Wind Finslerian structures:
from Zermelo’s navigation to
the causality of spacetimes
The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics (and Kropina ones) where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo’s navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The links between both interpretations as well as the possibility to improve the results on one of them using the other viewpoint are stressed.
The wind Finslerian structure is described in terms of two (conic, pseudo) Finsler metrics, and , the former with a convex indicatrix and the latter with a concave one. Notions such as balls and geodesics are extended to . Among the applications, we obtain the solution of Zermelo’s navigation with arbitrary time-independent wind, metric-type properties for (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat’s principle of independent interest for arbitrary spacetimes, as well as its applications to spacetimes and Zermelo’s navigation, are also provided.
Key words and phrases:Finsler spaces and generalizations, Killing vector field, Zermelo navigation problem, Kropina metric, spacetime
2010 Mathematics Subject Classification:53B40, 53C50, 53C60, 53C22
- 1 Introduction
- 2 Wind Finslerian structures
- 3 Fermat structures for splittings
- 4 The case of causal : Randers-Kropina metrics
- 5 The case of arbitrary : general wind Riemannian structures
- 6 Applications to wind Riemannian structures and navigation
Among the classic and recent applications of Finsler metrics, Randers ones can be linked to two quite different problems. The first one is Zermelo navigation problem that was considered for the first time in [Ze31]. It consists in determining the trajectories which minimize the flight time of an airship (or of any other object capable of a certain maximum speed and moving in a wind or a current). Zermelo determined the differential equations of the optimal trajectories in dimensions and (the so-called navigation equations). The problem was then considered by Levi-Civita, Von Mises, Caratheodory, Manià [Levi-C31, Mises31, Carath67, Mania37] becoming a classical problem in optimal control theory. Randers metrics turned then out to appear naturally in the problem of navigation under a mild time-independent wind [Sh03, BaRoSh04].
The second one is the description of the conformal geometry of spacetimes endowed with a timelike Killing vector field (the so-called standard stationary spacetimes). This is an important class of spacetimes: for example, the region outside the ergosphere in Kerr’s solution to Einstein’s equations is of this type and, more generally, the region outside the horizon of any black hole should be so, at sufficiently late times (see [ludvig, §14.4]). Also in this case, Randers metrics arise naturally on , encoding the causality of the spacetime [CapJavSan10].
In both cases, the interpretation of a Randers metric as a Riemannian one “with a displaced unit ball” becomes apparent: the displacement is caused by the vector field which represents the wind in the case of Zermelo’s problem, and which is constructed in a conformally invariant way from the lapse and the shift in the case of spacetimes. It is remarkable that Randers metrics provide a natural way to go from the navigation problem to spacetimes, and vice versa.
In both problems, however, there is a neat restriction: the wind must be mild () and, accordingly, the lapse of the spacetime must be positive (); otherwise, the displaced unit ball would not contain the zero vector, making to collapse the classical Finslerian description. Nevertheless, both problems are natural without such restrictions and, in fact, they become even more geometrically interesting then. Under a strong wind or current, the moving object (a Zeppelin or a plane in the air, a ship in the ocean, or even sound rays in the presence of a wind [GW10, GW11]) may face both, regions which cannot be reached and others that can be reached but must be abandoned by, say, the compelling wind. Analogously, the change in the sign of the lapse means that the causal character of the Killing vector field changes from timelike to spacelike and, so, one might find a Killing horizon, which is an especially interesting type of relativistic hypersurface [ChCoHe12, MarsReiris]. The correspondence between navigation and spacetimes becomes now even more appealing: although the description of the movement of the navigating object is non-relativistic, the set of points that can be reached at each instant of time becomes naturally described by the causal future of an event in the spacetime, and the latter may exhibit some of the known subtle possibilities in relativistic fauna: horizons, no-escape regions (black holes) and so on.
Our aim here is to show that both Zermelo navigation in the air or the sea, represented by a Riemannian manifold , with time-independent wind , and the geometry of a spacetime , with a non-vanishing Killing vector field , can still be described by a generalized Finsler structure , that we call wind Riemannian. Roughly, is the hypersurface of the tangent bundle which contains the maximum velocities of the mobile in all the points and all directions, i.e., each is obtained by adding the wind to the -unit sphere at , the latter representing the maximum possible velocities developed by the engine of the mobile at with respect to the air or sea.
By using this structure, we can interpret Zermelo navigation as a problem about geodesics whatever the strength of the wind is and we give sufficient conditions for the existence of a a solution minimizing or maximizing travel time (Theorem LABEL:compactcase). These are based on an assumption, called w-convexity which is satisfied if the wind Riemannian structure is geodesically complete. Clearly, this might hold also when is not compact, a case in that the so-called common compact support hypothesis in Filippov’s theorem, applied to the time-optimal control problem describing Zermelo navigation, does not hold (see [agrachev, Th. 10.1] and [userres, p. 52]). For example, our techniques can also be used to prove existence of a solution in a (possibly unbounded) open subset of a manifold , provided that the wind is mild in its boundary and the boundary is convex (Theorem LABEL:lake and Remark LABEL:unboundomain).
