scholarly journals Spacetime Causal Structure and Dimension from Horismotic Relation

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
O. C. Stoica
Keyword(s):  
Causal Structure ◽  
Differentiable Manifold ◽  
Conformal Structure ◽  
Lorentzian Manifolds ◽  
Causal Sets ◽  
Discretization Methods ◽  
Starting Point ◽  
Theories Of Gravity ◽  
Natural Way ◽  
Construction Works

A reflexive relation on a set can be a starting point in defining the causal structure of a spacetime in General Relativity and other relativistic theories of gravity. If we identify this relation as the relation between lightlike separated events (the horismos relation), we can construct in a natural way the entire causal structure: causal and chronological relations, causal curves, and a topology. By imposing a simple additional condition, the structure gains a definite number of dimensions. This construction works with both continuous and discrete spacetimes. The dimensionality is obtained also in the discrete case, so this approach can be suited to prove the fundamental conjecture of causal sets. Other simple conditions lead to a differentiable manifold with a conformal structure (the metric up to a scaling factor) as in Lorentzian manifolds. This structure provides a simple and general reconstruction of the spacetime in relativistic theories of gravity, which normally requires topological structure, differential structure, and geometric structure (which decomposes in the conformal structure, giving the causal relations and the volume element). Motivations for such a reconstruction come from relativistic theories of gravity, where the conformal structure is important, from the problem of singularities, and from Quantum Gravity, where various discretization methods are pursued, particularly in the causal sets approach.

scholarly journals On the Groups of Cobordism Ωk

1958 ◽  
Vol 13 ◽  
pp. 135-156 ◽  
Author(s):  
Masahisa Adachi
Keyword(s):  
Equivalence Relation ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Graded Algebra ◽  
Differentiable Manifolds ◽  
Free Part ◽  
Natural Way

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

Discrete quantum gravity and causal sets

2001 ◽  
Vol 79 (1) ◽  
pp. 1-16 ◽  
Author(s):  
D D Reid
Keyword(s):  
Quantum Gravity ◽  
High Energy Physics ◽  
Causal Structure ◽  
High Energy ◽  
Research Community ◽  
Small Scale ◽  
Causal Sets ◽  
Important Approach ◽  
Nonrelativistic Quantum Mechanics ◽  
Energy Physics

This paper provides a thorough introduction to the physical and conceptual need for a theory of quantum gravity; some knowledge of general relativity and nonrelativistic quantum mechanics is assumed. A theory of quantum gravity would have wide-ranging implications for high-energy physics, astrophysics, and cosmology. The paper goes on to describe an important approach to quantum gravity that is not well known outside of the quantum gravity research community — causal sets. The causal-set approach falls within the framework of discrete quantum gravity, which considers the possibility that the small-scale structure of spacetime might be discrete rather than continuous. Herein, I elucidate the arguments for why a discrete causal structure might be appropriate for a theory of quantum gravity. The logical and formal development of a causal-set theory as well as a few illuminating examples are also provided. PACS Nos.: 04.60-m, 04.60Nc

scholarly journals The collocation in learning vocabulary: the Problems and pedagogical suggestions

2018 ◽  
Vol 2 (3) ◽  
pp. 28-34
Author(s):  
Meylina Meylina
Keyword(s):  
Qualitative Research ◽  
Native Speakers ◽  
First Language ◽  
Advanced Students ◽  
Starting Point ◽  
Language Knowledge ◽  
Lexical Collocations ◽  
Native Speakers Of English ◽  
Natural Way ◽  
Descriptive Qualitative

This was a descriptive qualitative research which was administered to 20 students of STMIK Jayanusa Padang. The data from the observation shown that the lack of collocation competence of the students noticeable when non-native speakers of English need productive language knowledge. they only experienced the limited number of lexical collocations they know or under the influence of their first language “create” unnatural and farfetched collocations. In order to solve this problem, this investigation aimed to expose the students' collocation problems in vocabulary teaching by using collocation tests and questionnaire. The data found were used to offer some pedagogical suggestions that can be applied in class as a starting point, especially to advanced students. Then it is hoped that students will have properly developed and balanced in learning collocations which will help them speak and write English in a more natural way.

scholarly journals Observational effects of varying speed of light in quadratic gravity cosmological models

2018 ◽  
Vol 15 (05) ◽  
pp. 1850084
Author(s):  
Azam Izadi ◽  
Shadi Sajedi Shacker ◽  
Gonzalo J. Olmo ◽  
Robi Banerjee
Keyword(s):  
Causal Structure ◽  
Affine Connection ◽  
Structure Constant ◽  
Distance Modulus ◽  
Speed Of Light ◽  
Quadratic Gravity ◽  
Type Ia ◽  
Theories Of Gravity ◽  
Definition Of ◽  
Supernovae Type Ia

We study different manifestations of the speed of light in theories of gravity where metric and connection are regarded as independent fields. We find that for a generic gravity theory in a frame with locally vanishing affine connection, the usual degeneracy between different manifestations of the speed of light is broken. In particular, the space-time causal structure constant ([Formula: see text]) may become variable in that local frame. For theories of the form [Formula: see text], this variation in [Formula: see text] has an impact on the definition of the luminosity distance (and distance modulus), which can be used to confront the predictions of particular models against Supernovae type Ia (SN Ia) data. We carry out this test for a quadratic gravity model without cosmological constant assuming (i) a constant speed of light and (ii) a varying speed of light (VSL), and find that the latter scenario is favored by the data.

