Thermodynamics, Phase Transitions, and Geodesic Structure of $F(R)-$Phantom BTZ Black Holes

TL;DR

This paper studies the thermodynamics, phase transitions, and geodesic structure of $F(R)$-phantom BTZ black holes, revealing second-order phase transitions and the influence of phantom fields on stability and particle orbits.

Contribution

It provides a detailed analysis of thermodynamic behavior, phase transitions, and geodesic properties of $F(R)$-phantom BTZ black holes, highlighting the effects of phantom fields and $F(R)$ corrections.

Findings

Black holes exhibit second-order phase transitions confirmed by Ehrenfest equations.

Stable timelike and photon orbits exist only in the phantom regime for negative curvature backgrounds.

Phantom fields significantly influence the spacetime geometry and stability of particle orbits.

Abstract

This paper investigates phantom BTZ black holes within the high-curvature gravity theory framework, specifically using a special case of power-Maxwell theory, which functions as a nonlinear electrodynamics source called $F(R)-$conformally invariant Maxwell gravity. We examine how the phantom or anti-Maxwell field affects the structure of these black holes and how the theory's parameters influence their horizon structure. Additionally, we derive the conserved and thermodynamic potentials associated with these black holes, thereby establishing their conformance to the foundational first law of thermodynamics. Next, the stability characteristics of BTZ black holes endowed with phantom and Maxwell fields are explored under canonical and grand canonical ensemble conditions by inspecting their heat capacity and Gibbs free energy profiles. This assessment reveals how the phantom field and scalar curvature affect these stability regions. We then perform a rigorous analytical verification of the Ehrenfest equations to determine whether the critical behavior of the phantom BTZ black hole corresponds to a second-order phase transition. Our results demonstrate adherence to both Ehrenfest relations, thereby confirming the occurrence of a second-order phase transition within the black hole system concurrent with the critical point. Furthermore, we explore the geodesic structure of the obtained solutions to analyze the motion of massive and massless test particles in the $F(R)$-phantom BTZ spacetime. The analysis demonstrates that stable timelike circular orbits exist only in the phantom regime for negative curvature backgrounds, while the phantom configuration also allows for stable circular photon orbits. These results underscore the significant influence of the phantom field and the $F(R)$ correction on the spacetime geometry and orbital dynamics.