We discuss the topological nature of the boundary spacetime, the conformal
infinity of the ambient cosmological metric. Due to the existence of a
homothetic group, the bounding spacetime must be equipped not with the usual
Euclidean metric topology but with the Zeeman fine topology. This then places
severe constraints to the convergence of a sequence of causal curves and the
extraction of a limit curve, and also to our ability to formulate conditions
for singularity formation.Comment: 8 page

Antoniadis, Ignatios

Cotsakis, Spiros

English

arXiv

 We discuss the topological nature of the boundary spacetime, the conformal infinity of the ambient cosmological metric. Due to the existence of a homothetic group, the bounding spacetime must be equipped not with the usual Euclidean metric topology but with the Zeeman fine topology. This then places severe constraints to the convergence of a sequence of causal curves and the extraction of a limit curve, and also to our ability to formulate conditions for singularity formation. </jats:p

Ignatios Antoniadis

Spiros Cotsakis

Crossref

Modern Physics Letters A

Topology of the ambient boundary and the convergence of causal curves

We discuss the topological nature of the boundary spacetime, the conformal infinity of the ambient cosmological metric. Due to the existence of a homothetic group, the bounding spacetime must be equipped not with the usual Euclidean metric topology but with the Zeeman fine topology. This then places severe constraints to the convergence of a sequence of causal curves and the extraction of a limit curve, and also to our ability to formulate conditions for singularity formation

