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Causal Structures and Causal Boundaries 2 TOPICAL REVIEW Causal structures and causal boundaries Alfonso Garc´ıa-Parrado and Jos´eM. M. Senovilla Departamento de F´ısica Te´orica, Universidad del Pa´is Vasco. Apartado 644, 48080 Bilbao, Spain. Abstract. We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings. PACS numbers: 04.20.Gz, 04.20.Cv, 02.40.Ky Contents 1 Introduction 2 1.1 Plan of the paper with a brief description of contents . 7 1.2 ConventionsandNotation ......................... 8 2 Essentials of causality in Lorentzian Geometry 10 2.1 Thenullcone.Causaltensors . 11 2.2 Localcausality ............................... 12 2.3 Globalcausality............................... 13 2.3.1 Continuous causal curves on Lorentzian manifolds. 14 2.3.2 Basic sets used in causality theory. 15 3 Causality in abstract settings 17 3.1 TheKronheimer-Penrosecausalspaces . 17 arXiv:gr-qc/0501069v2 8 Mar 2005 3.1.1 Connectivityproperties . 20 3.2 Carter’sstudyofcausalspaces . 21 3.2.1 Etiological spaces. 24 3.3 Physically inspired axiomatic approaches . 26 3.3.1 Ehlers-Pirani-Schild-Woodhouse axiomatic construction. 26 3.3.2 Quantumapproaches. Causalsets. 29 4 Causal characterization of Lorentzian manifolds and their classifica- tion 31 4.1 Standard hierarchy of causality conditions . 32 4.2 Mappingspreservingcausalproperties . 35 4.2.1 Causal relationship. Isocausal Lorentzian manifolds. 37 4.3 Orderedchainsofcausalstructures . 40 Causal structures and causal boundaries 2 5 Causality and topology 41 5.1 Some classical results and their generalizations . 42 5.2 Splitting theorems for globally hyperbolic spacetimes . 44 5.3 Topologies on causal spaces and Lorentzian manifolds . 45 5.4 TopologyonLor(V). ............................ 47 6 Causal boundaries 48 6.1 Penroseconformalboundary. 49 6.2 Geroch, Kronheimer and Penroseconstruction. .. 53 6.3 Developments of the Geroch, Kronheimer and Penrose construction . 56 6.3.1 TheBudicandSachsconstruction. 56 6.3.2 R´acz’s generalizations . 57 6.3.3 The Szabados construction for strongly causal spacetimes. ... 57 6.3.4 TheMarolfandRossrecipe. 59 6.3.5 CaveatsontheGKP-basedconstructions. 60 6.3.6 Harris’ approach to the GKP boundary using categories. 61 6.4 Otherindependentdefinitions . 62 6.4.1 g-boundary.............................. 62 6.4.2 b-boundary ............................. 63 6.4.3 Meyer’smetricconstruction. 65 6.4.4 a-boundary.............................. 65 6.4.5 Causal relationship and the causal boundary. 68 6.5 Examplesandapplications. .. .. .. .. .. .. .. .. .. .. 70 6.5.1 Penrosediagrams. ......................... 70 6.5.2 Causalcompletionsandcausaldiagrams. 72 6.5.3 Causal properties of the Brinkmann spacetimes: pp-waves. 73 7 Causal transformations and causal symmetries 77 7.1 Causalsymmetries ............................. 78 7.2 Causal-preservingvectorfields . 79 7.3 Liesubsemigroups ............................. 80 8 Future perspectives 82 1. Introduction Causality is the relation between causes and their effects, or between regularly correlated phenomena. Causes are things that bring about results, actions or conditions. No doubt, therefore, causality has been a major theme of concern for all branches of philosophy and science for centuries. Perhaps the first philosophical statement about causality is due to the presocratic atomists Leukippus and Democritus: “nothing happens without the influence of a cause; everything occurs causally and by need”. This intuitive view was a matter of continuous controversy, eventually cleverly criticized by Hume, later followed by Kant, on the grounds that probably our belief that an event follows from a cause may be simply a prejudice due to an association of ideas founded on a large number of experiences where similar things happened in the same order. In summary, an incomplete induction. The task of science and particularly Physics, however, is establishing or unveiling relations between phenomena, with a main goal: predictability of repetitions or new Causal structures and causal boundaries 3 phenomena. Nevertheless, when we state that, for instance, an increase on the pressure of a gas reduces its volume, we could equally state that a decrease of the volume increases the pressure. The ambiguity of the couple cause-effect is even greater when we reverse the sense of time. And this is one of the key points. The whole idea of causality must be founded upon the basic a priori that there is an orientation of time, a time-arrow. This defines the future (and the past), at least on our neighbourhood and momentarily. Consequently, all branches of Physics, old and new, classical or modern, have a causality theory lying underneath. This