Causal completions as Lorentzian pre-length spaces

On the space of compact diamonds of Lorentzian length spaces

“…Thus, these spaces are not regularly localizable and can not have timelike curvature bounded from below (see [28]). Actually, R 1 1 × T R n does not have timelike curvature bounded from above either [4]. Non-timelike branching is an essential requirement in many important cases, like in the development of Lorentzian measure spaces [34].…”

“…In section 4 we prove that the hyperspace of compact subsets of a Lorentzian pre-length space can be endowed with such a structure. Finally, as application of this fundamental result, in section 5 we study the space of causal diamonds D(R 1 1 × T X) and prove that it admits a structure of globally hyperbolic Lorentzian length space, give an explicit parametrization for a family of maximal timelike curves and provide an explicit realization as a subspace of a Lorentzian uniform product.…”

“…As an example of the value of building a causal theory on axiomatic grounds we cite the causet approach to quantum gravity (see [46] and references therein), in which no spacetime metric is considered at all. The existence of a time separation function, as proposed in [28], opens the scope of applications to settings where no regular spacetime metric is given a priori, as is the case of Lorentzian manifolds with timelike boundary [2], causal completions of globally hyperbolic spacetimes [1] or even some class of contact structures [24].…”

“…In section 2 we state the basic definitions and establish the notation to be used throughout this work. In section 3 we describe the Lorentzian taxicab product of a Lorentzian pre-length space and a metric space; as well as the uniform Lorentzian product of two Lorentzian pre-length spaces, and study in more detail the case of the Lorentzian taxicab product R 1 1 × T X. In section 4 we prove that the hyperspace of compact subsets of a Lorentzian pre-length space can be endowed with such a structure.…”

The c-completion of Lorentzian metric spaces

“…A first step in the study of the causal boundary for pre-length spaces has been recently given by one of the authors in collaboration with Aké et al [2]. Although their work becomes useful in many situations, it is far from being totally general.…”