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The Alexandrov Topology in Sublorentzian Geometry

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The Alexandrov Topology in Sublorentzian Geometry The Alexandrov Topology in Sublorentzian Geometry Stephan Wojtowytsch June 6, 2012 Contents 1 Introduction 2 2 Basic Concepts 4 2.1 Lorentzian Geometry . 4 2.2 The Sub-setting . 7 3 Reachable Sets, Causality and the Alexandrov Topology 13 3.1 Reachable Sets . 13 3.2 The Alexandrov Topology . 17 3.3 Links to Causality . 18 3.4 The Time-Separation Topology . 21 3.5 The Alexandrov Topology in Sublorentzian Geometry . 23 3.6 Precausal Subspacetimes . 26 4 Geodesic Completeness and Connectedness 29 5 A Lorentzian Hopf-Rinow Theorem 31 6 Appendix 32 6.1 Notation . 32 6.2 Open Problems . 32 7 References 33 8 Addenda 34 1 2 Introduction 1 Introduction The main subject of the present work is SemiRiemannian (or PseudoRiemannian) manifolds, that are C1-smooth manifolds M equipped with a non-degenerate symmetric tensor g. The tensor defines a scalar product on the tangent space at each point. The quadratic form corresponding to the scalar product can have different number of negative eigenvalues. If the quadratic form is positively definite everywhere, the manifold is usually called the Riemannian one. The special case of one negative eigenvalue received the name the Lorentzian manifold. The Riemannian geometry is the oldest discipline, and it was motivated both by pure geometrical interests, as well as classical mechanics, where the positively definite quadratic form represents the kinetic energy. The Lorentzian geometry is the second most important instance of SemiRiemannian geometry, because it is the mathematical language of general relativity. It has its own deep and important results diffes it from the Riemannian geometry, such as, for instance, singularity theorems proved by Hawking and Penrose. A geometry associated with motions of systems under certain restrictions, for example, a car on a plane, that can not move aside, nevertheless can reach any position starting from any point, is represented by sub-SemiRiemannian manifolds. A sub-Riemannian manifold is a C1-smooth manifold M, a choice D (called distribution) of linear subspaces at each tangent space TxM smoothly varying with a point x 2 M, and a positively definite metric tensor (a Riemannian tensor) gD that measures vectors from chosen subspaces Dx ⊂ TxM. If we substitute in the above definition the positively definite tensor by a non-degenerate we receive general sub- SemiRiemannian manifolds (M; D; gD). A sub-Lorentzian manifold corresponds to the index one tensor. Since the metric tensor is defined only for vectors from Dx, one should restrict the study to admissible or so called horizontal curves, i. e., curves γ tangent to the subspaces everywhere: γ_ (t) 2 Dγ(t). In order to avoid the restriction of the geometry to submanifolds, it is assumed that the distribution D is non-integrable. This condition ensures, that every two points of a connected sub-SemiRiemannian manifold can be joint by a horizontal curve. The thesis mainly focuses on the study of peculiarities of sub-Lorentzian manifolds versus Lorentzian one in the same spirit as sub-Riemannian geometry differs from the Riemannian geometry. The main emphasis will be put on the existence in the sub-SemiRiemannian geometry of analogues for the Hopf- Rinow theorem from the Riemannian geometry and the Ball-Box theorem from the sub-Riemannian geometry. We remind both statements. Theorem 1.1 (Hopf-Rinow-Theorem). Let (M; g) be a connected Riemannian manifold and d: M × M ! R be the metric distance function defined by {Z } 1 p d(p; q) = inf g(_γ(t); γ_ (t)) dt γ(0) = p; γ(1) = q; γis piecewise smooth 0 Then the following conditions are equivalent. 1. The metric space (M; d) is complete. 2. All geodesics in M can be defined on the whole of R (or the manifold M is geodesically complete). 2 ! 3. For some x M the exponential map expx : TxM M is defined on the entire tangent space. 4. Every closed and bounded subset of M is compact. All of the above imply 6. Every two points p; q 2 M can be joined by a distance minimizing geodesic (geodesic connected- ness of M). Theorem 1.2 (Ball-Box Theorem). Let (M; D; gD) be a sub-Riemannian manifold. Then for ev- ery point p 2 M there exist coordinates (U; x) around p and constants c; C > 0 such that the sub- Riemannian distance function dsR defined by dsR(p; q) = infflength of curves γ : [0; 1] ! M; γ(0) = p; γ(1) = qg 2 Introduction 3 can be estimated by Xn Xn 1 1 i w i w c jx j i ≤ dsR(p; q) ≤ C jx j i ; x 2 U; i=1 i=1 where the constants wi are determined by the non-integrability condition of the distribution D. In particular, the metric topology coming from the sub-Riemannian distance dsR