Trajectories of Affine Control Systems and Geodesics of a Spacetime with a Causal Killing Vector Field

We study the geodesic connectedness of a globally hyperbolic spacetime (M,g) admitting a complete smooth Cauchy hypersurface S and endowed with a complete causal Killing vector field K. The main assumptions are that the kernel distribution D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal D$\end{document} of the one-form induced by K on S is non-integrable and that the gradient of g(K,K) is orthogonal to D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}. We approximate the metric g by metrics gε smoothly depending on a real parameter ε and admitting K as a timelike Killing vector field. A known existence result for geodesics of such type of metrics provides a sequence of approximating solutions, joining two given points, of the geodesic equations of (M,g) and whose Lorentzian energy turns out to be bounded thanks to an argument involving trajectories of some affine control systems related with D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document}.


Introduction
The existence of a shortest path between any two given points on a Riemannian manifold or more generally on a locally compact length space is a basic result in metric geometry.It is a consequence of Ascoli-Arzelá theorem once metric completeness is assumed (see, e.g., [6,Section 2.5]).In contrast with Riemannian ones, Lorentzian metrics do not define a length metric structure due to their indefiniteness as bilinear symmetric tensors and their length functional is defined only for causal curves and not on the whole set of rectifiable curves between two points.On the other hand, taking the square root of the absolute value of g( γ , γ ) allows one to consider all the absolutely continuous curves between two points but produces a length functional which might have minimum value equal to zero between any couple of points due the possible presence of piecewise smooth (not future-pointing) null curves between them.It is then not surprising that establishing the existence of a geodesic of a Lorentzian metric between any two points is a highly non-trivial problem.A generalization of the notion of length structure in the setting of causality theory and Lorentzian geometry has been recently proposed in [15].In particular, a classical existence result for futurepointing causal geodesics between two points admits an extension [15,Theorem 3.30].The assumption that replaces completeness in this case is global hyperbolicity as in the classical causality theory (see, e.g., [4,Theorem 6.1]).
It is quite surprising that global hyperbolicity which, on a non-compact spacetime (M, g) is equivalent to the compactness of its causal diamonds (see [14]), also plays a fundamental role in the proof, obtained in [7], of the full geodesic connectedness of a spacetime (M, g) when g admits a complete Killing vector field K, which is timelike (i.e., g(K, K) < 0), and there exists a smooth, spacelike, complete Cauchy hypersurface in M.An analogous result has been obtained in [3] when the complete Killing vector field K is everywhere lightlike (i.e., g(K, K) = 0 and K p = 0 for all p ∈ M).
In this work, we consider the case when K is causal, i.e., g(K, K) ≤ 0, K p = 0 for all p ∈ M. As far as we know, this case is open, apart from a recent result where a compact spacetime endowed with a causal Killing vector field satisfying the null generic condition and having globally hyperbolic universal covering is studied (see [2,Corollary 3.6]).
Let us give some further details on the geometric setting that we consider in connection with the ones in [3,7].
Let (M, g) be a globally hyperbolic spacetime endowed with a complete causal Killing vector field K and a (smooth, spacelike) Cauchy hypersurface S1 .Then there exists a diffeomorphism ϕ : S × R → M defined by the restriction of the flow K to S × R and the induced metric on S × R is where g 0 is the Riemannian metric induced by g on S that will be assumed to be complete, ω is the one-form metrically equivalent to the orthogonal projection of K on S, i.e., ω(v) = g(v, K) for all v ∈ T S and : S → R is the non-negative function on S defined as = −g(K, K)| S (see [7,Theorem 2.3], [3,Proposition 2.2]).Henceforth, we will identify (M, g) with the spacetime S × R endowed with the metric (1) that will be denoted with g as well.Moreover, we will assume that S (with the Riemannian metric g 0 ) is complete.
Notice that if K is timelike then (x) > 0 for all x ∈ S and the spacetime is called standard stationary; if K is lightlike then ≡ 0, ω does not vanish at any point and the metric on M becomes equal to A metric like (1) is a Lorentzian one if and only if being |ω x | 0 the g 0 -norm of ω x in T x S (see [10,Proposition 3.3]).
Before stating our main result, we observe that if ω x = 0 for all x ∈ S and D x := ker(ω x ), then with m = dim(S).In the following we will denote by d 0 the distance on S induced by the complete Riemannian metric g 0 and by ω the vector field g 0 -metrically equivalent to ω.
Theorem 1 Let (M, g) be a globally hyperbolic spacetime admitting a complete causal Killing vector field K and a smooth, spacelike, complete Cauchy hypersurface S. With the notations in (1), let us assume that Then (M, g) is geodesically connected.
