The order on the light cone and its induced topology

Papadopoulos, Kyriakos; Acharjee, Santanu; Papadopoulos, Basil K.

doi:10.1142/s021988781850069x

Cited by 7 publications

(12 citation statements)

References 13 publications

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“…In the sequence of papers, [13], [15], [16], [17] and [18], the authors aim to establish a common background for the topologisation problem of a space-time. This background is the Lorentz "metric" and the structure of the lightcone, where one can define the chronological order ≪, the causal order ≺, the relation horismos → and also the chorological order <; for the last one, see in particular [17] and for a complete list of relations R depending on the lightcone see [18].…”

Section: What Is (Or Should Be) the Role Of Spacetime Topology?mentioning

confidence: 99%

“…A basic-open set U in the weak interval topology T in is defined as U = A ∩ B, where A ∈ S + and B ∈ S − ; in other words, S + ∪ S − forms a subbase for T in . Such topologies where constructed in [13], [15] and [17], covering the cases of horismos, chronology, causality and chorology (which are lightlike, timelike, causal and spacelike relations, respectively). Such topologies belong to the class Z − G, as we have shown in [13].…”

Section: What Is (Or Should Be) the Role Of Spacetime Topology?mentioning

confidence: 99%

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Natural vs. Artificial Topologies on a Relativistic Spacetime

Papadopoulos¹

2020

Springer Optimization and Its Applications

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Consider a set M equipped with a structure * . We call a natural topology T * , on (M, * ), the topology induced by * . For example, a natural topology for a metric space (X, d) is a topology T d induced by the metric d and for a linearly ordered set (X, <) a natural topology should be the topology T < that is induced by the order <. This fundamental property, for a topology to be called "natural", has been largely ignored while studying topological properties of spacetime manifolds (M, g) where g is the Lorentz "metric", and the manifold topology T M has been used as a natural topology, ignoring the spacetime "metric" g. In this survey we review critically candidate topologies for a relativistic spacetime manifold, we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field, and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Göbel, Hawking-King-McCarthy and others, and we examine what should be meant by the term "natural topology" for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Theorem of Gao-Wald type of "time dilation" are such examples.

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Section: What Is (Or Should Be) the Role Of Spacetime Topology?mentioning

confidence: 99%

Section: What Is (Or Should Be) the Role Of Spacetime Topology?mentioning

confidence: 99%

Natural vs. Artificial Topologies on a Relativistic Spacetime

Papadopoulos¹

2020

Springer Optimization and Its Applications

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“…We have already expressed in [24] an idea of an evolving topology with respect to the class Z, so that different topologies of this class are assigned to each stage of the evolution as well as where the spacetime itself is subjected to singularities. It could be, for example, that the interval topology from horismos → (see [24]) could give a sufficient description to the transition from/to the Planck time and objects like black holes, while other topologies (where the LCT theorem holds for example) could explain the phase transition from locality to non-locality. Topologies like Z T are linked to a discrete space while Z S to a discrete time, while Z to a discrete light (these are actually remarks of Zeeman in [35], for their special relativistic analogues).…”

Section: A Is Hausdorffmentioning

confidence: 99%

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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“…We call Z the topology that is mentioned by Zeeman (see also the work of Papadopoulos et al) as an alternative topology for F . This topology is coarser than the fine Zeeman topology F , and it has a countable base of open sets of the form

Z_{ε} (x) = N_{ε}^{E} (x) \cap (T (x) \cup S (x)) .

…”

Section: On the Zeeman Z Topologymentioning

confidence: 99%

“…In particular, we show that these two topologies are intersection topologies (in the sense of the work of Reed) of the manifold topology (topology of

R^{4}

in the case of Minkoski spacetime) with (respectively) two order topologies (one induced by a chronological order and the other by a noncausal spacelike order). We have discussed about the remaining suggested topology by Zeeman in the last paragraph of his paper in the works of Papadopoulos et al and Antoniadis et al…”

Section: Introductory Survey Of the Terrain With Commentsmentioning

confidence: 99%

On two topologies that were suggested by Zeeman

Papadopoulos

2018

Math Methods in App Sciences

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The class of Zeeman topologies on spacetimes in the frame of relativity theory is considered to be of powerful intuitive justification, satisfying a sequence of properties with physical meaning, such as the group of homeomorphisms under such a topology is isomorphic to the Lorentz group and dilatations, in Minkowski spacetime, and to the group of homothetic symmetries in any curved spacetime. In this article, we focus on two distinct topologies that were suggested by Zeeman as alternatives to his fine topology, showing their connection with two orders, ie, a timelike and a (noncausal) spacelike one. For the (noncausal) spacelike order, we introduce a partition of the null cone, which gives the desired topology invariantly from the choice of the hyperplane of partition. In particular, we observe that these two orders induce topologies within the class of Zeeman topologies, while the two suggested topologies by Zeeman himself are intersection topologies of these two order topologies (respectively) with the manifold topology. We end up with a list of open questions and a discussion, comparing the topologies with bounded against those with unbounded open sets and their possible physical interpretation.

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The order on the light cone and its induced topology

Cited by 7 publications

References 13 publications

Natural vs. Artificial Topologies on a Relativistic Spacetime

Natural vs. Artificial Topologies on a Relativistic Spacetime

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

On two topologies that were suggested by Zeeman

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