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.