This is a question on differential geometry/topology of Lorentzian manifolds. It is motivated by Physics, but since I want a mathematicaly rigorous formulation I think the correct place to ask is here.
So let us state the problem:
Let Minkowski spacetime $(\mathbb{R}^4,\eta)$ be given with metric tensor $(\eta_{\mu\nu}) = \operatorname{diag}(-1,1,1,1)$. We want to talk about infinity as a place. So we want a bigger spacetime $(M,g)$ properly containing Minkowski spacetime and such that the boundary of Minkowski spacetime on this bigger manifold can be seen as the "points at infinity".
It turns out that one can discover that the right way to do this and have a correct picture of infinity is to require that the embedding of Minkowski spacetime on the bigger manifold be conformal.
So far so good. So what we do in practice is:
Introduce null spherical coordinates $u =t-r$ and $v = t+r$ with ranges $-\infty < u,v < +\infty$ and $u\leq v$ so that the metric tensor becomes $$\eta=-dudv+\frac{1}{4}(v-u)^2(d\theta^2+\sin^2\theta d\phi^2).$$
Compactify along these by defining $u = \tan U$ and $v = \tan V$ with ranges $-\pi/2 < U,V < \pi/2$ and $U\leq V$ so that the metric in these coordinates becomes $$\eta=\dfrac{1}{\cos^2 U\cos^2 V}(-dUdV+\frac{1}{4}\sin^2(U-V)(d\theta^2+\sin^2\theta d\phi^2))$$
Up to this point the coordinates $(U,V,\theta,\phi)$ with the ranges described in (2) together with the above metric, is just an awkward reparametrization of Minowski spacetime. What we want to do is to precisely describe the points $(U,V,\theta,\phi)$ with $U = \pm \pi/2$ or $V = \pm \pi/2$. These comprise the desired boundary.
It is also clear that if we find a bigger manifold $(N,g)$ with the metric in parenthesis, Minkowski spacetime can be conformally mapped to one open region of it.
We now turn to the description of the bigger manifold. The obvious thing to do is to allow the coordinates $U,V$ to extend further. In other words, the naive idea would be: *well, define $$g = -dUdV+\frac{1}{4}\sin^2(U-V)(d\theta^2+\sin^2\theta d\phi^2)$$ and then allow $U,V$ to run from $-\infty$ to $\infty$ together with the usual $(\theta,\phi)\in S^2$ to describe $(N,g)$.
Here comes the trouble. Proceeding naively like this, it seems to be no reason that $(-\pi/2,\pi/2,\theta,\phi)$ is not a sphere. It turns out that Roger Penrose states in an old work from 1964 that "because $\sin(U-V)=0$ on the metric in these coordinates, they all represent the same point. The coordinate system $(U,V,\theta,\phi)$ looses injectivity if we go to $U = -V = -\pi/2$. He says this is exactly the same as the $r = 0$ in polar coordinates in the plane. In other words: the metric places a constraint that the bigger manifold is such that the points with coordinates $(-\pi/2,\pi/2,\theta,\phi)$ are all the same point.
Now, I really have thought hard about that, but I fail to grasp the reasoning in (5). I mean, we in that case we do have the manifold $\mathbb{R}^2$ explicit and we see that the coordinate chart we propose geometricaly is not injective if we include the origin.
Here we don't have the manifold. All we have are coordinates and a metric and from this we want to find a manifold with this metric and with these coordinates containing properly the piece we already have. The fact that the $S^2$ part of the metric vanishes there seems to imply the extension is so that points which naively are $S^2$ are actually the same point.
Now why is that? How to understand this? And more importantly: how to state all of this discussion rigorously (I realize the talk about "to describe the bigger manifold just enlarge the ranges of coordiantes $U,V$" is not rigorous)?