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This paper examines the issue of the existence and nature of time-likegeodesics in asymptotically flat spacetimes and proposes a novel generalizedtopological criterion for the existence of time-like geodesics. Its validity isproved using theorems such as the Jordan-Brouwer Separation Theorem, theRaychaudhuri Equation, and key elements of Differential Geometry. Morespecifically, the proof primarily hinges on a closed, simply-connected subsetof the spacetime manifold and a continuous map, causing a non-trivial inductionon the first homology groups, from the boundary of this subset to a unitcircle. The mathematical analysis conclusively affirms the presence of thesegeodesics, intersecting transversally within the said subset of spacetime.Findings underscore these geodesics' significant implications for the structureof asymptotically flat spacetimes, including stability, and hypotheticalexistence of wormholes. The generalized topological criterion also hasimplications on the problem of obstructions for the existence of Lorentzianmetrics, and Einstein's Constraint Equations. Future research should extendthis topological criterion to other classes of spacetimes, including those withnon-trivial topologies or non-zero cosmological constants. Also, thecriterion's application to study complex dynamical systems, such asgravitational waves or rotating black holes, could offer significant insights.

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