Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed revisited, starting at seminal conformal constructions by R. Penrose. Our study covers: (1) adaptive possibilities techniques slicings, (2) global hyperbolicity of sliced spacetimes, (3) critical review on boundaries a spacetime, (4) procedures to compute temporal splitting using isocausal comparison static product. New simple counterexamples $\mathbb{R}^2$ illustrate variety related these splittings, such as logical independence (for normalized spacetimes) between completeness slices hyperbolicity, necessity uniform bounds in order ensure or insufficience computation boundary. A refinement one examples shows that space all (normalized, classes of) metrics smooth product manifold $\mathbb{R}\times S$ is not convex, even though it path connected means piecewise convex combinations.