The Alexandrov Topology in Sublorentzian Geometry Stephan Wojtowytsch June 6, 2012

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The Alexandrov Topology in Sublorentzian Geometry

Stephan Wojtowytsch June 6, 2012

1 Introduction

The main subject of the present work is SemiRiemannian (or PseudoRiemannian) manifolds, that are C^∞-smooth manifoldsM equipped with a non-degenerate symmetric tensorg. 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 aC^∞-smooth manifoldM, a choice D (called distribution) of linear subspaces at each tangent spaceTxM smoothly varying with a pointx∈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) ∈ D_γ(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 andd:M× M →Rbe the metric distance function defined by

d(p, q) = inf {∫ 1

0

√g( ˙γ(t),γ(t)) dt˙ γ(0) =p, γ(1) =q, γis piecewise smooth }

Then the following conditions are equivalent.

1. The metric space (M, d)is complete.

2. All geodesics inM can be defined on the whole ofR(or the manifoldM is geodesically complete).

3. For some x∈M the exponential mapexp_x: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 pointsp, q∈M can be joined by a distance minimizing geodesic (geodesic connected- ness ofM).

Theorem 1.2 (Ball-Box Theorem). Let (M, D, gD) be a sub-Riemannian manifold. Then for ev- ery point p ∈ M there exist coordinates (U, x) around p and constants c, C > 0 such that the sub- Riemannian distance functiondsR defined by

d_sR(p, q) = inf{length of curves γ: [0,1]→M, γ(0) =p, γ(1) =q}

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Introduction 3

can be estimated by

c

∑n i=1

|xⁱ|^wi¹ ≤ dsR(p, q) ≤ C

∑n i=1

|xⁱ|^wi¹ , x∈U,

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 distancedsRand 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 theAlexandrov 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.

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2 Basic Concepts

2.1 Lorentzian Geometry

Definition 2.1. LetM be a smooth manifold andgbe(0,2)-tensor onM such thatgp:TpM×TpM → R is a non-degenerate symmetric bilinear form of index one and p 7→ g_p(X_p, Y_p) 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 dropg 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 neighborhoodU of a pointpin a SemiRiemannian manifoldM is called normal, if for every pointq∈U there is exactly one geodesic curveσ_pq, such thatσ_pq(0) =p,σ_pq(1) =q, totally contained inU. We call an open setU uniformly normal or convex if it is a normal neighborhood of all its pointsp∈U.

The mappingexp :E ⊂T M →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∈ 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∈M,X∈TpM. We call the vectorX

• spacelike, ifg(X, X)>0 orX = 0,

• null or lightlike, ifg(X, X) = 0 andX̸= 0,

• timelike, ifg(X, X)<0,

• nonspacelike, if X null or timelike.

We call an absolutely continuous curveγ:I→M spacelike, null or timelike, ifγ˙ is spacelike, null or timelike, respectively, almost everywhere.

We want the0-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 curveγ:I→M is called

• future directed, ifg(T,γ)˙ <0almost everywhere,

• past directed, if g(T,γ)˙ >0 almost everywhere.

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Basic Concepts 5

Usually the symbolsg andT are droped in the notation of space-time and it is written justM for a space-time. We abbreviate timelike to t., nonspacelike to nspc., and future directed to f.d. from now on. Moreover, we will have the following convention in space-times: a curve is said to be timelike, if and only if it is t.f.d. or t.p.d., but not if it changes between those two. Having the time orientation on a space-timeM, it is possible to introduce a partial order between points onM.

Definition 2.7. Let M be a space-time andp, q∈M. We write

• p≤q ifp=qor there exists a piecewise smooth nspc.f.d. curve from ptoq,

• p≪q if there exists a piecewise smooth t.f.d. curve fromptoq.

We define the chronological past and futureI⁻, I⁺and the causal past and futureJ⁻, J⁺ of a point p∈M by

I⁺(p) ={q∈M|p≪q}, I⁻(p) ={q∈M |q≪p}, J⁺(p) ={q∈M|p≤q}, J⁻(p) ={q∈M|q≤p}.

Let U ⊂ M be open. We write I⁺(p, U)for the set I⁺(p) taken in U as a manifold in its own right. Note, that usuallyI⁺(p, U)̸=I⁺(p)∩U sinceU might lack convexity.

In Chapter 5, Proposition 34 [O’N83] it is shown that ifU is a normal neighborhood of p, then I⁺(p, U) = exp_p(I⁺(0)∩Vp) whereI⁺(0)⊂TpM is the Minkowski light cone andVp is the domain on whichexp_p is a diffeomorphism.

The definition of the order immediately givesp≪q⇒p≤q. Nevertheless it is possible to show even stronger property.

Lemma 2.8. If either p≤r, r≪q orp≪r, r≤q, thenp≪q.

Proof. The proof easily follows from the statement.

Lemma 2.9. Let γ: [0,1]→M be a nonspacelike curve, p=γ(0),q=γ(1). If γ is not a null curve that can be parameterized to be a geodesic, then there is a timelike curve σ: [0,1] → M such that σ(0) =pandσ(1) =q.