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arXiv:1505.04737v2  [hep-th]  24 Aug 2015Topology of the ambient boundary and theconvergence of causal curvesIgnatios Antoniadis1,2, Spiros Cotsakis3,4‡1 Albert Einstein Center, Institute for Theoretical PhysicsBern University, Sidlerstrasse 5 CH-3012 Bern, Switzerland2 LPTHE, UMR CNRS 7589Sorbonne Universite´s, UPMC Paris 6, 75005 Paris, France3 School of Applied Mathematics and Physical SciencesNational Technical University, 15780 Athens, GreeceE-mails: †antoniad@lpthe.jussieu.fr, ‡skot@aegean.grAugust 25, 2015AbstractWe discuss the topological nature of the boundary spacetime, the conformal infinityof the ambient cosmological metric. Due to the existence of a homothetic group,the bounding spacetime must be equipped not with the usual Euclidean metrictopology but with the Zeeman fine topology. This then places severe constraintsto the convergence of a sequence of causal curves and the extraction of a limitcurve, and also to our ability to formulate conditions for singularity formation.4On leave from the University of the Aegean, 83200 Samos, Greece.source: http://boris.unibe.ch/71321/ | downloaded: 13.3.20171 IntroductionFor half a century it has been known that general relativity has the remarkable propertyof generically leading to spacetime singularities where it must break down as a physicaltheory and together our ability to further predict the future or past of spacetime cometo a halt [1, 2]. The mathematical results that lead to the prediction of spacetimesingularities in general relativity, the so-called singularity theorems, rest on the importof topological methods in relativity, in particular, global methods of causality theory inLorentzian manifolds and the behaviour of geodesics in spacetime, and show that undervarious physically plausible and geometric conditions, spacetime must be geodesicallyincomplete [3, 4, 5, 6].Hence, general relativity as a physical theory predicts that there must be a collapsesingularity in the past of the universe’s history, known as the big bang, and also in thefuture, localized ‘inside’ black holes (assuming cosmic censorship), or in the form of anaked singularity. It is the existence of spacetime singularities as it follows from thesingularity theorems that is to a large measure responsible for the long-held view thatquantum gravitational effects may be directly required also for a restoration of the lossof classical predictability in the gravitational collapse to a black hole and in variousasymptotic questions of modern cosmology, in much the same way as in the ‘quantumrescue’ of classical newtonian evolution (cf. e.g., [7], various chapters).Recently we have examined the question of spacetime singularities in general relativityby introducing the notion of an ambient cosmological metric. In the resulting ‘ambientcosmology’, a number of questions such as the existence of spacetime singularities andcosmic censorship acquire new meaning and find novel interpretations. In this paper, wediscuss a consequence of the existence of a new principle in the boundary spacetime of theambient cosmological metric. This is related to the topological nature of the boundaryand leads to an impossibility of having converging families of causal curves.The plan of this paper is as follows. In the next section we review basic aspects of theambient construction including the basic asymptotic condition valid on the spacetime4-boundary M . In Section 3, we introduce the Zeeman topology and point out that it isthat which is uniquely adapted to the boundary spacetime in the ambient construction.1In Section 4, we show that due to the topological structure of the boundary, it is notpossible to have convergent families of causal curves onM and therefore the existence of alimit curve is lost in the bounding spacetime equipped with the Zeeman topology. In thelast Section, we discuss these results especially their connection with the impossibility offormulating singularity theorems for the conformal infinity of the ambient cosmologicalmetric.2 Ambient cosmologyThe basic idea of ambient cosmology [8] is to regard our 4-dimensional spacetime as akind of bounding hypersurface, the conformal infinity, of a new cosmological metric in 5-dimensional ‘ambient’ space. Although the ambient metric is a well-known constructionin conformal geometry [9], however, in our approach we use the word ‘ambient’ with adifferent meaning, and this leads to a completely new condition imposed on the boundary,instead of the usual uniqueness of the ambient 5-metric corresponding to two conformallyrelated 4-metrics on the boundary.Let us briefly explain some of the required ideas, referring to [8] for complete proofs.We start with a 4-dimensional spacetimeM on which a conformal structure [g4] is defined,that is we have a family of conformally related 4-metrics on M , much like as in the con-formal method, cf. [10], chap. 9. Our ambient metric g+ is then defined at each point p =(xµ, w) of the ‘ambient’ space V = M×R, where R parametrized by w denotes the fifth di-mension, and can be brought in the normal form g+ = w−n (σ2(w)g4(xµ) + dw2) , n ∈ Q+,as w → 0, with σ(w) a smooth (infinitely differentiable) function such that σ(0) is anonzero constant.In this asymptotic sense, the first