# Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $$d$$-dimensional spacetime $$(M,g)$$ (or simply just for $$d=4$$):

1. Does the conformal compactification of $$(M,g)$$ always exist?
2. Is it unique? Regarding the topology and the conformal structure of the compactified spacetime.

Example: unless I’m confused, the Euclidean plane $$\mathbb{R}^2$$ (which is not Lorentzian) has at least two conformal non-equivalent compactifications: the closed unit ball $$\overline{B_2(1)}\subset \mathbb{R}^2$$ or the 2-sphere $$S^2 \subset \mathbb{R}^3$$.

1. Does it always define a non-empty conformal boundary? In the example above, the first compactification has a non-empty conformal $$S^1$$ boundary, but the second one is a manifold without boundary. Or, do I have to consider the added north pole point as the conformal boundary of such compactification?
2. Is the signature of the conformal boundary always Lorentzian?
3. Is the conformal boundary always connected? Should I have to impose this condition?
4. In some cases, like in AdS space, the spacetime has closed timelike curves. We can solve this issue using the universal covering of such spacetime. After that, we can consider the conformal compactification of the universal covering. Is the conformal compactification of the universal covering spacetime equals the universal covering of the conformal compactification of the original spacetime? What I’m asking, if the operations of taking “universal covering” and “conformal compactification” commute?