Stable Causality Research Articles

A Generalization of Strong Causality

In this paper we are going to define strong Tn - causality for n ∈ N. These are levels in the causal ladder of spacetime between stable causality and strong causality. A characterization of them is presented. The newly proposed levels are compared with Carter’s levels of causal virtuosity. It is also proved that strong T 1- causality implies compact stable causality.

Time Functions as Utilities

Every time function on spacetime gives a (continuous) total preordering of the spacetime events which respects the notion of causal precedence. The problem of the existence of a (semi-)time function on spacetime and the problem of recovering the causal structure starting from the set of time functions are studied. It is pointed out that these problems have an analog in the field of microeconomics known as utility theory. In a chronological spacetime the semi-time functions correspond to the utilities for the chronological relation, while in a K-causal (stably causal) spacetime the time functions correspond to the utilities for the K + relation (Seifert’s relation). By exploiting this analogy, we are able to import some mathematical results, most notably Peleg’s and Levin’s theorems, to the spacetime framework. As a consequence, we prove that a K-causal (i.e. stably causal) spacetime admits a time function and that the time or temporal functions can be used to recover the K + (or Seifert) relation which indeed turns out to be the intersection of the time or temporal orderings. This result tells us in which circumstances it is possible to recover the chronological or causal relation starting from the set of time or temporal functions allowed by the spacetime. Moreover, it is proved that a chronological spacetime in which the closure of the causal relation is transitive (for instance a reflective spacetime) admits a semi-time function. Along the way a new proof avoiding smoothing techniques is given that the existence of a time function implies stable causality, and a new short proof of the equivalence between K-causality and stable causality is given which takes advantage of Levin’s theorem and smoothing techniques.

On the global structure of the Pomeransky–Senkov black holes

We construct analytic extensions of the Pomeransky-Senkov metrics with multiple Killing horizons and asymptotic regions. We show that, in our extensions, the singularities associated to an obstruction to differentiability of the metric lie beyond event horizons. We analyze the topology of the non-empty singular set, which turns out to be parameter-dependent. We present numerical evidence for stable causality of the domain of outer communications. The resulting global structure is somewhat reminiscent of that of Kerr space-time.

Widening the light cones on subsets of spacetime: some variations to stable causality

By definition a spacetime is stably causal if it is possible to widen the light cones all over the spacetime without spoiling causality. We prove that if the spacetime is at least non-total imprisoning then it is stably causal provided the light cones can be widened outside any arbitrarily large compact set, i.e. in a neighborhood of infinity, without spoiling causality. Furthermore, we prove that the new causality level ‘compact stable causality’ can be obtained as the antisymmetry condition of a new causal relation which we identify, but it cannot be obtained as a causal stability condition with respect to a topology on metrics. The difference between stable causality and compact stable causality is shown to follow from the fact that Geroch's interval topology on the space of conformal metrics of M is not Fréchet–Urysohn (in fact it is not even T-sequential). In particular we prove that (compact) stably causal metrics are those in the (sequential) interior of the set of chronological metrics. Finally, contrary to previous claims it is shown that stable causality with respect to the C0 fine topology on metrics leads to the usual notion of stable causality.

K-Causality Coincides with Stable Causality

It is proven that K-causality coincides with stable causality, and that in a K-causal spacetime the relation K^+ coincides with the Seifert's relation. As a consequence the causal relation "the spacetime is strongly causal and the closure of the causal relation is transitive" stays between stable causality and causal continuity.

The causal ladder and the strength of K-causality: II

Hawking's stable causality implies Sorkin and Woolgar's K-causality. The work investigates the possible equivalence between the two causality requirements, an issue which was first considered by H Seifert and then raised again by R Low after the introduction of K-causality. First, a new proof is given that a spacetime is stably causal iff the Seifert causal relation is a partial order. It is then shown that given a K-causal spacetime and having chosen an event, the light cones can be widened in a neighborhood of the event without spoiling K-causality. The idea is that this widening of the light cones can be continued leading to a global one. Unfortunately, due to some difficulties in the inductive process the author was not able to complete the program for a proof as originally conceived by H Seifert. Nevertheless, it is proved that if K-causality coincides with stable causality then in any K-causal spacetime the K+ future coincides with the Seifert future. Explicit examples are provided which show that the K+ future may differ from the Seifert future in causal spacetimes.

