  For spacetimes containing quiescent singularity hypersurfaces we propose a
general notion of junction conditions based on a prescribed singularity
scattering map, as we call it, and we introduce the notion of a cyclic
spacetime (also called a multiverse) consisting of spacetime domains bounded by
spacelike or timelike singularity hypersurfaces, across which our scattering
map is applied. A local existence theory is established here while, in a
companion paper, we construct plane-symmetric cyclic spacetimes. We study the
singularity data space consisting of the suitably rescaled metric, extrinsic
curvature, and matter fields which can be prescribed on each side of the
singularity, and for the class of so-called quiescent singularities we
establish restrictions that a singularity scattering map must satisfy. We
obtain a full characterization of all scattering maps that are covariant and
ultralocal, in a sense we define and, in particular, we distinguish between, on
the one hand, three laws of bouncing cosmology of universal nature and, on the
other hand, model-dependent junction conditions. The theory proposed in this
paper applies to spacelike and timelike hypersurfaces and without symmetry
restriction. It encompasses bouncing-cosmology scenarios, both in string theory
and in loop quantum cosmology, and puts strong restrictions on their possible
explicit realizations.
