We define the class of weakly regular spacetimes with T2 symmetry, and
investigate their global geometry structure. We formulate the initial value
problem for the Einstein vacuum equations with weak regularity, and establish
the existence of a global foliation by the level sets of the area R of the
orbits of symmetry, so that each leaf can be regarded as an initial
hypersurface. Except for the flat Kasner spacetimes which are known explicitly,
R takes all positive values. Our weak regularity assumptions only require that
the gradient of R is continuous while the metric coefficients belong to the
Sobolev space H1 (or have even less regularity).Comment: 5 page

LeFloch, Philippe G.

Smulevici, Jacques

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1231–1233Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical PhysicsGlobal geometry of T 2-symmetric spacetimes with weak regularityGéométrie globale des espaces–temps T 2-symétriques de faible régularitéPhilippe G. LeFloch a, Jacques Smulevici ba Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4, place Jussieu, 75252 Paris cedex 05, Franceb Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germanya r t i c l e i n f o a b s t r a c tArticle history:Received 18 June 2010Accepted 9 September 2010Presented by Yvonne Choquet-BruhatWe define the class of weakly regular spacetimes with T 2-symmetry, and investigatetheir global geometrical structure. We formulate the initial value problem for the Einsteinvacuum equations with weak regularity, and establish the existence of a global foliation bythe level sets of the area R of the orbits of symmetry, so that each leaf can be regarded asan initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly,R takes all positive values. Our weak regularity assumptions only require that the gradientof R is continuous while the metric coefficients belong to the Sobolev space H1 (or haveeven less regularity).© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous définissons la classe des espaces–temps à symétrie T 2 de faible régularité, et nousétudions leur géométrie globale. Nous formulons le problème de données initiales pour leséquations d’Einstein sous une faible régularité. Nous établissons l’existence d’un feuilletageglobal par les surfaces de niveau de la fonction d’aire R des surfaces de symétrie, desorte que chaque feuille induit une hypersurface initiale. A l’exception des espaces–tempsplats de Kasner (connus explicitement), la fonction R prend toutes valeurs positives. Noshypothèses imposent seulement que le gradient de R est continu et que les coefficientsmétriques sont dans l’espace de Sobolev H1 (ou sont moins réguliers).© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionWe are interested in spacetimes satisfying the Einstein vacuum equations and we provide a description of the globalgeometry of the development of an initial data set, having only weak regularity but enjoying certain symmetry properties.We consider a manifold that is diffeomorphic to the torus T 3 and initial data that are invariant under an action of the Liegroup T 2. This requirement characterizes the so-called T 2-symmetric spacetimes on T 3. A large literature is available on suchspacetimes when sufficiently high regularity is imposed; cf. Moncrief [13], Chruściel [4], Berger, Chruściel, Isenberg, andMoncrief [1], and Isenberg and Weaver [6] and, for further references, [14–16].The present Note is motivated by recent work by LeFloch and Stewart [11,12] and LeFloch and Rendall [9], who treatedGowdy-symmetric matter spacetimes described by the Einstein–Euler equations. It was recognized that, due to the formationof shock waves in the fluid and by virtue of the Einstein equations, only weak regularity on the geometry should be allowed.E-mail addresses: pgLeFloch@gmail.com (P.G. LeFloch), jacques.smulevici@aei.mpg.de (J. Smulevici).1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.09.0091232 P.G. LeFloch, J. Smulevici / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1231–1233Here, we carry out this strategy further, and encompass (vacuum) spacetimes under less restrictive regularity and sym-metry conditions. Importantly, we present a fully geometric framework within a class of weakly regular spacetimes andprecisely identify the weak regularity conditions allowing us to, both, establish the existence of such spacetimes and tacklethe analysis of their global geometric structure.Even in the smooth class, there remain challenging open problems for these spacetimes, such as the question of geodesiccompleteness in the expanding direction or the structure of singularities in the contracting