As mentioned above, wind Riemannian structures allow us to describe also the causal structure of a spacetime endowed with a non-vanishing Killing vector field which is everywhere transverse to the spacelike hypersurfaces . We name this type of spacetimes standard with a space-transverse Killing vector field, abbreviated in splitting. They are endowed with a -independent metric
(see Definition 3.2 and Proposition 3.3 for accurate details), so the Killing vector field is . Even though splittings are commonly used in General Relativity (see for example [MarsReiris] and references therein), we do not know any previous systematic study of their causal structure, so, this is carried out here with full depth. Of course, splittings include standard stationary spacetimes (i.e. the case in that is timelike or, equivalently, is positive) and also asymptotically flat spacetimes admitting a Killing vector field which is only asymptotically timelike (which, sometimes in the literature on Mathematical Relativity, are also called stationary spacetimes, see for example [ludvig, Definition 12.2]). The spacetime viewpoint will be crucial to solve technical problems about wind Riemannian structures.
The point at which Zermelo navigation and the causal geometry of an splitting are more closely related is Fermat’s principle. We prove here a Fermat’s principle in a very general setting which is then refined when the ambient spacetime is an splitting. Classical Fermat’s principle, as established by Kovner [Kovner90] and Perlick [Pe90], characterizes lightlike pregeodesics as the critical points of the arrival functional for smooth lightlike curves joining a prescribed point and a timelike curve . However, the case when is not timelike becomes also very interesting for different purposes. First, of course, this completes the mathematical development of the problem. In particular, the proof of the result here, Theorem LABEL:lema:fermatprinc (plus further extensions there), refines all previous approaches. However, this result and its strengthening to spacetimes (Theorem LABEL:fp, Corollary LABEL:fptimelike), admit interpretations for Zermelo’s navigation, as well as for spacetimes (arrival at a Killing horizon) and even for the classical Riemannian viewpoint (Remark LABEL:eriemann). Concretely, about Zermelo’s navigation, the case when the arrival curve is not timelike corresponds to a target point which lies in a zone of critical or strong wind (). Thus, Fermat’s principle can be interpreted as a variational principle for a generalized Zermelo’s navigation problem, in the sense that navigation paths are the critical (rather than only local minimum) points of the time of navigation.
About the technical framework of variational calculus, we would like to emphasize that the travel time minimizing paths between two given points are the curves connecting to which minimize the functional
is a signature changing tensor on which is Riemannian on the region of mild wind, Lorentzian on the region of strong wind, while in the region of critical wind (i.e., at the points where ), it is degenerate. On the region of critical or strong wind, this functional is defined (and positive) only for curves whose velocities belong to a conic sub-bundle of (see Proposition 2.57 and Proposition 2.58). This constraint on the admissible velocities plus the signature changing characteristic of make it difficult the use of direct methods. Actually, we are able to prove the existence of a minimum by using Lorentzian results about the existence of limit curves (see Definition 4.4, Lemma 5.7) in the splitting that can be associated with a data set for Zermelo navigation (Theorem 3.10). What is more, focusing only on the minimizing problem (or the optimal time control problem) is, in our opinion, somehow reductive of the rich geometrical features of Zermelo navigation. For example, Caratheodory abnormal geodesics [Carath67, §282] (see Section LABEL:ss6.3) are interpreted here as both, lightlike pregeodesics of (up to a finite number of instants where the velocity vanishes) or exceptional geodesics of the wind Riemannian structure (Definition 2.44).
In our study, we will proceed even from a more general viewpoint. We will move the indicatrix of any Finsler metric by using an arbitrary vector field and call the so-obtained hypersurface a wind Finslerian structure. We provide a thorough study of such a structure, which is then strengthened for wind Riemannian structures thanks to the correspondence with conformal classes of splittings. Of course, wind Finslerian structures generalize the class of all Finsler manifolds because the zero vector is allowed to belong to or to be outside each hypersurface . Remarkably, the correspondence between splittings and wind Riemannian structures allows us to study the latter, including some“singular” Finslerian geometries (such as the well-known Kropina metrics, where the vector belongs to the indicatrix ) in terms of the corresponding (non-singular) splitting.
Next, we give a brief description of each section, which may serve as a guide for the reader. In Section 2, we start by introducing wind Finslerian structures on a manifold. These will be defined in terms of a hypersurface of , satisfying a transversality condition which provides a strongly convex compact hypersurface at each point , called wind Minkowskian structure. This structure plays the role of indicatrix, although it might not surround the origin . An obvious example appears when the indicatrix bundle of a Finsler manifold is displaced along a vector field and any such can be constructed from some (clearly not univocally determined, even though a natural choice can be done), see Proposition 2.15. The intrinsic analysis of shows:
Any wind Finslerian structure can be described in terms of two conic pseudo-Finsler metrics and , the former (resp. the latter ) defined on all (resp. in the region of strong wind, i.e., whenever the zero vector is not enclosed by ) with:
(i) domain at each (resp. each equal to the interior of the conic region of determined by the half lines from the origin to , and
(ii) indicatrix the part of that is convex (resp. concave ) with respect to the position vector —so that becomes a conic Finsler metric and a Lorentzian Finsler metric (Proposition 2.5, Figure 1).
