Thermodynamic properties of the Kerr Black-hole in non-linear electrodynamics with cosmological constant

This paper investigates the thermodynamic properties and horizon structure of a slowly rotating, magnetically charged Kerr black hole within nonlinear electrodynamics and a cosmological constant, demonstrating that these factors significantly modify the black hole's mass profile, horizon configurations, and key thermodynamic parameters such as temperature, angular velocity, and entropy.

The Gist

Imagine the universe as a giant, complex machine. For a long time, scientists have been trying to understand the most extreme parts of this machine: Black Holes. These are regions where gravity is so strong that nothing, not even light, can escape.

This paper is like a detailed repair manual for a specific type of black hole—a spinning one called a Kerr Black Hole—but with a few special "upgrades" added to the standard model. The authors, Vinayak Pawar and Siba Prasad Das, are asking: What happens if we change the rules of electricity and magnetism inside the black hole, and also add a "cosmic push" (the cosmological constant) to the universe?

Here is a breakdown of their findings using simple analogies:

1. The Setup: A Spinning Top in a Pushy Room

Think of a black hole as a heavy, spinning top.

• The Spin: The paper focuses on "slowly rotating" tops (they spin, but not wildly fast).
• The Room (Cosmological Constant): The universe isn't empty; it has a background energy.
  • In a de-Sitter (dS) universe, imagine the room is expanding and pushing the top outward (like a balloon inflating).
  • In an Anti-de-Sitter (AdS) universe, imagine the room is a box with walls that pull the top inward (like a gravitational trap).
• The Upgrade (Non-Linear Electrodynamics): Standard physics says electric and magnetic fields act like simple springs. The authors use a new rule called Non-Linear Electrodynamics (NLED). Think of this as replacing the simple spring with a smart, stretchy rubber band. This rubber band behaves differently when squeezed very tightly (near the center of the black hole).

2. Fixing the "Singular Core"

In old models, the center of a black hole was a "singularity"—a point of infinite density where physics breaks down, like a math error in the universe's code.

• The Paper's Claim: By using the "smart rubber band" (NLED), the authors show that the center of the black hole isn't a broken point anymore. Instead, the mass is spread out like a smooth, dense cloud.
• The Result: They calculated how this mass is distributed. They found that no matter how you tweak the magnetic charge or the "stretchiness" of the rubber band, the mass eventually levels off (reaches a "plateau") near the edge of the universe. It's like filling a bucket with water; eventually, it stops rising and stays at a steady level.

3. The "Sharkfin" Map: Where Can Black Holes Exist?

The authors drew a map (called a "sharkfin diagram") to show which combinations of spin and mass allow a black hole to actually exist without falling apart.

• The Pushy Room (dS): Because the universe is pushing outward, it's harder to keep a black hole together. The "safe zone" on their map is smaller. If the push is too strong, the black hole can't form distinct boundaries.
• The Pulling Room (AdS): Because the universe is pulling inward, it's easier to hold a black hole together. The "safe zone" on the map is much larger.
• The Limit: They found a critical "tipping point." If the universe's push/pull is too weak or too strong, the black hole's boundaries (horizons) disappear or merge into a single, extreme state.

4. The Three (or Two) Layers of the Onion

A spinning black hole in this model has layers, like an onion:

1. The Inner Horizon (Cauchy): A deep inner shell. In a non-spinning black hole, this disappears. But because this one spins, this inner shell exists, though it's very small.
2. The Event Horizon: The main "point of no return" we usually think of.
3. The Cosmological Horizon: (Only in the "Pushy Room" / dS). A far-out boundary where the universe's expansion becomes so strong that even light can't reach the black hole.

The Finding: The "Pushy Room" (dS) has all three layers. The "Pulling Room" (AdS) only has the inner and event horizons because the walls of the box prevent a third, outer boundary from forming.

5. Temperature and Heat

The paper calculates how "hot" these different layers are.