The proof of Lemma 2.9 is based on calculus of variations, for details see [O’N83]. Lemma 2.8 can also be proved directly, at least for piecewise smooth curves, as done for example in [Pen72].

The Lorentzian manifolds are differed according to different types of causality.

Definition 2.10. We call a space-time M

• chronological, if there is nop∈M such that p≪p,

• causal, if there are no two pointsp̸=q∈M such thatp≤q≤p,

• strongly causal, if for every pointpand every open neighborhood U ofpthere is a neighbourhood V ⊂U of p, such that no nspc. curve that leaves this neighborhood ever returns to it,

• globally hyperbolic, if it is strongly causal and for any two pointsp, q∈M the setJ⁺(p)∩J⁻(q) is compact.

Theorem 2.11. [BEE96] Each requirement in Definition 2.10 is stronger than the preceding one, and no two requirements are equivalent.

The causality ladder has even more levels: causal, chronological, distinguishing, strongly causal, stably causal, causally continuous, causally simple, globally hyperbolic. Here, each statement is strictly weaker than the preceding one. It is however not of particular importance to our topic. For more details see e. g. [HE73, BEE96].

Remark 2.12. It is not clear that the causality ladder stays intact for sub-Lorentzian space-times in the most part. It would be an interesting topic of research for future.

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Lemma 2.13. Let U ⊂M be a convex neighborhood with compact closure. Then as a space-time the manifoldU is strongly causal.

Proof. Actually, the set U is even causally simple, but since it is not interesting for our purposes, we prove simpler result. Take the neighborhoodU so small that there is a function f:U →Rsuch that the gradientgrad(f)is timelike past directed. Choose a pointp∈U and a neighborhoodW of pin U. Fix a convex neighborhoodV ⊂W of pand a pointq ∈V such thatq≪ p. Consider the neighborhood

Uq,ϵ(p) :=I⁺(q)∩f⁻¹(−∞, f(p) +ϵ).

We want to show that for q close enough to pand sufficiently small ϵ we obtainU_q,ϵ(p) ⊂W. To achieve the result we use the compactness ofU and an auxiliary Riemannian metrich. Then

∃δ >0 such that d

dtf◦γ(t)≥δ ∀γ t.f.d. and parametrized by h-arclength.

Thus the functionf increases along t.f.d. curves before they leaveV and it increases by more than δ·dist_h(q, ∂V). Therefore it suffices to chooseqsuch that

f(p)−f(q)<disth(q, ∂V)·δ 2

and setϵ := ^δ₂. This is possible since the distance function of q on the right hand side cannot be arbitrary small whenqapproachesp.

Note that U, in general, is not globally hyperbolic. Indeed, suitable diamonds in the Minkowski space are globally hyperbolic, while cubes with boundaries parallel to the coordinate axes are not.

Finally, let us introduce the time separation function, sometimes refereed to as Lorentzian distance function¹

T^S(p, q) = sup {∫ 1

0

√−g( ˙γ,γ) dt˙ |γ∈Ω_p,q }

,

where the spaceΩ_p,q consists of future directed nonspacelike curves defined on the unit interval and joiningpwithq. In the caseΩp,q =∅we declare the supremum is equal to0. The quantity

L(γ) =

∫ 1 0

√−g( ˙γ,γ) dt˙

is called arc length of the curveγ. It is invariant under parameter transformation, hence the normal- ization to the unit interval is admissible. It follows immediately from the definition thatT^S satisfies the inverse triangle inequality

T^S(p, q)≥T^S(p, r) + T^S(r, q)

for all pointsp, q, r∈M. However, the distance function fails to be symmetric, possibly even to be finite, and it vanishes outside the causal future set. Even worse, if bothT^S(p, q)>0andT^S(q, p)>0 they both will be infinite. Indeed, from the definition of the distance function there are nonspacelike future directed curvesγ, γ^′ takingpto qand vice versa such that both have a positive length. The nonspacelike future directed curve αn := γ ⋆(γ ⋆ γ^′)ⁿ has length n·(L(γ) +L(γ^′)) +L(γ). As n approaches infinity, so does the arclength of αn. Since T^S(p, q) ≥ L(αn) for any n ∈ N, we have T^S(p, q) =∞and the same forT^S(q, p)andT^S(p, p).

1In Minkowski space the time separation of two pointsp, qis0if and only if there is a Lorentz transformation to a system in whichpandqhave the same time coordinate. Therefore it is appropriate to call it time separation function, while it does not have too much in common with distance functions.

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Basic Concepts 7

2.2 The Sub-setting

The setting we are going to explore now is a generalization of SemiRiemannian geometry. In sub- SemiRiemannian manifolds a metric, that is a (non-degenerate) scalar product at each point, is defined on a subspace of the tangent space, but not necessarily the whole tangent space. If the subspace is proper, those manifolds may behave quite differently from SemiRiemannian ones.

Definition 2.14. A smooth distribution D on a manifoldM is a map that assigns a subspace Dx⊂ TxM to every point xsuch that around every point x∈M there exists an open neighborhoodU and smooth vector fieldsX1, ..., Xk such that Dy= span{X1y, ..., Xky} for ally∈U.