Moreover, admits general notions of lengths and balls (Definitions 2.20, 2.26), which allows us to define geodesics (Definitions 2.35, 2.44), recovering the usual geodesics for and (Theorem 2.53).
Remarkably, we introduce the notion of c-ball in order to define geodesics directly for . These balls are intermediate between open and closed balls. They make sense even in the Riemannian case (Example 2.28), allowing a well motivated notion of convexity, namely, w-convexity (Proposition 2.34, Definition 2.45).
Especially, we focus on the case when is a wind Riemannian structure (Section 2.6). The link with Zermelo’s problem becomes apparent: describes the maximum velocity that the ship can reach in each direction and the minimum one. In this case, the conic pseudo-Finsler metrics can be described naturally in terms of the data and (Proposition 2.57), and a generalization of the Zermelo/Randers correspondence is carried out: now Randers metrics appear for mild wind (), the pair for strong wind (), and Kropina metrics for the case of critical wind (). In particular, becomes a Randers-Kropina metric in the region of non-strong wind (Definition 2.59, Proposition 2.58).
In Section 3, our aim is to describe the correspondence between the wind Riemannian structures and the (conformal classes) of splittings. The existence of a unique Fermat structure, i.e., a wind Riemannian structure naturally associated with the conformal class of an splitting, is characterized in Theorem 3.10. Moreover, the equivalence between these conformal -classes, and the description of a wind Riemannian structure either with Zermelo-type elements (i.e., in terms of a Randers-Kropina metric or a pair of metrics ) or with its explicit Riemannian metric and wind (i.e., the pair ) is analyzed in detail, see the summary in Fig. 6. In Subsection 3.4 we identify and interpret the (signature-changing) metric in (1), which becomes Riemannian when , Lorentzian of coindex when and degenerate otherwise. In particular, on its causal (timelike or lightlike) vectors in , it holds
(see (34), Corollary 3.19). As mentioned above, the metric will turn out essential for describing certain solutions of the Zermelo navigation problem. We emphasize that, even though has a natural interpretation from the spacetime viewpoint (Proposition 3.18), its importance would be difficult to discover from the purely Finslerian viewpoint (that is, from an expression such as (2)). Summing up:
Any wind Riemannian structure becomes equivalent to an conformal class . The spacetime interpretation allows us to reveal elements (as the metric in (1), (2), (34)) and to find illuminating interpretations which will become essential for the analysis of Finslerian properties as well as for the solution of technical problems there.
We end with a subsection where the fundamental tensors of and are computed explicitly and discussed —in particular, this makes possible to check the Finslerian character of the former and the Lorentzian Finsler one of the latter.
About Sections 4 and 5, recall first that the main theorems of this paper deal with an exhaustive correspondence between the causal properties of an splitting and the metric-type properties of wind Riemannian structures. These theorems will become fundamental from both, the spacetime viewpoint (as important relativistic properties are characterized) and the viewpoint of navigation and wind Riemannian structures (as sharp characterizations on the existence of critical points/ geodesics are derived by applying the spacetime machinery). For the convenience of the reader, they are obtained gradually in Sections 4 and 5.
In Section 4, we consider the case when the Killing field of the spacetime is causal or, consistently, when the Fermat structure has (pointwise) either mild or critical wind. In this case, the Lorentzian Finsler metric is not defined, and the conic Finsler metric becomes a Randers-Kropina one. We introduce the -separation in a way formally analogous to the (non-necessarily symmetric) distance of a Finsler manifold. But, as the curves connecting each pair of points must be admissible now (in the Kropina region, the velocity of the curves must be included in the open half tangent spaces where can be applied), one may have, for example, for some . In any case, the chronological relation of the splitting can be characterized in terms of (Proposition 4.1), and this allows us to prove that is still continuous outside the diagonal (Theorem 4.5). The main result, Theorem 4.9, yields a full characterization of the possible positions of the splitting in the so-called causal ladder of spacetimes in terms of the properties of . This extends the results for stationary spacetimes in [CapJavSan10], and they are applicable to relativistic spacetimes as the pp-waves (Example 4.11). A nice straightforward consequence is a version of Hopf-Rinow Theorem for the -separation of any Randers-Kropina metric (Corollary 4.10).
In Section 5 the general case when there is no restriction on (i.e., a strong wind is permitted) is considered. Now, there is definitively no any element similar to a distance. However, our definitions of balls and geodesics are enough for a full description of the causal ladder of the spacetime. In fact, the chronological and causal futures, , of any -point can be described in terms of the -balls and c-balls in (Proposition 5.1). Moreover, the horismotically related points (those in ) are characterized by the existence of extremizing geodesics (Corollary 5.3). This leads to a complete description of the geodesics of an splitting in terms of the geodesics of its Fermat structure (Theorem 5.5, Corollary 5.6, see also Fig. LABEL:geos). In order to characterize the closedness of (Proposition 5.8), as well as to carry out some other technical steps, we require a result of independent interest about limit curves (Lemma 5.7). This machinery allows us to prove our structural Theorem 5.9 which, roughly, means:
Any splitting is stably causal and it will have further causality properties when some appropriate properties of the balls or geodesics of the corresponding Fermat structure hold. In particular, is causally continuous iff a natural property of symmetry holds for the closed balls of , it is causally simple iff is w-convex and it is globally hyperbolic iff the intersections between the forward and backward closed -balls are compact. Moreover, the fact that the slices are Cauchy hypersurfaces is equivalent to the (forward and backward) geodesic completeness of .
