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In my black holes course, we are looking at the Penrose diagram for 1+1 D Minkowski space. My notes don't specifically describe $i^{\pm}$ (future/past timelike infinity) but do say all timelike curves end there. However, when looking at 1+1 Rindler space, we have observers that approach the speed of light, but never reach it. In my notes, the Penrose diagram for Rindler space is a subdiagram of that for Minkowski space. However, I noticed that the worldlines of the Rindler observers do not end up at $i^{\pm}$, even though their worldlines are timelike.

Is there a description of what $i^{\pm}$ that explains this?

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All timelike geodesics in Minkowski spacetime start at past timelike infinity and end at future timelike infinity. The worldlines of Rindler observers are not geodesics, whereas the worldlines of Minkowski observers are.

Heuristically think of a flat Euclidean plane. There are plenty of inextendible curves that don't go to infinity, but all geodesics start and end at infinity.

Wald pages 271-273 defines the conformal infinity of Minkowski space as the conformal mapping of Minkowski spacetime to a region of an Einstein static Universe. Conformal infinity is the boundary of this region and the past and future timelike infinities are special points on this boundary.

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