direction, and we believe thatthe new framework and estimates developed in the present work will be useful to tackle some of these issues.The initial value problem in general relativity (in the vacuum case) is formulated as follows. An initial data set for thevacuum Einstein equations is a triple (Σ,h, K ) such that (Σ,h) is a Riemannian 3-manifold and K is a symmetric 2-tensorfield on Σ , satisfying the so-called Einstein’s constraint equations. A solution to the initial value problem with initial data(Σ,h, K ) is a (3 + 1)-dimensional Lorentzian manifold (M, g) satisfying the vacuum Einstein equations (that is, the Ricciflat condition Ric = 0), together with an embedding φ : Σ → M such that φ(Σ) is a Cauchy hypersurface of (M, g) andthe pull-back of its first and second fundamental forms coincides with h and K , respectively.Recall that the existence of a unique (up to diffeomorphism) maximal globally hyperbolic development (M, g) was es-tablished in pioneering work by Choquet-Bruhat and Geroch [5,3] (reviewed in [2] together with extensive recent material).The optimal existence result as far as the regularity of the initial data is concerned is currently given in Klainerman andRodnianski [7] in the Sobolev space H2+ε (with ε > 0).2. Statement of the main resultBy restricting attention to the class of T 2-symmetric spacetimes, our regularity assumptions will go far below the regu-larity covered in previous works. We assume here that the area R of the orbits of T 2-symmetry is a C1-function. Introducinga frame of commuting vector fields (T ,Θ, X, Y ) where X, Y are Killing fields, the remaining components of the metric be-long to the Sobolev space H1 on every spacelike slice of a foliation, except for the cross terms between the Killing fieldsX, Y and the vector field Θ which have lower regularity. (See Section 3, below.) Importantly, not all Christoffel symbols andcurvature components make sense as distributions, but only those that are relevant to formulate the Einstein equations ina weak sense. To do otherwise, additional regularity would be necessary, as recognized in [8].Under these regularity conditions, our main result establishes the existence of a weakly regular development, and pro-vides detailed information about its global geometry. We refer to the next section and [10] for the precise terminology usedin the following theorem:Theorem 2.1.• If (Σ,h, K ) is a weakly regular T 2-symmetric triple, then the Einstein constraint equations (suitably reformulated) make senseas equality between distributions. Similarly, if (M, g) is a weakly regular T 2-symmetric Lorentzian manifold, then the Einsteinequations (suitably reformulated) make sense as equality between distributions.• For all weakly regular T 2-symmetric initial data set (Σ,h, K ) with topology T 3 and constant area function R, there exists a weaklyregular vacuum spacetime with T 2-symmetry on T 3 , (M, g), which is a development of (Σ,h, K ) and admits a global foliationby the level sets of the area R ∈ (R∗,∞) (for some R∗  0). Moreover, unless (M, g) is a flat Kasner spacetime, one has R∗ = 0.The compactness arguments developed in [10] for the existence proof are very different from the classical techniquesused in the high regularity case [13,1]. In the rest of this Note, we discuss in further detail the class of initial data andpresent our new geometric formulation which is an essential contribution of [10].3. Weakly regular T 2-symmetric manifoldsWhile all topological manifolds under consideration are of class C∞ , the metric structures of interest here have lowregularity. For instance, the Lie derivative L Z h of a measurable and locally integrable 2-tensor h is defined in the distributionsense, for any C1-vector fields X , T , Z , by(L X h)(T , Z) := X(h(T , Z)) − h(L X T , Z) − h(T , L X Z), (1)in which the last two terms are classically defined as locally integrable functions, but the first one is defined in the distri-butional sense, only.Definition 3.1. A weakly regular T 2-symmetric Riemannian manifold is a compact C∞-manifold Σ endowed with a tensorfield h which enjoys the following regularity and symmetry properties:(i) Riemannian structure. The field h is a Riemannian metric with components in L∞(Σ), and the associated volume formis continuous.P.G. LeFloch, J. Smulevici / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1231–1233 1233(ii) Symmetry property. On the Riemannian manifold (Σ,h), there exists an action of the Lie group T 2 generated by two(linearly independent, commuting) Killing fields X, Y of class C∞ (with closed orbits), satisfying in particular L X h = 0and LY h = 0 understood in the distributional sense (1).