Causal Structures in Linear Spaces

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin's more general framework and I characterise what Maudlin's topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin's theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin's theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin's framework. This fact poses a challenge for the proponents of Maudlin's theory to prove it fruitful.

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.

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.

2016

Let X be a topological space equipped with a collection P of continuous paths. Associated to the pair (X,P) we define a path topology that generalizes the P-topology of Hawking et al. [1] Moreover, we axiomatize a collection P that generates an order that we call an NRT full order. This in turn generalizes the chronology order of spacetime defined via timelike paths. We also investigate the associated Alexandrov topology using ideas of Penrose and Kronheimer[2]. We conclude by defining the notion of a strongly path ordered space (X, τ, P ) that generalizes the concept of strongly causal spacetimes and prove an equivalence involving these topologies.

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.

2019

To a significant extent, the metrical and topological properties of spacetime can be described purely order-theoretically. The K relation has proven to be useful for this purpose, and one could wonder whether it could serve as the primary causal order from which everything else would follow. In that direction, we prove, by defining a suitable order-theoretic boundary of K(p), that in a K-causal spacetime, the manifold-topology can be recovered from K. We also state a conjecture on how the chronological relation I could be defined directly in terms of K.

We present a deductive theory of space-time which is realistic, objective, and relational. It is realistic because it assumes the existence of physical things endowed with concrete properties. It is objective because it can be formulated without any reference to knowing subjects or sensorial fields. Finally, it is relational because it assumes that space-time is not a thing, but a complex of relations among things. In this way, the original program of Leibniz is consummated, in the sense that space is ultimately an order of coexistents, and time is an order of successives. In this context, we show that the metric and topological properties of Minkowskian space-time are reduced to relational properties of concrete things. We also sketch how our theory can be extended to encompass a Riemannian space-time.

Bulletin of the Australian Mathematical Society, 1991

An axiomatic approach to relativity theory is proposed which is based entirely on a single reflexive relation called signalling. This generalises and simplifies earlier schemes of Kronheimer and Penrose and of Carter which are based on causality relations. Many of the features of space-times, such as photon paths and topology, have their counterparts in general signal spaces.

We present a deductive theory of space-time which is realistic, objective, and relational. It is realistic because it assumes the existence of physical things endowed with concrete properties. It is objective because it can be formulated without any reference to cognoscent subjects or sensorial fields. Finally, it is relational because it assumes that space-time is not a thing but a complex of relations among things. In this way, the original program of Leibniz is consummated, in the sense that space is ultimately an order of coexistents, and time is an order of succesives. In this context, we show that the metric and topological properties of Minkowskian space-time are reduced to relational properties of concrete things. We also sketch how our theory can be extended to encompass a Riemmanian 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.