Joule-Thomson Effect and Geodesic Structure of Charged AdS Black Holes in f(R,T) Coupled with Nonlinear Electrodynamics

Technical Summary: Joule-Thomson Effect and Geodesic Structure of Charged AdS Black Holes in f(R, T) Coupled with Nonlinear Electrodynamics

Problem Statement
While General Relativity (GR) has been experimentally confirmed through gravitational wave detection and black hole imaging, it faces challenges in explaining cosmological phenomena such as dark energy and cosmic inflation, and it predicts spacetime singularities. Modified gravity theories, specifically $f(R, T)$ gravity (where the action depends on the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$), offer a framework to address these issues by introducing non-minimal matter-curvature coupling. Concurrently, Nonlinear Electrodynamics (NLED) provides a mechanism to resolve central singularities, leading to Regular Black Hole (RBH) solutions. Despite existing studies on RBHs in GR and separate investigations into $f(R, T)$ gravity, a systematic analysis of the thermodynamic behavior (specifically Joule-Thomson expansion) and geodesic structures of RBHs arising from the coupling of $f(R, T)$ gravity with NLED remains largely unexplored. This work addresses that gap.

Methodology
The authors investigate a static, spherically symmetric charged Anti-de Sitter (AdS) black hole solution derived from an action combining $f(R, T)$ gravity and a power-law NLED source.

• Theoretical Framework: The model adopts a linear $f(R, T) = R + \beta T$ and a power-law NLED Lagrangian $L_{NLED} = f_0 + F + \alpha F^p$. The study focuses on the specific case where the power parameter $p=6$, which allows for the resolution of singularities. The metric function $A(r)$ is derived, incorporating an effective cosmological constant $\Lambda_{eff}$ and higher-order correction terms dependent on the charge $Q$, the NLED nonlinearity parameter $\alpha$, and the modified gravity coupling $\beta$.
• Thermodynamic Analysis: The study operates within the extended phase space thermodynamics framework, treating the black hole mass as enthalpy. The authors derive expressions for mass, Hawking temperature, entropy, Gibbs free energy, and specific heat. They perform both global stability analysis (via Gibbs free energy) and local stability analysis (via the Hessian matrix of the mass function).
• Joule-Thomson (JT) Expansion: The JT expansion process is analyzed by calculating the JT coefficient ($\mu_{JT}$) and deriving inversion curves ($T_i$ vs. $P_i$) and isenthalpic trajectories. This characterizes the cooling and heating regimes of the black hole during expansion.
• Geodesic Analysis: The motion of massive (timelike) and massless (null) particles is studied using the Lagrangian formalism. The authors analyze the effective potential to determine conditions for stable circular orbits, Innermost Stable Circular Orbits (ISCO), photon spheres, and scattering/capture regions.

Key Contributions and Results

1. Thermodynamic Stability:

   • The Gibbs free energy analysis reveals swallowtail structures characteristic of first-order phase transitions.
   • The electric charge ($Q$) and the cosmological constant ($\Lambda$) are found to have the most significant impact on the global thermodynamic structure, suppressing the swallowtail feature and shifting phase transition boundaries.
   • The NLED parameter ($\alpha$) and the modified gravity coupling ($\beta$) introduce only minor quantitative corrections to the global stability, preserving the overall phase behavior.
   • Local stability analysis via the Hessian matrix identifies a critical radius separating stable and unstable branches. Increasing $Q$ and $\Lambda$ delays the onset of stability to larger horizon radii, whereas increasing $\alpha$ and $\beta$ enhances stability, allowing stable configurations at smaller radii.

2. Joule-Thomson Expansion:

   • The inversion temperature curves contain a single branch separating cooling ($\mu_{JT} > 0$) and heating ($\mu_{JT} < 0$) domains, lacking critical turning points, similar to Reissner-Nordström-AdS spacetimes.
   • Charge Dominance: The electric charge exerts the strongest influence on JT behavior. Increasing $Q$ enlarges the cooling domain and raises the inversion temperature.
   • Correction Effects: Stronger NLED ($\alpha$) and modified gravity ($\beta$) contributions tend to reduce the extent of the cooling domain. While $\alpha$ and $\beta$ shift the inversion curves upward (increasing inversion temperature), the isenthalpic trajectories show that these parameters simultaneously reduce the peak temperature and compress the cooling region, thereby suppressing overall cooling efficiency.
   • The inversion curves exhibit Van der Waals-like behavior, where inversion temperature increases monotonically with pressure.

3. Geodesic Structure and Orbital Dynamics:

   • Dominance of Charge: The electric charge significantly modifies the effective potential, elevating barrier heights and shifting peaks inward. This strongly influences orbital precession and the critical impact parameter for photon capture.
   • Suppressed Corrections: The contributions from NLED ($\alpha$) and modified gravity ($\beta$) appear as higher-order corrections scaling as $O(r^{-22})$. Consequently, these parameters have negligible influence on the macroscopic geodesic structure outside the near-horizon region.
   • Orbital Features: The system supports stable circular orbits, bound orbits exhibiting perihelion precession, unstable photon spheres, and distinct scattering and capture regions.
   • Asymptotic Behavior: At astrophysically relevant distances, the geometry closely reproduces the behavior expected from a Reissner-Nordström-AdS black hole, as the negative cosmological constant provides an effective confining effect at large radial distances.

Significance and Claims
The paper claims to provide new insights into the interplay between modified gravity, nonlinear electromagnetic fields, and black hole chemistry. A central finding is the establishment of a direct physical correlation between the thermodynamic phase boundaries and the surrounding orbital dynamics, as both are governed by the metric function $A(r)$ and its derivative.

The authors assert that the regularisation mechanisms required to resolve central singularities in this framework do not destabilize the exterior trajectory framework. The spacetime maintains stable circular orbits and standard thermodynamic behaviors. Furthermore, the work suggests that because large-scale kinematics decouple from core modifications while remaining highly sensitive to charge and cosmological parameters, classical testing methods (such as analyzing photon sphere boundaries and thermal transitions) remain valid tools for probing quantum-inspired regimes. The results offer a theoretical basis for interpreting future observational data from gravitational wave interferometers and event horizon imaging arrays within the context of extended gravity models.