Geodesics Structure and Thermodynamic Properties of Gaussian Black Hole in Quadratic Ricci Scaler Gravity

This paper investigates and compares the geodesic motion of test particles and the thermodynamic stability of Gaussian Black Holes in both Einstein and modified quadratic Ricci scalar gravities, concluding that while differences exist in both aspects, the modified gravity model aligns more closely with physical reality, particularly in its thermodynamic behavior.

The Gist

Imagine the universe as a giant, complex video game. For decades, the best "physics engine" we had to explain how gravity works was Einstein's General Relativity. It's a fantastic engine that explains most things, but recently, scientists noticed some glitches. The universe isn't just moving; it's speeding up (expanding), and there's a lot of invisible "Dark Energy" and "Dark Matter" that the old engine struggles to explain perfectly.

To fix these glitches, physicists are trying out new "patches" or modified gravity theories. One of these patches is called $R^2$ gravity (or Quadratic Ricci Scalar gravity). It's like adding a new layer of rules to the game that handles extreme situations better.

This paper is a comparison between two versions of a specific cosmic object: a Gaussian Black Hole (GBH). Think of a Gaussian Black Hole not as a sharp, pointy singularity (a mathematical "infinity" that breaks the game), but as a "fuzzy" black hole. Instead of all its mass being crushed into a single, infinitely small dot, the mass is smeared out like a drop of ink in water, following a smooth, bell-shaped curve.

The authors, M. Haditale and B. Malekolkalami, asked: "If we put this fuzzy black hole into Einstein's old rules versus the new $R^2$ rules, how does it behave?" They looked at two main things: how things move around it (Geodesics) and how it feels "hot" or "stable" (Thermodynamics).

Here is what they found, explained simply:

1. The Movement of Particles (The "Roller Coaster" Test)

Imagine dropping a marble (a particle) and a beam of light (a photon) near this fuzzy black hole.

• The Old Rules (Einstein): The marble rolls down the hill and spirals in.
• The New Rules ($R^2$): The marble also rolls down and spirals in, but it does it faster and takes a shorter path.

The Analogy: Think of the new gravity theory as a steeper, more slippery slide. Even though the shape of the slide looks similar in both versions, the new one pulls things in with a bit more "grip." The authors found that in the new theory, gravity is slightly stronger, dragging particles into the black hole more aggressively.

2. The "Fuzzy" Mass Limit (The Backpack Analogy)

In the old theory, a black hole can theoretically keep getting heavier and heavier forever, like a backpack that never gets full.

• The New Theory: The authors found a "cap" on the backpack. As the black hole gets bigger, its mass stops growing and hits a maximum limit. It can't get infinitely heavy.
• Why it matters: The authors argue this is more realistic. In the real world, things usually have limits. A theory that says a black hole can grow without bound feels a bit "broken" to them, whereas the new theory puts a natural ceiling on it.

3. Temperature and "Cooling Down"

Black holes aren't just cold, dark pits; they actually have a temperature and can radiate energy (like a hot stove cooling down).

• The Finding: The new theory predicts that these fuzzy black holes are cooler than the ones in Einstein's theory.
• The Real-World Connection: We don't see black holes in our current universe blasting out massive amounts of radiation. The authors suggest that the new theory is a better fit for reality because it predicts lower temperatures, which explains why these black holes are "quiet" and not evaporating rapidly right now.

4. Stability and the "Tipping Point"

The authors checked if these black holes are stable or if they might fall apart.

• Einstein's Version: The black hole is always "stable" in a global sense. It's like a ball sitting at the very bottom of a bowl; it never wants to move.
• The New Version: The black hole has a "tipping point." There are specific sizes where the black hole becomes unstable and wants to radiate energy (like a ball balancing on a hilltop that might roll down).
• Why it matters: The authors think this is more realistic. In the real universe, things change phases (like water turning to ice). The new theory allows for these "phase changes" in black holes, whereas the old theory says they are stuck in one state forever.

5. The "Negative" Entropy Mystery

Entropy is a measure of disorder or "messiness." Usually, things get messier over time (positive entropy).

• The Twist: In the new theory, the "messiness" of the black hole can actually be negative or zero for a while.
• The Analogy: Imagine a messy room that, for a brief moment, becomes less messy than before without anyone cleaning it. This sounds weird, but the authors suggest this might be a better way to describe how information is preserved in black holes, potentially solving some of the "information paradox" puzzles that physicists have been stuck on for years.

The Bottom Line

The paper concludes that while the movement of particles looks roughly the same in both theories (just a bit faster in the new one), the thermodynamic properties (mass limits, temperature, and stability) are very different.

The authors argue that the $R^2$ modified gravity version of the Gaussian Black Hole is a better match for our physical world. It has natural limits on mass, predicts cooler temperatures (matching our observations of quiet black holes), and allows for complex stability changes that feel more like the dynamic universe we live in, rather than the rigid, infinite behavior of the old Einstein model.