• The Surprise: The inner layer (Cauchy horizon) is extremely hot—much hotter than the main event horizon.
• The Analogy: Imagine a campfire (the event horizon) and a tiny, super-heated ember deep inside the logs (the inner horizon). The paper shows that in a spinning black hole, this inner ember is blazing with intense heat, while the outer fire is much cooler.
• Entropy (Disorder): They also measured the "disorder" (entropy) of the black hole. They found that the more the black hole spins, the lower its entropy becomes, and vice versa.

Summary of the Main Takeaway

The authors didn't just look at a black hole; they looked at a black hole with new physics (NLED) inside a changing universe (Cosmological Constant).

Their main conclusion is that these two factors change the black hole's personality significantly:

1. They remove the "broken point" at the center, making the black hole smooth and regular.
2. They change the number of boundaries the black hole has (3 in an expanding universe, 2 in a trapping one).
3. They create a massive temperature difference between the inner and outer layers, suggesting that these black holes are complex, non-equilibrium systems that are very different from the simple black holes we usually imagine.

In short, the paper argues that if you tweak the rules of magnetism and the expansion of the universe, the black hole transforms from a simple, singular object into a complex, multi-layered structure with a smooth core and distinct thermal zones.

Technical Summary

1. Problem Statement

The paper addresses the thermodynamic properties of slowly rotating Kerr black holes (BHs) in the presence of Non-Linear Electrodynamics (NLED) and a Cosmological Constant ($\Lambda$).

• Context: While the thermodynamics of standard Kerr-Newman black holes are well-studied, the interplay between rotation, NLED (which regularizes singularities), and the cosmological constant (de-Sitter or Anti-de-Sitter backgrounds) in a slowly rotating regime ($a \lesssim 0.1$) has not been systematically investigated.
• Gap: Existing literature often treats these factors separately. The authors aim to fill the gap by analyzing how NLED modifies the mass distribution and how this, combined with $\Lambda$, alters the horizon structure and thermodynamic variables (Temperature $T$, Angular Velocity $\Omega_h$, Entropy $S$) of a magnetically charged Kerr black hole.

2. Methodology

The authors employ a semi-analytical and numerical approach within the framework of General Relativity (GTR) coupled with NLED.

• Theoretical Framework:

  • Action: The study utilizes the action for NLED-AdS theory, combining the Einstein-Hilbert term with a cosmological constant and a specific NLED Lagrangian:
    $$\mathcal{L}(F) = -\frac{F}{1 + \sqrt{2|\beta F|}}$$
    where $F$ is the electromagnetic invariant and $\beta$ is the non-linearity parameter.
  • Metric: They assume a nearly spherical symmetric limit due to slow rotation ($a \leq 0.1$), utilizing the Boyer-Lindquist coordinates for the Kerr-(A)dS metric.
  • Charge Configuration: The analysis focuses on magnetically charged black holes ($q_m \neq 0$) with zero electric charge ($q_e = 0$) to avoid singularities associated with electrically charged models in the weak-field limit.

• Key Calculations:

  1. Mass Profile Derivation: The authors derive the radial mass function $M(r)$ by integrating the energy density $\rho(r)$, which includes contributions from both the NLED field and the cosmological constant. They explicitly exclude the divergent cosmological term from the mass integration to ensure a finite mass profile.
  2. Horizon Analysis: They solve the horizon equation $\Delta(r) = 0$ to identify the existence of Cauchy (inner), Event (outer), and Cosmological horizons.
  3. Parameter Space Mapping: "Sharkfin diagrams" are constructed in the $a-M$ plane to visualize the allowed regions for physical black hole solutions (where the discriminant $D > 0$) for various values of the cosmological length $L$.
  4. Thermodynamic Quantities: Numerical values for Entropy ($S$), Hawking Temperature ($T$), and Angular Velocity ($\Omega_h$) are calculated at the specific horizon radii for both de-Sitter (dS, $\Lambda > 0$) and Anti-de-Sitter (AdS, $\Lambda < 0$) backgrounds.