As already recalled, the case when K is lightlike everywhere has been studied in [3,Theorem 1.2], where it is proved that any couple of points p 0 = (x 0 , t 0 ), p 1 = (x 1 , t 1 ) in S × R can be connected by a geodesic provided that there exists a C 1 curve σ on S between x 0 and x 1 such that ω( σ ) is constant.We notice here that the existence of a curve σ between any two points in S satisfying ω( σ ) = const.(in particular 0) follows by assuming the nonintegrability of the distribution defined pointwise by the kernel of the one-form ω thanks to Chow-Rashevskii theorem (see, e.g., [1,Theorem 3.31]).On the other hand, [3, Example (c), p.22] shows that the integrability of ω is quite a natural obstruction to the existence of a geodesic between any couple of points of a spacetime endowed with a lightlike Killing vector field.
A class of examples satisfying the assumptions in Theorem 1 is the following.Let us consider a product manifold S × R where S = R 3 is endowed with spherical coordinates (r, θ, φ).Let g be the Lorentzian metric on S × R defined as dr 2 + 2a(r, θ)(dθ + dφ)dt + r 2 (dθ 2 + sin 2 θdφ 2 ) − (r)dt 2 , where = (r) is a smooth, non-negative, bounded function which is 0 on the interval [0, R] and a is a bounded function such that a(r, θ) ≥ ν 1 > 0, having nowhere vanishing partial derivative a r .The vector field ∂ t is a causal Killing vector field which is lightlike on [0, R] × S 2 × R and timelike otherwise.The one-form ω is given by a(dθ + dφ) and, being ω ∧ dω = −a r adr ∧ dθ ∧ dφ, it is a contact form on S and then its kernel distribution is non-integrable.Notice that ∇ = (r)∂ r , thus it is contained in the kernel of ω.We notice also that (S × R, g) is globally hyperbolic with complete Cauchy hypersurfaces S × {t}, t ∈ R, see Remark 7.
We emphasize that Theorem 1 extends [3, Theorem 1.2] and its proof is independent of it.On the other hand, the idea of approximating the metric g with metrics g ε , ε > 0, such that the vector field K is Killing and timelike for each metric g ε , is the same as in [3] and in [8].A novelty of the present work is the use of some affine control systems associated with D and drifts depending on ε, on the t-components t 0 , t 1 ∈ R of the fixed points and collinear with the vector field g 0 -metrically equivalent to ω. Thanks to appropriate curves constructed by concatenating solutions of these control systems, we get a bound from above of the critical values of some special (in a sense that will be explained in Section 3) connecting geodesics of the approximating metrics g ε .We mention that similar affine control systems (but with a fixed drift) have been recently used in [9] to study multiplicity of geodesics between two points on some singular Finsler spaces.
The paper is organized as follows: in Section 2 we introduce some preliminary remarks involving the distribution D and the causality of M.Then, in Section 3 we adapt control systems to stationary perturbations (M, g ε ), ε > 0, of (M, g).Exploiting the result in [7], we consider a special family γ ε = (ρ ε , t ε ) of geodesics of g ε joining two points (x 0 , t 0 ), (x 1 , t 1 ) ∈ M and we construct, by means of a control system having the drift smoothly depending on ε, a family σ ε of curves connecting x 0 to x 1 and having bounded g 0 -energy.In Section 4 we show that the family {γ ε } is bounded in the C 1 -topology, so that, up to pass to a subsequence, for any sequence ε n → 0, {γ ε n } converges (in the C ∞ -topology) to a geodesic of (M, g) joining (x 0 , t 0 ) and (x 1 , t 1 ).
Let us finally specify some notations.We do not explicitly write the point where a vector field or a tensor is applied, except for some cases where the point might appear as an index (as, e.g., ω x ).If we look at a vector field X on a manifold S as a vector field along a curve σ , then we write X(σ ).An exception is when it is clear from the context that a vector field must be restricted to a given curve (as, e.g., in the expression g 0 (∇ ρ ρ, ω ), where ρ is a curve).On the other hand we always write the evaluation of a function on S at a point x (as, e.g., (x)) and of a one-form or a (1, 1)-tensor field at a vector v ∈ T S (as, e.g., in (v)).Analogously, for a function defined on T S, as the Finsler type functions F ± in (6), we write F ± (v) without specifying the point x ∈ S where v is applied.