Section 6 is devoted to the applications of the SSTK viewpoint to the geometry of wind Riemannian structures; here, the role of the spacetime viewpoint becomes crucial and allows us to solve technical problems as well as the use of clarifying interpretations as a guide. Subsection 6.1 develops direct consequences of the previous results: (1) a full characterization of the -geodesics as either (a) geodesics for or , or (b) lightlike pregeodesics of in the region of strong wind, up to isolated points of vanishing velocity (Theorem 6.3; the last possibility refines the result for any wind Finslerian structure in Theorem 2.53) and (2) a characterization of completeness and w-convexity in the spirit of Hopf-Rinow theorem (Proposition 6.6). However, in Subsection LABEL:ss6.2 a subtler application on is developed. Indeed, the same spacetime may split as an in two different ways (Lemma LABEL:fsplitting), yielding two different Fermat structures (Proposition LABEL:changedf). These structures share some properties intrinsic to the spacetime and their consequences for the wind Riemannian structures associated with each splitting are analyzed. In Subsection LABEL:ss6.3 we introduce a relation of weak precedence (resp. precedence ) between pairs of points in defined by the existence of a connecting wind curve (resp. an -wind curve), namely, a curve with velocity included in the region (resp. the interior of the region) allowed by . Such a relation can be characterized as the projection of the causal (resp. chronological) relation on the corresponding (Proposition LABEL:esalva). This allows us to prove results on existence of minimizing and maximizing connecting geodesics (Theorems LABEL:compactcase and LABEL:lake, Theorem LABEL:maxZermelo) and of closed geodesics for (Theorem LABEL:prop:closedgeo). In particular, Theorems LABEL:compactcase, LABEL:maxZermelo and Corollaries LABEL:compactcase2, LABEL:rsummaryZ provide the full solution to Zermelo navigation problem:
For any wind Riemannian structure, the solutions of Zermelo problem are pregeodesics of . The metric in (34) defines a natural relation of weak precedence (resp. precedence ) which determines if a point can be connected with a second one by means of a wind (resp. -wind) curve; when the wind is strong, i.e. , becomes Lorentzian on all and the relation of weak precedence (resp. precedence) coincides with the natural causal (resp. chronological) of . Then:
(a) if , , and the c-balls are closed (i.e. is w-convex) then there exists a geodesic of of minimum -length from to (which is also a lightlike pregeodesic of when );
(b) if , , the wind is strong and is globally hyperbolic on all then there exists a geodesic of of maximum -length from to (which is also a lightlike pregeodesic of if ).
The possibility of the existence of maximal solutions as well as of solutions which are limits of minimal and maximal ones was pointed out by Caratheodory in [Carath67] (see the discussion at part (2) below Corollary LABEL:rsummaryZ). We stress that our result interprets geometrically all of them as geodesics. In particular, the limits of minimal and maximal ones correspond (up to isolated points) to the lightlike pregeodesics of . We would like to emphasize that the accuracy of most of our results for wind Riemannian structures relies on their correspondence to splittings (see, e.g. Proposition 6.2). Nevertheless, some of these results might be extendible to general wind Finslerian ones.111 Indeed, in the case of the correspondence of Randers metrics with stationary spacetimes already developed in [CapJavSan10], some of the properties obtained by using the spacetime viewpoint could be extended to any Finslerian manifold (see for example [Matvee13] or compare [CapJavSan10, Th. 4,10] with [TanSab12, Theorem A]). Thus, the results for the wind Riemannian case might serve as a guide for a further development of wind Finslerian structures.
In Section LABEL:further1, we discuss Fermat’s principle, which constitutes a topic of interest in its own right. After an introductory motivation in Subsection LABEL:Fermatprinciple1, in Subsection LABEL:Fermatprinciple2 we prove our Generalized Fermat’s principle valid for causally arbitrary arrival curves (Theorem LABEL:lema:fermatprinc). Moreover, we also develop an extension to the case when the trial curves are timelike with a prescribed proper time (instead of lightlike with necessarily 0 proper time, Corollary LABEL:timelikefermat) as well as a first application to two purely Riemannian variational problems (Corollary LABEL:cor:appRieFermat). In Subsection LABEL:Fermatprinciple3 Generalized Fermat’s principle is refined for Zermelo trajectories in SSTK spacetimes, providing a variational interpretation of the geodesics of any wind Riemannian structure (Theorem LABEL:fp, Corollary LABEL:fptimelike).