and the manifold topology agree. In the Lorentzian and sub-Lorentzian cases we can not obtain a metric distance function from the given indefinite scalar product. The closest analogue is the time separation function, for which Lorentzian geodesics maximize it instead of minimizing. Therefore, the time separation function satisfies an inverse triangle inequality. The other specific feature is a causality structure. In a sense, causality theory is the natural substitution of the metric geometry of Riemannian manifolds in the Lorentzian case. From causal relations one can obtain a new topology, called the Alexandrov topology. It is known that in the Lorentzian manifold the Alexandrov topology can be obtained also from the time separation function. We showed that for sub-Lorentzian manifolds it not always can be done. Our main interest is the studying of these two topologies, their similarities and differences from the Lorentzian Alexandrov topologies. We proved that, in the contrast to the Riemannian or sub-Riemannian geometries, the Alexandrov and "time separation" topologies on a sub-Lorentzian manifolds not always coincide with the manifold topology. The thesis is organized in the following way. Chapter 2 collects all basic concepts. Thus a reader who is not very familiar with the sub-SemiRiemannian geometry can get the necessary vocabulary and definitions in order to follow the exposition. Then we investigate properties of reachable sets that leads us to the Alexandrov topology. We discuss its connection to causality structures and have a short look at its features in the well behaved case. We present examples and counterexamples of the new properties occurring in sub-Lorentzian geometry and are not inherited from Lorentzian or sub-Riemannian geometries. To be able to give an analogue to the Hopf-Rinow theorem, we briefly overview the existence of maximizers without entering into details, specifically we do not tauch the question whether those maximizers can be derived as solutions of Hamiltonian system. 3 4 Basic Concepts 2 Basic Concepts 2.1 Lorentzian Geometry Definition 2.1. Let M be a smooth manifold and g be (0; 2)-tensor on M such that gp : TpM ×TpM ! R is a non-degenerate symmetric bilinear form of index one and p 7! gp(Xp;Yp) is smooth for all smooth vector fields X; Y in the set of smooth sections Γ(M). We call the pair (M; g) a Lorentzian manifold. If there can be no confusion, we will drop g in the notation of the Lorentzian manifold. In the aspects related to the presence of the symmetric Levi-Civita connection the SemiRiemannian and, particularly, the Lorentzian geometry is a lot like its Riemannian analogue. We remind some necessary definitions. Definition 2.2. A neighborhood U of a point p in a SemiRiemannian manifold M is called normal, if for every point q 2 U there is exactly one geodesic curve σpq, such that σpq(0) = p, σpq(1) = q, totally contained in U. We call an open set U uniformly normal or convex if it is a normal neighborhood of all its points p 2 U. The mapping exp: E ⊂ TM ! M × M and the inverse function theorem implies the existence of a convex neighborhood of an arbitrary point. Theorem 2.3. [O’N83] In a SemiRiemannian manifold M every point p 2 M has an arbitrarily small convex neighborhood. SemiRiemannian manifolds possess a causal structure, that is a division of all tangent vectors into three classes according to the sign of the quadratic form. Namely, Definition 2.4. Let p 2 M, X 2 TpM. We call the vector X • spacelike, if g(X; X) > 0 or X = 0, • null or lightlike, if g(X; X) = 0 and X =6 0, • timelike, if g(X; X) < 0, • nonspacelike, if X null or timelike. We call an absolutely continuous curve γ : I ! M spacelike, null or timelike, if γ_ is spacelike, null or timelike, respectively, almost everywhere. We want the 0-vector to be spacelike to avoid nonspacelike curves standing still. Timelike curves correspond to test particles moving slower than the speed of light, null curves to test particles moving at the speed of light. A curve that is standing still does not make any physical sense. Among SemiRiemannian manifolds, a large subclass of the Lorentzian ones admits an orientation in time direction. Definition 2.5. Let (M; g) be a Lorentzian manifold and T be a globally defined timelike vector field. Such a vector field is called time orientation of (M; g). The triplet (M; g; T ) received the name space-time or time oriented manifold. Every Lorentzian manifold is either time orientable or admits a twofold time orientable cover (see [BEE96]). Definition 2.6. Let (M; g; T ) be space-time. An absolutely continuous curve γ : I ! M is called • future directed, if g(T; γ_ ) < 0 almost everywhere, • past directed, if g(T; γ_ ) > 0 almost everywhere.
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