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In flat space, Penrose diagrams look something like this:

penrose diagram

Geodesics are then curved lines and null curves are straight lines at $45^{\circ}$. What would a vertical line at $r\ne 0$ represent? I think it would be a timelike curve (since it is not null and does not cross any null curves), but it reaches null infinity.

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Yes. It represents a class of timelike worldlines (noting you are suppressing a couple coordinates). It may involve arbitrary acceleration however, depending on the spacetime.

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