In Section LABEL:further2, we go further in the description of causal elements of splittings. Indeed, in subsection LABEL:develops, Cauchy developments and horizons of subsets included in a slice are described accurately in terms of the Fermat structure (Proposition LABEL:cauchydevhor). As a nice consequence, in subsection LABEL:s8.2 the results on differentiability of horizons for spacetimes can be now applied to obtain results on smoothability of the Randers-Kropina separation to a subset (Proposition LABEL:smoothdF), so extending results in [ChFuGH02] for the Riemannian case and in [CapJavSan10, §5.4] for the Randers one. In the last part (subsection LABEL:ss6.5), we also introduce and develop the notion of -horizon for any wind Finslerian structure. In particular, such horizons allow us to describe the regions where the ship in Zermelo’s navigation cannot enter (or from where it cannot escape). Accordingly, from the spacetime viewpoint, it provides a description of black hole regions from the Killing initial data (KID) on a Riemannian manifold for any splitting (see [CaMars, BeCh, Mae, MarsReiris]). Notice that these data appear naturally in the initial value problem for the Einstein equation, and include our and (usually denoted and in Physics literature, the latter regarded eventually as a vector field). Given the initial data, the splitting is called its (infinite) Killing development [MarsReiris, Definition 2]. When the initial data are well posed (namely, they satisfy conditions of compatibility with matter in the sense of [CaMars, Definition 2]), the Cauchy development of will include the unique maximal globally hyperbolic spacetime obtained as a solution of the Einstein equation. Our results on Cauchy developments make it possible to determine these regions, as well as possible black hole horizons, in terms of the Fermat structure.
Finally, in Section LABEL:conclusions some conclusions and prospects are summarized. Due to the big number of notions here introduced, an appendix containing a list of symbols and definitions used throughout the paper is given for the reader’s convenience.
Acknowledgements The authors would like to thank Lorenzo D’Ambrosio for questions and conversations about the existence of a solution to the Zermelo navigation problem in the compact case.
2. Wind Finslerian structures
2.1. Wind Minkowskian structures on vector spaces
Let us begin by recalling the classical notion of Minkowski norm.
Let be a real vector space of finite dimension . We say that a continuous non-negative function is a Minkowski norm if
it is positive and smooth away from the zero vector,
it is positively homogeneous, namely, for every and ,
for any , its fundamental tensor , defined as
for any , is positive definite.
The indicatrix of is defined as the subset . Observe that is a strongly convex smooth hypersurface embedded in , in the sense that its second fundamental form with respect to one (and then all) transversal vector field is definite —in the remainder, we choose the orientation of the transverse so that will be positive definite, as usual. Notice that, in general, any (connected) compact, strongly convex hypersurface embedded in must be a topological sphere (the Gauss map with respect to any auxiliary scalar product would yield a diffeomorphism) and both, and the bounded region determined by , are strictly convex in the usual sense (i.e. touches every hyperplane tangent to it only at the tangency point and lies in one of the two half-spaces determined by the hyperplane, and satisfies that the segment between any two points in is contained in , except at most its endpoints). When , a Minkowski norm is uniquely determined having as indicatrix just by putting for all , where is the unique positive number such that (see for example [JavSan11, Prop. 2.3]).
If the indicatrix of a given Minkowski norm is translated, one obtains another strongly convex smooth hypersurface that determines a new Minkowski norm whenever still belongs to the new bounded region . As explained in the Introduction, this process of generating Minkowski norms is used pointwise in Zermelo’s navigation problem and one obtains (see Fig. 1):
Let be the indicatrix of a Minkowski norm. The translated indicatrix defines a Minkowski norm if and only if .
This is a restriction of “mild wind” in Zermelo’s problem; so, let us consider now the case in that . In this case, the zero vector is not contained in the open bounded region delimited by the translated indicatrix and, as a consequence, does not define a classical Finsler metric. Indeed, not all the rays departing from the zero vector must intersect and, among the intersecting ones, those intersecting transversely will cross twice, and those intersecting non-transversely will intersect only once, see Fig. 1.
The above discussion motivates the following definition.
A wind Minkowskian structure on a real vector space of dimension (resp. ) is a compact, connected, strongly convex, smooth hypersurface embedded in (resp. a set of two points , , for some ). The bounded open domain (resp. the open segment ) enclosed by will be called the unit ball of the wind Minkowskian structure.
As an abuse of language, may also be said the unit sphere or the indicatrix of the wind Minkowskian structure. In order to study wind Minkowskian structures, it is convenient to consider the following generalization of Minkowski norms (see [JavSan11] for a detailed study).
Let be an open conic subset, in the sense that if , then for every .222Notice that, if an open conic subset contains the zero vector then . As we will be especially interested in the case , in the remainder the vector will be always removed from for convenience. For comparison with the results in [JavSan11], notice that will be always convex in the following sections, even though one does not need to assume this a priori. We say that a function is a conic pseudo-Minkowski norm if it satisfies and in Definition 2.1 (see [JavSan11, Definition 2.4]). Moreover if, for any , the fundamental tensor defined in (3) is positive definite then is said a conic Minkowski norm while if it has coindex then is said a Lorentzian norm.
Of course, any conic pseudo-Minkowski norm can be extended continuously to whenever does not lie in the closure in of the indicatrix and this is natural in the case ; in particular, Minkowski norms can be seen as conic pseudo-Minkowski norms.
According to these definitions, there are three different possibilities for a wind Minkowskian structure.
Let be a wind Minkowskian structure in and its unit ball.
If , then is the indicatrix of a Minkowski norm.
If , then is the indicatrix of a conic Minkowski norm with domain equal to an (open) half vector space.
