Endless geodesics in closed spacetime

AI Thread Summary
A closed spacetime can indeed contain geodesics of infinite length, as demonstrated by certain models in general relativity. These geodesics can loop back on themselves, allowing for the possibility of infinite traversal without exiting the closed universe. The presence of such geodesics influences local curvature, as the mass-energy content of the universe shapes the spacetime geometry. Specifically, the curvature can lead to singularities, where paths of photons may ultimately converge. Thus, a closed universe can support infinite-length geodesics while also being shaped by its intrinsic curvature dynamics.
Loren Booda
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Can you show that a closed spacetime may embody one (or even an infinity of) geodesics with infinite length? How does this influence local curvature?
 
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That is, can a closed universe contain a geodesic infinite in length?

(I assume that the mass-energy curvature of a closed universe would eventually lead any interior photon to a singularity.)
 

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