If we require of those vector fields to be linearly independent, on a connectedn-dimensional manifold M we getdimDx ≡k, 2 ≤ k≤n. The case k =n is the well-known Lorentzian manifold and for k= 1we cannot introduce a Lorentzian metric on D.

Anadmissible or horizontal curveγ: I →M is an absolutely continuous curve such thatγ(t)˙ ∈ D_γ(t). We are interested in connectivity property of two arbitrary points by an admissible curve. In general it might not be possible, as the Frobenius theorem states.

Theorem 2.15(Frobenius Theorem). [Lan99] LetDbe a distribution such that[X, Y]∈Dwhenever X, Y ∈D. Then around everyp∈M there exists an imbedded submanifoldN ofM such thatp∈N, TpN =Dp. We say thatD is integrable.

Note that one dimensional distributions are always integrable. Let us see on an opposite situation, known as a non-integrable, or completely non-holonomic distributions.

Definition 2.16. A distribution D satisfies the bracket generating hypothesis, if the flag D⁰_x:=Dx, D^k+1_x := [D, D^k]x⊕D^k_x

eventually spans the whole tangent spaceTxM. Note that "eventually" means that the number of steps need not be the same for all points of manifold and that D_x^k need not be a smooth distribution. The sequence

λk := dim(D^k_x)−dim(D_x^k⁻¹)

is called the growth vector of D. A regular point is a point where λis continuous. A vector field X such that Xx∈Dx for allx∈M is called horizontal.

Remark 2.17. The expression[D, D^k]x has to be understood precisely as iterated commutators of all vector fields from D. Note that the distribution

X= ∂

∂y +x⁴ ∂

∂z, Y = ∂

∂x

is five step-generating on all R³, as we need four commutators to get a nonzero derivative of x⁴ and finally span the whole tangent space along theyz-plane. If on the other hand we consider sections of the space already spanned, we would end up with

D_x¹= {

T_xM x̸= 0 span{∂y, ∂z} x= 0,

where the section x∂x lies inD_x¹ everywhere, but it is not a vector field which we can generate from D with just one commutator! Therefore, it seems more precise to consider vector fields as the givens.

The terminology then changes a little, a family (not even necessarily a module) F of vector fields is called Lie-generating, if the Lie algebra created by the vector fields from F is the Lie algebra of all vector fields in a sense that at each pointx∈M it coincides withTxM.

The sufficient condition of the connectivity by admissible curves is given by the following theorem.

Theorem 2.18 (The Chow-Rashevskii Theorem). Let M be a connected manifold with a bracket- generating distribution D. Then any two points x, y∈M can be connected by a horizontal curve.

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Proof. Proof can be found, for instance in [Mon02]. Even though the proof is given in the context of sub-Riemannian manifolds, only the bracket generating hypothesis and the inverse function theorem are important, thus it remains valid in any sub-SemiRiemannian geometry.

We emphasize that the bracket generating is the sufficient condition for the connectivity, but not necessary. To show this, we consider the distribution given by the kernel of the one formdz− α1(x, y)dx−α2(x, y)dyonR³. The distribution will be bracket generating away from the vanishing locus of∂xα2−∂yα1. However, if the vanishing locus is not the whole R³, then any two points can be connected by a horizontal curve. For details see [Mon02].

Note also, that for four fold commutators, there are two different types: [[X, Y],[Z, W]] and [X,[Y,[Z, W]]] that agree due to the Jacobi identity. As we look at the vector spaces they generate, we will make use only one of them.

There is a stronger version of Chow’s theorem in subRiemannian geometry, the ball-box theorem, that inspired some of the following work.

Definition 2.19. We call coordinatesx:U ⊂M →V ⊂Rⁿlinearly adapted toDat a pointp∈U, if the differentialsdx^λⁱ⁺¹, ...,dx^λⁿannihilateDⁱat the pointp. Hereλis the growth vector andx(p) = 0.

Theλ-weightedϵ-box is the set

Box^λ(ϵ) ={y∈Rⁿ| |y_i|< ϵλ_i, i= 1, ..., n}.

Theorem 2.20(Ball-Box Theorem). [Mon02] There are linearly adapted coordinatesxand constants c, C, ϵ0>0 such that

x⁻¹(

Box^λ(cϵ))

⊂ B(p, ϵ) ⊂ x⁻¹(

Box^λ(Cϵ)) , whereB(p, ϵ)is the sub-Riemannian ϵ-ball aroundp.

Particularly, we obtain that the manifold topology coincides with the metric topology since neighborhood bases agree. The consequences are even stronger, since it gives an estimate on the distance function. A first approach to a similar estimate on sub-Lorentzian geometry was found in [Gro05].

Roughly speaking, the theorem states that we not only connect two points by a horizontal curve, but if the points are close, we can connect them by a relatively short curve and don’t have to run around the whole manifold.

Definition 2.21. A sub-SemiRiemannian manifold is a triple(M, D, g)whereM is a manifold with a smooth bracket generating distributionDand a non-degenerate symmetric bilinear formg: D×D→R of arbitrary constant index. The terms sub-Riemannian and sub-Lorentzian are used for indices0and 1.