If , then define as the interior of the set which includes all the rays starting at 0 and crossing ; then is a (convex) conic open set and, when , two conic pseudo-Minkowski norms with domain can be characterized as follows:
each one of their indicatrices is a connected part of , and
is a conic Minkowski norm and , a Lorentzian norm.
Moreover, on all , both pseudo-Minkowski norms can be extended continuously to the closure of in and both extensions coincide on the boundary of .
We will say that in each one of the previous cases is, respectively, a Minkowski norm, a Kropina type norm or a strong (or proper) wind Minkowskian structure.
Parts and are an easy consequence of [JavSan11, Theorem 2.14]. For part , if a ray from zero meets transversely, it will cut in two points whereas if it is tangent to there will be a unique cut point. Then we can divide in three disjoint regions , where and are the sets of the points where the rays departing from cut transversely, first in and then in , and is the set of points where the rays from zero are tangent to (see Fig. 1). The rays cutting generate the open subset ; recall that the compactness and strong convexity of imply both, the arc-connectedness of and , and the convexity of , ensuring . Moreover, defines a Lorentzian norm , since the restriction of its fundamental tensor to the tangent hypersurface to is negative definite and -orthogonal to [JavSan11, Prop. 2.2] (recall that this restriction coincides, up to a negative constant, with the second fundamental form of with respect to the opposite to the position vector, [JavSan11, Eq. (2.5)]). Analogously, defines a conic norm (thus completing ) and, by the choice of , one has . Finally, observe that the points of lie necessarily in the boundary of since the rays from zero are tangent to (which is strictly convex, in particular); moreover, lies in the boundary of both and , which ensures the properties of the extension. ∎
Observe that, in general, a converse of Proposition 2.5 (namely, whether a wind Minkowski norm is determined by a conic Minkowski norm and a Lorentzian norm defined both in an open conic subset , such that and can be continuously extended to and the extensions coincide) would require further hypotheses in order to ensure that the closures in of the indicatrices of and glue smoothly at their intersection with the boundary of .
As a limit case of Proposition 2.5 and, thus, , one has naturally a Minkowski norm or a Kropina norm (the latter identifiable to a norm with domain only a half line) when or , resp. When , choose and assume . Then, define (resp. ), as the indicatrix of a conic Minkowski norm, which will also be regarded as Lorentzian norm in the case of ( are clearly independent of the chosen vector ).
2.2. Notions on manifolds and characterizations
Let be a smooth -dimensional manifold333 Manifolds are always assumed to be Hausdorff and paracompact. However, the latter can be deduced from the existence of a Finsler metric (as then the manifold will admit a reversible one, and will be metrizable) as well as from the existence of a wind Finsler structure (as in this case the centroid vector field is univocally defined, and will admit a Finsler metric, see Proposition 2.15 below). , its tangent bundle and the natural projection. Let us recall that a Finsler metric in is a continuous function smooth away from the zero section and such that is a Minkowski norm for every . Analogously, a conic Finsler metric, conic pseudo-Finsler metric or a Lorentzian Finsler metric is a smooth function , where is a conic open subset of (i.e., each is a conic subset) such that is, respectively, a conic Minkowski norm, a conic pseudo-Minkowski norm or a Lorentzian norm.
A smooth (embedded) hypersurface is a wind Finslerian structure on the manifold if, for every : (a) defines a wind Minkowskian structure in , and (b) for each , is transversal to the vertical space in . In this case, the pair is a wind Finslerian manifold. Moreover, we will denote by the unit ball of each ; while the (open) domain of the wind Finslerian structure will be the union of the sets , where is defined as if and by parts and of Proposition 2.5 otherwise.
For a standard Finsler structure , the indicatrix is a wind Finslerian structure. In fact, (a) follows trivially, and (b) holds because, otherwise, being smooth on , would lie in the kernel of , in contradiction with the homogeneity of in the direction . Notice that this property of transversality (b) also holds for the indicatrix of any conic Finsler or Lorentzian Finsler metric defined on (while (a) does not).
R.L. Bryant [Bryant02] defined a generalization of Finsler metrics also as a hypersurface. The proof of Proposition 2.12 below shows that this notion is clearly related to the notion of conic Finsler metric used here (even though, among other differences, in his definition must be radially transverse and it may be non-embedded and non-compact).
The wind Finslerian structure is closed as a subset of , and foliated by spheres. Moreover, the union of all the unit balls , as well as , are open in . If is connected and (resp. ), then is connected (resp. has two connected parts, each one naturally diffeomorphic to ).
For the first sentence, recall that the property (a) of Definition 2.8 implies that is foliated by topological spheres and each admits a neighborhood such that is compact and homeomorphic to . Indeed, for each chart around some , one can take the natural bundle chart and choose a vector inside the inner domain of . We can assume by taking smaller if necessary that is in the inner domain of for all , where the superscript means the associated linear coordinates on . Then the one-to-one map:
is a homeomorphism because of the invariance of domain theorem. Now, for each there exists a unique such that and varies continuously with and . Thus, as is a topological sphere, the required foliation of is obtained. For the last assertion, notice that, otherwise, any two non-empty disjoint open subsets that covered would project onto open subsets of with a non-empty intersection , in contradiction with the connectedness of at each (for , admits a non-vanishing vector field , so that each two points in can be written now as , with on all , thus yield the required diffeomorphisms with ). ∎
Let be a wind Finslerian manifold. The region of critical wind (resp. mild wind) is
and the properly wind Finslerian region or region of strong wind is
The (open) conic domain of the associated Lorentzian Finsler metric is
Let be the 0-section of . The extended domain of is
Any wind Finslerian structure in determines the conic pseudo-Finsler metrics and in and respectively (the latter when ) characterized by the properties:
is a conic Finsler metric with indicatrix included in ,
is a Lorentzian Finsler metric with indicatrix included in
Moreover, on , both and can be extended continuously to the boundary of in (i.e., ), and both extensions coincide in this boundary.
