On the final definition of the causal boundary and its relation with the conformal boundary

Understand

The notion of causal boundary $\partial M$ for a strongly causal spacetime $M$ has been a controversial topic along last decades: on one hand, some attempted definitions were not fully consistent, on the other, there were simple examples where an open conformal embedding $i:M\hookarrow M_{0}$ could be defined, but the corresponding conformal boundary $\partial_{i}M$ disagreed drastically with the causal one.

• Nevertheless, the recent progress in this topic suggests a definitive option for $\partial M$, which is developed here in detail.
• Our study has two parts: (I) To give general arguments on a boundary in order to ensure that it is admissible as a causal boundary at the three natural levels, i.e., as a point set, as a chronological space and as a topological space.
• Then, the essential uniqueness of our choice is stressed, and the relatively few admissible alternatives are discussed.