A sub-space-time is a quadruple(M, D, g, X)where(M, D, g)is a sub-Lorentzian manifold andX is a horizontal timelike vector field, i. e. g(X, X)<0.

Unfortunately, the concept of the Levi-Civita connection does not well developed for the sub- SemiRiemannian case. To sketch the difficulty we assume that a torsion free connection is given.

Then ∇XY − ∇YX = [X, Y], but due to bracket generating condition, in general [X, Y] ∈/ D, so neither will∇XY be inD.

It is always possible, at least locally, to extend the scalar product from a distribution to a scalar product of same index to the entire tangent space. Let a generalized orthonormal frame{X1, ..., Xk} ofD in a neighborhoodU be given. Then we extend it to a frame {X1, ..., Xk, Xk+1, ..., Xn} of the tangent bundle overU. This works for small enough U. Declare the new frame to be orthonormal with positive value for the last coordinates. In the following theorem we see, that the extension can be given even globally.

However, this changes the geometry of our space too much, therefore we should abstain from thus adding new geometric information.

Theorem 2.22. A sub-Lorentzian metric can always be extended to a Lorentzian metric over the whole manifold.

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Basic Concepts 9

Proof. Let (M, D, g, T) be a sub-Lorentzian manifold. Any smooth manifold admits a Riemannian metric. Choose, therefore, a Riemannian metric h on M. DeclareVx := (Dx)^⊥^h Find an open set U such thatD|U = span{X0, ..., Xd}. Using the Gram-Schmidt algorithm, the frameX0, ..., Xd can be made h-orthonormal. We can extend this (at least over some smaller neighborhood U) to an orthonormal frame {X0, ..., Xn} of T M|U. Then V|U = span{Xd+1, ..., Xn}, and we conclude that V is also a smooth distribution, that we call the vertical distribution. Therefore, the h-orthogonal projections

πD:T M →D, πV :T M →V are smooth. Now define a tensorg˜by

˜

g(v, w) =g(πDv, πDw) +h(πVv, πVw).

Obviously ˜gis symmetric and non-degenerate. Indeed, takev∈TxM and writev=vD+vV. Then

˜

g(v, w) =g(v_D, w_D) +h(v_V, w_V).

IfvD̸= 0orvV ̸= 0, then we can easily chose a horizontal or, respectively,h-vertical vectorwto make the scalar product not 0. If v_D =v_V = 0, then also v = 0. Clearly, ˜g has index1 andg˜|D×D =g.

Thus we have extended our metric.

The notion of causality from Lorentzian geometry carries directly to the sub-Lorentzian case, simply by requesting additionally that tangent vectors to curves be horizontal, so that we can define spacelikeness and timelikeness.

A continuous horizontal curveγ:I→M, that need not be smooth, can be called timelike future directed ifs < t⇒γ(s)≪γ(t)for alls, t∈I and similar for past directed and non-spacelike curves.

Lemma 2.23. Letγ: I→M be a nspc.f.d. horizontal curve. Thenγ is locally Lipschitz with respect to an auxiliary Riemannian metric.

Proof. Let M be a sub-Lorentzian manifold and γ:I →M be a non-spacelike future directed horizontal curve. Choose an arbitrary extension of the metric tensor to a Lorentzian metric. Now take t ∈ [0,1], p = γ(t). Then we can find a neighborhood U of p and coordinates x: U → V, x= (x0, x1, . . . , xn)with an open setV ⊂Rⁿ⁺¹such that the gradient∇x0is timelike. The Lorentzian metric is represented as

g=

∑n i,j=0

gijdxⁱ⊗dx^j

By a linear transformation we can achieve that at the pointpthe scalar productg_pis represented by the Minkowski scalar product gp =−(dx⁰)²+∑n

i=1(dxⁱ)². Then all the other directions will be spacelike in a neighbourhood which without loss of generality we also call V. We denote by η the auxiliary metric expressed by η =−(dx⁰)²+∑n

i=1(dxⁱ)² andξ=g−η. Take a pointx∈U and a vectorv∈TxM, then

∑n i,j=0

ξ_ijdxⁱ⊗dx^j(v, v) =

∑

i,j

ξ_ijvⁱv^j ≤∑

i,j

|ξ_ij| |vⁱv^j|

≤1 2

∑

i,j

|ξij| ((

vⁱ)2

+( v^j)2)

≤ (

Ξ∑

i

dxⁱ⊗dxⁱ )

(v, v),

whereΞ :=nmax{|ξij(y)| : i, j= 0, ..., n, y∈V}does not depend onxorv. Sinceξvanishes atpit is possible to chooseV so small thatΞ<1. Now assume thatv is nonspacelike

0≥g(v, v) =η(v, v) +ξ(v, v)

≥η(v, v)−Ξ

∑n i=0

(vⁱ)² =−(1 + Ξ) (v⁰)²+ (1−Ξ)

∑n i=1

(vⁱ)²

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then also (

−Kdx⁰⊗dx⁰+

∑n i=1

dxⁱ⊗dxⁱ )

(v, v)≤0

whereK := ^1+Ξ₁₋_Ξ. Define the auxiliary Lorentzian metric g0 and the auxiliary Riemannian metric h by

g0:=−Kdx⁰⊗dx⁰+

∑n i=1

dxⁱ⊗dxⁱ, h:= dx⁰⊗dx⁰+

∑n i=1

dxⁱ⊗dxⁱ

The metricg0 has wider lightcones than g at every point. Consider a non-spacelike future directed with respect to the metric g curve γ, then γ is non-spacelike also with respect to g0. Since γ is nspc.f.d., the coordinate functionγ⁰must be strictly increasing. By a change of parameter we obtain

γ(t) = (t, γ¹(t), ..., γⁿ(t)), t=γ⁰. Sinceγis nspc. forg0 we have

−K+

∑n i=1

˙

γ_i²(t)≤0.

