Limit curve theorems in Lorentzian geometry

Minguzzi, E.

doi:10.1063/1.2973048

Cited by 59 publications

(115 citation statements)

References 12 publications

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“…The next result joins two theorems, one by Kriele [14, Theorem 4] who improved previous results by Tipler [29] and the other by the author [15].…”

Section: The Non-chronological Casementioning

confidence: 74%

“…For the proof that the setsĊ α are disjoint I refer the reader to [15]. Instead, I elaborate on Kriele's argument by giving a slightly different proof that the boundaries B αk are non-compact.…”

Section: The Non-chronological Casementioning

confidence: 99%

“…Take two pairs (x, y) ∈ A + and (y, z) ∈ A + and two sequences of causal curves σ n of endpoints (x n , y n ) → (x, y), and γ n of endpoints (y ′ n , z n ) → (y, z). Apply the limit curve theorem [15] to both sequences, and consider first the case in which the limit curve in both cases does not connect the limit points. By the limit curve theorem, σ n has a limit curve σ which is a past inextendible causal curve ending at y. Analogously γ n has a limit curve γ which is a future inextendible causal curve starting from y.…”

Section: Theorem 5 If a Spacetime Does Not Have Lightlike Lines Thenmentioning

confidence: 99%

“…Several versions of the limit curve theorem will be repeatedly used, particularly those referring to sequences of g n -causal curves, where the metrics in the sequence g n may differ. The reader is referred to [15] for a sufficiently strong formulation.…”

Section: Introductionmentioning

confidence: 99%

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Chronological Spacetimes without Lightlike Lines are Stably Causal

Minguzzi

2009

Commun. Math. Phys.

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The statement of the title is proved. It implies that under physically reasonable conditions, spacetimes which are free from singularities are necessarily stably causal and hence admit a time function. Read as a singularity theorem it states that if there is some form of causality violation on spacetime then either it is the worst possible, namely violation of chronology, or there is a singularity. The analogous result: "Non-totally vicious spacetimes without lightlike rays are globally hyperbolic" is also proved, and its physical consequences are explored.

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“…The next result joins two theorems, one by Kriele [14, Theorem 4] who improved previous results by Tipler [29] and the other by the author [15].…”

Section: The Non-chronological Casementioning

confidence: 74%

Section: The Non-chronological Casementioning

confidence: 99%

Section: Theorem 5 If a Spacetime Does Not Have Lightlike Lines Thenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Chronological Spacetimes without Lightlike Lines are Stably Causal

Minguzzi

2009

Commun. Math. Phys.

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“…There is a sequence of closed timelike curves that accomulates on r. Each element of the sequence can be considered as an inextendible curve, it suffice to choose the parametrization so as to cover the image of the curve several times. By the limit curve theorem [18,19] there is an inextendible causal curve which passes through r and stays in [b]. It is easy to prove that either the half-curve in the future of r is achronal and contained in ∂[b] or so is the half-curve in its past.…”

Section: Kriele's Theoremmentioning

confidence: 99%

Chronology violations and the origin of time

Minguzzi

2013

EPJ Web of Conferences

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Abstract. We review some results which relate chronology violations to singularities, and explain how the absence of both pathologies implies the existence of a cosmological time. Building on these mathematical ideas we then propose a causality argument in order to solve the homogeneity and entropy problems of cosmology. The solution is based on the replacement of the spacelike Big Bang boundary with a null boundary behind which stays a chronology violating region. This solution requiring a tilting of the light cones near the null boundary is based more on the behavior of the light cones and hence on causality, than on the behavior of the scale factor (expansion). The philosophical connection of this picture with Augustine of Hyppo famous discussion on time and creation is commented.

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Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously

Papadopoulos

Scardigli

2019

Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

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Why is the manifold topology in a spacetime taken for granted? Why do we prefer to use Riemann open balls as basic-open sets, while there also exists a Lorentz metric? Which topology is a best candidate for a spacetime; a topology sufficient for the description of spacetime singularities or a topology which incorporates the causal structure? Or both? Is it more preferable to consider a topology with as many physical properties as possible, whose description might be complicated and counterintuitive, or a topology which can be described via a countable basis but misses some important information? These are just a few from the questions that we ask in this Chapter, which serves as a critical review of the terrain and contains a survey with remarks, corrections and open questions. AMSC: 83XX, 83F05, 85A40, 54XX keywords: Zeeman -Göbel topologies, topologising a spacetime, spacetime singularities, causal topologies, manifold topology 1 in the presence of Riemannian basic-open balls fail as well. Zeeman's main arguments against the Euclidean R 4 topology for Minkowski spacetime M (extended by Göbel for curved spacetimes) can be summarised as follows: 1. The 4-dimensional Euclidean topology is locally homogeneous, whereas M is not; every point has associated with it a light cone, separating space vectors from time vectors. 2. The group of all homeomorphisms of 4-dimensional Euclidean space is vast, and of no physical significance.Heathcote's antilogue belongs to (sic) a realist view of spacetime topology as against the instrumentalist position. A realist point of view divides the space intro structural levels, such as metric tensor field, affine connection, conformal structure, differentiable manifold and topology. Heathcote highlights that the manifold topology is present as long as the structure of manifold is present, and there are two "untenable" possibilities for a replacement of the manifold topology, in both cases by finer topologies (see [13], page 255, for more details).

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Limit curve theorems in Lorentzian geometry

Cited by 59 publications

References 12 publications

Chronological Spacetimes without Lightlike Lines are Stably Causal

Chronological Spacetimes without Lightlike Lines are Stably Causal

Chronology violations and the origin of time

Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously

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