Causal completions as Lorentzian pre-length spaces

Hau, Luis Aké; Burgos, Saul; Solis, Didier A.

doi:10.1007/s10714-022-02980-x

Cited by 3 publications

(22 citation statements)

References 35 publications

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“…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].…”

Section: Lorentzian Taxicab Productmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 98%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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On the space of compact diamonds of Lorentzian length spaces

Barrera,

Montes de Oca,

Solis

2024

Class. Quantum Grav.

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In this work we introduce the taxicab and uniform products for Lorentzian pre-length spaces. We further use these concepts to endow the space D(R ×T X) of causal diamonds with a Lorentzian length space structure, closely relating its causal properties with its geometry as a metric space furnished with its associated Hausdorff distance. Among the general results, we show that this space is geodesic and globally hyperbolic for a complete length space (X, d).

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Section: Lorentzian Taxicab Productmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the space of compact diamonds of Lorentzian length spaces

Barrera,

Montes de Oca,

Solis

2024

Class. Quantum Grav.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

The c-completion of Lorentzian metric spaces

Burgos,

Flores,

Herrera

2023

Class. Quantum Grav.

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Inspired by some Lorentzian versions of the notion of metric and length spaceintroduced by Kunzinger and S¨amman, and more recently, by Müller, andMinguzzi and Sühr, we revisit the notion of Lorentzian metric space in order tolater construct the c-completion of these general objects. We not only prove thatthis construction is feasible in great generality for these objects, including spacetimesof low regularity, but also endow the c-completion with a structure of Lorentzianmetric space by itself. We also prove that the c-completion constitutes a well-suitedextension of the original space, which really completes it in a precise sense andbecomes sensible to certain causal properties of that space.

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A Lower Semicontinuous Time Separation Function for $$C^0$$ Spacetimes

Ling

2024

Ann. Henri Poincaré

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The time separation function (or Lorentzian distance function) is a fundamental tool used in Lorentzian geometry. For smooth spacetimes it is known to be lower semicontinuous, and, in fact, continuous for globally hyperbolic spacetimes. Moreover, an axiom for Lorentzian length spaces—a synthetic approach to Lorentzian geometry—is the existence of a lower semicontinuous time separation function. Nevertheless, the usual time separation function is not necessarily lower semicontinuous for $$C^0$$ C 0 spacetimes due to bubbling phenomena. In this paper, we introduce a class of curves called “nearly timelike” and show that the time separation function for $$C^0$$ C 0 spacetimes is lower semicontinuous when defined with respect to nearly timelike curves. Moreover, this time separation function agrees with the usual one when the metric is smooth. Lastly, sufficient conditions are found guaranteeing the existence of nearly timelike maximizers between two points in a $$C^0$$ C 0 spacetime.

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Causal completions as Lorentzian pre-length spaces

Cited by 3 publications

References 35 publications

On the space of compact diamonds of Lorentzian length spaces

On the space of compact diamonds of Lorentzian length spaces

The c-completion of Lorentzian metric spaces

A Lower Semicontinuous Time Separation Function for $$C^0$$ Spacetimes

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