Thus

h( ˙γ,γ)˙ = 1²+

∑n i=1

˙

γ²_i ≤ K+ 1, which implies the Lipschitz condition

d_h(γ(t₁), γ(t₂)) =

∫ t2

t₁

√h( ˙γ,γ) ds˙ ≤

∫ t2

t₁

√K+ 1 ds≤L|t₂−t₁|

forL:=√

K+ 1 with respect toh.

Corollary 2.24. A horizontal nspc.f.d. curve γ:I → M is differentiable almost everywhere and absolutely continuous with a locally square integrable weak derivative with respect to any Riemannian metric onM. By absolute continuity we mean that given any chart we can reconstruct the curveγby integrating its derivative:

φ◦γ(t) =φ◦γ(s) +

∫ t s

dφ_γ(r)γ(r) dr˙

Since coordinate changes are diffeomorphisms, this condition does not depend on the chart.

Proof. Denote byW^k,p(U)the Sobolev space ofk-times weakly differentiable functions with derivatives of order0≤j ≤kthat arep-integrable on U and writeH^k =W^k,2. TakeW_loc^k,p(U)to be the space, where integrability holds on every compact subset V ⊂ U ⊂ Rⁿ. We will only be interested in first derivatives and their integrability, since the integral of the curve itself stems from a choice of coordinates and has no deeper meaning.

Thei-th coordinate functionγⁱ is locally Lipschitz with respect to some Riemannian metric. Take some chart (U, φ) and a compact intervall J such that γ(J) ⊂ U. Then we can work on φ(U), where along the curve the standard metric ofRⁿ is bi-Lipschitz equivalent to any other metric tensor.

Thereforeγⁱ|Jis Lipschitz with respect to the standard norm ofRⁿ, which means thatγⁱ|J ∈W^1,^∞(J) with respect to any chart for any compact J ⊂ I whose image is contained in one map domain.

Combining these (and the metrics with a partition of unity), we findγⁱ ∈ W_loc^1,^∞(I). This in turn impliesγⁱ∈W_loc^1,1(I).

Since the parameter space is one-dimensional, the space of absolutely continuous functions agrees with the Sobolev spaceW_loc^1,1(I). The inclusionγⁱ∈W_loc^1,^∞(I)also impliesγⁱ ∈H_loc¹ (I). This gives us local integrability with respect to some special Riemannian metric. Since along a curveγthe functions g₁( ˙γ,γ)˙ andg₂( ˙γ,γ)˙ (for any two Riemannian metricsg₁andg₂) will be locally Lipschitz related, we receive the result for all Riemannian metrics.

The necessary background in functional analysis can be found, for example, in [Eva10].

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Basic Concepts 11

Hence from now on, every curve is assumed horizontal and absolutely continuous with square integrable weak derivatives in all Riemannian metrics on M. As we have just shown, for the curves we are interested in this is not a restriction.

Unfortunately, there exist examples of timelike future directed curves that are non-differentiable on a dense subset of their domain. To give the example, takeϵ >0and an ordering ofQ∩[0,1]given by(qn)^∞_n=1. Define

U :=

∪∞ n=1

( qn− ϵ

n², qn+ ϵ n²

)

Then0< λ(U)≤ ^π₃²^ϵ, whereλis the Lebesgue measure. If we takeγ to be theH¹-solution to

˙ γ(t) =

{

∂₀+∂₁ t∈U

∂0 t∈U^c, γ(0) = 0,

in the Minkowski space, we find a non-spacelike curve of positive length, that is timelike on some no open set and it is non-differentiable at a dense number of points. So, we cannot restrict our attention to piecewise smooth curves. We need a sensible way to distinguish curves and express convergence.

This will be realized in theC⁰-topology.

The Lorentzian or sub-Lorentzian manifolds do not carry a natural metric which would allow a natural topology on curves in the manifold. Also, we do not much care about parametrization, as (monotonous) reparametrization does not influence causal character. Therefore we define the C⁰ topology in the following way.

Definition 2.25. LetI= [a, b]be an interval U, V, W be open sets inM such thatV, W ⊂U. Then we define the set

B_U,V,W,a,b={γ∈C(I, M)|γ(a)∈V, γ(b)∈W, γ(I)⊂U} and to eliminate the need to fix parametrization

BU,V,W = ∪

a<b

BU,V,W,a,b

The C⁰ topology on curves is the topology generated by the basis B:={BU,V,W |U, V, W ∈τ, V, W ⊂U}

TheC⁰topology is constructed in such a way that curvesγn: [a, b]→M converge toγ: [a, b]→R if and only if

γn(a)→γ(a), γn(b)→γ(b), ∀U ∈τ :γ⊂U∃N ∈N:γn ⊂U∀ n≥N.

