A new criterion for inextendibility of expanding cosmological models with
symmetry is presented. It is applied to derive a number of new results and to
simplify the proofs of existing ones. In particular it shows that the solutions
of the Einstein-Vlasov system with $T^2$ symmetry, including the vacuum
solutions, are inextendible in the future. The technique introduced adds a
qualitatively new element to the available tool-kit for studying strong cosmic
censorship.Comment: 7 page

Alan D Rendall

Chruściel P T

Dafermos M Rendall A D

Hawking S W

Mihalis Dafermos

Rendall A D

Tegankong D

English

arXiv

A new criterion for inextendibility of expanding cosmological models with symmetry is presented. It is applied to derive a number of new results and to simplify the proofs of existing ones. In particular, it shows that the solutions of the Einstein–Vlasov system with T2 symmetry, including the vacuum solutions, are inextendible in the future. The technique introduced adds a qualitatively new element to the available tool-kit for studying strong cosmic censorship

Dafermos, Mihalis

Rendall, Alan D.

Max Planck Society eDoc Server

Inextendibility of expanding cosmological models with symmetry

Crossref

Dafermos, M.

Rendall, A.

MPG.PuRe

arXiv:gr-qc/0509106 v1   27 Sep 2005Inextendibility of expanding cosmologicalmodels with symmetryMihalis DafermosUniversity of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsWilberforce Road, Cambridge CB3 0WB, UKandAlan D. RendallMax Planck Institute for Gravitational PhysicsAlbert Einstein InstituteAm Mühlenberg 1D-14476 Golm, GermanyAbstractA new criterion for inextendibility of expanding cosmological mod-els with symmetry is presented. It is applied to derive a number ofnew results and to simplify the proofs of existing ones. In particu-lar it shows that the solutions of the Einstein-Vlasov system with T 2symmetry, including the vacuum solutions, are inextendible in the fu-ture. The technique introduced adds a qualitatively new element tothe available tool-kit for studying strong cosmic censorship.1 IntroductionA central open question in mathematical relativity is that of strong cosmiccensorship [10, 7]. It is convenient to formulate this in terms of the maximalCauchy development [5]. Intuitively, this is the largest globally hyperbolicspacetime satisfying the Einstein equations which evolves from prescribedinitial data on a Cauchy surface. Implicit in this is the assumption that1the Einstein equations are coupled to some suitable equations describingthe motion of matter and that the matter model chosen does not sufferfrom unphysical singularities. (See [12], section 4 for a discussion of thelatter point.) The strong cosmic censorship hypothesis says that the maximalCauchy development should be inextendible. In other words it should notbe possible to embed the maximal Cauchy development non-trivially in anyother regular spacetime, at least for generic initial data.There are some ambiguities in this formulation as just stated. When wesay an extension do we mean only that the geometry can be extended, orthat this should also hold for the matter fields? Should the extended metricand matter fields satisfy the Einstein equations and the equations of motioncoming from the given matter model? To say that the extension is regularmeans that it satisfies certain differentiability properties. What degree ofdifferentiability is to be chosen? This is not the place to give a generaldiscussion of these questions - in the following they will only be answeredinsofar as this is directly relevant to the main discussion.When discussing strong cosmic censorship for solutions which are suitablefor being used as cosmological models there are two typical regimes whichcome up. These are the approach to a singularity and an epoch of unendingexpansion. This Letter is chiefly concerned with the second of these regimesalthough some remarks will also be made on the first. The approach to thesingularity will be considered in detail in a related paper [8].How can inextendibility be shown? A standard approach [7] is to showthat either causal geodesics are complete or that, if they are incomplete, somecurvature invariant blows up along any incomplete direction. Usually thisrequires detailed information about the asymptotic behaviour of solutionsof the Einstein-matter equations in the given asymptotic regime and this ishard, even under strong symmetry assumptions on the solutions considered.This paper presents a method which in many cases is much simpler. It appliesto spacetimes with Killing vectors.2 Extensions of Killing vectorsLet Xα be a Killing vector field on a Lorentzian manifold (M, gαβ). It satisfiesthe well-known relation∇α∇βXγ = RαβγδXδ. (1)2Next suppose that the original spacetime can be embedded isometrically inanother spacetime (M̃, g̃αβ). Let H be the boundary of the image of Min M̃ . Suppose further that if p ∈ H there is a local coordinate system{x1, x2, x3, x4} such that p is at the origin, the part of H covered by thesecoordinates coincides with the set defined by the equation x4 = f(x1, x2, x3)for some continuous function f and the part of M covered coincides withthe set defined by x4 < f(x1, x2, x3). We now think of the equation (1) asreferring to components