Conditional homotopy retracts and topological structures of Reissner–Nordström wormholes

Abstract

This paper establishes a novel bridge between differential geometry and topology by introducing the concepts of conditional retraction and conditional homotopy retract in the context of Reissner–Nordström wormhole spacetimes. We develop a rigorous mathematical framework that reveals previously unexplored topological properties of wormhole structures, demonstrating how these exotic spacetime configurations can be understood through the lens of modern homotopy theory. By analyzing the Lagrangian formulation of geodesics within the wormhole metric, we prove the existence of two distinct types of geodesic conditional retractions in four-dimensional Reissner–Nordström wormhole spacetime, each with unique topological characteristics. We extend these results to n-dimensional wormholes and establish a fundamental theorem showing that the limit chain of conditional retracts converges to a zero-dimensional wormhole spacetime, which provides fresh perspectives on the deep connection between dimensionality reduction and topological invariants in curved spacetimes. Furthermore, we demonstrate that while conditional retractions preserve certain homeomorphic properties, they exhibit asymmetric behavior under homeomorphisms, a finding with profound implications for understanding the fundamental structure of spacetime near singularities. These results advance our theoretical understanding of wormhole topology and also provide a mathematical foundation for exploring quantum gravitational effects in highly curved spacetimes, potentially offering new approaches to resolving the information paradox in black hole physics.