We analyse the causal structure of the ambient boundary, the conformal
infinity of the ambient (Poincar\'e) metric. Using topological tools we show
that the only causal relation compatible with the global topology of the
boundary spacetime is the horismos order. This has important consequences for
the notion of time in the conformal geometry of the ambient boundary.Comment: 7 page

Antoniadis, Ignatios

Cotsakis, Spiros

Papadopoulos, Kyriakos

English

arXiv

We analyze the causal structure of the ambient boundary, the conformal infinity of the ambient (Poincaré) metric. Using topological tools, we show that the only causal relation compatible with the global topology of the boundary spacetime is the horismos order. This has important consequences for the notion of time in the conformal geometry of the ambient boundary

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arXiv:1603.08117v1  [hep-th]  26 Mar 2016The causal order on the ambientboundaryIgnatios Antoniadis1,2∗, Spiros Cotsakis†3‡Kyriakos Papadopoulos3§,1LPTHE, UMR CNRS 7589, Sorbonne Universite´s, UPMC Paris 6,75005 Paris, France2 Albert Einstein Center for Fundamental Physics, ITP,Bern University,Sidlerstrasse 5 CH-3012 Bern, Switzerland3Department of Mathematics,American University of the Middle EastP. O. Box 220 Dasman, 15453, KuwaitMarch 29, 2016AbstractWe analyse the causal structure of the ambient boundary, the conformal infinity ofthe ambient (Poincare´) metric. Using topological tools we show that the only causalrelation compatible with the global topology of the boundary spacetime is the horis-mos order. This has important consequences for the notion of time in the conformalgeometry of the ambient boundary.∗antoniad@lpthe.jussieu.fr†On leave from the University of the Aegean, 83200 Samos, Greece.‡skot@aegean.gr§Kyriakos.P@aum.edu.kw1source: https://doi.org/10.7892/boris.86329 | downloaded: 13.3.20171 IntroductionAmbient cosmology is an attempt to describe the structure of the universe in the neighbor-hood of the spacetime singularities predicted by the singularity theorems of general relativity.This idea was introduced in Refs. [1, 2, 3] where it was shown that singularities are absent,cosmic censorship becomes a valid proposition by construction, and in fact the method ofproof of the theorems predicting singularities becomes inapplicable in this framework.The main motivation of the ambient approach to the problem of singularities in cosmologyis that the infinities that are inherent in the spacetime metric according to the singularitytheorems indicate the necessity of a conformal geometry of metrics, g → Ω2g, to absorb them,not a breakdown of general relativity. Any regular metric will be mapped to a different,inequivalent one, so only singularities may give rise to a conformal symmetry.In the ambient approach, we imagine that our universe M becomes a conformal 4-boundary at infinity of another metric, a conformal metric on an ambient 5-space V . Thisarises because starting from the cone metric g5 = a2(y)g4+dy2 on V , an asymptotic analysisof solutions to the Einstein equations with a fluid source in the ambient space implies a veryrich structure at the infinity of V : It is possible to express the wrap factor a(y) as a formalpower series converging to a smooth function σ(w), giving rise to an ‘ambient’ (Poincare´)5-metric g+ = ω−n (σ2(w)g4(xµ) + dw2) , n ∈ Q+. This allows us to extract a regular met-ric g˚ whose restriction to the conformal boundary at infinity g˚|M becomes non-degenerate,and due to a subtle asymptotic condition, it gives a conformal structure to the boundaryspacetime M , the ambient boundary, cf. [1].Because of the existence of a conformal geometry on the ambient boundary there isno unique metric there, and so all physically important quantities have to be conformallyinvariant. This places certain constraints on the various causality relations which couldin principle be defined there. Another important feature tightly connected to the wholegeometry of the model is the existence of a unique topology on the ambient boundary, theZeeman topology [2].In Section 2, we show that among all possible causal order relations defined on theambient boundary, only a horismotic one may be fully compatible with the Zeeman topologyon the boundary. This fact has important consequences for the causality structure of the2ambient boundary, a fact we discuss in Section 3.2 The interval topology from horismosIn spacetime geometry, it is standard to introduce three order relations, namely, the chrono-logical order ≪, the causal order ≺ and the order horismos →, and these can be mean-ingfully extended to any event-set, a set (X,≪,≺,→) equipped