with respect to these local coordinates.It will be assumed that the metric g̃αβ is C2, so that the components of itscurvature are continuous. Define Yαβ = ∂αXβ. Then (1) implies, by writingout the covariant derivatives explicitly in terms of Christoffel symbols, therelation:∂4Yβγ = AστβγYστ + BδβγXδ (2)for some continuous coefficients Aστβγ and Bδ. Moreover ∂4Xα = Y4α. Hencealong the coordinate lines of x4 the collection (Xα, Yαβ) satisifies a linearsystem of ordinary differential equations with coefficients which are contin-uous up to and including H . It follows that the components Xα and theirfirst order partial derivatives are bounded as H is approached. Hence thesecomponents are uniformly continuous and as a consequence they extend con-tinuously to H within the given coordinate system. To sum up, under thegiven assumptions, Killing vectors of gαβ have continuous local extensions toH .3 Symmetric spacetimesConsider first the case of a Lorentzian metric with T 2-symmetry. In this caseit is assumed that that there are two commuting spacelike Killing vectorswithout fixed points. Call them X and Y . It is possible to choose localcoordinates so that they take the form ∂/∂x and ∂/∂y. Then the componentsof the metric in these coordinates are independent of x and y. It is commonto define a quantity R as a constant times the square root of the determinantof the metric induced on an orbit of the local group action generated by theKilling vectors. This can also be defined in a coordinate-invariant way as thesquare root of the determinant of the matrix of inner products of the Killingvectors X and Y . With the latter definition it is clear from the discussion ofthe last section that when the metric has a suitable C2 extension the functionR extends continuously to the boundary H of the extension.3The information about R just obtained will now be compared with theresults of [2]. If gαβ is the metric defined by the maximal globally hyperbolicdevelopment of some Cauchy data then the boundary H of this spacetime inany extension is called the Cauchy horizon. The boundary is known to satisfythe conditions required in the previous section - the function f can even betaken to be Lipschitz continuous [9]. Thus R extends continuously to H . Itfollows from Theorem 2 of [2] that if the spacetime concerned is a solutionof the Einstein-Vlasov system which is initially expanding then R tends toinfinity along any future-directed causal curve. As a consequence, no future-directed causal curve can tend to a point of H and no future extension of theoriginal spacetime is possible. Thus the part of cosmic censorship referringto the future is settled for these spacetimes. To say it another way, it hasbeen proved that the future Cauchy horizon is empty. Note that this result inparticular covers vacuum spacetimes with this symmetry. With the methodof the present paper it follows immediately from [3]. This result will now beformulated as a theorem:Theorem The maximal globally hyperbolic development of data for theEinstein-Vlasov equation with T 2 symmetry which are expanding cannot beextended in a C2 way so as to allow an inextendible future-directed causalcurve in the original spacetime to be continued.The inextendibility result just derived was not previously known for gen-eral T 2-symmetric vacuum spacetimes or for plane-symmetric solutions ofthe Einstein-Vlasov equations, a very special case. It was known for Gowdyspacetimes but the proof was based on a detailed determination of the asymp-totics [13] which was very complicated. Of course the asymptotics is in itselfof great interest but the point of this paper is to show that the question ofinextendibility can be handled with much simpler methods.Consider next the case of spherical or hyperbolic symmetry. In thesesymmetry types there are three Killing vectors but these do not all commuteand they have fixed points. Hence the approach used for T 2-symmetry mustbe modified. To do this we use the fact that there are three Killing vectorsX, Y and Z on the standard sphere with the property that the sum ofthe squares of their lengths is constant. This can be seen by consideringthe standard embedding of the unit sphere in R3 for which these Killingvectors can be taken as x∂y − y∂x, y∂z − z∂y and z∂x − x∂z. There arecorresponding Killing vectors on the spacetime and locally each of themextends continuously to H . The sum of the squares of the lengths of X,Y and Z is proportional to R and so it can be concluded that R extends4continuously to H . It follows that a spherically symmetric spacetime whichis a maximal globally hyperbolic development where R goes to infinity alongany future-directed causal geodesic cannot be extended to the future. In factspherically symmetric spacetimes tend to recollapse but in the presence of apositive cosmological constant this result can be applied. In the case where itis known that r tends to infinity geodesic completeness has also been proved[14]. Nevertheless it represents a simplified proof of inextendibility in thatcase and related applications are likely to come up in the future.Spacetimes with hyperbolic symmetry can be handled in a similar way.This time we choose Killing vectors X, Y and Z on hyperbolic space withthe property that