Causal Spacetime Research Articles

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.

The Causal Boundary of Spacetimes Revisited

We present a new development of the causal boundary of spacetimes, originally introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime (or, more generally, a chronological set), we reconsider the GKP ideas to construct a family of completions with a chronology and topology extending the original ones. Many of these completions present undesirable features, like those appeared in previous approaches by other authors. However, we show that all these deficiencies are due to the attachment of an ``excessively big'' boundary. In fact, a notion of ``completion with minimal boundary'' is then introduced in our family such that, when we restrict to these minimal completions, which always exist, all previous objections disappear. The optimal character of our construction is illustrated by a number of satisfactory properties and examples.

Spacetime Causality in the Study of the Hankel Transform

We study Hilbert space aspects of the Klein-Gordon equation in two-dimensional spacetime. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. We apply this to give a novel proof, solely based on the causality of this spatio-temporal wave propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel pairs with a support property. We recover their isometric expansion as an application of Riemann’s general method for solving Cauchy-Goursat problems of hyperbolic type.

Foliations and2+1causal dynamical triangulation models

The original models of causal dynamical triangulations construct space-time by arranging a set of simplices in layers separated by a fixed time-like distance. The importance of the foliation structure in the 2+1 dimensional model is studied by considering variations in which this property is relaxed. It turns out that the fixed-lapse condition can be equivalently replaced by a set of global constraints that have geometrical interpretation. On the other hand, the introduction of new types of simplices that puncture the foliating sheets in general leads to different low-energy behavior compared to the original model.

Conventionality of Simultaneity: Malament’s Result Revisited

Some authors have given the impression that David Malament’s proof of the uniqueness of standard simultaneity in causal Minkowski spacetime also shows uniqueness in Minkowski spacetime proper. This paper has two aims. First, to show that this idea is actually false and, secondly, to present a result which proves that, under certain “natural” conditions, standard simultaneity is also unique in Minkowski spacetime.

The Stability of Abstract Boundary Essential Singularities

The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a non-regular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-times to causal geodesic incompleteness. Linking this with the work of Beem on the C r -stability of geodesic incompleteness allows proof of the stability of these singularities. Here I present this result stating the conditions under which essential singularities are C 1-stable against perturbations of the metric.

Nonperturbative lorentzian path integral for gravity

We construct a well-defined regularized path integral for Lorentzian quantum gravity in terms of dynamically triangulated causal space-times. Each Lorentzian geometry and its action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time and, for finite lattice volume, the associated transfer matrix is self-adjoint, bounded, and strictly positive. The degenerate geometric phases found in dynamically triangulated Euclidean gravity are not present.

(Quantum) spacetime as a statistical geometryof lumps in random networks

In the following we undertake to describe how macroscopic spacetime(or rather, a microscopic protoform of it) is supposed to emerge as asuperstructure of a web of lumps in a stochastic discrete networkstructure. As in preceding work (mentioned below), our analysis isbased on the working philosophy that both physics and thecorresponding mathematics have to be genuinely discrete on theprimordial (Planck scale) level. This strategy is concretelyimplemented in the form of cellular networks and randomgraphs. One of our main themes is the development of the conceptof physical (proto)points or lumps as densely entangledsubcomplexes of the network and their respective web, establishingsomething like (proto)causality. It may perhaps be said thatcertain parts of our programme are realizations of some early ideasof Menger and more recent ones sketched by Smolin a couple of yearsago. We briefly indicate how this two-storey concept ofquantum spacetime can be used to encode the (at least in ourview) existing non-local aspects of quantum theory without violatingmacroscopic spacetime causality!

Holographic view of causality and locality via branes in AdS/CFT correspondence

We study dynamical aspects of the holographic correspondence between d=5 anti-de Sitter supergravity and d=4 super Yang–Mills theory. We probe causality and locality of ambient space-time from super Yang–Mills theory by studying transmission of low-energy brane waves via an open string stretched between two D3-branes in the Coulomb branch. By analyzing two relevant physical threshold scales, we find that causality and locality is encoded in the super Yang–Mills theory in terms of an infinite tower of long supermultiplet operators. The massive W-boson and dual magnetic monopole behave more properly as extended, bilocal objects. We also study the causal time-delay of a low-energy excitation on a heavy quark or meson and find excellent agreement between the anti-de Sitter supergravity and super Yang–Mills theory descriptions. We observe that the strong 't Hooft coupling dynamics and the holographic scale-size relation thereof play a crucial role to the agreement of dynamical processes.

