Shadow and Thermodynamics of deformed Schwarzschild-AdS black hole with a Cloud of Strings embedded in Perfect Fluid Dark Matter

This paper investigates the optical and thermodynamic properties of a deformed Schwarzschild-AdS black hole coupled to a cloud of strings and perfect fluid dark matter, revealing that while geometric deformation and dark matter influence the black hole shadow, they do not induce new phase transitions, with criticality emerging only when the deformation parameter mimics an electric charge in a Reissner-Nordström-AdS-like regime.

The Gist

Imagine the universe as a giant, cosmic ocean. For over a century, we've used a map called General Relativity to navigate it. This map has been incredibly accurate, helping us predict everything from the wobble of planets to the ripples in space-time caused by colliding black holes. But, like any old map, it has some blank spots. It doesn't explain the "dark stuff" (Dark Matter and Dark Energy) that makes up most of the universe, and it doesn't quite mesh with the tiny world of quantum mechanics.

To fill in these blanks, scientists are testing new theories. This paper is like a detective story where three researchers (Faizuddin Ahmed, Carlos Romero-Figueroa, and Hernando Quevedo) investigate a very specific, strange type of black hole to see how it behaves under these new conditions.

Here is the story of their investigation, broken down into simple concepts.

1. The Setting: A Black Hole with "Accessories"

Usually, we imagine a black hole as a simple, lonely sphere of gravity. But in this study, the scientists dressed up the black hole with three strange "accessories" to see how they change the game:

• The Cloud of Strings: Imagine the black hole is wrapped in a giant, invisible net made of one-dimensional strings (like a cosmic fishing net). This net pulls on space-time in a unique way, creating a "gap" or a missing slice in the geometry of space.
• Perfect Fluid Dark Matter: Think of this as a thick, invisible fog surrounding the black hole. Unlike normal gas, this "fog" has special rules about how it moves and presses against things. It's the invisible stuff that holds galaxies together.
• Geometric Deformation: Imagine the black hole isn't a perfect sphere but is slightly squashed or stretched, like a balloon that's been squeezed. This represents a "deformation" in the fabric of space-time itself.

2. The Optical Test: The "Shadow" and the "Ring of Fire"

The first thing the team looked at was Optics. How does light behave near this dressed-up black hole?

• The Photon Sphere (The Ring of Fire): Imagine a race track right next to the black hole where photons (light particles) can run in circles. This is the "photon sphere." If you were standing there, you'd see a ring of light.
  • The Finding: The "Dark Matter Fog" (the fluid) makes this ring bigger. It acts like a magnifying glass, pushing the light out. However, the "String Net" and the "Squashed Shape" tend to shrink or shift this ring.
• The Shadow (The Silhouette): If you look at a black hole from far away (like the Event Horizon Telescope does), you see a dark circle (the shadow) surrounded by a bright ring.
  • The Finding: The size of this dark shadow changes based on the accessories. The "String Net" creates a deficit in the angle of view, making the shadow look slightly different than a standard black hole. The Dark Matter fog makes the shadow appear larger.

Analogy: Think of the black hole as a lighthouse. The "Cloud of Strings" and "Dark Matter" are like different types of fog or glass lenses placed around the lighthouse. They don't turn the light off, but they change how big the beam looks and how far the light can reach before getting trapped.

3. The Thermodynamic Test: The "Weather Report"

Next, the scientists looked at the Thermodynamics. This is basically the "weather report" of the black hole: its temperature, pressure, and energy.

• The Phase Transition (The Boiling Water): In the world of black holes, there's a famous phenomenon called the Van der Waals transition. Imagine water boiling. As you heat it, it turns from liquid to gas. Similarly, black holes can switch between a "Small Black Hole" phase and a "Large Black Hole" phase, just like water boiling.
• The Big Question: Do these new accessories (Strings, Dark Matter, Deformation) create new types of weather? Do they create new kinds of phase transitions?
• The Finding: No. The scientists found that the "weather" of this complex black hole is almost exactly the same as a standard, charged black hole (called Reissner-Nordström-AdS).
  • The "Deformation" and "Dark Matter" parameters act like a dial that slightly shifts the temperature or pressure, but they don't invent a new type of storm. The critical point (where the phase transition happens) is still driven by the "charge" (the squashed shape acting like electricity), not by the strings or the dark matter fog.

Analogy: Imagine you have a pot of water. You add salt (Dark Matter) and you squeeze the pot (Deformation). The water might boil at a slightly different temperature, but it's still just water turning into steam. You haven't turned the water into "fire" or "ice" just by adding those ingredients. The fundamental nature of the boiling remains the same.