the combination |X|2 + |Y |2 − |Z|2 is constant, where themodulus denotes the length with respect to the given metric. To see thatvectors of this kind exist, consider the standard embedding of hyperbolicspace in three-dimensional Minkowski space with metric dx2 + dy2 − dz2 asthe hyperboloid x2 + y2 − z2 = −1 and take the vectors x∂z + z∂x, y∂z + z∂yand x∂y − y∂x. Now take the corresponding Killing vectors on spacetimeand put them into the quadratic form introduced above. This quantity isproportional to R. Following the same procedure as above gives a criterion forinextendibility in this case too. Combining it with a result of [1] shows thatsolutions of the Einstein-Vlasov system with hyperbolic symmetry cannotbe extended to the future. This was previously known only in a small datasituation [11].Using the results of Tegankong [15] it can be shown that maximal globallyhyperbolic developments are inextendible in the future in the case of solutionsof the Einstein-Vlasov-scalar field system with plane or hyperbolic symmetry.The methods used here can be extended to the case of spacetimes with onlyone spacelike Killing vector (spacetimes with U(1) symmetry). In that casethe analogue of R is the length of the Killing vector. An inextendibilityresult for vacuum spacetimes with small initial data follows from [6]. In factin that case a lot more control on the asymptotics is available and geodesiccompleteness can be proved [4].4 Concluding remarksStrong cosmic censorship is a hard problem and for this reason it is commonto study restricted problems for spacetimes with symmetry. In this papera powerful method has been presented for proving inextendibility of cosmo-5logical spacetimes with symmetry in the future. No symmetry is assumed ofpossible extensions. Straightforward consequences include a number of newresults on classes of spacetimes frequently studied in the literature togetherwith major simplifications of proofs of known theorems.It should be noted that the method presented here only involves thegeometry of an extension. Properties of matter fields play no role. On theother hand the control of the quantity R which is necessary in order for themethod to be applicable in any given example relies on a previous analysisof the coupled Einstein-matter equations.The results which have been presented here shift the emphasis in provingcosmic censorship to the analysis of the initial singularity. For that casethe procedure of this paper says that at any potential singularity on theboundary of the maximal Cauchy development is such that the Killing vectorsextend continuously (for any of the symmetry types considered here), as doesthe function R. It is known in certain cases by other arguments that theextension of R is constant on the boundary. The limiting constant valuemay be positive or zero. The further insights which may be obtained in thisway on the structure of singularities will be discussed elsewhere [8].Acknowledgment ADR thanks the Isaac Newton Institute, Cambridge, forhospitality while this work was being done.References[1] Andréasson H, Rein G and Rendall A D 2003 On the Einstein-Vlasovsystem with hyperbolic symmetry. Math. Proc. Camb. Phil. Soc. 134529-549[2] Andréasson H, Rendall A D and Weaver M 2004 Existence of CMC andconstant areal time foliations in T 2 symmetric spacetimes with Vlasovmatter. Commun. PDE 29 237-262[3] Berger B, Chruściel P T, Isenberg J and Moncrief V 1997 Global foli-ations of vacuum spacetimes with T 2 isometry. Ann. Phys. (NY) 260117-148[4] Choquet-Bruhat Y and Cotsakis S 2002 Global hyperbolicity and com-pleteness. J. Geom. Phys. 43 345-3506[5] Choquet-Bruhat Y and Geroch R 1969 Global aspects of the Cauchyproblem in general relativity. Commun. Math. Phys. 14 329-335[6] Choquet-Bruhat Y and Moncrief V 2001 Future global in time Ein-steinian spacetimes with U(1) isometry group. Ann. H. Poincaré 2 1007-1064[7] Chruściel P T 1991 On uniqueness in the large of solutions of the Ein-stein equations (strong cosmic censorship). it Proc. Centre Math. Appl.27 Australian National University Press, Canberra.[8] Dafermos M and Rendall A D Strong cosmic censorship for surface sym-metric cosmological spacetimes with collisionless matter. In preparation.[9] Hawking S W and Ellis G F R 1973 The large-scale structure of spacetime(Cambridge University Press, Cambridge)[10] Moncrief V and Eardley D M 1981 The global existence problem andcosmic censorship in general relativity. Gen. Rel. Grav. 13 887-892[11] Rein G 2004 On future geodesic completeness for the Einstein-Vlasovsystem with hyperbolic symmetry. Math. Proc. Camb. Phil. Soc. 137237-244[12] Rendall A D 2005 The nature of spacetime singularities. Preprintgr-qc/0503112 (to appear in A Ashtekar (ed.) Space-time structure: Ein-stein and beyond)[13] Ringström H 2004 On a wave map arising in general relativity. Commun.Pure Appl. Math. 57 657-703[14] Tchapnda S B and Noutchegueme N 2005 The surface-symmetricEinstein-Vlasov system with cosmological constant. Math. Proc. Camb.Phil. Soc. 138 541-553[15] Tegankong D 2005 Global existence and future asymptotic behaviour forsolutions of the Einstein-Vlasov-scalar field system with surface symme-try. Class. Quantum Grav. 22 2381-23927

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