For general space-times or sub-space-times this notion of convergence might not be too powerful, some information on that may be found in [BEE96]. However, it becomes useful for strongly causal space-times.

Lemma 2.26. If nspc.f.d. horizontal curves γn: [a, b] → M converge to γ: [a, b] → M in the C⁰- topology on curves in a strongly causal sub-space-time then γ is horizontal nspc.f.d.

Proof. We cover γ by finitely many causally convex relatively compact neighborhoodsUα such that Uα∩Uβ̸=∅ for only two indicesβ=α±1: the preceding and the following neighborhood. Here we use strong causality, otherwise it would not work. We take U to be one such neighborhoods with a frameX0, ..., XnofT M|U and such thatX0, ..., XdspanDU. Then we can extend the sub-Lorentzian metric to a Lorentzian one overU by taking

g_λ(X_i, X_j) =











0 i̸=j

−1 i=j= 0 1 1≤i=j≤d λ² d < i=j.

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We can assume that almost all curvesγn run entirely in∪

Uα, and they must get from the first to the last open set. Since curvesγn are nspc.f.d., they must enter and leave everyUαexactly once, save the first and last neighborhoods. The boundary∂Uα is always compact, so the sequences of entering and leaving points have convergent subsequencesp¹_αn→p¹_α, p²_αn →p²_α. The limit pointsp¹, p²∈γby C⁰-convergence.

The neighborhoodU is convex with respect to the metricgλ, and we can even assume the convexity up to the boundary by taking a slightly bigger convex neighborhood. Hence there are geodesics contained inUαconnectingp¹_αntop²_αnwhich will be nspc.f.d. by assumption and a geodesic connecting p¹ to p². The question is essentially reduces to vectors vαn ∈ Tp¹_αnM, vα ∈ Tp¹_αM that express initial velocities of geodesics. Since the functionv 7→g_λ(v, v)is continuous on T M|U, we have that g_λ(v_α, v_α) = lim_n_→∞g_λ(v_αn, v_αn)≤ 0, so that the geodesic connecting p¹_α and p²_α will be nspc.f.d.

with respect to the metricg_λ.

Decreasing the size ofU_βas a collection insideU_αapplying the same argument with a fixed metric g_λ we see that there is a dense subset of causally related points on γ∩U_α. Taking two arbitrary pointsq, r ∈ γ∩Uα we find sequences of causally related points qn → q, rn →r. Then, as before, usinggλ(v, v)we find that q≤rrelative to gλ inUα. We conclude that with respect togλ the curve γis locally nspc.f.d., and hence belongs also toH¹. We expand the weak derivative as

˙ γ=

∑n i=0

˙

γⁱXi, gλ( ˙γ,γ) =˙ −( ˙γ⁰)²+

∑d i=1

( ˙γⁱ)²+λ²

∑n i=d+1

( ˙γⁱ)²≤0

Asγis nspc.f.d. for allg_λ, the last term must be0, else it will go to infinity sooner or later asλ→ ∞. That means

˙ γ=

∑d i=0

˙ γⁱX_i,

i. e. γ is horizontal. It is therefore globally nspc.f.d. with respect to the original sub-Lorentzian metric.

Like in Lorentzian geometry, in sub-Lorentzian geometry we have relations≤, ≪, and again we obtain the following properties.

• ≤,≪ are partial orders. (But Lemma 2.8 does not hold anymore, see Example 3.20. It will however hold with a different proof for a certain class of sub-space-times, the precausal sub- space-times.)

• Causality conditions are defined just like in the Lorentzian case. For the conditions we are intersted in, nothing new happens.

• The setsJ^±, I^± are defined just as before.

We will introduce one new causal relation in the sub-Lorentzian case.

Definition 2.27. A sub-space-time M is called strongly chronological if every point has arbitrarily small neighbourhoods to which no timelike curve ever returns after leaving it.

In the Lorentzian case strongly chronological space-times are strongly causal, since if we have a nspc.f.d. curveγleavingU and returning to it, we can make it a little longer attaching at one end a t.f.d. curve segment and then using Lemma2.9 we can make the whole curveσ⊕γ timelike. We do not know of any example of a strongly chronological but not strongly causal sub-space-time, but it seems prudent to assume that there should exist one due to the existence of rigid curves.

Remind that convex sets in sub-Lorentzian geometry were defined to be convex sets with respect to an arbitrary extension of the sub-Lorentzian metric.

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Reachable Sets, Causality and the Alexandrov Topology 13

3 Reachable Sets, Causality and the Alexandrov Topology

3.1 Reachable Sets

As was mentioned above, the manifold topology and metric topologies an Riemannain and sub- Riemannian manifolds are equivalent due to Hopf-Rinow and ball-box theorems. The causality structure of Lorentzian manifolds allows introduce a new topology, called the Alexandrov topology that equivalent to the manifolds topology. We are interested to compare analogous of the Alexandrov topology for sub-Lorentzian manifolds and the initial manifold topology. We start from reviewing the background for Lorentzian manifolds.