Non-Integrability of ω and Causality
In [5,11] the homotopy properties of the trajectories of an affine control system on a manifold S are studied.A trajectory σ : [0, b] → S is an absolutely continuous curve solving the system for some functions u = (u 1 , . . ., u d ) called controls, where V , X 1 , . . ., X d are vector fields, with V playing the role of a drift (which in some cases -as in the sub-Riemannian oneis the null vector field) and X 1 , . . ., X d satisfy the bracket generating condition (see, e.g., [1,Definition 3.1]).The regularity assumption on the controls determines the topology on the space of the trajectories.Henceforth, we will consider L 2 controls and, called u the i.e., F associates to each trajectory its endpoint.The set is the set of trajectories joining x 0 to x.Now let {X 1 , . . .X d } be a set of globally defined smooth vector fields on S, with d ≥ rank D, which generate D as in (3) (see [1,Corollary 3.27]) and W a smooth vector field on S. Let us consider the affine control system ( 5 ) Remark 2 Being D non-integrable and of rank m − 1 by (iii), for all x 1 ∈ S there exist controls u 1 , . . ., u d ∈ L 2 ([0, 1], R) and a solution of ( 5) parametrized on [0, 1] which is a curve in This is a consequence of a far more general result [5, Theorem 5]; we notice that by (iii) in Theorem 1 and since the rank of D is m − 1, the exponent p c in [5, Theorem 5] is equal to +∞ and then p = 2 is allowed in our setting (see last remark at the end of the proof of Proposition 2 in [5]).
We recall that on a Lorentzian manifold (M, g) a tangent vector w ∈ T M is timelike (resp.lightlike; spacelike; causal) if g(w, w) < 0 (resp.g(w, w) = 0 and w = 0; g(w, w) > 0 or w = 0; w is either timelike or lightlike).It is well known that the set of causal vectors at each tangent space has a structure of double cone called causal cones.In the spacetime it is smooth and strictly increasing when composed with any future-pointing causal curve in (S × R, g).The notion of being future-pointing for a vector or a curve is related to the opposite of the gradient of the function (x, t) ∈ S × R → t ∈ R. In fact, it can be proved that −∇t is timelike and then it gives a time-orientation to (S × R, g) in the sense that it allows us to choose, continuously and globally, one of the two causal cones at T p (S × R), p ∈ S × R. The selected ones (containing −∇t) constitute the set of future-pointing causal vectors in T (S × R); with our convention on the signature of the metric g, they are non-zero vectors w ∈ T (S × R), such that g(w, w) ≤ 0 and dt (w) > 0, so that ∂ t is future-pointing as well.Thus, a causal vector w ∈ T (S × R) is future-pointing if and only if g(w, ∂ t ) ≤ 0.
Remark 3 Let M = S × R be endowed with a metric g as in (1).Taking W equal to one of the two vector fields x ∈ D x for all x ∈ S. By Remark 2, there exists a solution σ ± of ( 5) with W = W ± .Thus, ω( σ ± ) = −ω W ± ) = ∓1.By [10, Proposition 3.12, Corollary 3.16] any trajectory σ ± of (5) with W = W ± can be lifted to a future-pointing (resp.past-pointing) lightlike curve γ ± starting from a p 0 = (x 0 , t 0 ) ∈ S × R and given by where Notice that F − (v) = F + (−v), hence F + , F − are defined on T x S if (x) > 0, while if (x) = 0, F + and F − are respectively defined on those vectors v ∈ T x S such that ω x (v) < 0 and ω x (v) > 0. This implies that any point p 0 and any integral line of ∂ t can be joined by at least one causal curve.
Recall that, given p, q ∈ M, we say that p is in the causal past of q, and we write p < q, if there exists a future-directed causal curve from p to q.Moreover, we denote by p ≤ q either p < q or p = q.For each p ∈ M, the causal past J − (p) and the causal future J + (p) are defined as Moreover, fixed p 0 = (x 0 , t 0 ) ∈ M = S × R and x 1 ∈ S, let us define which are non-empty by Remarks 2 and 3. Let then set Remark 4 Notice that being the line t ∈ R → (x 1 , t) ∈ S × R causal and future-pointing, (x 1 , t) ∈ J + (x 0 , t 0 ) for all t > + (x 0 , x 1 ) and (x 1 , t) ∈ J − (x 0 , t 0 ) for all t < − (x 0 , x 1 ).Moreover, since a globally hyperbolic spacetime is causally simple (see, e.g., [4, Proposition 3.16]), if (M, g) is globally hyperbolic then J ± (x 0 , t 0 ) are closed and x 1 , ± (x 0 , x 1 ) ∈ J ± (x 0 , t 0 ).
The following proposition holds.
