Definition of $i^+,i^-,i^0, \mathscr{I}^+,\mathscr{I}^-$ in a general spacetime I've seem sometimes a construction being carried out specificaly in Minkowski spacetime:


*

*One picks the standard metric tensor $$g = -dt^2 + dr^2 +  r^2 d\Omega^2$$
an introduces two new coordinate functions $T,R$ defined by
$$T=\arctan(t-r)+\arctan(t+r),\quad R=\arctan(t+r)-\arctan(t-r)$$
these coordinates have finite ranges $0\leq R< \pi$ and $|T|+R<\pi$. The metric tensor aquires then the form
$$g=\omega^{-2}(T,R)(-dT^2+dR^2+\sin^2 R d\Omega^2).$$
with $\omega(T,R) = \cos T + \cos R$.

*After this, one pictures $M$ sitting inside a bigger manifold described by the full range $-\infty < T < \infty$ and $0\leq R \leq \pi$ which is in this case $\mathbb{R}\times S^3$. Inside $\mathbb{R}\times S^3$ we have a boundary for $M$ which is decomposed as
$$\partial M = i^+\cup i^0\cup i^{-}\cup \mathscr{I}^+\cup \mathscr{I}^-$$
where we have the pieces in coordinates:
$$i^+ = \{p\in \mathbb{R}\times S^3 : T(p)=\pi, R(p)=0\} \\ i^0 = \{p\in \mathbb{R}\times S^3 : T(p)=0, R(p)=\pi\}, \\ i^- = \{p\in \mathbb{R}\times S^3 : T(p)=-\pi, R(p)=0\}, \\ \mathscr{I}^+ = \{p\in \mathbb{R}\times S^3 : T(p)=\pi - R(p), 0 < R(p) < \pi\}, \\ \mathscr{I}^- = \{p\in \mathbb{R}\times S^3 : T(p)=-\pi + R(p), 0 < R(p) <\pi\}$$


This allows for a precise way to talk about "infinity". Intuitively it seems like $i^+,i^-$ are respectively the far future and far past for massive particles $(t\to \pm \infty)$, while $\mathscr{I}^+,\mathscr{I}^-$ the analogue for massless particles and $i^0$ is the spatial infinity $(r\to \infty)$.
Unfortunately I've never seem this done in a more general and perhaps coordinate free setting. 
My question is: given a general spacetime $(M,g)$ how does one define $i^0,i^+,i^-,\mathscr{I}^-,\mathscr{I}^+$? 
 A: 
given a general spacetime $(M,g)$ how does one define $i^0,i^+,i^-,\mathscr{I}^-,\mathscr{I}^+$? 

A generic spacetime does not admit such extensions. These extension exist provided a larger spacetime $(M',g')$ exists equipped with a suitable embedding $\psi: M\to M'$ and a map $\Omega : M' \to \mathbb R$ such that $M$ exists in $M'$ as a manifold with boundary and parts of the boundary $\partial M$ identify to $i^0,i^+,i^-,\mathscr{I}^-,\mathscr{I}^+$. The map $\Omega$ connects the physical metric $g$ the the unphysical one $g'$ as a (singular) conformal factor $g' = \Omega^2 \psi_* g$ and $\Omega$ smoothly vanishes on $\partial M$ and satisfies some further technical properties (see quoted references) essentially saying that the added "infinity" completion looks like the infinity completion of Minkowski spacetime.  
By definition a spacetime is asymptotically flat at null infinity if admits null-like completion $\mathscr{I}^\pm$ as said above. An analogous definition is valid for spatial asymptotical flatness.  
The definition is a direct extension of the picture of Minkowski spacetime and its conformal extension including $i^0,i^+,i^-,\mathscr{I}^-,\mathscr{I}^+$.
The precise definitions can be found for instance in Wald's textbook (from page 275 on) or also in my recently co-authored book  and references therein.
