Thermodynamics and Optical Properties of Charged Black Holes in Bumblebee gravity Sourced by a Cloud of Strings

Abstract: In theories where the Lorentz symmetry of gravity is spontaneously broken, a non-minimally coupled bumblebee vector field acquires a nonzero vacuum expectation value, leading to modifications of standard General Relativity (GR). In this work, we investigate exact solutions describing static and spherically symmetric charged black holes surrounded by a cloud of strings within the framework of bumblebee gravity. We begin by analyzing the thermodynamic properties of these black hole solutions, including their mass, temperature, and entropy, highlighting how Lorentz-violating effects alter standard results. Next, we examine the optical properties of the spacetime, focusing on the photon sphere, the resulting black hole shadow, and the deflection of light, thereby providing potential observational signatures of Lorentz violation. Finally, we explore the impact of Lorentz-violating parameters on classical gravitational tests within the Solar System, such as advance of perihelion precession, in order to set observational constraints. Our analysis provides a comprehensive investigation of the interplay between Lorentz violation, black hole physics, and cloud of strings, offering a framework to probe new physics beyond GR.