factor in the cartesian product, M , is located atthe limiting position w = 0 (see also below), and everything we do may be described as‘expanding around w = 0’. The metric g+ solves the 5-dim Einstein equations GAB =TAB on V , where A,B = 0, 1, 2, 3, 4, and TAB is the stress tensor of an analog of a perfectfluid with equation of state P = γρ, where the ‘pressure’ P and the ‘energy density’ ρdepend only on the extra spacelike dimension w, and the density ρ+(w) has the genericform ρ+ = w−zθ(w), z ∈ Z and θ(w) smooth.2Our ambient metric g+ has two further properties of special significance in the presentcontext. Firstly, g+ on V has M as its conformal infinity equipped with the manifestlyregular, non-degenerate metric at w = 0, denoted by g˚|M(0), where g˚ = wng+ = σ2g4 +dw2. In this sense the bounding spacetime M has become the conformal infinity of theambient space V , with a 4-metric given by g˚|M = σ2(w)g4.The second property of the ambient metric g+ is what we call ‘the asymptotic condi-tion’, and it basically reveals the presence of a homothetic symmetry on the boundaryM .It may be described as follows (cf. [8], Section 6 for a proof). Consider two conformallyrelated 4-metrics g1, g2 in the conformal geometry of M , g1 being the ‘bad’ (roughlymeaning ‘singular’) and g2 the ‘good’ metric on the boundary (these are technical termsdefined precisely in [8]). Then their ambient metrics g˚1|M , g˚2|M , although not equal as inthe Fefferman-Graham construction [9], differ by a homothetic transformation, that isg˚2|M(0) = c g˚1|M(0), c : const. (1)Therefore, starting from a conformal geometry on the spacetime M , the ambient cosmo-logical metric returns a 4-geometry on M (its conformal infinity) that has a homotheticsymmetry. In [8], we have given conclusive evidence that because of these two properties,namely, the fact that M is the conformal infinity of g+ and that it also possesses a ho-mothetic symmetry, no spacetime singularities can be developed on M , that is (M, g˚|M)must be free of cosmological and naked singularities, hence providing a major improve-ment over the standard situation in general relativity.However, this raises the intriguing question of whether or not one may still formulatefor M the conditions of the standard singularity theorems of general relativity. Thepurpose of the remaining of this paper is to give a negative answer to this question.The arguments advanced below will also help us understand why one was originally ableto formulate the conditions necessary for the validity of the singularity theorems as isusually done in general relativity.33 Uniqueness of the Zeeman topologySince the mathematical techniques involved in global causality theory and the singularitytheorems are topological in nature, the first question we have to face is how to connectthe homothetic symmetry on the conformal boundary M (as a result of the ambientcosmological 5-metric on V ) with its topological properties. There is a very neat answerto this question, provided long ago by a remarkable mathematical theorem of C. Zeeman.In 1967 he published a paper [11] in which showed that for Minkowski spacetime the groupof homothetic symmetries (that is Lorentz transformations with dilatations) coincideswith the group of all homeomorphisms of M provided that its topology is not the usualEuclidean metric topology (that is M is locally Euclidean) but a new one, called the finetopology Z.The Zeeman fine topology Z being strictly finer than the Euclidean metric topology(it is actually the finest topology that can be defined on M , finest meaning the onehaving the most open sets), possesses various improved properties over the latter whichmirror precisely the spacetime nature ofM , rather than importing onM properties fromthe Euclidean space R4 unphysical for a spacetime.To describe Z, we say that for x ∈ M , an open ball in Z has the form BZ(x; r) =(BE(x; r) \ N(x)) ∪ {x}, where BE(x; r) is the Euclidean-open ball, and N(x) the nullcone at x (we remove N(x) and put back only the point x). Then BZ(x; r) is Z-open,but not E-open. Hence, a set A ⊂ M is Z-open if A ∩B as a subset of B is E-open, forevery spacelike plane and timelike line B.Zeeman also conjectured in [11] that an extension to the curved spacetimes of generalrelativity should be possible, that is for a general spacetime the homothetic group mustbe isomorphic to the homeomorphism group of the Zeeman topology, a conjecture thatwas shown to be correct by Go¨bel [12] (see also [13]). It was also shown in those referencesthat amongst all possible generalized topologies, the Zeeman topology is the unique onehaving this property (all others have homeomorphism groups isomorphic to the conformalgroup). Hence, for any spacetime M in general relativity we have the freedom to chooseeither the standard Euclidean metric topology, giving M the usual manifold topology, orthe Zeeman topology. It is of course the former that is used in all standard discussions4of relativity.However, for our bounding spacetime M , the conformal infinity of the ambient spaceV , it appears we do not have this freedom. Since we have shown the existence of ahomothetic symmetry on M , it follows from the above discussion that it would be quitewrong to assume the usual Euclidean metric topology for M , and