Lemma 3.1. Let (M, g, X) be a space-time (not a sub-space-time). ThenI⁺(p)and I⁻(p)are open in the manifold topology for all points p∈M.

Proof. It suffices to deal with the case ofI⁺(p). Since any tangent space is isometric to the Minkowski space and the set

I⁺(0) ={

(x⁰, ..., xⁿ)∈Rⁿ⁺¹| −(x⁰)²+ (x¹)²+...+ (xⁿ)² < 0}

is obviously open, the set I⁺(0, W) =I⁺(0)∩W is also open for any convex neighborhood of the origin ofTpM. Let nowσ: [0,1]→M be a t.f.d. curve such thatσ(0) =p,σ(1) =q. Chooset∈[0,1], a convex neighborhood U ofq such thatr=σ(t)∈U. Obviouslyq∈I⁺(r, U)⊂I⁺(r)⊂I⁺(p).

We choose a small enough convex open set V ⊂ T_rM such that exp_r: V ⊂ T_rM → U be a diffeomorphism. Owing to the Gauss-Lemma we get exp_r(I⁺(0)∩V) ⊂ I⁺(r, U) and that the set exp_r(I⁺(0)∩V)is open in U. SinceU is open in M, we obtain thatexp_r(I⁺(0)∩V)is open inM. Moreover,

q∈exp_r(I⁺(0)∩V)⊂I⁺(r, U)⊂I⁺(r)⊂I⁺(p).

So the pointqhas an open neighborhood inM that is totally contained in I⁺(p). This meansI⁺(p) is open.

Let us present two examples, showing that sets I⁺(0), I⁻(0) may not be open in the case of sub-Lorentzian manifolds.

Example 3.2. The author of [Gro05] gives the example of the sub-space-time M =R³, D= span

{ T= ∂

∂y +x² ∂

∂z, Y = ∂

∂x }

with the metric satisfyingg(T, T) =−1,g(T, Y) = 0,g(Y, Y) = 1, and time-orientationT, where the set I⁺(0, U)is not open for any neighborhood of the origin.

To show this we first check that the distributionDis bracket generating. Indeed, we have

∂

∂x =Y, ∂

∂y =T+x 2[T, Y],

∂

∂z =−1

2[Y,[T, Y]]

Let us show that no matter how small the neighborhoodU of0, the point(0, θ,0)will be contained inI⁺(0, U)for some smallθbut(0, θ,−a)will not be inI⁺(0, U)for any choice ofa >0, that precisely means that I⁺(0, U) is not open for any neighborhood U of the origin. The curve γ(t) = (0, t,0) is timelike future directed since γ(t) =˙ Xγ(t). Assume that there is a horizontal nspc.f.d. curve σ: [0,Θ]→M,σ(t) = (x(t), y(t), z(t))from0to (0, θ,−a)for somea >0. Then

˙

σ(t) =α(t)X(σ(t)) +β(t)Y(σ(t)) = (β(t), α(t), x²(t)α(t)) withα(t)>0a. e. due to future directedness. This means that

−a=z(Θ) =

∫ Θ 0

˙ z(t)dt =

∫ Θ 0

α(t)x²(t)dt

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since the integrand is non-negative this cannot be. Hence(0, θ,−a)∈/I⁺(0)for anya >0.

The curveγ lies on the boundary ofI⁺(0). Using the same argument as before we can also show, that we cannot use any variation with fixed endpoints. The curveγ is the only nspc.f.d. curve from 0to(0, θ,0). Such kind of curves received the namerigid curves. We see

0 =z(θ) =

∫ T 0

α(t)x²(t)dt ⇒ x= 0 a.e. ⇒z= 0 a.e.

which means that any other curve connecting0 with(0, T,0) is only a reparametrization ofγ.

The next example shows that there are sub-space-times for which Lemma3.1 holds. We did not find yet the precise criterium for sub-space-times that ensures the openness of chronological future and past, and we think it is interesting open question.

Example 3.3. Consider the sub-space-time withM =R³,D= span{X, Y}, where X = ∂

∂x+1 2y ∂

∂z, Y = ∂

∂y −1 2 x ∂

∂z,

the metricg(X, X) = −g(Y, Y) = −1, g(X, Y) = 0, and the vector field X as a time orientation.

This sub-space-time is called the Lorentzian Heisenberg group. In this sub-space-time the setsI⁺(p), I⁻(p)are open for allp∈M.

The details of the proof can be found in [Gro05], but we sketch it for completeness. The chronological future set for the origin

I⁺(0) ={

(x, y, z)| −x²+y²+ 4|z|<0, x >0}

was calculated in [Gro05]. The setI⁺(0) is obviously open. We apply the Heisenberg group multi- plication ofp= (x, y, z)byp0= (x0, y0, z0)to translate the setI⁺(0)to the chronological future set I⁺(p0). Thus the map

Φ(x, y, z) = (

x−x0, y−y0, z−z0+1

2(y x0−x y0) )

takes p0 to 0, preserves the distribution D, the inner product on D, and, therefore, it is a sub- Lorentzian isometry. SinceΦis also a diffeomorphism of R³and maps I⁺(p0)toI⁺(0), we conclude thatI⁺(p)is open for allp∈R³.

