Lorentzian Distance

Beem, John K.; Ehrlich, Paul E.; Easley, Kevin L.

doi:10.1201/9780203753125-4

Cited by 27 publications

(73 citation statements)

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“…We shall assume that every continuous causal (timelike) curve is equipped with a parametrization so that it is also a locally Lipschitz curve. That such a parametrization always exists is proven on page 75 of [8].…”

Section: Common Definitionsmentioning

confidence: 92%

“…Throughout this paper we restrict our attention to n-dimensional, paracompact, connected, Hausdorff, ∞ C manifolds  which are Lorentzian and time orientable. Standard definitions, e.g., of strong causality, the maximal extension of a manifold and of the generalized affine parameter, are taken from [8]. We do, however, deviate from [8] by using the 'total imprisonment' of [2] rather than the 'imprisonment' of [8].…”

Section: Common Definitionsmentioning

confidence: 99%

“…Standard definitions, e.g., of strong causality, the maximal extension of a manifold and of the generalized affine parameter, are taken from [8]. We do, however, deviate from [8] by using the 'total imprisonment' of [2] rather than the 'imprisonment' of [8]. The two definitions are equivalent; we simply prefer the terminology of [2].…”

Section: Common Definitionsmentioning

confidence: 99%

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Generalizations of the abstract boundary singularity theorem

Whale¹,

Ashley²,

Scott³

2015

Class. Quantum Grav.

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The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from C 1 curves to locally Lipschitz curves.

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Section: Common Definitionsmentioning

confidence: 92%

Section: Common Definitionsmentioning

confidence: 99%

See 1 more Smart Citation

Generalizations of the abstract boundary singularity theorem

Whale¹,

Ashley²,

Scott³

2015

Class. Quantum Grav.

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“…By the time the second edition of their book was published, Beem and Ehrlich had worked to salvage the stability of geodesic (in)completeness by restricting attention to special cases. Consider the following proposition, which is representative of this effort (Beem et al 1996).…”

Section: Stabilitymentioning

confidence: 99%

“…Here we review what is known concerning the (in)stability of spacetime properties within this framework. After considering some foundational results concerning causal properties (Hawking 1969; Geroch 1970a) and a fascinating drama concerning geodesic (in)completeness (Beem et al 1996), we focus on the property of spacetime inextendibility, about which very little is known. Because inextendibility is defined relative to a background “possibility space” in the form of a standard collection of spacetimes, one can naturally consider variant definitions relative to other collections.…”

Section: Introductionmentioning

confidence: 99%

On the (In?)Stability of Spacetime Inextendibility

Manchak

2023

Philos. sci.

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Examples of cosmological spacetimes without CMC Cauchy surfaces

Ling,

Ohanyan

2024

Lett Math Phys

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CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models.

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Lorentzian Distance

Cited by 27 publications

References 0 publications

Generalizations of the abstract boundary singularity theorem

Generalizations of the abstract boundary singularity theorem

On the (In?)Stability of Spacetime Inextendibility

Examples of cosmological spacetimes without CMC Cauchy surfaces

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