exist in two different sizes at the same time, and it suddenly "jumps" from one size to the other, just like water boiling.
The ModMax field shifts where this jump happens, and the Massive Gravity determines if the jump happens at all.

Summary

In short, this paper treats a black hole like a complex machine. They found that:

ModMax (the electric field) acts like a dimmer switch that lets the black hole cool down more easily.
Sharma-Mittal (the entropy math) acts like a stabilizer for the black hole's internal temperature fluctuations.
Massive Gravity (the heavy graviton) acts as the architect, determining the overall shape of the black hole's behavior.

The most important takeaway is that these three forces work together but have distinct roles. You can tweak the "local" stability (entropy) without changing the "global" rules (gravity), and you can change the "cooling range" (ModMax) without breaking the structure of the universe around it. This helps scientists understand how different theories of physics might fit together to describe the most extreme objects in the universe.

Technical Summary: Extended Thermodynamics and Throttling Process of Charged AdS Black Holes in ModMax-dRGT Massive Gravity with Sharma-Mittal Entropy

Problem Statement
This paper investigates the thermodynamic behavior of four-dimensional charged anti-de Sitter (AdS) black holes within a complex theoretical framework combining three distinct extensions of standard physics:

ModMax Nonlinear Electrodynamics: A conformally invariant, one-parameter nonlinear extension of Maxwell's theory.
dRGT-like Massive Gravity: A ghost-free theory where the graviton possesses a non-zero mass, utilizing a singular reference metric.
Sharma-Mittal (SM) Entropy: A generalized, non-extensive entropy formulation intended to capture statistical correlations that standard Boltzmann-Gibbs entropy may miss in systems with exotic matter or long-range interactions.

While the thermodynamic properties and phase transitions of black holes in these individual or paired frameworks have been explored, the specific dynamics of the Joule-Thomson (JT) expansion (throttling process) and the interplay of these three specific modifications in an extended phase space remained largely unaddressed. The study aims to derive exact analytical expressions for the modified Hawking temperature, specific heat, JT coefficient, and inversion boundaries to understand how these parameters influence the heating-cooling domains and phase stability of the black hole.

Methodology
The authors operate within the extended phase space formalism, where the negative cosmological constant ( $\Lambda$ ) is interpreted as a thermodynamic pressure ( $P = -\Lambda/8\pi$ ) and the black hole mass ( $M$ ) is identified with thermodynamic enthalpy.

Metric and Equation of State: Starting from the static, spherically symmetric line element for ModMax-dRGT-like massive gravity, the authors derive the metric function $\psi(r)$ . This incorporates the ModMax parameter ( $\gamma$ ), electric charge ( $q$ ), and massive gravity parameters ( $m_g, c_1, c_2, C$ ).
Generalized Entropy: Instead of the standard Bekenstein-Hawking area law, the study employs the Sharma-Mittal entropy ( $S_{SM}$ ), defined by two non-extensive parameters ( $\delta, R$ ). This modifies the Hawking temperature ( $T_{SM}$ ) via the first law of thermodynamics ( $T = \partial M / \partial S$ ).
Joule-Thomson Analysis: The authors derive the JT coefficient ( $\mu_{SM}$ ) for an isenthalpic process ($dM=0$). The inversion curve, separating the cooling ( $\mu > 0$ ) and heating ( $\mu < 0$ ) regimes, is determined by the condition $\mu = 0$ . Exact analytical expressions for the inversion pressure ( $P_i$ ) and temperature ( $T_i$ ) are derived as functions of the horizon radius ( $r_+$ ).
Criticality and Stability: The study analyzes $P-V$ criticality by solving for inflection points in the equation of state. Global thermodynamic stability is assessed via the Gibbs free energy ( $G$ ), looking for "swallow-tail" structures indicative of first-order phase transitions.
Numerical Analysis: The theoretical results are validated through numerical simulations across two regimes of massive gravity (Weak and Strong) while systematically varying the ModMax parameter ( $\gamma$ ) and SM parameters ( $\delta, R$ ).

Key Contributions and Results

Modified Thermodynamic Quantities: The paper provides exact analytical expressions for the modified Hawking temperature and the equation of state, explicitly showing how the ModMax parameter $\gamma$ exponentially suppresses the electromagnetic contribution ( $q^2 e^{-\gamma}$ ) and how SM parameters introduce a scaling factor $f(r_+) = (1 + \delta \pi r_+^2)^{1-R/\delta}$ .
Specific Heat and Local Stability:
- The ModMax parameter $\gamma$ acts as a stabilizing agent. In regimes of strong massive gravity where standard electrodynamics yields volatile specific heat profiles with multiple divergences, increasing $\gamma$ smooths these instabilities, promoting continuous thermodynamic evolution.
- The SM parameters exhibit competing effects: increasing $\delta$ suppresses thermal fluctuations at large radii, while increasing $R$ amplifies them, potentially driving the system toward divergence in strong gravity regimes.
Joule-Thomson Expansion and Inversion Curves:
- ModMax Influence: Increasing $\gamma$ shifts the inversion transition to smaller horizon radii. Crucially, this shifts the global inversion curve ( $T_i$ vs. $P_i$ ) downward. Since the cooling region lies above the inversion curve in standard AdS thermodynamics, this downward shift implies that ModMax nonlinearities expand the physically accessible cooling domain for the black hole.
- Decoupling of Scales: A significant finding is the physical decoupling between microscopic statistical effects and macroscopic phase boundaries. While SM parameters ( $\delta, R$ ) critically govern local specific heat and the precise location of the inversion radius, the global inversion curve is virtually insensitive to variations in $\delta$ and $R$ . The global heating-cooling boundary is dictated primarily by the massive graviton background and the ModMax field, not the non-extensive statistical distribution of microstates.
- Single Branch: Unlike classical van der Waals fluids which often exhibit dual inversion branches, the black hole inversion curve retains a single, monotonically increasing branch.
Phase Transitions and Criticality:
- The system exhibits a van der Waals-like first-order phase transition between small and large black hole phases, terminating at a second-order critical point.
- The Gibbs free energy analysis reveals a characteristic swallow-tail structure for subcritical pressures ( $P < P_c$ ), confirming the coexistence of phases. As pressure approaches $P_c$ , the swallow-tail shrinks and vanishes, indicating a continuous transition.
- The ModMax parameter shifts the critical point by suppressing electromagnetic interactions, while massive gravity parameters determine the overarching macroscopic phase landscape.

Significance and Claims
The paper claims that the combined framework of ModMax electrodynamics, massive gravity, and Sharma-Mittal entropy offers a robust theoretical laboratory for probing black hole thermodynamics. The study highlights a distinct physical decoupling in the critical regime:

ModMax nonlinearities modify the critical phase boundary by suppressing electromagnetic interactions.
Sharma-Mittal parameters dictate the relative thermal stability of competing phases and local microstate correlations.
Massive gravity governs the overarching macroscopic phase landscape and the global inversion boundary.

The authors conclude that thermodynamic phase phenomena, particularly the Joule-Thomson expansion and critical behavior, serve as sensitive diagnostic tools. They can distinguish between nonlinear electrodynamics, non-extensive statistical corrections, and massive gravity effects, demonstrating that these modifications leave distinct, separable signatures on the black hole's thermodynamic geometry.