# 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:

1. 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 $\omega(T,R) = \cos T + \cos R$.
2. 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 $i^0,i^+,i^-,\mathscr{I}^-,\mathscr{I}^+$?

• Isn't the infinity just an artifact of your coordinate choice anyways? The real physical object is the spacetime manifold; the infinites in the coordinate system for the 3 sphere occur only because it can't be covered by a single chart. For pedagogy the presentation is done in reverse but i think that isnt really the proper way to think about it. – Aaron Oct 19 '17 at 3:51
• This may help : arxiv.org/abs/gr-qc/0501069 – Slereah Oct 19 '17 at 12:53
• It is obviously not possible (or, anyway, relevant) if your spacetime is already compact in some sense (perhaps only spatially compact). – tfb Oct 19 '17 at 22:14
• @Aaron if the Kretcschmann Scalar: $K =R^{abcd} R_{abcd} \to \infty$, then, this is a real, physical singularity. It is NOT a coordinate singularity, and cannot be removed by a coordinate transformation. – Dr. Ikjyot Singh Kohli Oct 19 '17 at 23:12

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.

• Thanks for the answer and references! I'm quite interested in QFT on curved spacetimes, so your book will certainly be useful. I gave a look on them. Wald talks of this in the context of assymptotic flatness, and if I understood correctly, he says that the "flatness" part is contained in the condition imposed on the deriatives of $\Omega$. So, relaxing said condition allows for definition of $i^0,i^+,i^-\mathscr{I}^+,\mathscr{I}^-$ even when the spacetime isn't assymptoticaly flat? Or identifying these places is only possible when the spacetime is assymptoticaly flat? – user1620696 Oct 22 '17 at 18:01
• The existence of those types of infinity is the definition of the corresponding asymptotic flatness, at least for the first and the last two types you listed. Future and past timelike infinity have a different nature. – Valter Moretti Oct 22 '17 at 18:22
• I confess that my intuition on this failed then. At least intuitively I thought that with just the existence of the embedding you mention (1) one could define these five objects and then (2) one could further define assymptotic flatness by imposing extra conditions on them. So the conclusion is that one can only define these infinities when the spacetime is assymptoticaly flat, so that just the embedding isn't enough to identify the various infinities? – user1620696 Oct 23 '17 at 19:19
• That is the definition(s) of asymptotic flatness, you can accept it or not. A discussion on this definition can be found in Wald's book. As a matter of fact these definitions are not very intuitive. – Valter Moretti Oct 23 '17 at 19:26
• In a sense, these spacetimes look like Minkowski spacetime at infinity since they can be completed similarly to it there. – Valter Moretti Oct 23 '17 at 19:34