is specially the case after the unification process initiated by Maxwell with Electromagnetism and partly culminated by Einstein with the theories of Relativity, where time and space are intimately inter- related. Soon after, Minkowski was able to unite these two concepts in one single entity: spacetime, a generalization of the classical three-dimensional Euclidean space by adding an axis of time. A decisive difference with Euclidean space arises, though: time has a different status, it mingles with space but keeping its identity. It has, so to speak, a “different sign”. This immediately leads to the most basic and fundamental causal object, the cone which embraces the time-axis, leaving all space axes outside. And upon this basic fact, by naming the two halves of the cone (future and past), we may erect a whole theory of causality and causal structure for spacetimes. Four-dimensional spacetimes (see definition 2.1) are the basic arena in General and Special Relativity and their avatars, and almost any other theory trying to incorporate the gravitational field or the finite speed of propagation of signals will have a (possibly n-dimensional with n ≥ 4) spacetime at its base. The maximum speed of propagation is represented in this picture by the angle of the cone at each point. And this obviously determines the points that may affect, or be influenced by, other points. In Physics, spacetimes represent the Universe, or that part of it, we live in or want to model. Thus, the notion of spacetime is pregnant with “causal” concepts, that is, with an inherent causality theory. Causality theory has played a very important role in the development of General Relativity. It is fundamental for all global formalisms, for the theory of radiation, asymptotics, initial value formulation, mathematical developments, singularity theorems, and many more. There are books dealing primarily with causality theory ([10, 88]) and many standard books about General Relativity contain at least a chapter devoted to causality [77, 130, 197]. Recently, modern approaches to “quantum gravity” have borrowed concepts widely used in causality theory such as the causal boundary (AdS/CFT correspondence [90]) or abstract causal spaces (quantum causal sets [17]). Roughly speaking the causal structure of a spacetime is determined at three stages: primarily, by the mentioned cones—called null cones—, at each event. This is the algebraic level. Secondly, by the the connectivity properties concerning nearby points, which is essentially determined by the null cone and the local differential structure through geodesic arcs. This is the local stage. And thirdly, by the connectivity at large, that is, between any possible pair of points, be they close or not. This connection must be achieved by sequences of locally causally related events, usually by causal curves —representing the paths of physical small objects. This is the global level. These three stages taken together determine the causal structure of the spacetime. In short, the causal structure involves the sort of relations, properties, and constructions arising between events, or defined on tensor objects, which depend essentially on the existence of the null cones, that is to say, on the existence of a sole time in front of all spatial dimensions. Unfortunately, a precise definition of Causal structures and causal boundaries 4 “causal structure” is in general lacking in the literature, as it is more or less taken for granted in the formalisms employed by each author. One of the aims of this review is to provide an appropriate, useful and rigorous definition of “causal structure” of sufficient generality. From a mathematical viewpoint, a spacetime is a Lorentzian manifold: an n- dimensional semi-Riemannian manifold (V, g) where at each point x ∈ V the metric tensor g|x, which gives a local notion of distances and time intervals, has Lorentzian signature [138], the axis with the different sign (n> 2) indicating “time” (cf. definition 2.1 for more details). Even though a great deal of research has been performed in four- dimensional Lorentzian manifolds for obvious reasons, almost all the results do not depend on the manifold dimension and we will always work in arbitrary dimension n. This is also adapted to more recent advances such as String Theory, Supergravity, et cetera. The field equations or physical conditions that the metric tensor g must satisfy in order to lead to an acceptable representation of an actual spacetime are in principle outside the scope of this review. There is an entire branch of Mathematics called Lorentzian geometry whose subject is the study of Lorentzian manifolds and it encompasses
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