Thermodynamic topology, photon spheres, and evidence for weak gravity conjecture in charged black holes with perfect fluid within Rastall theory

Abstract

This study explores the Weak Gravity Conjecture within the framework of charged black holes surrounded by a perfect fluid, analyzed under Rastall theory. A key focus is the behavior of photon spheres, which serve as critical indicators of whether these objects qualify as true black holes or devolve into naked singularities. The investigation sheds light on the deep interplay between quantum forces and gravity, pointing toward new links between high-energy physics and alternative theories of gravity. We find that the structure of these black holes is highly sensitive to variations in the intensity of the surrounding fluid and the parameters specific to Rastall gravity. In the case of a dust-like environment, the black hole satisfies the conditions required by the Weak Gravity Conjecture (WGC) when it reaches an extremal state, and the existence of a photon sphere confirms its black hole nature. However, pushing the gravitational parameters further or weakening the fluid intensity can lead to the disappearance of the photon sphere, signaling a transition to a singularity rather than a black hole. The analysis extends to other types of cosmic fluids, including those representing radiation, quintessence, and phantom energy. Across various conditions, we identify scenarios where the Weak Gravity Conjecture continues to hold, reinforcing its broader applicability. Our numerical method reveal distinct regions in the parameter space where the conjecture is validated, offering insights that bridge quantum-scale physics with cosmological structures. Additionally, we examine the topological properties of these black holes using a theoretical approach that incorporates a generalized version of Helmholtz free energy and topological current theory. Our findings indicate that, regardless of the fluid type considered, the overall topological classification remains unchanged an show the behaviour of a classical black hole solutions (Reissner-Nordström black holes) with a total topological charges $W=0$.