Technical Summary

Technical Summary: Geodesics Structure and Thermodynamic Properties of Gaussian Black Hole in Quadratic Ricci Scaler Gravity

Problem Statement
General Relativity (GR) successfully describes gravitational interactions but faces challenges regarding the accelerating expansion of the universe, the nature of Dark Energy, and the existence of singularities. Modified gravity theories, such as $R + R^2$ (Quadratic Ricci Scaler gravity), offer alternatives that avoid ghost-like states and provide scale-invariant frameworks. Concurrently, the concept of Regular Black Holes (RBHs), specifically Gaussian Black Holes (GBHs) derived from Noncommutative Geometry, addresses the issue of spacetime singularities by replacing point-like structures with smeared Gaussian matter distributions. While GBHs have been studied within standard Einstein gravity, their behavior and thermodynamic consistency within the $R^2$ modified gravity framework remain under-explored. This paper aims to compare the geodesic structures and thermodynamic properties of GBHs in $R^2$ gravity against those in standard Einstein gravity to determine which theory offers a more physically consistent description.

Methodology
The authors employ a comparative analytical and numerical approach within the framework of $F(R)$ gravity, specifically setting $F(R) = R^2$.

1. Metric Derivation: Starting from the action for $F(R)$ gravity coupled with a Gaussian matter distribution (Energy-Momentum tensor derived from Noncommutative Geometry), the authors derive the field equations. They solve these equations numerically to obtain the metric function $f(r)$ for the GBH in $R^2$ gravity, ensuring the solution satisfies asymptotic flatness and regularity conditions. This is contrasted with the known metric function for GBH in Einstein gravity.
2. Geodesic Analysis: The motion of test particles (massive and massless) is analyzed using the effective potential method and the full geodesic equations. The authors numerically integrate the equations of motion in the equatorial plane to plot geodesic paths. They also derive expressions for the orbital frequency of circular orbits.
3. Thermodynamic Analysis: The authors calculate key thermodynamic quantities: Mass ($M$), Entropy ($S$), and Hawking Temperature ($T$).
   • Mass is determined by solving $f(r_+) = 0$ for the horizon radius $r_+$.
   • Entropy is calculated using the Wald entropy formula adapted for $F(R)$ gravity ($S \propto F'(R)$).
   • Temperature is derived from the surface gravity ($T \propto f'(r_+)$).
4. Stability Analysis: Local and global stability are assessed using Heat Capacity ($C$) and Gibbs Free Energy ($G = M - TS$). The sign of $C$ indicates local stability, while the sign of $G$ and its roots (Hawking-Page transition points) indicate global stability and phase transitions.

Key Contributions and Results

• Geodesic Structure:

  • The effective potential and geodesic paths in $R^2$ gravity are qualitatively similar to those in Einstein gravity, featuring stable circular orbits inside the horizon and unstable ones outside.
  • Quantitative Difference: Numerical results indicate that gravity is stronger in the $R^2$ framework. Test particles fall into the horizon via shorter paths in $R^2$ gravity compared to Einstein gravity.
  • Orbital Frequency: The orbital frequency of circular orbits decreases asymptotically with radius in both theories. However, the frequency is quantitatively higher in $R^2$ gravity, reinforcing the conclusion of stronger gravitational interaction in the modified theory.

• Thermodynamic Properties:

  • Mass: In both theories, a minimum mass is required for black hole formation. However, a significant divergence occurs at large horizon radii: in Einstein gravity, mass grows indefinitely, whereas in $R^2$ gravity, the mass asymptotically approaches a finite maximum value. The authors argue the latter is more physically plausible.
  • Entropy: Unlike Einstein gravity, where entropy is strictly positive and increases quadratically, the entropy in $R^2$ gravity can be positive, zero, or negative. The authors suggest that negative entropy changes align better with the information paradox and the evaporation process via Hawking radiation.
  • Temperature: Both theories predict a maximum temperature. Quantitatively, $R^2$ gravity predicts lower temperatures for the same horizon radius compared to Einstein gravity. The authors posit that this lower temperature is more consistent with the observed lack of black hole radiation in the current universe.
  • Stability (Heat Capacity): Both theories exhibit similar qualitative behavior, allowing for continuous (Type I) and discontinuous (Type II) phase transitions. However, near the singularity ($r_+ \to 0$), the heat capacity in $R^2$ gravity tends to $\pm \infty$, whereas it remains finite (negative) in Einstein gravity. For large horizons, both behave like the Schwarzschild black hole (unstable).
  • Stability (Gibbs Energy): In Einstein gravity, the Gibbs energy is always positive, implying the black hole is always globally stable and never enters a radiative phase. In contrast, $R^2$ gravity predicts two Hawking-Page transition points. Between these points, the black hole is globally unstable (radiative phase), while it is stable outside this range.

Significance and Claims
The paper concludes that while the geodesic structures of GBHs in $R^2$ and Einstein gravities are qualitatively similar, the thermodynamic properties reveal substantial differences. The authors claim that the $R^2$ modified gravity model provides a more consistent description of the physical world for several reasons:

1. It imposes a natural upper limit on black hole mass, avoiding the unphysical infinite mass growth predicted by Einstein gravity.
2. It allows for negative entropy changes, offering a more complete framework for understanding information preservation during evaporation.
3. It predicts lower temperatures, which better aligns with the observational reality that black holes in the present universe do not exhibit significant radiation.
4. It permits global instability and phase transitions (Hawking-Page transitions), which are absent in the standard Einstein GBH model, making the thermodynamic behavior more realistic.

Ultimately, the study suggests that the $R^2$ quadratic gravity framework, when applied to regular Gaussian black holes, yields results that are physically more robust and plausible than those derived from standard Einstein gravity.