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Spacelike Hypersurfaces of Constant Mean Curvature in Spacetimes with Symmetries

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Spacelike Hypersurfaces of Constant Mean Curvature in Spacetimes with Symmetries Spacelike Hypersurfaces of Constant Mean Curvature in Spacetimes with Symmetries Luis J. Al¶³as1, Alfonso Romero2 and Miguel Sanchez¶ 2 1 Departamento de Matem¶aticas, Universidad de Murcia, 30100-Espinardo, Murcia, Spain 2 Departamento de Geometr¶³ay Topolog¶³a,Universidad de Granada, 18071-Granada, Spain. Abstract The role of some symmetries in General Relativity is discussed, espe- cially those involving a timelike conformal vector ¯eld. The importance of spacelike hypersurfaces of constant mean curvature in the study of the Einstein equation is explained. Uniqueness results of compact spacelike hypersurfaces of constant mean curvature in spacetimes which admits a timelike conformal vector ¯eld are given by means of several integral formulas. As an application, several uniqueness consequences for the spacelike constant mean curvature di®erential equation are obtained. 1 Introduction Our purpose is to explain some uniqueness results of spacelike constant mean curvature hypersurfaces in certain ambient spacetimes; these hypersurfaces become relevant for the study of the Einstein equation. Our ambient space- times are not so restrictive as, for example, Lorentzian space forms, which have been widely studied from a purely mathematical point of view. Nevertheless, many of them contain some general symmetries such that the hypersurfaces can be seen as solutions to a partial di®erential equation. In the remainder of this section we explain the role of symmetries in General Relativity and de¯ne our ambient spacetimes; in Section 2 we see how spacelike hypersurfaces of constant mean curvature are useful to simplify Einstein equation; in Section 3 the associated di®erential equation and variational problem is introduced; in Section 4 we write some integral formulas which are our basic tool; in Section 5 the main results, essentially contained in [2], [3], [4], are explained. The concept of symmetry is basic to Physics. In General Relativity, sym- metry is usually based on a local one-parameter group of isometries generated 2 Spacelike Hypersurfaces of Constant Mean Curvature by a Killing or, more generally, conformal, vector ¯eld. In fact, the main simpli¯cation for the search of exact solutions to the Einstein equation, is to assume, a priori, the existence of such symmetries [10], [12]. Even more, the use of a±ne and a±ne conformal vector ¯elds has been useful to obtain new exact solutions [11]. We remark that a completely general approach to symmetries in General Relativity has been developed in [31]. In the above mentioned references, the causal character of the Killing or conformal vector ¯eld is not always pre¯xed. However, it is natural to assume that this vector ¯eld is timelike. This is supported by very well-known examples of exact so- lutions. At the same time, under this assumption, the integral curves of such vector ¯eld provide a privileged class of observers (in the sense of [24]) or test particles in spacetime. A Lorentzian manifold which admits a timelike Killing vector ¯eld K is called a stationary spacetime. Locally, a set of coordinates (t; x1; : : : ; xn) can be chosen such that K = @=@t, each @=@xi is spacelike and the metric com- ponents are independent of the coordinate t. Such vector ¯elds have shown to be useful in a purely mathematical area, in fact, they are used to classify Lorentzian manifolds of constant sectional curvature. So, in [15], a Lorentzian space form is called standard if its time-orientable double cover is station- ary. In [25] the geometry of stationary spacetimes is analysed from several viewpoints, some of them of physical interest. When K is irrotational (its or- thogonal distribution is involutive) the stationary spacetime is called static.A standard static spacetime is a warped product, in the sense of [21] (see also [7]), M¹ =M £I, with base a Riemannian manifold (M; g), ¯ber (I; ¡dt2) and warp- ing function h 2 C1(M), h > 0, so that the Lorentzian metric on M¹ is writen ¤ 2 ¤ 2 as ¼M g ¡ h(¼M ) ¼I (dt ). If we add in this last metric cross terms which are independent of t, the spacetime is standard stationary. Locally, every static or stationary spacetime looks like the corresponding standard one. The simplest stationary spacetimes are the Lorentzian products of a de¯nite negative real line (I; ¡dt2) and a Riemannian manifold (M; g). Under several geometric assumptions a stationary spacetime is such a Lorentzian product. In fact, it was proven in [6] that a geodesically complete simply connected stationary spacetime with non-negative timelike sectional curvatures is the Lorentzian product of (R; ¡dt2) and a geodesically complete Riemannian manifold. Nev- ertheless, recall that, in this result, the