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

General Relativity and Quantum Cosmology