Causal structure of spacetime and Scott topology

Classical and Quantum Gravity, 2007

The topology of the causal boundary for standard static spacetimes---spacetimes time-invariantly conformal to a metric product of the Lorentz line and a Riemannian manifold---is studied in depth. As this is given in terms of a set of real-valued functions on the Riemannian factor, one could use a function-space topology, but physical reasons recommend a chronological topology instead. The function-space topology has a simple product structure, while the chronological topology might not. This paper examines when the chronological topology coincides with the function-space topology and when it has a simple product structure. A class of standard static spacetimes is examined, all of which yield a simple product structure for the causal boundary; the conformal class of these spacetimes includes classical spacetimes such as external Schwarzschild or Reissner Nordström. J L Flores was supported in part by MCyT-FEDER Grant BFM2001-2871-C04-01, MECyD Grant EX-2002-0612 and MEC Grant RyC-200...

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.

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, 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

2012

We consider Lorentzian manifolds as examples of partially ordered measure spaces, sets endowed with compatible partial order relations and measures, in this case given by the causal structure and the volume element defined by each Lorentzian metric. This places the structure normally used to describe spacetime in geometrical theories of gravity in a more general context, which includes the locally finite partially ordered sets of the causal set approach to quantum gravity. We then introduce a function characterizing the closeness between any two partially ordered measure spaces and show that, when restricted to compact spaces satisfying a simple separability condition, it is a distance. In particular, this provides a quantitative, covariant way of describing how close two manifolds with Lorentzian metrics are, or how manifoldlike a causal set is.

Classical and Quantum Gravity, 2002

Given a (d + 1)-dimensional spacetime (M, g), one can consider the set N of all its null geodesics. If (M, g) is globally hyperbolic then this set is naturally a smooth (2d − 1)-manifold. The sky of an event x ∈ M is the subset

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.

Using K-causal relation introduced by Sorkin and Woolgar [1], we generalize results of Garcia-Parrado and Senovilla [2,3] on causal maps. We also introduce causality conditions with respect to K-causality which are analogous to those in classical causality theory and prove their inter-relationships. We introduce a new causality condition following the work of Bombelli and Noldus [4] and show that this condition lies in between global hyperbolicity and causal simplicity. This approach is simpler and more general as compared to traditional causal approach and it has been used by Penrose et al in giving a new proof of positivity of mass theorem. C 0 -space-time structures arise in many mathematical and physical situations like conical singularities, discontinuous matter distributions, phenomena of topology-change in quantum field theory etc.

Fortschritte der Physik/Progress of Physics, 1992

At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables.

Matemática contemporânea, 2005

After the heroic epoch of Causality Theory, problems concerning the smoothability of time functions and Cauchy hypersurfaces remained as unanswered folk questions. Just recently solved, our aim is to discuss the state of the art on this topic, including self-contained proofs for questions on causally continuous, stably causal and globally hyperbolic spacetimes.