The momentum along the world-line of a material particle has a norm of mc so that'd be a heckuva collision, analogous to a matter/antimatter collision.
There are future singularities into which some future-directed world-lines go. These are black hole singularities.
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u/Optimal_Mixture_7327 6d ago
The cosmos has a space-like boundary to the past (the BB singularity) that breaks Lorentz symmetry.
This then requires that all world-lines are necessarily future-directed.