Thermodynamics and null-geodesic of the Kerr-Newman black hole surrounded by quintessence and a cloud of string

This paper investigates the thermodynamic properties, stability, and null-geodesic structure of a Kerr-Newman black hole surrounded by quintessence and a cloud of strings, specifically analyzing how these features are modified by the inclusion of a modified dispersion relation (MDR).

The Gist

Imagine a black hole not just as a cosmic vacuum cleaner, but as a complex, spinning machine sitting in a very strange neighborhood. This neighborhood isn't empty; it's filled with two unusual things: Quintessence (a mysterious, ghostly energy that pushes the universe apart) and a Cloud of Strings (a fog made of tiny, vibrating cosmic threads).

This paper asks: "What happens to this machine if we also tweak the fundamental rules of how energy and speed work at the tiniest possible scales?"

Here is a breakdown of their findings using simple analogies:

1. The Setting: A Spinning Machine in a Weird Fog

Think of the Kerr-Newman black hole as a giant, charged, spinning top.

• Quintessence is like a wind blowing outward, trying to push the top away from itself.
• The Cloud of Strings is like a thick, sticky fog surrounding the top, adding extra weight and tension to the space around it.

2. The Twist: The "Speed Limit" Breaks (MDR)

In our normal world, there are strict rules about how fast things can go and how energy relates to speed (Einstein's Special Relativity). But the authors imagine a scenario where, at the tiniest scales (the Planck scale), these rules get a little "glitchy." This is called the Modified Dispersion Relation (MDR).

Think of it like driving a car. Normally, if you press the gas, you go faster. But in this "glitchy" world, the relationship between the gas pedal and speed changes slightly when you are driving at near-light speeds. The paper asks: How does this glitch affect the black hole's temperature and stability?

3. The Thermodynamics: The Black Hole's "Fever" and "Stomach"

The authors studied how the black hole behaves like a hot object that radiates heat (Hawking radiation).

• The Temperature (The Fever):
  Normally, a black hole has a specific temperature based on its size and spin. The authors found that because of the "glitchy" rules (MDR), the black hole's temperature changes.

  • The Analogy: Imagine the black hole is a feverish patient. The Quintessence and String Cloud are like different medicines or environmental factors. The "glitchy" rules act like a new, strange thermometer that gives a different reading than usual. They found that if you get too close to the center (the event horizon), the temperature reading goes wild, creating a "singularity" (a point where the math breaks down).

• The Heat Capacity (The Stomach):
  Heat capacity tells us how much energy a black hole needs to change its temperature.

  • The Discovery: Without the "glitch," the black hole has one stable state. But with the glitch, it has two distinct states (a "dual phase transition"). It's like water that can suddenly act like both ice and steam at the same time under specific conditions.
  • The Remnant: Eventually, as the black hole evaporates, it stops shrinking at a certain size. It leaves behind a tiny, stable "remnant" (like a seed that won't rot). The study found that the "fog" (strings) and the "wind" (quintessence) decide how big this seed is, but the "glitchy" rules (MDR) don't change the size of the seed, just how the black hole gets there.

• The Equation of State (The Pressure):
  They calculated the pressure inside this system. They found that the "glitchy" rules create a point of infinite pressure (a singularity) if you try to squeeze the black hole too small. It's like trying to compress a spring that suddenly turns into a brick.

4. The Light Show: Photons on a Rollercoaster

The second half of the paper looks at how light (photons) moves around this black hole. Light doesn't just fall in; it can orbit the black hole in circles, like a satellite.

• The Effective Potential (The Rollercoaster Track):
  Imagine the space around the black hole as a rollercoaster track. Light wants to roll down the track.

  • Quintessence (The Wind): Makes the track slightly flatter, making it easier for light to escape the "trap."
  • The String Cloud (The Sticky Fog): Makes the track steeper and higher. It acts like a stronger cage, trapping the light more tightly.
  • Spin (The Rotation): Because the black hole is spinning, it drags space with it (like a spinning spoon in honey).
    • Light spinning with the black hole (prograde) gets dragged closer to the center.
    • Light spinning against it (retrograde) gets pushed further out.

