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Constraint Equations for 3 + 1 Vacuum Einstein Equations with a Translational Space-Like Killing Field in the Asymptotically Flat Case Cécile Huneau

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Constraint Equations for 3 + 1 Vacuum Einstein Equations with a Translational Space-Like Killing Field in the Asymptotically Flat Case Cécile Huneau Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case Cécile Huneau To cite this version: Cécile Huneau. Constraint equations for 3 + 1 vacuum Einstein equations with a translational space- like Killing field in the asymptotically flat case. Annales Henri Poincaré, Springer Verlag, 2016. hal-01709667 HAL Id: hal-01709667 https://hal.archives-ouvertes.fr/hal-01709667 Submitted on 15 Feb 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case Cécile Huneau November 19, 2014 Ecole Normale Supérieure, 45 rue d’Ulm, 75005 Paris, email : [email protected] Abstract We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum space- time with a space-like translational Killing field. The presence of a space-like transla- tional Killing field allows for a reduction of the 3 + 1 dimensional problem to a 2 + 1 dimensional one. Vacuum Einstein equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history (see [3], [1], [4]). In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on R2. The main difficulty in that case is due to the delicate inversion of the Laplacian on R2. In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations. 1 Introduction Einstein equations can be formulated as a Cauchy problem whose initial data must satisfy compatibility conditions known as the constraint equations. In this paper, we will consider the constraint equations for the vacuum Einstein equations, in the particular case where the space-time possesses a space-like translational Killing field. It allows for a reduction of the 3+1 dimensional problem to a 2+1 dimensional one. This symmetry has been studied by Choquet-Bruhat and Moncrief in [8] (see also [6]) in the case of a space-time of the 1 form Σ × S × R, where Σ is a compact two dimensional manifold of genus G ≥ 2, and R is 1 the time axis, with a space-time metric independent of the S coordinate. They prove the existence of global solutions corresponding to perturbation of particular expanding initial data. 2 In this paper we consider a space-time of the form R × R × R, symmetric with respect to the third coordinate. Minkowski space-time is a particular solution of vacuum Einstein equations which exhibits this symmetry. Since the celebrated work of Christodoulou and Klainerman (see [10]), we know that Minkowski space-time is stable, that is to say asymp- totically flat perturbations of the trivial initial data lead to global solutions converging to Minkowski space-time. It is an interesting problem to ask whether the stability also holds in the setting of perturbations of Minkowski space-time with a space-like transla- tional Killing field. Let’s note that it is not included in the work of Christodoulou and Klainerman. However, it is crucial, before considering this problem, to ensure the existence of compatible initial data, i.e. the existence of solutions to the constraint equations. This is the purpose of the present paper. 1 In the compact case, if one looks for solutions with constant mean curvature, as it is done in [8], the issue of solving the constraint equations is straightforward. Every metric on a compact manifold of genus G ≥ 2 is conformal to a metric of scalar curvature −1. As a consequence, it is possible to decouple the system into elliptic scalar equations of the form ∆u = f(x; u) with @uf > 0, for which existence results are standard (see for example chapter 14 in [18]). The asymptotically flat case is more challenging. First, the definition of an asymptot- ically flat manifold is not so clear in two dimension. In [3], [1], [4] radial solutions of the 2 + 1 dimensional problem with an angle at space-like infinity are constructed. In partic- ular, these solutions do not tend to the Euclidean metric at space-like infinity. Moreover, 2 the behaviour of the Laplace operator on R makes the issue of finding solutions to the constraint equations more intricate. 