Space-Time Topology (II)—Causality, the Fourth Stiefel–Whitney Class and Space-Time as a Boundary

We show that stable causality is related to the vanishing of the top Stiefel–Whitney class of a space-time manifold M, and that if M is a stably causal space-time manifold, then it is the boundary of a five-dimensional space-time. We then propose a scheme for making it both a necessary and sufficient condition.

Causal inheritance in plane wave quotients

We investigate the appearance of closed timelike curves in quotients of plane waves along spacelike isometries. First we formulate a necessary and sufficient condition for a quotient of a general space-time to preserve stable causality. We explicitly show that the plane waves are stably causal; in passing, we observe that some pp waves are not even distinguishing. We then consider the classification of all quotients of the maximally supersymmetric ten-dimensional plane wave under a spacelike isometry, and show that the quotient will lead to closed timelike curves iff the isometry involves a translation along the u direction. The appearance of these closed timelike curves is thus connected to the special properties of the light cones in plane wave space-times. We show that all other quotients preserve stable causality.

Topology change without any pathology

A class of (noncompact) singularity-free spacetimes is found, where topology changes without violating the weak energy condition (wec) or the stable causality condition.

Mathematical Structures of Space-Time

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. Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christoffel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Thus, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work should be compared with important previous papers. There are some relevant differences, because we rely on a different definition of

Stably causal Gödel-type models

The problem of finding stationary Gödel-type cosmological space-times endowed with stable causality is solved in general relativity. These models are generated by spin fluids with strong spin-vorticity coupling, and present interesting physical properties. Their intrinsic characterization is made by means of an invariant algebraic classification. A simple technique to test the stable causality is applied to new and old distinct stationary space-times.

On the topology of stable causality

A topological description of the stable causality condition on a spacetime is given. The structure of the subsets that, for a given point, control the fulfillment of strong and stable causality conditions at that point are studied. Separation properties of the relevant topologies are also analyzed.

Distinguishing properties of causality conditions

We show that certain causality conditions, such as causality, (future-, past-) distinguishing, strong causality, and stable causality have the common feature that there exists a map from the space-time manifoldM to the power setP(M) which is injective if and only if the right causality condition holds onM.

On almost causality

The almost causal precedence concept proposed by Woodhouse and a causality axiom based on it are analyzed. Properties of the almost future are examined and the above causality axiom is shown to be deducible for a causally continuous space–time. It is shown to be coinciding with known causality conditions for a reflecting space–time. Also the equivalence of various causality conditions for a reflecting space–time and a reduction of the causal continuity axiom are obtained. It is next shown that the above causality axiom is not stable under metric perturbations. Then the Seifert future concept has been analyzed and the Woodhouse causality is proved to be strictly weaker than the stable causality condition. It is concluded that as far as the uncertainties in the form of metric perturbations are concerned, the Seifert future is useful but not the almost future.

Some considerations on negative impedance circuits Causality analysis with modified signal flowgraphs

A modified signal flowgraph approach is proposed to give a simple and systematic method for clarifying the difference between current controlled and voltage controlled negative resistances in linear domains. New rules for signal flowgraphs are introduced and discussed to predict a stable causality as the proper solution in the analysis of active circuits including negative resistances and/or controlled sources.

A plane gravitational wave: Global structure, completeness, stable causality

A plane gravitational wave: Global structure, completeness, stable causality

Lorentz cobordism. II

It is shown that ‘changes of topology’ (of spacelike sections) in the spacetime of classical general relativity are consistent with the following requirements: (i) stable causality, (ii) future causal geodesic completeness, and (iii) finite, positive energy density. This amounts to showing that the framework of classical general relativity encompasses ‘changes of topology’.

The existence of cosmic time functions

It is shown that stable causality is the necessary and sufficient condition that there should exist a cosmic time function which increases along every future directed timelike or null curve. Stable causality means that there are no closed timelike or null curves in any Lorentz metric that is sufficiently close to the space-time metric.