• The Instability (The Wobble):
  These circular orbits are unstable. If a photon gets nudged, it either falls in or flies away. The authors measured this "wobble" using something called the Lyapunov exponent.

  • The Finding: The faster the black hole spins, the more unstable the "against-the-spin" orbit becomes. However, the presence of the String Cloud actually calms the system down, making the orbits slightly more stable (less wobbly).

Summary: What Did They Learn?

1. Everything is Connected: The temperature, stability, and light behavior of a black hole aren't just about the black hole itself. They are heavily influenced by the "glitchy" quantum rules, the dark energy (quintessence), and the string cloud surrounding it.
2. The Black Hole Doesn't Vanish Completely: Instead of disappearing entirely, it likely leaves behind a tiny, stable remnant. The size of this remnant depends on the surrounding "fog" and "wind," not the quantum glitch.
3. Light Behaves Differently: The "fog" of strings makes it harder for light to escape, while the black hole's spin drags light in different directions depending on which way it's spinning.

In a nutshell: This paper is like a simulation of a cosmic engine running on a slightly different operating system. It shows us that if the universe's fundamental rules are tweaked, black holes don't just get hotter or colder; they change their entire personality, their stability, and how they interact with light.

Technical Summary

1. Problem Statement

The paper investigates the physical properties of a Kerr-Newman black hole (a rotating, charged black hole) immersed in two specific external fields: quintessence (a dynamic form of dark energy) and a cloud of strings. The study aims to understand how these external fields, combined with Modified Dispersion Relations (MDR) arising from quantum gravity theories (specifically Doubly Special Relativity), alter:

• The thermodynamic properties of the black hole (Hawking temperature, entropy, heat capacity, equation of state, and remnant formation).
• The null geodesic structure (photon motion, effective potential, circular orbits, and orbital instability).

The core problem addresses the gap in understanding how quantum corrections (via MDR) interact with complex astrophysical environments (quintessence and string clouds) to modify black hole evaporation and photon dynamics.

2. Methodology

The authors employ a combination of analytical derivation and numerical analysis:

• Metric Formulation: They utilize the metric for a Kerr-Newman black hole surrounded by quintessence and a cloud of strings, defined by parameters: mass ($M$), spin ($a$), charge ($Q$), quintessence strength ($\alpha$), equation of state ($\omega$), and cloud of string parameter ($b$).
• Modified Dispersion Relations (MDR): Two specific MDR models are applied to the energy-momentum relation:
  1. $p^2 = E^2 - \mu^2 + \eta_1 l_p E^3$
  2. $p^2 = E^2 - \mu^2 + \eta_2 l_p^2 E^4$
     These are used to derive corrections to the Heisenberg uncertainty principle and subsequently the black hole thermodynamics.
• Thermodynamic Derivation:
  • Hawking Temperature ($T_H$): Calculated using surface gravity ($\kappa$) and the area-entropy relation, corrected by MDR terms.
  • Entropy ($S$): Derived by integrating the first law of thermodynamics with MDR-corrected area changes.
  • Heat Capacity ($C$): Calculated as $C = \partial M / \partial T$ to analyze stability and phase transitions.
  • Equation of State: Derived by relating pressure ($P$) to the quintessence parameter and thermodynamic volume ($V$).
• Null Geodesic Analysis:
  • The Hamilton-Jacobi formalism is used to separate variables and derive the equations of motion for photons.
  • Effective Potential ($V_{eff}$): Constructed to analyze radial motion.
  • Circular Orbits: Radii for prograde and retrograde orbits are determined by solving $V_{eff} = 0$ and $V'_{eff} = 0$.
  • Lyapunov Exponent ($\lambda$): Calculated to quantify the instability timescale of circular photon orbits.

3. Key Contributions

• MDR-Corrected Thermodynamics: The paper provides explicit analytical expressions for the Hawking temperature, entropy, and heat capacity of a Kerr-Newman black hole in the presence of quintessence and a string cloud, incorporating two distinct MDR correction orders.
• Phase Transition Analysis: It demonstrates that MDR corrections induce a dual phase transition in the heat capacity (two singularities), whereas the standard case (without MDR) exhibits only a single phase transition.
• Remnant Formation: The study identifies conditions under which the heat capacity vanishes, signaling the formation of a black hole remnant. It clarifies that while quintessence and string parameters influence remnant mass, MDR parameters do not affect the existence of the remnant, only the thermodynamic path to it.
• Geometric Dynamics: The paper maps the influence of the quintessence parameter ($\omega$), spin ($a$), and string cloud ($b$) on the effective potential and the stability of photon orbits, offering a comprehensive view of how these fields compete or cooperate in shaping spacetime geometry.