Proposition 5 Let M = S × R be endowed with a metric g as in (1).Assume that (M, g) is globally hyperbolic and (iii) in Theorem 1 holds.Then for all p 0 = (x 0 , t 0 ) ∈ M and Proof We notice that by the first part of Remark 4, if x 1 = x 0 then + (x 0 , x 1 ) ≤ t 0 and − (x 0 , x 1 ) ≥ t 0 ; since the function t : S × R → R is strictly increasing (resp.decreasing) along all the future-pointing (resp.past-pointing) causal curves we then get ± (x 0 , x 1 ) = t 0 .By Remark 3, + (x 0 , x 1 ) ∈ [t 0 , +∞) when x 0 = x 1 .Now we notice that + (x 0 , x 1 ) cannot be equal to t 0 , otherwise by the second part of Remark 4 x 1 , + (x 0 , x 1 ) ∈ J + (x 0 , t 0 ) and there would exist a future-pointing causal curve between (x 0 , t 0 ) and (x 1 , t 0 ), in contradiction with the strict monotonicity of the function t along future-pointing causal curves.A similar reasoning holds also for − (x 0 , x 1 ).

Control Systems Adapted to Stationary Approximations
The function in (1) is non-negative, thus (x) + ε > 0 for all ε > 0, x ∈ S and the corresponding metric g ε on S × R has larger future-causal cones than g (g ≺ g ε , for all ε > 0); moreover, g ε ≺ g ε for all 0 < ε < ε .In particular, the vector field ∂ t becomes timelike for g ε , remaining a Killing vector field, thus (S × R, g ε ) is a standard stationary spacetime for each ε > 0 (see Section 1).By computing the Euler-Lagrange equation of the energy functional (defined on the space of piecewise smooth curves parametrized on [0, 1] and connecting two given points p 0 , p 1 ∈ S × R), it follows that a curve γ ε = (ρ ε , t ε ) is a geodesic of the metric g ε if and only if it is smooth and satisfies the following system of differential equations: for some constant C γ ε ,ε , where ∇ is the covariant derivative associated to the Levi-Civita connection of metric g 0 and is the (1, 1)-tensor field g 0 -metrically equivalent to := dω.We point out that the second equation in ( 8) is equivalent to the conservation law h( γ , K) = const.that any geodesic of a pseudo-Riemannian metric h endowed with a Killing vector field K must satisfy.

Remark 6
In particular, for ε = 0 the above equations give the geodesic ones for the metric g.
Our aim is to prove that a subsequence γ ε n , ε n → 0, of these connecting geodesics converges in C ∞ -topology to a geodesic of the metric g between (x 0 , t 0 ) and (x 1 , t 1 ) (see Section 4).In order to do this, here we seek for a family of curves σ ε connecting x 0 to x 1 and having bounded g 0 -energy: these curves will be used to control from above the minimum values of the functionals J ε .To this end, we modify the control system (5) introducing a family of drifts smoothly depending on the parameter ε.For ε ≥ 0, let W ε be the vector field on S defined as We notice that if t 1 − t 0 = 0 then, for all ε > 0, (W ε ) x ∈ D x , while for ε = 0, (W 0 ) x ∈ D x at those x where (x) = 0 and, if t 1 = t 0 , then W ε ≡ 0 for all ε ≥ 0. Let us then consider for ε ≥ 0 the control systems with control functions u = (u 1 , . . ., u d ) ∈ L 2 ([0, 1/2], R d ), so that the trajectories belong to the Sobolev manifold of absolutely continuous curves τ : [0, 1/2] → S between x 0 and τ (1/2) with 1/2 0 g 0 ( τ , τ ) ds < +∞.For any ε ≥ 0 we denote by ε the set of trajectories of ( 14) endowed with the H 1 -topology and by F ε the associated end-point map, hence For the following result we use some ideas contained in the proof of [18,Proposition 3.1].
Let {s l }, l ∈ {0, . . ., h}, be a partition of the interval [0, 1/2] such that τ 0 ([s l−1 , s l ]) ⊂ U l for each l ∈ {1, . . .h}. Identifying then the vector fields W ε , X i with their images by dϕ l on ϕ l (U l ) we have that they all are bounded and Lipschitz on ϕ l (U l ).Let L be a common Lipschitz constant for the above vector fields on all the subsets ϕ l (U l ).Thus, following the proof of [16,Lemma D.3], thanks to the uniform bound ( 16) and the equi-Lipschitz property, we see that, for each ε ∈ (0, 1], the trajectories τ ε of ( 14) are defined on the same interval [0, b), b < 1/2, and contained in U 1 .Being such trajectories uniformly 1 2 -Hölder continuous on [0, b) (as it can be easily seen by using the integral representation of the solutions of ( 14)), they can be extended at b.