3. Key Contributions

• Regularization of the Core: The study demonstrates that the NLED model replaces the central singularity of the Kerr black hole with a finite, smooth energy distribution. The mass profile $M(r)$ is shown to be regular and convergent.
• Mass Plateau Phenomenon: A novel finding is that the mass profile $M(r)$ attains a plateau at radial distances close to the cosmological length scale ($L$), regardless of the specific combinations of magnetic charge ($q_m$) and non-linearity parameter ($\beta$).
• Horizon Structure Dependence: The paper establishes that the horizon structure (number and position of horizons) is critically dependent on the interplay between the spin parameter $a$, the cosmological length $L$, and the mass profile $M(r)$.
• Critical Length Threshold: The authors identify a critical cosmological length ($L_{cr} \approx 4.08$ for their specific parameters) below which real event horizons cannot form in the dS background due to the dominance of cosmological repulsion.

4. Key Results

A. Mass Distribution ($M(r)$)

• Behavior: For small $\beta$, the mass profile rises sharply near the center, indicating high energy density concentration. For larger $\beta$, the rise is gradual, reflecting a more diffuse energy distribution.
• Saturation: The total mass becomes finite as $r \to \infty$ (or approaches $L$), confirming the regular nature of the black hole solution.
• Parameters: Increasing magnetic charge $q_m$ increases the total mass, while increasing the non-linearity parameter $\beta$ decreases the energy density and total mass accumulation.

B. Horizon Structure (Sharkfin Diagrams)

• de-Sitter (dS) Background ($\Lambda > 0$):
  • Three distinct horizons exist: Inner (Cauchy), Outer (Event), and Cosmological.
  • The allowed parameter space in the $a-M$ plane is restricted compared to AdS due to the repulsive nature of positive $\Lambda$.
  • As $L$ increases, the repulsive effect weakens, expanding the allowed parameter space toward the standard Kerr limit.
• Anti-de-Sitter (AdS) Background ($\Lambda < 0$):
  • Only two horizons exist: Inner (Cauchy) and Outer (Event). The cosmological horizon is absent due to the attractive nature of negative $\Lambda$.
  • The allowed parameter space is larger than in the dS case, as the negative $\Lambda$ enhances gravitational confinement.
  • In the non-rotating limit ($a=0$), the inner horizon vanishes (collapsing to the singularity), but reappears with a finite radius for any $a > 0$.

C. Thermodynamic Properties

• Temperature ($T$):
  • dS: The event horizon and cosmological horizon have distinct temperatures, indicating a non-equilibrium thermodynamic state.
  • AdS: The inner horizon exhibits an extremely high temperature ($T_- \gg T_+$) compared to the event horizon, especially for small rotation parameters.
• Entropy ($S$) and Angular Velocity ($\Omega_h$):
  • Entropy is primarily influenced by the horizon area and the cosmological constant.
  • There is an inverse relationship between entropy and angular velocity for fixed parameters: larger entropy corresponds to lower angular velocity.
  • In AdS, the inner horizon has very low entropy but extremely high angular velocity and temperature.

5. Significance and Outlook

• Theoretical Implications: The work provides a consistent framework for studying regular black holes where NLED resolves the central singularity, while the cosmological constant dictates the global horizon structure.
• Thermodynamic Stability: The presence of multiple horizons with different temperatures (especially in dS) and the extreme temperature gradients in AdS suggest complex non-equilibrium thermodynamic behaviors that warrant further study regarding stability and phase transitions.
• Future Directions: The authors suggest extending the analysis beyond the slow-rotation approximation, investigating the magnetic potential as a thermodynamic variable, and exploring observational signatures such as black hole shadows, geodesic motion, and gravitational wave emissions from these NLED-corrected black holes.

In conclusion, the paper successfully demonstrates that the inclusion of NLED and a cosmological constant significantly modifies the internal structure and thermodynamic properties of Kerr black holes, offering a more regular and physically diverse model than classical General Relativity alone.