4. Key Results

A. Thermodynamic Properties

• Hawking Temperature: The corrected temperature depends on MDR parameters ($\eta_1, \eta_2$), quintessence ($\alpha$), and string cloud ($b$). As the event horizon radius ($r_h$) decreases, MDR corrections lead to a singularity in temperature, suggesting a halt in evaporation.
• Entropy:
  • The corrected entropy depends only on MDR parameters, not on quintessence or string clouds.
  • The first MDR model ($S_1$) yields a linear correction term, while the second ($S_2$) introduces a logarithmic correction term ($\ln A$), a feature common in various quantum gravity theories.
• Heat Capacity & Stability:
  • Dual Phase Transition: MDR introduces a second phase transition point in the heat capacity graph ($C$ vs. $r_h$).
  • Remnants: The heat capacity vanishes at specific $r_h$ values determined by $\omega$, $\alpha$, and $b$. This indicates the black hole stops evaporating and leaves a stable remnant. The MDR parameters do not alter the remnant mass but affect the temperature profile leading up to it.
• Equation of State: The $P-V$ isotherms show that MDR deformation parameters ($\eta_i$) introduce singularities at very small volumes ($V$), indicating a breakdown of the classical equation of state at the quantum scale.

B. Null Geodesics and Photon Dynamics

• Effective Potential ($V_{eff}$):
  • Quintessence ($\omega$): Decreasing $\omega$ (making it more negative) lowers the potential barrier peak and shifts it slightly, weakening gravitational trapping.
  • Spin ($a$): Increasing spin significantly lowers the potential barrier height and shifts the peak to larger radii, weakening photon trapping.
  • String Cloud ($b$): Increasing $b$ raises the potential barrier height and shifts the peak to smaller radii, significantly strengthening gravitational confinement.
• Circular Photon Orbits:
  • Prograde vs. Retrograde: Prograde orbits (moving with rotation) are closer to the horizon due to frame dragging; retrograde orbits are further out.
  • Spin Effect: Increasing $a$ decreases the prograde orbit radius ($r_-$) and increases the retrograde radius ($r_+$).
  • String Cloud Effect: Increasing $b$ causes both prograde and retrograde orbit radii to increase monotonically, showing no directional preference.
• Lyapunov Exponent (Instability):
  • Spin ($a$): Increases the instability (Lyapunov exponent) of retrograde orbits but decreases it for prograde orbits.
  • String Cloud ($b$): Increasing $b$ suppresses the instability for both prograde and retrograde orbits (Lyapunov exponent decreases), making the circular orbits more stable against perturbations.

5. Significance

This work bridges the gap between quantum gravity corrections (MDR) and complex astrophysical environments.

1. Quantum-Gravity Interface: It demonstrates that MDRs significantly alter black hole thermodynamics, specifically by introducing logarithmic entropy corrections and dual phase transitions, which are crucial for understanding the final stages of black hole evaporation.
2. Astrophysical Modeling: By incorporating quintessence and string clouds, the paper offers a more realistic model of black holes in the current universe (dominated by dark energy) and early universe scenarios (string theory contexts).
3. Observational Implications: The findings on the Lyapunov exponent and effective potential provide theoretical constraints for future observations of black hole shadows (e.g., by the Event Horizon Telescope). The results suggest that the presence of a string cloud could stabilize photon orbits and alter the size and shape of the shadow, while quintessence and rotation have opposing effects on photon trapping.
4. Remnant Physics: The identification of conditions for black hole remnants supports the hypothesis that black holes may not evaporate completely, potentially solving information paradox issues.

In conclusion, the paper establishes that the interplay between quantum corrections (MDR), rotation, charge, and external fields (quintessence/strings) creates a rich and complex thermodynamic and dynamical landscape for Kerr-Newman black holes, fundamentally altering their stability and evaporation signatures.