From Proposition 2.5, we have to prove just the smoothability of in , by using both, the smoothness of and its transversality. Let , and consider the ray (recall that ). This ray is transversal to and, because of the property of transversality of , it is transversal to in too. This property holds also for some open connected neighborhood of in , where will be either strongly convex (thus defining ) or strongly concave (defining ) towards , for all . Moreover, the map:
is injective and smooth. Even more, is bijective at each point , because of transversality, and it is also bijective at any because the homothety maps in the hypersurface which is also transversal to the radial direction. Summing up, is a diffeomorphism onto its image , and the inverse
maps each in either or in , depending on the convexity or concaveness of , , proving consistently the smoothness of or . ∎
Let and be, resp., a wind Finslerian structure and a (smooth) vector field on . Then, is a wind Finslerian structure on .
The translation , is a bundle isomorphism of ; so, it preserves the properties of smoothness and transversality of . ∎
In particular, the translation of the indicatrix of any standard Finsler metric along is a wind Finslerian structure . In this case, the associated conic pseudo-Finsler metrics and can be determined as follows.
Let be a Finsler metric and be a smooth vector field on . Then the translation of the indicatrix of by is a wind Finslerian structure whose conic pseudo-Finsler metrics are determined as the solutions of the equation
Clearly equation (4) corresponds to a translation by of the indicatrix of (see also the definition of the Zermelo metric in [Sh03]). The convexity of the indicatrix of implies that this equation will have a unique positive solution for any if , no solution or only a positive one if , no solution or two positive ones if . ∎
Any wind Finslerian structure can be obtained as the displacement of the indicatrix of a Finsler metric along some vector field . Moreover, can be chosen such that each is the centroid of .
Even if this proof can be carried out by choosing a family of vector fields defined in some open subset with this property, whose existence is trivial, and then doing a convex sum in all the manifold with the help of a partition of unity, we will prove in fact that the vector field provided by the centroid is smooth. For this aim, we can actually assume that is the indicatrix of a standard Finsler metric defined on some open subset of (notice that (i) the smoothability of is a local property, (ii) if a vector belongs to the open ball enclosed by , this property will hold for any vector field extending in some neighborhood of , so that Propositions 2.13 and 2.12 can be claimed, and (iii) the translation also translates the centroids). Let be the canonical unit sphere in with volume element . So, the natural -coordinate of the centroid is computed as:
and its smoothness follows from the smooth variation of the integrands with . ∎
Example 2.16 (Role of transversality).
The smoothness of relies on the smoothness of in (5) and, thus, the transversality of imposed in the assumption (b) of Definition 2.8 becomes essential. Figure 2 shows a -dimensional counterexample if the transversality condition is not imposed. Notice also that, as the absence of transversality would lead to non-smooth metrics, then this would lead to non-smooth splittings in the next Section 3. The well-known exotic properties of the chronological and causal futures and pasts of spacetimes with non-smooth metrics (see for example [ChrGra12]) would be related to exotic properties of .
Let be a wind Finslerian structure on . Then,
is the reverse wind Finslerian structure of .
Obviously, is a wind Finslerian structure too and, from the definition, one gets easily the following.
Given a wind Finslerian structure , the conic Finsler metric and the Lorentzian Finsler one associated with the reverse wind Finslerian structure are the (natural) reverse conic pseudo-Finsler metrics of and , that is, the domains of and are, respectively, and and they are defined as for every and for every .
2.3. Wind lengths and balls
In order to deal with curves, the following conventions will be useful.
For any wind Finslerian structure we extend and to as follows. First, consistently with Proposition 2.12, and are regarded as continuously extended to the boundary of in . is extended as equal to on in the regions of mild and critical wind i.e. on the set (that is, is equal to on the vectors where has been defined and has not). Finally, we define and as equal to on the set of critical wind zeroes (i.e., the set , which was included in the definition of , Definition 2.11). Notice that neither this choice of and on the critical wind region nor any other can ensure their continuity; however, and are continuous on . We also use natural notation such as , .
To understand this choice, recall first that the necessity to extend to in the critical and strong wind regions comes from the fact that all the indicatrices should be contained in . In the critical region, lies in and, so, in the domain of . Therefore, it is not strange to include in so that is defined on this vector and, obviously, the choice comes from the fact that lies in the indicatrix and in the boundary of . A further support for these choices will come from the viewpoint of spacetimes, as the vectors in are those which can be obtained as the projection of a lightlike vector in the spacetime.
