⚛️ general relativity

Geodesics and thermodynamics of a $\kappa$ -deformed anti-de Sitter black hole surrounded by a quintessence field

This paper investigates a $\kappa$ -deformed Schwarzschild-anti-de Sitter black hole surrounded by quintessence, revealing that the mass-dependent deformation creates an inner Cauchy horizon and intensifies the central singularity while necessitating modified entropy or mass rescaling to restore thermodynamic consistency, and uniquely decoupling Joule-Thomson inversion from van der Waals criticality alongside a significant suppression of Hawking emission.

Original authors: Faizuddin Ahmed, Ahmad Al-Badawi, Izzet Sakalli, Erdem Sucu

Published 2026-09-03

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CC BY 4.0

Original authors: Faizuddin Ahmed, Ahmad Al-Badawi, Izzet Sakalli, Erdem Sucu

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

Black holes are often imagined as simple, all-consuming pits in space, but modern physics suggests they are far more complex. They are not just objects of gravity; they are thermodynamic systems that possess temperature, entropy, and a capacity to radiate energy. For decades, scientists have studied how these objects behave when placed in a universe with a cosmological constant, a form of energy that pushes space apart, creating a background known as anti-de Sitter space. Recently, researchers have begun to ask what happens when these black holes are subjected to the strange rules of quantum gravity, where the smooth fabric of spacetime might be replaced by a grainy, deformed structure. One such deformation, known as kappa-deformation, suggests that the geometry of space changes in a way that depends on the mass of the object itself. Understanding how this specific type of quantum correction alters the life and death of a black hole is crucial, because it could reveal whether the laws of thermodynamics hold up under the extreme conditions where gravity and quantum mechanics collide.

In a recent study, a team of physicists investigated a specific model of a black hole that has been modified by this kappa-deformation and is also surrounded by a field of quintessence, a theoretical form of dark energy that exerts a repulsive force. The researchers wanted to see how this unique combination of quantum deformation and dark energy would change the black hole's internal structure, its path through space, and its heat. They found that the deformation, which scales with the mass of the black hole, creates a hidden inner horizon inside the event horizon, a feature usually only seen in charged or spinning black holes. However, this new structure does not fix the central problem of black holes: the singularity. Instead of smoothing out the infinite density at the center, the deformation makes the curvature of space diverge even more violently, becoming ten times more severe than in a standard black hole.

The study also explored how light and matter move around this deformed object. By tracing the paths of photons, the researchers discovered that the quantum deformation pulls the photon sphere—the region where light orbits the black hole—closer to the center. This causes the black hole's shadow, the dark silhouette it casts against the background light, to shrink slightly. For a static observer far away, the angular size of this shadow decreases as the deformation grows stronger. Similarly, the orbits of massive particles, such as stars or gas clouds, are pulled inward, making the innermost stable orbit smaller. While these changes are measurable, the study suggests that the shadow's size is a much more sensitive indicator of this specific deformation than the orbital patterns of surrounding matter.

Perhaps the most significant finding concerns the black hole's temperature and how it obeys the laws of thermodynamics. The researchers found that the standard rule linking a black hole's size to its entropy breaks down when this deformation is present. To keep the fundamental laws of physics consistent, they had to propose a new, corrected formula for the black hole's entropy, which turns out to be larger than the standard value. This correction is necessary because the deformation changes the relationship between the black hole's mass and its surface area. Furthermore, the study revealed a surprising new behavior: unlike a standard black hole, which always cools down as it expands, this deformed black hole can actually heat up under certain conditions. This phenomenon, known as the Joule-Thomson effect, creates a specific boundary where the black hole switches from cooling to heating.

Interestingly, the researchers found that while this heating effect appears, the black hole does not exhibit the complex phase transitions seen in other quantum-corrected models. In many other theories, quantum corrections create a critical point where the black hole behaves like a gas transitioning into a liquid, but this specific deformation prevents that from happening. The study concludes that the quantum-gravity correction responsible for the heating effect is distinct from the one that creates critical phase transitions. Finally, the team calculated how the black hole radiates energy. They found that the deformation suppresses the peak emission of Hawking radiation by a factor of roughly four, making the black hole radiate much more slowly and more sparsely. This reduction is driven almost entirely by the drop in temperature caused by the deformation, rather than by changes in the black hole's size or shadow. The work provides a clear, concrete example of how quantum gravity can alter the thermodynamic fate of a black hole without necessarily resolving its central singularity.

Technical Summary: Geodesics and Thermodynamics of a $\kappa$ -Deformed Anti-de Sitter Black Hole Surrounded by a Quintessence Field

Problem Statement
This study investigates the structural, optical, and thermodynamic properties of a Schwarzschild-anti-de Sitter (AdS) black hole modified by a $\kappa$ -deformation and immersed in a Kiselev quintessence field. Unlike previous quantum-corrected black hole models (such as those based on Moyal noncommutative geometry or loop quantum gravity) where deformation parameters often act as independent "hairs," this model employs a mass-dependent lapse deformation. Specifically, the deformation amplitude scales as $\hat{a}M^3$ , where $\hat{a}$ is a dimensionless parameter. The authors aim to determine how this specific scaling affects the horizon structure, singularity resolution, geodesic motion, and thermodynamic consistency, particularly regarding the Joule-Thomson (JT) expansion and the existence of van der Waals-like criticality.

