Linear structures, causal sets and topology

1997

Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for both the finite-dimensional and infinite-dimensional case. In applications to physics, partial ordering is interpreted as the causal structure. Both Newtonian and the special relativity causal structures are studied, and some other possible types of causality are discussed. Linear topological spaces with pseudometric which satisfies the time inequality instead of the triangle inequality are studied (3 axioms). Pseudometric (which is determined by a pseudonorm) is shown to define a topology on a linear space, it being a continuous mapping in this topology. Proved that for a space with pseudometric to be a linear kinematics it is necessary and sufficient that mapping of multiplication by -1 (i.e. time reversion) be continuous. Minkovskii space of the...

11, 10] proved many interesting results. Especially after good deal of effort, Minguzzi proved that K -causality condition is equivalent to stably causal condition. More recently, K.Martin and Panangaden making use of domain theory, a branch of theoretical computer science, proved fascinating results in the causal structure theory of space-time. The remarkable fact about their work is that only order is needed to develop the theory and topology is an outcome of the order. In addition to this consequence, there are abstract approaches, algebraic as well as geometric to the theory of cones and cone preserving mappings. Use of quasi-order (a relation which is reflexive and transitive) and partial order is made in defining the cone structure. Such structures and partial orderings are used in the optimization problems [13], game theory and decision making etc . The interplay between ideas from theoretical computer science and causal structure of space-time is becoming more evident in the recent works . Keeping these developments in view , in this article, we present a review of geometric and algebraic approach to causal cones and describe cone preserving transformations and their relationship with causal structure. We also describe certain implications of these concepts in special and general theory of relativity related to causal structure and topology of space-time. Thus in section 2, we begin with describing Lie groups, especially matrix Lie groups, homogeneous spaces and then causal cones. We give an algebraic description of cones by using quasi-order. Furthermore, we describe cone preserving transformations. These maps are generalizations of causal maps related to causal structure of spacetime which we shall describe in section 3. We then describe explicitly Minkowski space as an illustration of these concepts and note that some of the space-time models in general theory of relativity can be described as homogeneous spaces. In section 3, we describe causal structure of space time, causality conditions, Kcausality and hierarchy among these conditions in the light of recent work of S. Janardhan and R.V. Saraykar and E.Minguzzi and M.Sanchez [8,. We also describe geometric structure of causal group, a group of transformations preserving causal structures or a group of causal maps on a space-time.

The article discusses the concept of causality. The issues of the passage of time determining the sequence of events in the chain of cause-consequence relationship, quantization of continuous physical processes into events, the process of transformation of cause into effect and the topology of space within the elementary link of cause-consequence relationship are considered. The problems of connectivity of the event space are analyzed. The problems of modern physics and the expediency of returning to the idea of ether as the basis of matter in all its manifestations are discussed.

2013

An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using “thickened antichains” is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or “Hauptvermutung ” of causal set theory. 1

Journal for General Philosophy of Science, 2012

More often than not, recently popular structuralist interpretations of physical theories leave the central concept of a structure insufficiently precisified. The incipient causal sets approach to quantum gravity offers a paradigmatic case of a physical theory predestined to be interpreted in structuralist terms. It is shown how employing structuralism lends itself to a natural interpretation of the physical meaning of causal sets theory. Conversely, the conceptually exceptionally clear case of causal sets is used as a foil to illustrate how a mathematically informed rigorous conceptualization of structure serves to identify structures in physical theories. Furthermore, a number of technical issues infesting structuralist interpretations of physical theories such as difficulties with grounding the identity of the places of highly symmetrical physical structures in their relational profile and what may resolve these difficulties can be vividly illustrated with causal sets.

Journal of Mathematical Physics, 2007

An important question that discrete approaches to quantum gravity must address is how continuum features of space-time can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the space-time continuum is a locally finite partial order. A new topology on causal sets using "thickened antichains" is constructed. This topology is then used to recover the homology of a globally hyperbolic space-time from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or "Hauptvermutung" of causal set theory.

Gravitation and Cosmology

Principles of construction of a causal space-time theory are discussed. A system of axioms for Special Relativity Theory, which postulates the macrocausality and continuity of time order, is considered. The posibilities of a topos-theoretic approach to the foundations of Relativity Theory are investigated. Construction of a causal theory of space-time is one of the most attractive tasks of science in the 20th century. From the viewpoint of mathematics, partially ordered structures should be considered. The latter is commonly understood as a set V with a specified reflexive and transitive binary relation. A primary notion is actually not that of causality but rather that of motion (interaction) of material objects. Causality is brought to the foreground since an observer detects changes of object motion or state. It is this detection that gives rise to the view of a particular significance of causes and effects for a phenomenon under study, along with the conviction that causal connections are non-symmetric. Causality is treated as such a relation in the material world that plays a key role in explaining the topological, metric and all other world structures.

arXiv preprint gr-qc/0407093, 2004

Abstract: We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. This provides an abstract mathematical setting in which one can study causality independent of geometry and differentiable structure.

Bulletin of the American Mathematical Society, 1989

2011

We show that there exists a canonical topology, naturally connected with the causal site of J. D. Christensen and L. Crane, a pointless algebraic structure motivated by quantum gravity. Taking a causal site compatible with Minkowski space, on every compact subset our topology became a reconstruction of the original topology of the spacetime (only from its causal structure). From the global point of view, the reconstructed topology is the de Groot dual or co-compact with respect to the original, Euclidean topology. The result indicates that the causality is the primary structure of the spacetime, carrying also its topological information.