4. The Microstructure: The "Social Network" of Atoms

Finally, they used a tool called Geometrothermodynamics (GTD). This is a fancy way of looking at the "social network" of the black hole's microscopic parts.

• Curvature as Interaction: In this framework, the "curvature" of the black hole's thermodynamic map tells us how its internal particles interact.
  • Positive Curvature: Particles are pushing each other away (repulsive).
  • Negative Curvature: Particles are pulling each other together (attractive).
• The Finding:
  • Small Black Holes: They have high curvature, meaning their internal parts are fighting and pushing against each other (strong interactions).
  • Large Black Holes: They have almost zero curvature, meaning their internal parts are calm and barely interacting, like a gas in a room (weak interactions).
  • The transition from Small to Large is like going from a crowded, chaotic mosh pit to a quiet, empty park.

The Bottom Line

This paper is a stress test for our understanding of the universe. The researchers asked: "If we wrap a black hole in strings, surround it with dark matter, and squish its shape, does it become a totally new kind of object?"

The Answer: Not really.
While these accessories change the size of the shadow and the exact temperature at which things happen, they don't change the fundamental rules. The black hole still behaves like a standard, charged black hole. The "Dark Matter" and "Strings" are just minor tweaks to the volume knobs, not new instruments in the orchestra.

This is good news for physicists! It means that even with all these complex additions, the core laws of black hole thermodynamics are robust and consistent. The universe is complicated, but it's not chaotically complicated; the underlying patterns remain clear.

Technical Summary

1. Problem Statement

The paper addresses the need to understand how geometric deformations and exotic matter distributions influence the observable properties and thermodynamic stability of black holes. Specifically, the authors investigate a static, spherically symmetric deformed Schwarzschild-AdS black hole that is simultaneously:

• Coupled to a Cloud of Strings (CS).
• Embedded in a Perfect Fluid Dark Matter (PFDM) background.

The study aims to determine how these factors modify:

1. Optical observables: The photon sphere radius and the black hole shadow size.
2. Thermodynamic behavior: Local stability, phase transitions (specifically Van der Waals-like behavior), and the Hawking-Page transition.
3. Microstructure: The underlying thermodynamic interactions via Geometrothermodynamics (GTD).

2. Methodology

A. Spacetime Construction

The authors construct the metric for the system by solving Einstein's field equations with the energy-momentum tensors of:

• Perfect Fluid Dark Matter (PFDM): Modeled with an equation of state $p/\rho = \epsilon$, leading to a logarithmic term in the metric function.
• Cloud of Strings (CS): Modeled via the Nambu-Goto action, introducing a solid angle deficit parameter.
• Geometric Deformation: A deformation function involving parameters $\alpha$ and $\beta$ is introduced, which generalizes the Reissner-Nordström (RN) solution.

The resulting lapse function $f(r)$ is:
$$f(r) = 1 - \gamma - \frac{2M}{r} + \alpha \left[ \frac{\beta^2}{3r(\beta+r)^3} + \frac{1}{(\beta+r)^2} \right] - \frac{\Lambda}{3}r^2 + \frac{\lambda}{r} \ln \frac{r}{|\lambda|}$$
Where:

• $M$: Mass.
• $\gamma$: String cloud parameter.
• $\lambda$: PFDM parameter.
• $\alpha, \beta$: Deformation parameters ($\alpha$ acts analogously to electric charge squared, $Q^2$).
• $\Lambda$: Cosmological constant.

B. Optical Analysis

• Null Geodesics: The authors use the Lagrangian formalism to derive the effective potential $V_{eff}$ for photon motion.
• Photon Sphere: Determined by solving $d/dr(f(r)/r^2) = 0$ numerically.
• Shadow Radius: Calculated using the critical impact parameter $b_c$ and accounting for the solid angle deficit caused by the string cloud, leading to the relation $R_{sh} = \sqrt{1-\gamma} b_c$.

