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What is the exact definition of the conformal infinities in a conformal compactification of a spacetime (not necessarily asymptotically flat)? I want to say that it's something of the type (for a time-oriented spacetime) :

The future and past timelike infinity $i^+$ and $i^-$ corresponds to the image of the set of future and past inextendible timelike curves of infinite half-length at $\pm \infty$, on the boundary $\mathscr I$ .

with similar definitions for null and spacelike infinity. Is that a valid definition for it? I'm having trouble finding an actual definition for it independent of any specific spacetime.

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This topic is called "boundary constructions." There are multiple ways of defining a boundary, including Geroch's g boundary, Schmidt's b boundary, and the Geroch-Kronheimer-Penrose boundary. There have been seemingly pointless religious wars over which is the right one. None really seems to have the complete list of correct properties, including coordinate-independence.

Note that we don't just want to define the boundary as a set of idealized points, we also want to define a topology on it. E.g., a problem with the b boundary is the fact that the topology comes out non-Hausdorff for both FRW and Schwarzschild.

A review article on this topic is Parrado and Senovilla, Causal Structures and Causal Boundaries, http://arxiv.org/abs/gr-qc/0501069 .

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