(iii) Regularity of the metric on the orbits. The functions h(X, X), h(X, Y ), and h(Y , Y ) belong to the Sobolev spaceH1(Σ;h).(iv) Regularity of the area function. The orbit area functionR2 := h(X, X)h(Y , Y ) − h(X, Y )2 (2)(which then lies in W 1,1(Σ;h)) is required to belong to C1(Σ).(v) Regularity of the metric on the orthogonal of the orbits. There exists a smooth vector field Θ defined on Σ such that(Θ, X, Y ) form a basis of commuting vector fields and the vector fieldZ := Θ + aX + bY , Z ∈ {X, Y }⊥, L X Z = LY Z = 0 (3)implies the regularity h(Z , Z) ∈ H1(Σ;h).We refer to such a triple (X, Y , Z) as an adapted frame on Σ . We emphasize that no regularity is required on thederivatives of the “cross-terms” h(X,Θ) and h(Y ,Θ). We also need to consider the second fundamental form of the initialslice.Definition 3.2. A weakly regular T 2-symmetric triple (Σ,h, K ) is a weakly regular T 2-symmetric Riemannian manifold(Σ,h) with adapted frame (X, Y , Z), satisfying the following conditions:(i) Integrability. The field K is a symmetric 2-tensor field on Σ such that its components in the basis (X, Y , Z) lie inL2(Σ).(ii) Symmetry property. Moreover, K is invariant under the action of the Lie group T 2 generated by (X, Y ), that is, L X K =LY K = 0 understood in the sense of distributions (1).(iii) Additional regularity (associated with the time derivative of the area function):Tr(2)(K ) := h(X, X)K (X, X) + 2h(X, Y )K (X, Y ) + h(Y , Y )K (Y , Y ) ∈ C(Σ).Our notion of initial data set follows the above two definitions, as we define the curvature of the manifold and imposethe constraint equations in the distribution sense. Such a statement requires to take into account our symmetry assumptionin order for the curvature terms to make sense as distributions.AcknowledgementsThe first author (P.L.F.) was partially supported by the Agence Nationale de la Recherche (ANR) through the grant 06-2-134423 entitled “Mathematical Methods in General Relativity” (Math-GR). The second author (J.S.) thanks the Albert EinsteinInstitute for financial support. This work was done when the second author (J.S.) was visiting the Laboratoire Jacques-LouisLions with the support of the ANR.References[1] B.K. Berger, P. Chruściel, J. Isenberg, V. Moncrief, Global foliations of vacuum spacetimes with T 2 isometry, Ann. Phys. 260 (1997) 117–148.[2] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford Math. Monographs, Oxford Univ. Press, 2009.[3] Y. Choquet-Bruhat, R. Geroch, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys. 14 (1969) 329–335.[4] P. Chruściel, On spacetimes with U (1) × U (1) symmetric compact Cauchy surfaces, Ann. Phys. 202 (1990) 100–150.[5] Y. Foures-Bruhat, Théorèmes d’existence pour certains systèmes d’équations aux dérivées partielles non-linéaires, Acta Math. 88 (1952) 141–225.[6] J. Isenberg, M. Weaver, On the area of the symmetry orbits in T 2-symmetric spacetimes, Classical Quantum Gravity 20 (2003) 3783–3796.[7] S. Klainerman, I. Rodnianski, Rough solutions of the Einstein vacuum equations, Ann. of Math. 161 (2005) 1143–1193.[8] P.G. LeFloch, C. Mardare, Definition and weak stability of spacetimes with distributional curvature, Port. Math. 64 (2007) 535–573.[9] P.G. LeFloch, A. Rendall, A global foliation of Einstein–Euler spacetimes with Gowdy symmetry on T 3, preprint, arXiv:1004.0427.[10] P.G. LeFloch, J. Smulevici, Global geometry of future expanding T 2-symmetric spacetimes with weak regularity, in preparation.[11] P.G. LeFloch, J.M. Stewart, Shock waves and gravitational waves in matter spacetimes with Gowdy symmetry, Port. Math. 62 (2005) 349–370.[12] P.G. LeFloch, J.M. Stewart, The characteristic initial value problem for plane symmetric spacetimes with weak regularity, preprint, arXiv:1004.2343.[13] V. Moncrief, Global properties of Gowdy spacetimes with T 3 × R topology, Ann. Phys. 132 (1981) 87–107.[14] A.D. Rendall, Existence of constant mean curvature foliations in spacetimes with two-dimensional local symmetry, Comm. Math. Phys. 189 (1997)145–164.[15] H. Ringström, On a wave map equation arising in general relativity, Comm. Pure Appl. Math. 57 (2004) 657–703.[16] J. Smulevici, On the area of the symmetry orbits in spacetimes with toroidal or hyperbolic symmetry, Analysis and PDE, submitted for publication.

Global geometry of $ {T}^{2}$-symmetric spacetimes with weak regularity

Philippe G. LeFloch

Jacques Smulevici

Comptes Rendus Mathématique

http://www.numdam.org/item/10.1016/j.crma.2010.09.009.pdf

Global geometry of T2 symmetric spacetimes with weak regularity

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