Probing f(R) AdS Black Hole via Hawking Evaporation, Shadows and Thermal Fluctuations

Technical Summary: Probing f(R) AdS Black Hole via Hawking Evaporation, Shadows and Thermal Fluctuations

Problem Statement
This study investigates the physical properties of Anti-de Sitter (AdS) black holes (BHs) within the framework of $f(R)$ gravity, specifically incorporating a global monopole. The research addresses three primary areas: the dynamics of Hawking evaporation, the optical properties of BH shadows (including infalling accretion matter), and the thermodynamic stability of the system under thermal fluctuations. The authors aim to determine how the $f(R)$ coupling parameter ($\psi_0$), the global monopole parameter ($\lambda$), and the AdS radius ($\ell$) influence these phenomena, particularly in the context of modified gravity theories which offer alternatives to General Relativity (GR) for resolving cosmic problems.

Methodology
The authors employ a combination of analytical derivations and numerical simulations to analyze the static AdS BH solution in $f(R)$ gravity.

• Metric and Field Equations: The study utilizes the action for $f(R)$ gravity coupled with a global monopole field. The resulting static metric is derived, yielding a metric function $A(r)$ dependent on mass $M$, the AdS radius $\ell$, the global monopole parameter $\lambda$, and the $f(R)$ parameter $\psi_0$.
• Hawking Evaporation: The evaporation process is analyzed using the Stefan-Boltzmann law. The authors calculate the critical impact parameter ($b_c$) associated with unstable photon orbits and the Hawking temperature ($T$). The mass loss rate ($dM/dt$) is computed numerically to determine the BH lifetime.
• Shadow Analysis: The visual appearance of the BH is examined by calculating the angular radius of the shadow ($\beta_{sh}$) and the shadow radius ($R_{sh}$) using null geodesics. The study also employs a "backwards ray tracing" procedure to simulate the specific intensity of optically sparse, radially infalling accretion matter, generating 2D intensity maps in celestial coordinates.
• Thermodynamics: The authors derive expressions for various thermodynamic quantities, including entropy, Helmholtz free energy, internal energy, enthalpy, Gibbs free energy, and specific heat. Crucially, they incorporate thermal fluctuation corrections to the entropy (logarithmic corrections) to study the system's stability and phase transitions.

Key Contributions and Results

1. Hawking Evaporation and Lifetime:

   • Numerical analysis reveals that the BH lifetime is generally infinite, implying that the black hole does not evaporate completely but instead forms a stable remnant in the late stages. This finding is consistent with the third law of black hole thermodynamics.
   • The evaporation rate is highly sensitive to the AdS radius ($\ell$) and the coupling parameters ($\lambda$ and $\psi_0$). Specifically, for certain parameter sets (e.g., large $\ell$), the BH may evaporate in a finite time, whereas for others, the process is suppressed as the mass and temperature approach zero.

2. Shadow and Optical Properties:

   • Shadow Radius: The study finds an inverse relationship between the $f(R)$ parameter $\psi_0$ and the shadow radius; as $\psi_0$ increases, the shadow radius decreases. Conversely, the global monopole parameter $\lambda$ exhibits a direct relationship, where increasing $\lambda$ leads to an increase in the shadow radius.
   • Photon Sphere: The effective potential for photon orbits is analyzed, showing that $\psi_0$ increases the potential barrier while $\lambda$ decreases it.
   • Accretion Signatures: Simulations of infalling accretion matter show that while parameter variations influence the central dark region, the most significant visual change occurs in the bright photon ring located at the photon sphere. The ring's position shifts in accordance with the changes in the shadow radius driven by $\psi_0$ and $\lambda$.

3. Thermodynamics and Stability:

   • Entropy: Corrected entropy (including logarithmic terms from thermal fluctuations) shows that small horizon radii often correspond to negative trends, indicating instability and potential phase transitions. The parameter $\sigma$ (related to the correction term) significantly alters the entropy behavior.
   • Free Energies: The Helmholtz and Gibbs free energies indicate that the system can transition between stable and unstable phases. Negative free energy values generally suggest thermodynamic stability, while positive values indicate instability. The parameter $\sigma$ introduces distinct behaviors, particularly regarding phase transitions at fixed horizon radii.
   • Specific Heat: The specific heat is found to be negative across the domain of the horizon radius for all parameter variations, suggesting that the BH configuration is thermodynamically unstable and prone to phase transitions.
   • Pressure and Enthalpy: The corrected pressure remains positive under variations of $\lambda$ and $\psi_0$ but becomes negative with $\sigma$. The enthalpy analysis suggests a transition from unstable small BH phases to stable large BH phases as the horizon radius increases.

Significance of the Work
The paper claims that its analysis provides a comprehensive understanding of how modified gravity parameters ($f(R)$ and global monopole) alter the fundamental behavior of AdS black holes. The results highlight that:

• The lifetime of such black holes is not necessarily finite, potentially resolving the information loss paradox by suggesting the existence of remnants.
• The optical signatures (shadows and accretion rings) are sensitive to the specific values of $\psi_0$ and $\lambda$, offering potential observational tests to distinguish between GR and $f(R)$ gravity models using future telescope data.
• Thermal fluctuations play a critical role in the thermodynamic stability of these systems, particularly in the microscopic (small radius) regime where quantum effects are non-negligible.

The authors conclude that modified gravity theories, specifically $f(R)$ gravity, offer a robust framework for exploring the intricate dynamics of black holes, providing deeper insights into both their shadow formation and thermodynamic behavior within the AdS background. They suggest that future work could extend this analysis to rotating black holes within the same framework.