scholarly journals Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes

2018 ◽  
Vol 52 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Daniele Boffi ◽  
Daniele A. Di Pietro
Keyword(s):  
Mixed Method ◽  
Degrees Of Freedom ◽  
Recent Literature ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Discretization Methods ◽  
Polyhedral Meshes ◽  
Starting Point ◽  
Primal Method ◽  
Polytopal Meshes

We propose in this work a unified formulation of mixed and primal discretization methods on polyhedral meshes hinging on globally coupled degrees of freedom that are discontinuous polynomials on the mesh skeleton. To emphasize this feature, these methods are referred to here as discontinuous skeletal. As a starting point, we define two families of discretizations corresponding, respectively, to mixed and primal formulations of discontinuous skeletal methods. Each family is uniquely identified by prescribing three polynomial degrees defining the degrees of freedom, and a stabilization bilinear form which has to satisfy two properties of simple verification: stability and polynomial consistency. Several examples of methods available in the recent literature are shown to belong to either one of those families. We then prove new equivalence results that build a bridge between the two families of methods. Precisely, we show that for any mixed method there exists a corresponding equivalent primal method, and the converse is true provided that the gradients are approximated in suitable spaces. A unified convergence analysis is carried out delivering optimal error estimates in both energy- and L2-norms.

scholarly journals Non-Shannon inequalities in the entropy vector approach to causal structures

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 57 ◽  
Author(s):  
Mirjam Weilenmann ◽  
Roger Colbeck
Keyword(s):  
Causal Structure ◽  
Conditional Entropy ◽  
Classical Case ◽  
Conditional Mutual Information ◽  
Entropic Inequalities ◽  
Starting Point ◽  
Outer Approximations ◽  
Open Question ◽  
Causal Structures ◽  
Vector Approach

A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.

Existence of geodesics in static Lorentzian manifolds with convex boundary

2000 ◽  
Vol 130 (1) ◽  
pp. 189-215 ◽  
Author(s):  
P. Piccione
Keyword(s):  
Variational Principle ◽  
Fixed Points ◽  
Smooth Boundary ◽  
Differentiable Manifold ◽  
Lorentzian Manifold ◽  
Multiplicity Result ◽  
Lorentzian Manifolds ◽  
Global Hyperbolicity ◽  
Convex Boundary ◽  
Convexity Assumption

We study some global geometric properties of a static Lorentzian manifold Λ embedded in a differentiable manifold M, with possibly non-smooth boundary ∂Λ. We prove a variational principle for geodesics in static manifolds, and using this principle we establish the existence of geodesics that do not touch ∂Λ and that join two fixed points of Λ. The results are obtained under a suitable completeness assumption for Λ that generalizes the property of global hyperbolicity, and a weak convexity assumption on ∂Λ. Moreover, under a non-triviality assumption on the topology of Λ, we also get a multiplicity result for geodesics in Λ joining two fixed points.

scholarly journals Reconstruction of Lorentzian Manifolds from Boundary Light Observation Sets

2017 ◽  
Vol 2019 (22) ◽  
pp. 6949-6987
Author(s):  
Peter Hintz ◽  
Gunther Uhlmann
Keyword(s):  
Open Subset ◽  
Lorentzian Manifold ◽  
Conformal Structure ◽  
Conjugate Points ◽  
Lorentzian Manifolds ◽  
Convexity Assumption ◽  
Nonempty Boundary ◽  
Light Rays ◽  
Snell’S Law ◽  
Snell's Law

Abstract On a time-oriented Lorentzian manifold (M, g) with nonempty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets S ⊂ M of sources is uniquely determined by measurements of the intersection of future light cones from points in S with a fixed open subset of the boundary of M; here, light rays are reflected at ∂M according to Snell’s law. Our proof is constructive, and allows for interior conjugate points as well as multiply reflected and self-intersecting light cones.

Metric and connections in theories of gravity. The role of equivalence principle

2016 ◽  
Vol 13 (08) ◽  
pp. 1640007 ◽  
Author(s):  
Salvatore Capozziello ◽  
Mariafelicia De Laurentis
Keyword(s):  
General Relativity ◽  
Equivalence Principle ◽  
Causal Structure ◽  
Quantum Level ◽  
Space Experiments ◽  
Theories Of Gravity ◽  
Geodesic Structure ◽  
Einstein’S General Relativity

Fundamental issues underlying gravitational physics and some of the shortcomings of Einstein’s general relativity (GR) are discussed. In particular, after taking into account the role of the two main objects of relativistic theories of gravity, i.e. the metric and the connection fields, we consider the possibility that they are not trivially related so that the geodesic structure and the causal structure of the spacetime could be disentangled, as supposed in the Palatini formulation of gravity. In this perspective, the equivalence principle (EP), in its weak and strong formulations, can play a fundamental role in discriminating among competing theories. The possibility of its violation at quantum level could open new perspectives in gravitational physics and in unification with other interactions. We shortly debate the possibility of EP measurements by ground-based and space experiments.

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