with all three of theseorders having no metric [4, 5]. In this context we say that the event x chronologicallyprecedes event y, written x ≪ y if y lies inside the future null cone of x, x causallyprecedes y, x ≺ y, if y lies inside or on the future null cone of x, and x is at horismoswith y, written x → y, if y lies on the future null cone of x. The chronological and ho-rismotic orders are anti-reflexive, while the causal order is reflexive. Then the notationsI+(x) = {y ∈ M : x ≪ y}, J+(x) = {y ∈ M : x ≺ y} will be used for the chronological andthe causal futures of x respectively (and with a minus instead of a plus sign for the pasts),while the future null cone of x will be denoted by N+(x) ≡ ∂J+(x) = {y ∈ M : x → y},and dually for the null past of x, cf. [5]. Our metric signature is timelike, (+,−,−,−).The above definitions of futures and the pasts of a set can be trivially extended tothe situation of any partially ordered set (X,<), and the so-called orderability problem isconcerned with the conditions under which the topology T< induced by the order < is equalto some given topology T on X , and conversely, the possible topologies induced on it by anygiven order, cf. [6], [7], [8].In a purely topological context this is usually done by passing to the so-called upper (i.e.,future) and lower (i.e., past) sets which in turn lead to the future and past topologies (see[6]). A subset A ⊂ X is a past set if A = I−(A) and dually for the future. Then, the futuretopology T + is generated by the subbase S+ = {X \ I−(x) : x ∈ X} and the past topologyT − by S− = {X \ I+(x) : x ∈ X}. The interval topology Tin on X then consists of basic setswhich are finite intersections of subbasic sets of the past and the future topologies. This isin fact the topology that fully characterizes a given order of the poset X , since it is solelygenerated by the order properties and nothing else.For reasons explained in the Introduction, we will particularly focus on the developmentof the orderability problem in the context of the Zeeman fine topology F , defined as the finesttopology with the property that on M it induces the 3-dimensional Euclidean topology on3every space axis and the 1-dimensional Euclidean topology on every time axis, cf. [9, 10]. Weshall use ME to denote M equipped with the 4-dimensional, standard (manifold), locally-Euclidean topology, and MF to mean that M is equipped with the Zeeman fine topology.In fact, we shall consider a somewhat subtler version of the fine topology, namely, thetopology given by the basic sets of the form BFǫ (x), where x ∈M and ǫ > 0. This is coarserthan MF but finer than ME (see [9]). We denote this topology by Z, and call it the Zeemantopology. In particular, BFǫ (x) =(BEǫ (x)−N(x))∪{x}, where BEǫ (x) is the Euclidean openball and N (x) the nullcone at x. In other words, we remove N (x) and put back only thepoint x. Obviously, the sets BFǫ (x) are not Euclidean-open.What is the relation between the Zeeman topology Z and that induced from horismos?From the definition of horismos, we have that x → y, if x ≺ y but not x ≪ y. For theconstruction of the interval topology T→ induced from→, we first consider the complementsof the sets N±(x), that is the future and past nullcones of x,M \ N+(x) = {y ∈M : y ≪ x or x ≺ y or y ⊀ x}, (1)and dually for M \ N−(x). Then we observe that the intersection of the subbasic setsM \N+(x) and M \N−(x) gives a Zeeman basic set because for each event x, M \N+(x)∩M \N−(x) gives a Euclidean neighbourhood with the nullcone removed, but x is kept since→ is anti-reflexive. We have thus shown the following result.Theorem 2.1. The order horismos induces the Zeeman topology on M .It is interesting that neither the chronological order ≪ nor the causal order ≺ induce atopology equal to the Zeeman topology. In particular, for an event x, the interval topologyT≪ from ≪ gives basic sets containing x, the light cone of x as well as all events y wherey ⊀ x, but does not include any event inside the future or the past nullcone of x. In a similarfashion, the interval topology T≺ from ≺ gives basic open sets containing y and all events xsuch that x ⊀ y, without the light and time cones of y.Further, by its construction, the Alexandrov topology [5] does not agree with the Zeemantopology: For every event in spacetime, an Alexandrov-open set is the union of sets of theform < u, v >= {w : u≪ w ≪ v}), different from T≪.4Also any order different than horismos fails to give the Zeeman topology. This is eventrue for the horismotic versions of the causality and chronology relations constructed byusing only the properties of the horismos order as in [4], as we show next.Starting from the