As usual, a piecewise smooth curve will be defined in a compact interval , and it will be smooth except in a finite number of breaks , , where it is continuous and its one-sided derivatives are well defined444Even though typically, all the curves will be defined on a compact interval , when necessary all the following notions can be used for non-compact . In this case, one assumes that the restriction of to compact subintervals of satisfies the stated property, and it is natural to impose additionally that the images of the breaks do not accumulate.; its reparametrizations will be assumed also piecewise smooth and with positive one-sided derivatives (so that, for example a piecewise smooth geodesic with proportional one-sided derivatives at each break pointing in the same direction can be reparametrized as smooth geodesics), unless otherwise specified.
Let be a wind Finslerian structure with associated pseudo-Finsler metrics and and consider a piecewise smooth curve , .
(i) is -admissible if its left and right derivatives belong to at every . Analogously, is -admissible if , for each . Accordingly, a vector field on is -admissible (resp. -admissible) if for each , (resp. for each ).
(ii) A -admissible curve is a wind curve if
and an -admissible wind curve will be called just -wind curve.
(iii) A -admissible curve is a regular curve if its one-sided derivatives can vanish only at isolated points (which can be regarded as break points, even though the curve may be smooth there), and it is a strictly regular curve if its one-sided derivatives (and, thus, its velocity outside the breaks) cannot vanish at any point.
(iv) The wind lengths of a -admissible curve (not necessarily a wind curve) are defined as
Obviously, from (6) we get:
If is a wind curve then
We will use this and other natural properties (as the fact that the concatenation of two wind curves such that is another wind curve) with no further mention.
Wind curves collect the intuitive idea of Zermelo’s navigation problem, namely: the possible velocities attained by the moving object are those satisfying the inequalities in (6) (observe that in the region , the inequalities in (6) reduce to ). These velocities never include if and must include if , which happens iff , even though, by convenience, we have excluded from if and included it in the extended domain when . The reason to exclude from when is just to emphasize the different role of the zero vector in this region and in (as well as avoiding problems of differentiability with ).555If the reader felt more comfortable, he/she could redefine by adding with no harm. In the part of spacetimes, the so redefined subset could be interpreted as the set which contains the projections of all the (future-pointing) timelike vectors, and as the set which contains the projections of the causal vectors. However, the reader should take into account that the fundamental tensor of a pseudo-Finsler metric is not well-defined in the zero section. In fact, in order to connect points by means of curves included in , one can avoid to use velocities that vanish (and this may be convenient for purposes such as reparametrizing the curve at constant speed; such an assumption is frequent in Riemannian Geometry too). However, as in the case of Riemannian Geometry, the vanishing of the velocity in subsets with accumulation points leads to bothering problems about its reparametrizations. So, we will consider the solutions of Zermelo’s problem as regular wind curves (allowing the velocity to vanish in isolated points), and we will ensure the existence of such solutions (see Corollary LABEL:rsummaryZ). Observe also that the continuity of and has to be checked only when is equal to a zero of the critical region (see Proposition 2.30-(ii)) and, in this case, and are defined as equal to there. A further explanation of this choice is provided in Example 2.23 below, where two paradigmatic examples of curves with Kropina’s zeroes in the derivatives are given.
Let be endowed with the Kropina norm defined in . Then the curve , , satisfies that and . Clearly, the reparametrization of this curve as an -unit curve is not differentiable at . In fact, this kind of curves was excluded in the mild region. However, consider the indicatrix of as a curve, take the part which is -admissible and reparametrize it as an -unit curve. In such a way, you get a curve whose derivative is zero in the two end-points, but with constantly equal to . This second kind of curves is the main reason for including the zero in the Kropina region in the domains of and . Observe that if you want to exclude the first kind of curves, it is enough to require the continuity of in the definition of wind curves in every smooth piece.
Let and let us denote by (resp. , ) the set of the wind curves (resp. -wind curves, -admissible curves) between and (each curve defined in a possibly different interval ).
Following [JavSan11], we introduce the following notions.
Given a conic pseudo-Finsler metric , the Finslerian separation, also called -separation, is defined as if otherwise . By using the Finslerian separation two families of subsets of can be introduced: for any and , set and . Moreover a conic pseudo-Finsler metric is said Riemannianly lower bounded on an open subset of if there exists a Riemannian metric on such that , for all .
As and are continuously extendible to , we immediately get, by homogeneity, that they are Riemannianly lower bounded on, respectively, and . By [JavSan11, Proposition 3.13], the collections of a Riemannianly lower bounded conic pseudo-Finsler constitute a basis for the topology of , thus we have:
The collections of (resp ) constitute a basis for the topology of (resp. ).
Some cautions, however, must be taken. For example, the Finslerian separation of the conic Finsler metric may be discontinuous; in fact, the conic Finsler metric in [JavSan11, Example 3.18] exhibits this property (see also Section 4 below). We refer to [JavSan11, Section 3.5] for a summary of the properties satisfied by the Fnslerian separation.
In order to work with the full geometry associated with we also introduce the following new collections of subsets of .
Let and . The forward (resp. backward) wind balls of center and radius associated with the wind Finslerian structure are:
|being the closed balls their closures. Moreover, the (forward, backward) c-balls are defined as:|