assumed inequality for the curvature implies an inequality for the Ricci tensor which is opposite to the physically relevant timelike convergent condition (see Section 5). More generally, we will deal with Lorentzian manifolds which admit a timelike conformal vector ¯eld. As in [4], we call them conformally stationary spacetimes (CS spacetimes). If (M;¹ g¹) is an (n + 1)-dimensional CS spacetime, Luis J. Al¶³as, Alfonso Romero and Miguel Sanchez¶ 3 then K will denote a timelike conformal vector ¯eld on M¹ , and ½ the smooth function on M¹ given by LK g¹ = 2½g¹. The concept of CS spacetime is motivated in part by the following simple fact, which can be easily shown from the last equality: if a Lorentzian metricg ¹ admits a timelike conformal vector ¯eld K, then K is a timelike Killing vector ¯eld for a new Lorentzian metric pointwise conformally related tog ¹, concretely (¡1=g¹(K; K))¹g. If K is a conformal vector ¯eld and its corresponding ½ is constant on all the spacetime, then K is said to be homothetical. From a physical viewpoint, for any timelike vector ¯eld K on a spacetime, the normalized vector ¯eld Z := p 1 K is a reference frame; ¡g¹(K;K) i.e. a vector ¯eld each of whose integral curves is an observer [24, De¯nition 2.3.1]. If K is a timelike conformal vector ¯eld for (M;¹ g¹), a direct computation gives div(Z) = p n¡1 ½, where div is the divergence with respect to the ¡g¹(K;K) Lorentz metricg ¹. Thus, if ½ > 0 on an open set U, then div(Z) > 0 on U, which indicates that the observers in Z are on average spreading apart in U. Similarly, ½ < 0 on U indicates that the observers come together [24, p. 121]. An interesting subclass of CS spacetimes is the family of Generalized Robertson-Walker spacetimes (GRW spacetimes). For a GRW spacetime we mean a product manifold M¹ = I £ M with the Lorentzian metricg ¹ = ¤ 2 2 ¤ ¡¼I (dt ) + f(¼I ) ¼M (g) where ¼I and ¼M denote the projections onto I and M, respectively and g is a Riemannian metric on M. Then (M;¹ g¹) is a warped product, with base (I; ¡dt2), ¯ber (M; g) and warping function f. The no- tion of GRW spacetime is taken from [2],[3] (see also [1], [13], [23], [26], [27] for a systematic study of the geometry of such Lorentzian manifolds). Let us remark that, in this case, the timelike conformal vector ¯eld K may be 0 chosen as f(¼I )@=@t, and so ½ = f (¼I ). Note that, in our de¯nition of GRW spacetime the ¯ber (M; g) is not assumed to be of constant sectional curva- ture, in general. When this holds and n = 3 the GRW spacetime is a (clas- sical) Robertson-Walker spacetime. Thus, GRW spacetimes widely extend to Robertson-Walker spacetimes and include, for instance, Einstein-de Sitter spacetime, Friedmann cosmological models, the static Einstein spacetime and the de Sitter spacetime. Note that conformal changes of the metric of a GRW spacetime, with a conformal factor which only depends on t, produce new GRW spacetimes. Moreover, small deformations of the metric on the ¯ber of Robertson-Walker spacetimes also ¯t into the class of GRW spacetimes. Thus, a GRW spacetime is not necessarily spatially homogeneous, as in the classical cosmological models. Recall that spatial homogeneity seems appropriate just as a rough approach to consider the universe in the large. However, in order to consider it in a more accurate scale, this assumption could not be realistic and GRW spacetimes could be suitable spacetimes to model universes with inhomogeneous spacelike geometry [22]. On the other hand, the family of CS 4 Spacelike Hypersurfaces of Constant Mean Curvature spacetimes is general enough to include GRW spacetime as well as stationary spacetimes. Moreover, recall that a globally conformal Lorentzian manifold to a CS spacetime is itself a CS spacetime. Therefore, CS spacetimes include Lorentzian manifolds which are globally conformal to GRW spacetimes as well as to stationary spacetimes. We will show how the timelike conformal vector ¯eld can be used as a tool to study compact spacelike hypersurfaces of constant mean curvature in CS spacetimes (note that if a GRW spacetime admits a compact spacelike hypersurface then its ¯ber is also compact [2]). The topological requirement compact is assumed, on one hand, because we will use a simple but powerful tool: the classical divergence theorem; on the other, to include in our study spatially closed spacetimes. 2 Spacelike hypersurfaces as initial hypersurfaces in Relativity Next, we will explain how spacelike hypersurfaces of constant mean curvature are interesting to study the Einstein equation. Let (M;¹ g¹) be a 4-dimensional spacetime. Let us denote by Ric and R(¹g) its Ricci tensor and its scalar curvature, respectively. Consider a stress-energy tensor ¯eld T on M¹ , that is, a 2{covariant symmetric tensor which is assumed to satisfy some reasonable conditions from a physical viewpoint (say, T (v; v) ¸ 0 for any timelike vector v, see for example [24, Section 3.3]).
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