K-causal structure of space-time in general relativity

Classical and Quantum Gravity, 1996

We recast the tools of "global causal analysis" in accord with an approach to the subject animated by two distinctive features: a thoroughgoing reliance on order-theoretic concepts, and a utilization of the Vietoris topology for the space of closed subsets of a compact set. We are led to work with a new causal relation which we call K + , and in terms of it we formulate extended definitions of concepts like causal curve and global hyperbolicity. In particular we prove that, in a spacetime M which is free of causal cycles, one may define a causal curve simply as a compact connected subset of M which is linearly ordered by K +. Our definitions all make sense for arbitrary C 0 metrics (and even for certain metrics which fail to be invertible in places). Using this feature, we prove for a general C 0 metric, the familiar theorem that the space of causal curves between any two compact subsets of a globally hyperbolic spacetime is compact. We feel that our approach, in addition to yielding a more general theorem, simplifies and clarifies the reasoning involved. Our results have application in a recent positive energy theorem [1], and may also prove useful in the study of topology change [2]. We have tried to make our treatment

Pre-print UFES-DF-MMS.1993/4, 1993

We discuss the level of implementation of causality in field theory, highlight features of a proposed [1] new model of spacetime structure with a stricter causality implementation which produces an infinityfree formalism with a linear and conformally invariant dynamics. This is relevant since it extends to 4-dimensions results known, up to now, only in bi-dimensional theories. It is particularly interesting that even in a still purely classical formulation it brings, naturally, the idea of quantum of a field, its particle likeness. The standard formalism with all its infinities and divergencies is retrieved in terms of average fields. The singularity of a field is shown to be just a consequence of this averaging in its definiton.

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.

Classical and Quantum Gravity, 2005

2002

We start from the well-known form of the interval of the special relativity, stare it, and build up an attempt to implement the causality from it. Some features appear to be new, they involve the mass of the particle and the structure of space-time. erev Tishrei, 10th.

Communications in Mathematical Physics, 2008

The classical linking number lk is defined when link components are zero homologous. In [13] we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the study of causality in Lorentzian manifolds. Let M m be a spacelike Cauchy surface in a globally hyperbolic space-time (X m+1 , g). The spherical cotangent bundle ST * M is identified with the space N of all null geodesics in (X, g). Hence the set of null geodesics passing through a point x ∈ X gives an embedded (m− 1)-sphere Sx in N = ST * M called the sky of x. Low observed that if the link (Sx, Sy) is nontrivial, then x, y ∈ X are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply the machinery of knot theory to the study of causality. The spheres Sx are isotopic to the fibers of (ST * M) 2m−1 → M m. They are nonzero homologous and the classical linking number lk(Sx, Sy) is undefined when M is closed, while alk(Sx, Sy) is well defined. Moreover, alk(Sx, Sy) ∈ Z if M is not an odd-dimensional rational homology sphere. We give a formula for the increment of alk under passages through Arnold dangerous tangencies. If (X, g) is such that alk takes values in Z and g is conformal to b g that has all the timelike sectional curvatures nonnegative, then x, y ∈ X are causally related if and only if alk(Sx, Sy) = 0. We prove that if alk takes values in Z and y is in the causal future of x, then alk(Sx, Sy) is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x. We show that x, y in a nonrefocussing (X, g) are causally unrelated if and only if (Sx, Sy) can be deformed to a pair of S m−1fibers of ST * M → M by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact.

Noûs, 2023

We develop a new version of the causal theory of spacetime. Whereas traditional versions of the theory seek to identify spatiotemporal relations with causal relations, the version we develop takes causal relations to be the grounds for spatiotemporal relations. Causation is thus distinct from, and more basic than, spacetime. We argue that this non-identity theory, suitably developed, avoids the challenges facing the traditional identity theory.

Classical and Quantum Gravity, 1992

We investigate the causal structure of (1 + 1)-dimensional spacetimes. For two sets of field equations we show that at least locally any spacetime is a solution for an appropriate choice of the matter fields. For the theories under consideration we investigate how smoothness of their black hole solutions affects time orientation. We show that if an analog to Hawking's area theorem holds in two spacetime dimensions, it must actually state that the size of a black hole never increases, contrary to what happens in four dimensions. Finally, we discuss the applicability of the Penrose and Hawking singularity theorems to two spacetime dimensions.

Communications in Mathematical Physics, 1974

Causally continuous general relativistic spacetimes are defined and analyzed. In a causally continuous spacetime, the past and future of a local observer behave continuously under small perturbations of the metric or small changes in his location. Causally simple spacetimes are causally continuous; causally continuous spacetimes are causally stable. Possible reasons for taking causal continuity as a basic postulate in macrophysics are briefly discussed.

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.