Existence of Time-like Geodesics in Asymptotically Flat Spacetimes: A Generalized Topological Criterion

This paper examines the issue of the existence and nature of time-like geodesics in asymptotically flat spacetimes and proposes a novel generalized topological criterion for the existence of time-like geodesics. Its validity is proved using theorems such as the Jordan-Brouwer Separation Theorem, the Raychaudhuri Equation, and key elements of Differential Geometry. More specifically, the proof primarily hinges on a closed, simply-connected subset of the spacetime manifold and a continuous map, causing a non-trivial induction on the first homology groups, from the boundary of this subset to a unit circle. The mathematical analysis conclusively affirms the presence of these geodesics, intersecting transversally within the said subset of spacetime. Findings underscore these geodesics' significant implications for the structure of asymptotically flat spacetimes, including stability, and hypothetical existence of wormholes. The generalized topological criterion also has implications on the problem of obstructions for the existence of Lorentzian metrics, and Einstein's Constraint Equations. Future research should extend this topological criterion to other classes of spacetimes, including those with non-trivial topologies or non-zero cosmological constants. Also, the criterion's application to study complex dynamical systems, such as gravitational waves or rotating black holes, could offer significant insights.