Schwarzschild–AdS black hole with a cloud of strings in non-commutative geometry: Geodesics, thermodynamics and scalar perturbations

In this work, we study geodesic motion of massless (photon) and massive particles in a modified Schwarzschild–anti-de Sitter black hole spacetime incorporating non-commutative geometry and a surrounding string cloud. We analyze the combined effects on the effective potential, photon sphere, black hole shadow, photon deflection, and particle dynamics, revealing significant deviations from classical behavior. We also investigate thermodynamic properties, deriving the Hawking temperature, entropy, Gibbs free energy, and specific heat, showing that non-commutativity and string clouds shift phase transitions and alter stability. Finally, we examine scalar field perturbations via the massless Klein–Gordon equation, obtaining a Schrödinger-like form for the effective potential, which is markedly influenced by the modified geometry and string cloud.