The above examples leads to the consideration of special type of sub-space-times, that we introduce in the following definition.

Definition 3.4. A sub-space-time, in which I^±(p)are open for all p∈M is called precausal.

Notice that as we have seen in the proof of Lemma 3.1, a sub-space-time is precausal if sets I⁺(p, U)are open for all convex neighborhoodsU ofp. Remind that convex neighborhood is defined with respect to an arbitrarily extended metric.

Although it is a tempting thought and it is true in the Minkowski space, in general J⁺(p) and J⁻(p)will not be closed. For example, if we remove the point (1,1) in two dimensional Minkowski space, thenJ⁺(0,0)andJ⁻(2,2)will not be closed. However, the following topological properties of J⁺(p)andJ⁻(p)in a sub-space-time can be obtained.

Theorem 3.5. Let(M, D, g, T)be a sub-space-time,p∈M andU a convex neighbourhood ofp. Then 1. int(I⁺(p, U))^U =J⁺(p, U). In particular,intI⁺(p, U)̸=∅ andJ^±(p, U)is closed inU.

2. int(I⁺(p, U)) = int(J⁺(p, U)).

3. ∂I˜ ⁺(p, U) = ˜∂J⁺(p, U).

where A^U is the closure of Arelative to U and∂A˜ is the boundary of Arelative to U.

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Reachable Sets, Causality and the Alexandrov Topology 15

Proof. REVISION NEEDS!!!! We start from proving the first statement, that is essentially follows from Krener’s theorem, see [AS04]. Krener’s theorem states that for Lie generating family of vector fields F the attainable setR_F(q0)fromq0satisfies

RF(q0)⊂ int(RF(q0)). Remined that the attainable set of a familyF is the set

RF(q₀) ={q∈M | ∃γ: [0, θ]→M such thatγ(0) =q₀, γ(θ) =q,γ˙ ∈F}.

If we show that t.f.d. vector fields are Lie generating then we conclude thatI⁺(p)⊂int(I⁺(p)). Let F be the family of vector fields coming from all sections of the bracket generating distributionD, that by definition contains the time orientationT. Choose an arbitrary vector fieldZ ∈F. Then

g(Z+λT, Z+λT) =g(Z, Z) + 2λ g(T, Z) +λ²g(T, T), g(Z+λT, T) =g(Z, T) +λg(T, T)

Since the lead coefficients are negative, the vector fieldZ+λT is t.f.d. for positive big enoughλ. Thus Z = (Z+λT)−λT is the difference between two t.f.d. vector fields and we conclude that any vector field from the family F is a linear combination of t.f.d. vector fields. In other words the t.f.d. vector fields are Lie generating.

Now we want to show the inclusionJ⁺(p, U)⊂I⁺(p, U). LetU be small enough to have a local frame X0, ..., Xd for the distribution D over U. If q∈J⁺(p, U)then there is a curve γ: [0, θ]→U, such thatq=γ(θ)andγ solves

γ(0) =p, γ˙ =−X0+

∑d k=1

u^kXk

for some square integrable functionsu: [0, θ]→R^d and∑d

k=1u²_k ≤1. We need to show thatqis the limit point for some sequence from I⁺(p, U). We use an auxiliary norm∥ · ∥onU, since the normed topology agrees with the restriction of the manifold topology on U.

Consider the sequence of pointsγϵ(θ)∈I⁺(p)defined as final points of curves γϵ(0) =p, γ˙ϵ=−X0+ (1−ϵ)

∑d k=1

u^kXk,

where u^k are the same as before. THERE MIGHT BE SOME PROBLEMS WITH DOMAINS OF DEFINITION HERE, MAYBE JUST CITE ODE-THEORY FROM SOMEONE AND DON’T USE GRONWALL DIRECTLY TO AVOID A LENGTHY WORK? First we show that curvesγϵconverges to γasϵ→0in the auxiliary norm. We get

||γ(t)−γϵ(t)||=

∫ t 0

˙

γ(s)−γ˙ϵ(s) ds

≤

∫ t 0

||γ(s)˙ −γ˙ϵ(s)||ds

=

∫ t 0

∑d i=0

uⁱ(X_{i γ(s)}−X_{i γ}_ϵ_(s)) +ϵ

∑d i=0

uⁱX_{i γ}_ϵ_(s) ds

≤

∫ t 0

∑d i=0

X_{i γ(s)}−X_{i γ}_ϵ_(s)+ϵ

∑d i=0

||uⁱX_{i γ}_ϵ_(s)||ds

≤

∫ t 0

∑d i=0

Lⁱ||γ(s)−γϵ(s)||ds +ϵ

∫ t 0

∑d i=0

||uⁱXi||ds

≤

∫ t 0

L||γ(s)−γϵ(s)||ds +ϵAt.

The Alexandrov Topology in Sublorentzian Geometry Stephan Wojtowytsch June 6, 2012