Exploring geodesics, quantum fields and thermodynamics of Schwarzschild-AdS black hole with a global monopole in non-commutative geometry

Abstract

This work presents a comprehensive analysis of the geodesic motion, scalar field perturbations, and thermodynamic behavior of a static, spherically symmetric black hole solution arising within the framework of non-commutative geometry and incorporating the presence of a global monopole. The study is motivated by the interplay between quantum gravity effects (modeled via non-commutative geometry) and topological defects such as global monopoles, both of which contribute to significant deviations from classical black hole solutions. We begin by formulating the effective potential governing the motion of test particles-both massive and massless-in the modified spacetime. The effects of the non-commutative parameter and the global monopole are systematically examined in the context of various key features: the photon sphere radius, light deflection angle, stability of circular orbits, and the innermost stable circular orbit (ISCO). Our results show that the presence of the non-commutative structure enlarges the photon sphere radius and increases the deflection of light, while simultaneously destabilizing circular orbits. Furthermore, we observe that the ISCO radius is highly sensitive to the global monopole parameter, with a noticeable increase for certain ranges of the non-commutative parameter. These findings suggest that non-commutative geometry and topological defects significantly affect the observable properties of black hole environments. In the second part of the study, we explore linear perturbations of a massless scalar field by solving the Klein-Gordon equation in the background spacetime. The resulting effective potential for scalar perturbations is shown to be deeply influenced by both the non-commutative geometry and the global monopole, modifying the scattering and decay characteristics of the scalar field. By employing the eikonal approximation, we compute the complex quasinormal mode (QNM) frequencies and analyze how their real and imaginary parts evolve with changes in the underlying parameters. The results provide insight into the stability of the black hole under perturbations and offer potential observational signatures of non-commutative and monopole-induced corrections. Finally, the thermodynamic properties of the black hole are investigated. We derive the expressions for the Hawking temperature, specific heat, and Gibbs free energy, showing how these quantities deviate from their counterparts in the standard Schwarzschild solution. Notably, we find that the introduction of a non-commutative structure and global monopole leads to non-trivial modifications in the thermodynamic phase structure. In particular, the black hole may exhibit modified thermal stability and phase transitions depending on the values of the non-commutative and monopole parameters.