We will actually prove that Using that {W ε } ε∈(0,1] are equi-Lipschitz, Then for all r ∈ [0, b] and for C : Hence, by the Gronwall inequality where h(s) , as ε → 0, we can repeat the above reasoning, starting at s = b and, in a finite number of steps, we obtain that the trajectories τ ε uniformly converge to τ 0 on [0, s 1 ] and do not leave U 1 .Then we can repeat the same argument on the interval [s 1 , s 2 ] and so on, covering the whole interval [0, 1/2].
In order to concatenate the curve τ ε in Lemma 8 with trajectories of ( 14) with W ε ≡ 0 and starting from x 1 , we consider the system For any ε > 0, we denote by ν ε : [0, 1  2 ] → S a trajectory of (18) [5,Theorem 5]).Since the end-point map is continuous in the H 1 -topology (see, e.g., [1, Proposition 8.5]) we get the following result: Lemma 9 For all δ > 0 there exists ε > 0 such that for all ε ∈ (0, ε): Proof We can modify the fields X i outside a given compact subset of S containing x 1 and the points x ε , for ε small enough, in order to get bounded vector fields Xi .Then we consider the system By the continuity of the end-point map, there exists a neighborhood in L 2 ([0, 1/2], R d ) of the zero control and ε > 0 such that, for any ε ∈ (0, ε), a control u ε in such a neighborhood and an associated trajectory ν ε of (20) connecting x 1 to x ε do exist.We then have where M := max i∈{1,...,d} max x∈S | Xi (x)| 0 ).This implies that (19) holds and the curves ν ε are in a small compact set containing x 1 , hence they are also trajectories of (18).
Using, for each ε ∈ (0, ε), a curve τ ε as in Lemma 8 and one ν ε as in Lemma 9, we define the curves which connect x 0 to x 1 .
By Lemma 8, the trajectories τ ε are definitively contained in a compact subset K of S thus, recalling that W ε is g 0 -orthogonal to D, we obtain where M X := max i∈{1,...,d} max x∈K |X i (x)| 0 .
For all ε ∈ (0, ε) we pair the family σ ε in (21) with a function t : [0, 1] → R assuming values t 0 and t 1 , t 1 = t 0 , at the endpoints and so that (10) and (11) are satisfied for a zero constant: Let also σ be a trajectory of for some control functions ū1 , . . ., ūd parametrized on [0, 1] and connecting x 0 to x 1 .

Geodesic Connectedness
In this section we prove Theorem 1.Let us start by showing that thanks to Remark 12 the minimizers ρ ε constitute a family of bounded curves in x 0 x 1 (S).
Proof From ( 10) and (iv) in Theorem 1 we get and from ( 9) and Remark 12 we get that 1 0 | ρε | 0 ds is bounded w.r.t.ε ∈ (0, ε).Taking into account that the curves ρ ε connect the fixed points x 0 and x 1 , by Ascoli-Arzelà theorem we have that any sequence ε n → 0 admits a subsequence ε n k such that {ρ ε n k } uniformly converges to a continuous curve ρ connecting x 0 to x 1 .

Remark 14
We notice that the proof of Lemma 13 implies that the family 1 0 is bounded as well.
Let us now rewrite the geodesic (8) for the metrics g ε , ε > 0, and g (recall Remark 6) as a system of second-order differential equations in normal form.
Proposition 15 Let ε ≥ 0; a curve γ : [0, 1] → M, γ (s) = ρ(s), t (s) , is a geodesic of the metric g ε if ε > 0, or of the metric g if ε = 0, if and only if t satisfies the following equation and ρ satisfies the first equation in (8) with ẗ replaced by the expression in (25).
Proof Let us assume that γ is a geodesic of g ε or g.Taking the product of the first equation in (8) by the vector field ω along ρ (recall also Remark 6), we obtain: For the other implication, we notice that by assumption ρ satisfies the first equation in (8).
For the second one, by ( 27) and ( 26 i.e., the second equation in ( 8) is satisfied too.
Remark 16 Notice that (25) is invariant by affine reparametrization (and then the first equation in (8) also remains invariant by affine reparametrization when ẗ is replaced with its value in (25)).
Remark 17 Equation (25) and the first equation in (8) with ẗ replaced by (25) make clear that the geodesic equations of the metrics g ε and g smoothly depend on ε on the interval [0, +∞).
We are now ready to prove Theorem 1.