horismos relation →, we may construct orders similar to chronologyand causality as follows, cf. [4]: x is chronologically related to y through horismos, x ≺U y,if and only if there exists a finite sequence u1, · · · , un, such that x = u1 → · · · → un = y,while x is causally related to y through horismos, x ≪U y if and only if x ≺U y but notx → y. Then we get subbasic sets which are complements of sets ↑ {x} = {y : x ≺U y},and similarly for the set ↓ {x}. The sets ↑ {x}, for example, take the form {x : ∃ u1 =x, · · · , un = y,with x→ u2 → · · · → un−1 → y}, and so their complements will be unions ofZeeman-open balls. Similarly for ↓ {x}, and of course when we intersect unions of Zeeman-open balls we get Zeeman-open sets. This consequently does not guarantee that we will getthe whole topology Z.We have therefore shown the following result.Theorem 2.2. The topologies from the chronological and causal orders do not generate theZeeman topology Z. In addition, the interval topologies T≺U and T≪U are subsets of theZeeman topology Z.3 DiscussionIn this paper we have analyzed the causality structure on the ambient boundary by purelytopological considerations. We have shown that among all possible causal relations onM , theonly one compatible with the topology of the ambient boundary is the horismotic relation.This is the causal order permitted by the conformal geometry on the boundary.In fact, due to the presence of a conformal (homothetic) symmetry on the ambient bound-ary, no other structure except the null structure may have any physical significance. In par-ticular, there can be no objective notion of a timelike or spacelike curve on the boundary,and this manifests itself here in the compatibility of only the horismotic (null) relation to theunderlying topology of the ambient boundary amongst all other possible order relations. Asit was already noted in [1], this is related to the idea of masslessness inherent in the ambientboundary.5It is interesting that this gives rise to a novel resolution of the problem of time in thepresent context. Although time continues to be one of the coordinates in the ambient bound-ary, it loses any relation to the idea of causality as it is known in general relativity, and infact, in the conformal infinity of the ambient 5-space only null relations are permitted, com-patible with the ambient boundary conformal geometry and the topology from horismos.In a sense, time is not felt by any ambient boundary observer, while massless particles likephotons and gravitons move on the null cone.An important issue that then arises is how to define the idea of dynamical evolution onthe ambient boundary when only horismotic relations exist in it. The fact that we may infact do so is related to the ambient space and the extra dimension. Also we may use theresults of the present paper as a means to probe the topology of the ambient space. We shallreturn to these issues in a future publication.References[1] I. Antoniadis, S. Cotsakis, Eur. Phys. J. C. 75:35 (2015) 1-12, CERN-PH-TH/2014-176, arXiv:1409.2220.[2] I. Antoniadis, S. Cotsakis, Mod. Phys. Lett. A. 30 (2015) 1550161, arXiv:1505.04737.[3] I. Antoniadis, S. Cotsakis, The large-scale structure of the ambient boundary,arXiv:1512.09216.[4] E. H. Kronheimer and R. Penrose, Proc. Camb. Phil. Soc. 63 (1967) 481-501.[5] R. Penrose, Techniques of Differential Topology in Relativity (CBMS-NSF RegionalConference Series in Applied Mathematics, 1972).[6] G. Gierz et al, A compendium of continuous lattices (Springer-Verlag, 1980).[7] C. Good, K. Papadopoulos, Topology Appl 159 (2012) 1565-1572.[8] K. Papadopoulos, On the Orderability Problem and the Interval Topology, In: Topicsin Mathematical Analysis and Applications, T. Rassias and L. Toth Eds., (Optimizationand Its Applications Springer Series, Springer-Verlag, 2014).6[9] E. C. Zeeman, Topology 6 (1967) 161-170.[10] R. Gobel, Comm. Math. Phys. 46 (1976) 289-307.[11] E. C. Zeeman, J. Math. Phys. 5 (1964) 490-493.7

The causal order on the ambient boundary

 We analyze the causal structure of the ambient boundary, the conformal infinity of the ambient (Poincaré) metric. Using topological tools, we show that the only causal relation compatible with the global topology of the boundary spacetime is the horismos order. This has important consequences for the notion of time in the conformal geometry of the ambient boundary. </jats:p

Ignatios Antoniadis

Spiros Cotsakis

Kyriakos Papadopoulos

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Modern Physics Letters A

http://boris.unibe.ch/86329/8/1603.08117v1.pdf

The causal order on the ambient boundary

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