Some Splitting Theorems for Stably Causal Spacetimes

We prove a splitting theorem for stably causal spacetimes and another splitting theorem for finitely compact spacetimes admitting a proper time synchronizable reference frame.

Singularity versus splitting theorems for stably causal spacetimes

Splitting theorems for stable causal spacetimes admitting certain metric related reference frames are obtained in connection to timelike geodesic incompletenes.

An equivalent definition of the strong causality of space-time

An equivalent definition of the strong causality of space-time

Strong curvature singularities and causal simplicity

Techniques of differential topology in Lorentzian manifolds developed by Geroch, Hawking, and Penrose are used to rule out a class of locally naked strong curvature singularities in strongly causal space-times. This result yields some support to the validity of Penrose’s strong cosmic censorship hypothesis.

Discrete operator fields as carriers of causal spacetime structure. Internal symmetry

Discrete operator fields as carriers of causal spacetime structure. Internal symmetry

Geodesics in Gödel-type space-times

We investigate the geodesic curves of the homogeneous Godel-type space-times, which constitute a two-parameter (l and Ω) class of solutions presented to several theories of gravitation (general relativity, Einstein-Cartan and higher derivative). To this end, we first examine the qualitative properties of those curves by means of the introduction of an effective potential and then accomplish the analytical integration of the equations of motion. We show that some of the qualitative features of the free motion in GSdel's universe (l 2=2Ω2) are preserved in all space-times, namely: (a) the projections of the geodesics onto the 2-surface (r, φ) are simple closed curves (with some exceptions forl 2≥4Ω2), and (b) the geodesies for which the ratio of azimuthal angular momentum to total energy,γ, is equal to zero always cross the originr=0. However, two new cases appear: (i) radially unbounded geodesies withγ assuming any (real) value, which may occur only for the causal space-times (l 2≥4Ω2), and (ii) geodesies withγ bounded both below and above, which always occur for the circular family (l 2<0) of space-times.

Spaces of causal paths and naked singularities

The space of causal geodesics is considered, with reference to the natural topology and manifold structure (in the case where M is strongly causal). It is shown that conjugacy arises as a natural geometric concept in this context, and that non-Hausdorffness of the space of geodesics corresponds to a particularly weak kind of naked singularity. Next the space of smooth endless causal curves C is considered, and an appropriate topology put on the space. For a strongly causal spacetime M the author obtains the result that M is globally hyperbolic iff C is Hausdorff. The strength of the naked singularities which cause C not to be Hausdorff is defined in terms of how hard they are to escape from.

Lorentzian geometry of globally framed manifolds

A new class of globally framed manifolds (carrying a Lorentz metric) is introduced to establish a relation between the spacetime geometry and framed structures. We show that strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular framed structure. As examples, we present a class of spacetimes of general relativity, having an electromagnetic field, endowed with a framed structure and a causal spacetime with a nonregular contact structure. This paper opens a few new problems, of geometric/physical significance, for further study.

Wormholes, baby universes, and causality.

In this paper wormholes defined on a Minkowski signature manifold are considered, both at the classical and quantum levels. It is argued that causality in quantum gravity may best be imposed by restricting the functional integral to include only causal Lorentzian spacetimes. Subject to this assumption, one can put very tight constraints on the quantum behavior of wormholes, their cousins the baby universes, and topology-changing processes in general. Even though topology-changing processes are tightly constrained, this still allows very interesting geometrical (rather than topological) effects. In particular, the laboratory construction of baby universes is not prohibited provided that the "umbilical cord" is never cut. Methods for relaxing these causality constraints are also discussed.

Space time manifolds and contact structures

A new class of contact manifolds (carring a global non‐vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4‐dimensional spacetime of general relativity as a contact CR‐submanifold.

The Gross-Neveu model near the critical temperature. A mechanism for quark-hadron phase transition in relativistic ion collisions

It is argued that the longitudinal, critical evolution of the quark-gluon system, formed in relativistic ion collisions, can be reliably studied within the framework of the Gross-Neveu two-dimensional field theory at finite temperature. In particular, the Ginzburg-Landau approximation, near the critical temperature reveals some remarkable and measurable inhomogeneities in the rapidity density of the hadronic phase, when appropriate boundary conditions, dictated by space-time causality, are imposed on the classical solutions of the model. The relevance of the results for the experiments with relativistic ions is briefly discussed.