C. Thermodynamic Analysis

• Extended Phase Space: The cosmological constant is treated as thermodynamic pressure ($P = -\Lambda/8\pi$), and the black hole mass is interpreted as enthalpy.
• Quasi-Homogeneity: The system is analyzed as a quasi-homogeneous thermodynamic system of degree $w_M = 1/2$. The authors derive the fundamental equation $M(S, P, \gamma, \lambda, \alpha, \beta)$, the generalized Smarr relation, and the first law.
• Phase Transitions: Stability is analyzed via heat capacity ($C_X$) and Gibbs free energy ($G$) to identify Van der Waals (VdW) type phase transitions and Hawking-Page (HP) transitions.
• Geometrothermodynamics (GTD): The thermodynamic microstructure is probed using the GTD formalism (specifically metric $g_{II}$). The curvature scalar $R_{II}$ is calculated to identify phase transitions and characterize microscopic interactions (attractive vs. repulsive).

3. Key Contributions and Results

A. Optical Properties

• Photon Sphere: The radius of the photon sphere ($r_s$) increases with both the string cloud parameter ($\gamma$) and the magnitude of the PFDM parameter ($|\lambda|$). The deformation parameter $\beta$ also influences the size but acts differently than the matter terms.
• Effective Potential:
  • PFDM ($\lambda$): Increases the height of the potential barrier, strengthening photon confinement.
  • String Cloud ($\gamma$): Reduces the potential barrier globally due to the solid angle deficit.
• Shadow Radius: The presence of the string cloud creates a non-trivial relation between the critical impact parameter and the shadow radius ($R_{sh} \propto \sqrt{1-\gamma}$). Both CS and PFDM tend to enlarge the shadow radius.
• Singularity: The PFDM parameter introduces a logarithmic divergence at the center ($r \to 0$), preventing the existence of regular black hole solutions unless $\lambda \to 0$ and specific constraints on $\alpha/\beta$ are met.

B. Thermodynamic Properties

• Critical Behavior: The system exhibits Van der Waals-like phase transitions (Small Black Hole $\leftrightarrow$ Large Black Hole).
• Role of Parameters:
  • The critical behavior is inherited from the RN-AdS sector. The deformation parameter $\alpha$ effectively plays the role of electric charge squared ($Q^2$).
  • No New Phases: Neither the geometric deformation ($\beta$), the string cloud ($\gamma$), nor the PFDM ($\lambda$) generates new independent critical structures. They only perturb the location of the critical point $(P_c, v_c, T_c)$ of the RN-AdS solution.
  • Universal Ratio: The critical ratio $P_c v_c / T_c$ remains $3/8$ at the leading order (RN-AdS value), receiving only subleading corrections from $\beta$ and $\lambda$. The string cloud parameter $\gamma$ does not affect this ratio at the linear level.
• Hawking-Page (HP) Transition:
  • The CS ($\gamma$) and PFDM ($\lambda$) contributions increase the HP transition temperature.
  • The deformation parameter $\alpha$ decreases the HP temperature.
  • The net shift is determined by the competition between these enhancing and suppressing effects.

C. Geometrothermodynamics (GTD)

• Curvature Singularities: The GTD scalar curvature $R_{II}$ diverges exactly where the heat capacity diverges, confirming that geometric singularities correspond to physical phase transitions.
• Microstructure Interpretation:
  • Small Black Holes (SBH): Exhibit positive curvature ($R_{II} > 0$), indicating repulsive thermodynamic interactions and a strongly interacting microscopic structure.
  • Large Black Holes (LBH): Exhibit negative or near-zero curvature ($R_{II} \le 0$), indicating attractive interactions or an ideal-gas-like regime with weak interactions.
  • This provides a geometric explanation for the SBH-LBH phase transition as a change in the nature of microscopic interactions.

4. Significance and Conclusion

The paper provides a comprehensive framework for understanding how matter distributions and geometric deformations interact in the strong-field regime of black holes.

• Observational Relevance: The results suggest that future observations of black hole shadows (e.g., by the Event Horizon Telescope) could potentially constrain the parameters of string clouds and dark matter profiles, as these significantly alter the shadow size and photon sphere.
• Theoretical Insight: The study clarifies that while external matter and deformations quantitatively shift thermodynamic quantities, they do not qualitatively alter the fundamental phase structure of AdS black holes. The critical phenomena remain driven by the effective charge-like parameter ($\alpha$), suggesting that the mechanism for phase transitions in these systems is fundamentally electromagnetic-like rather than induced by dark matter or string clouds.
• Microstructure: The application of GTD successfully links macroscopic phase transitions to microscopic interaction types (repulsive vs. attractive), offering a deeper geometric understanding of black hole thermodynamics.

In summary, the work demonstrates that while the "cloud of strings" and "perfect fluid dark matter" leave distinct imprints on optical observables and shift thermodynamic critical points, the underlying phase structure of the deformed black hole remains robustly tied to the RN-AdS solution.