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

TL;DR

This paper explores the geometric and topological features of spherically symmetric black holes in Einstein gravity, linking these properties to Bose-Einstein Condensation and establishing uniqueness results and interaction frameworks.

Contribution

It introduces a novel connection between black hole topology and Bose-Einstein Condensation, proving the uniqueness of certain solutions and analyzing black hole interactions mathematically.

Findings

Black hole solutions supporting BEC are unique.

Bose-Einstein Condensation cannot occur in Kerr black holes.

Mathematical framework links black hole collisions with scalar field solutions.

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.