Timelike infinity ($i^{\pm }$)

Future and past timelike infinity are ideal endpoints of asymptotically escaping timelike trajectories in a conformal completion. For the Minkowski conformal compactification, they are the vertices $(T,R)=(\pi ,0)$ and $(-\pi ,0)$; these vertices are not physical spacetime events.

This is the past endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(-\pi ,0)$. It is the time reversal of future timelike infinity and is reached at infinite physical proper time to the past.

This is the future endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(\pi ,0)$. The endpoint has finite unphysical coordinate time but infinite physical proper time. Accelerated timelike observers need not have this endpoint; a Rindler horizon can accompany an observer asymptotic to a null direction instead.