Methodology
The authors begin by constructing the line element for the $\kappa$ -deformed Schwarzschild-AdS black hole in a quintessence background, derived from a corrected energy-momentum tensor consistent with $\kappa$ -deformed Newtonian potentials. The metric function $f(r)$ includes terms for the mass $M$ , the cosmological constant $\Lambda$ , the quintessence parameters $(N, w)$ , and the deformation term $\hat{a}M^3/r^3$ .

The analysis proceeds through four main stages:

Geometric Analysis: The authors compute curvature invariants (Ricci scalar and Kretschmann scalar) to assess singularity behavior and determine the admissibility bounds for the deformation parameter by analyzing the horizon equation, which becomes a cubic in mass due to the deformation.
Geodesic Structure: Null and timelike geodesics are analyzed to determine the photon sphere radius, critical impact parameter, black hole shadow, Lyapunov exponents, and innermost stable circular orbits (ISCO).
Thermodynamics: The surface gravity is used to define the Hawking temperature. The authors test the consistency of the first law of thermodynamics ( $dM = TdS$ ) with the standard Bekenstein-Hawking area law. Finding a violation due to the mass-dependent deformation, they derive a corrected entropy function and an alternative mass-rescaling factor to restore consistency.
Joule-Thomson Expansion: Treating the mass as enthalpy in the extended phase space, the authors derive a closed-form expression for the JT coefficient $\mu_{JT} = (\partial T / \partial P)_M$ . They map the inversion curves and compare the phase structure with standard AdS black holes and other quantum-corrected models.

Key Contributions and Results

Horizon Structure and Singularity: The $\kappa$ -deformation introduces an inner Cauchy horizon even in the absence of charge or spin. However, contrary to some minimal-length models that regularize singularities, this deformation exacerbates the central singularity. The Kretschmann scalar diverges as $r^{-10}$ (compared to the Schwarzschild $r^{-6}$ ), with a coefficient growing quadratically with the deformation strength. The solution exists only within a bounded parameter space defined by $\hat{a} < \hat{a}_{max}$ , which depends on the horizon radius and quintessence parameters.
Optical Properties: Increasing the deformation parameter $\hat{a}$ shrinks both the photon sphere radius and the critical impact parameter. Consequently, the angular radius of the black hole shadow, as seen by a static observer, decreases. The Lyapunov exponent (governing instability) decreases, while the orbital frequency at the photon sphere increases, suggesting a higher ringdown frequency and longer damping time in the eikonal limit.
Thermodynamic Consistency: The mass-dependence of the deformation term causes the standard area law ( $S = \pi r_+^2$ ) to violate the first law of thermodynamics. The authors resolve this by deriving a modified entropy $S(r_+)$ that satisfies $dM = TdS$ . Alternatively, they show that keeping the area law intact requires rescaling the mass (enthalpy) by a factor $W$ . In the Schwarzschild limit, the corrected entropy is a simple rescaling of the area law rather than a logarithmic correction, a distinct feature of this mass-coupled deformation.
Joule-Thomson Expansion and Criticality: The study derives a compact closed-form for the JT coefficient, revealing that its zero locus (the inversion curve) is directly related to the photon sphere condition shifted by one derivative. Crucially, while the $\kappa$ -deformation induces a JT inversion curve (a heating region at small radii) in an otherwise neutral AdS black hole—mimicking effects seen in Moyal and loop quantum gravity models—it fails to generate a van der Waals critical point. The admissibility bound truncates the isotherms before a critical loop can form. This demonstrates that JT inversion and van der Waals criticality can be decoupled in quantum-gravity-corrected black holes.
Radiation Sparsity and Emission: The deformation lowers the Hawking temperature significantly. This leads to a substantial increase in the sparsity of Hawking radiation (the mean separation between quanta). The peak of the energy emission spectrum drops by a factor of roughly four as $\hat{a}$ increases from 0 to 6, an effect driven almost entirely by the temperature drop rather than changes in the geometric cross-section (shadow).

Significance and Claims
The paper claims that the mass-dependent nature of the $\kappa$ -deformation is the organizing principle behind all observed results. The primary significance lies in demonstrating that:

Quantum corrections can generate Cauchy horizons without charge or rotation but do not necessarily resolve singularities.
Thermodynamic consistency in such models requires either a correction to the entropy or a rescaling of the mass, leading to a modified Smarr relation where the deformation parameter does not appear as a conjugate variable (due to its dimensionless nature).
The presence of a Joule-Thomson inversion curve does not imply the existence of a van der Waals critical point. This geometry serves as a counterexample to the notion that these two phenomena are inextricably linked in quantum-gravity-corrected black holes.
The deformation suppresses Hawking emission intensity and increases flux sparsity, suggesting that such black holes would evaporate more slowly and "colder" than their classical counterparts, with the effect being observable in the emission spectrum rather than just the shadow size.

The authors conclude that the contraction of the shadow by approximately 7% across the admissible range is potentially within the reach of next-generation very-long-baseline interferometry, offering a potential observational constraint on the $\kappa$ -deformation scale.

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