1.1 Reduction of the Einstein equations Before discussing the constraint equations, we first briefly recall the form of the Einstein equations in the presence of a space-like translational Killing field. We follow here the (4) 2 exposition in [6]. A metric g on R ×R×R admitting @3 as a Killing field can be written (4) 2γ 3 α 2 g = ge + e (dx + Aαdx ) ; 1+2 1+2 where ge is a Lorentzian metric on R , γ is a scalar function on R , A is a 1-form on 1+2 α 1+2 R and x , α = 0; 1; 2, are the coordinates on R . Since @3 is a Killing field, g, γ and A do not depend on x3. We set F = dA, where d is the exterior differential. F is (4) (4) then a 2-form. Let also Rµν denote the Ricci tensor associated to g. Re αβ and De are respectively the Ricci tensor and the covariant derivative associated to ge. With this metric, the vacuum Einstein equations (4) Rµν = 0; µ; ν = 0; 1; 2; 3 α 3 α can be written in the basis (dx ; dx + Aαdx ) (see [6] appendix VII) (4) 1 2γ λ 0 = Rαβ = Re αβ − e Fα Fβλ − De α@βγ − @αγ@βγ; (1) 2 (4) 1 −γ 3γ β 0 = Rα3 = e De β(e Fα ); (2) 2 (4) −2γ 1 2γ αβ αβ αβ 0 = R33 = −e − e FαβF + g @αγ@βγ + g De α@βγ : (3) 4 e e The equation (2) is equivalent to d(∗e3γF ) = 0 3γ 3γ 1+2 where ∗e F is the adjoint one form associated to e F . This is equivalent, on R , to the existence of a potential ! such that ∗e3γF = d!: Since F is a closed 2-form, we have dF = 0. By doing the conformal change of metric −2γ ge = e g, this equation, together with the equations (1) and (3), yield the following system, 2 α g! − 4@ γ@α! = 0; (4) 1 γ + e−4γ@α!@ ! = 0; (5) g 2 α 1 R = 2@ γ@ γ + e−4γ@ !@ !; α; β = 0; 1; 2; (6) αβ α β 2 α β 1 where g is the d’Alembertian in the metric g and Rαβ is the Ricci tensor associated to g. We introduce the following notation u ≡ (γ; !); (7) together with the scalar product 1 @ u:@ u = 2@ γ@ γ + e−4γ@ !@ !: (8) α β α β 2 α β We consider the Cauchy problem for the equations (4), (5) and (6). As it is in the case for the 3 + 1 Einstein equation, the initial data for (4), (5) and (6) cannot be prescribed arbitrarily. They have to satisfy constraint equations. 1.2 Constraint equations We can write the metric g under the form 2 2 i i j j g = −N (dt) + gij(dx + β dt)(dx + β dt); (9) where the scalar function N is called the lapse, the vector field β is called the shift and g 2 is a Riemannian metric on R . 2 We consider the initial space-like surface R = ft = 0g. Let T be the unit normal to 2 R = ft = 0g. We set j e0 = NT = @t − β @j: We will use the notation @0 = Le0 = @t − Lβ; where L is the Lie derivative. With this notation, we have the following expression for the 2 second fundamental form of R 1 K = − @ g : ij 2N 0 ij We will use the notation ij τ = g Kij for the mean curvature. We also introduce the Einstein tensor 1 G = R − Rg ; αβ αβ 2 αβ αβ where R is the scalar curvature R = g Rαβ. The constraint equations are given by i G0j ≡ N(@jτ − D Kij) = @0u:@ju; j = 1; 2; (10) N 2 1 G ≡ (R − jKj2 + τ 2) = @ u:@ u − g gαβ@ u@ u; (11) 00 2 0 0 2 00 α β 1 g is the Lorentzian equivalent of the Laplace-Beltrami operator in Riemannian geometry. In a 1 αβ p coordinate system, we have gu = p @α(g jgj@β u). jgj 3 where D and R are respectively the covariant derivative and the scalar curvature associated to g (see [6] chapter VI for a derivation of (10) and (11)). Equation (10) is called the momentum constraint and (11) is called the Hamiltonian constraint. If we came back to the 3 + 1 problem, there should be four constraint equations. However, since the fourth would be obtained by taking α = 0 in (2), it is trivially satisfied if we set ∗e3γF = d!. 2λ 2 We will look for g of the form g = e δ where δ is the Euclidean metric on R .
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