we have no choice butto think of M equipped with the Zeeman topology Z. (This is quite apart from thewell-known arguments of Refs. [11, 12, 13] that it would be wrong to equip a spacetimewith the Euclidean rather than the Zeeman (or a similar) topology in the first place!)4 Impossibility of singularitiesIs there then any obstruction to the validity of the singularity theorems in a spacetimeMusing the Zeeman topology instead of the Euclidean metric one? One would expect thatthe answer would be ‘no’ because the singularity theorems are results that are known tobe only valid for spacetimes (manifolds with a Lorentzian signature) and are certainlynot true in Riemannian geometry (positive signature).However, it is a very surprising result that in fact the answer to the above questionis that it is impossible to formulate not only the singularity theorems but also the mostbasic causality results forM in the Zeeman topology. One notion that plays a key role inmany such theorems is the convergence of a sequence of causal curves. Looking carefullyat the proofs of various such results, we note that for a sequence of causal curves toconverge to a limit curve one uses in an essential way the Euclidean balls with theirEuclidean metric and their compactness to extract the necessary limits (cf. e.g., [6],Prop. 8.1.5 and Theorem 9.5.2, [4], pp. 266-71, etc).It follows that since the Zeeman topology is strictly finer, such sequences will beZeno sequences, that is their convergence in the Euclidean topology will not guaranteethe existence of a limit curve in the Zeeman topology. The impossibility of a limit curvein this sense is very important, for it forbids the formation of a basic contradictionpresent in the proofs of all singularity theorems.In any result of this type, a contradiction appears when assuming the existence of acurve of length greater than some maximum starting from a spacelike Cauchy surface Σ5(on which the mean curvature is negative) downwards to the past. In any typical proofof the singularity theorems (such as e.g., Theorem 9.5.1 in [6]), one uses the process ofextraction of a limit curve γ (which locally maximizes the length between Σ and an eventp) to arrive at a contradiction in the sense that no curve can have length greater than thatof the geodesic γ. If γ cannot be extracted as a suitable limit of a converging family ofcausal curves on M , the nonexistence of curves with length greater than the ‘maximum’length of the locally maximizing geodesic (without conjugate points) γ cannot be effected.Therefore one may not speak of incomplete geodesics on the boundary spacetime M ofthe present context.Hence, it follows that it may not be possible to formulate the same sufficient condi-tions for geodesic incompleteness for the Zeeman topology for M as it is usually doneusing the manifold topology.5 DiscussionIn this paper we have used the topological nature of the boundary spacetime of theambient cosmological 5-metric to examine the question of whether or not one can formsufficient conditions for singularity formation using convergent families of causal curves.We reviewed the ambient cosmological construction which implies an asymptoticrestriction, the existence of a homothetic symmetry on the boundary, the conformalinfinity of the ambien metric. This in turn implies that the topology of the boundarymust necessarily be the Zeeman fine topology instead of the standard Euclidean one.Armed with the topological information of the boundary spacetime, we proceeded toanalyze the question of singularities through the focusing of families of causal curves.We found that it is not possible to extract a limit curve from any family of causal curvesof M . This implies that, given the homothetic symmetry on M , one cannot arrive ata contradiction by constructing curves of larger than some maximum length as in thestandard singularity theorems.The arguments advanced in this paper provide another proof of the impossibility ofspacetime singularities on the boundary spacetime surrounding the ambient 5-space onwhich the Einstein equations with fluid sources hold.6References[1] R. Penrose, Phys. Rev. Lett. 14 (1965) 57.[2] S. W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A314 (1970) 529.[3] R. Penrose, Techniques of Differential Topology in Relativity (SIAM, 1972).[4] S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-Time (Cam-bridge University Press, 1973).[5] B. O’ Neill, Semi-Riemannian Geometry (Academic Press, 1983).[6] R. M. Wald, General Relativity (Chicago University Press, 1984).[7] R. Penrose, The Road to Reality, A Complete Guide to the Laws of the Universe(Broadley Head, 2004).[8] I. Antoniadis and S. Cotsakis, Eur. Phys. J. C. 75:35 (2015) 1-12; arXiv:1409.2220.[9] C. Fefferman and C. Robin Graham, The Ambient Metric (Princeton UniversityPress, 2012), arXiv:0710.0919.[10] R. Penrose and W. Rindler, Spinors and Space-time, Vol. 2: Spinor and twistormethods in space-time geometry (Cambridge University Press, 1986).[11] C. Zeeman, Topology 6 (1967) 161-170.[12] R. Go¨bel, Comm. Math. Phys. 46 (1976) 289-307.[13] S. W. Hawking, A. R. King and P. J. McCarthy, J. Math. Phys. 17 (1976) 174-181.7

Topology of the ambient boundary and the convergence of causal curves

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