Topological and Geometric Properties of Spherically Symmetric Black Hole Metrics: Connections to Bose-Einstein Condensation and Uniqueness in Einstein Gravity

Abstract: This paper investigates the interplay between the geometric and topological properties of spherically symmetric black hole metrics within Einstein gravity, emphasizing implications for Bose-Einstein Condensation (BEC). By analyzing metric functions, scalar fields, and the cosmological constant, we reveal how these black hole solutions are intrinsically linked to the underlying spacetime topology. We establish the uniqueness of a general black hole solution that supports BEC and demonstrate the impossibility of BEC in Kerr black holes. Additionally, through Laurent series expansions, residue calculations, winding numbers, and contour integrals, we confirm the algebraic and dimensional consistency between double Kerr black hole collisions and specific scalar field black hole solutions. This work uncovers fundamental connections in black hole interactions, providing a robust mathematical framework for understanding the dynamics of complex black hole systems and their interactions with scalar fields.