Finsler metrics and relativistic spacetimes
aa r X i v : . [ m a t h . DG ] S e p September 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2
International Journal of Geometric Methods in Modern Physicsc (cid:13)
World Scientific Publishing Company
Finsler metrics and relativistic spacetimes
Miguel A. Javaloyes
Departamento de Matem´aticas, Universidad de Murcia,Facultad de Matem´aticas, Campus de Espinardo s/nMurcia, E-30100, [email protected]
Miguel S´anchez
Departamento de Geometr´ıa y Topolog´ıa, Universidad de Granada.Facultad de Ciencias, Campus de Fuentenueva s/n.E-18071 Granada (Spain)[email protected]
Received (Day Month Year)Revised (Day Month Year)
Submitted to the Special Issue for the XXII IFWGP Evora, with Associated EditorsR. Albuquerque, M. de Leon and M. C. Munoz Lecanda.
Recent links between Finsler Geometry and the geometry of spacetimes are briefly re-visited, and prospective ideas and results are explained. Special attention is paid togeometric problems with a direct motivation in Relativity and other parts of Physics.
Keywords : Finsler spacetime; Randers metrics; stationary spacetimes.
1. Introduction
There has been a recent interest in the links between Finsler Geometry and thegeometry of relativistic spacetimes. A well-known reason comes from the viewpointof applicability: Finsler metrics are much more general than Riemannian ones and,accordingly, if one replaces the Lorentz metric of a spacetime by a Finslerian coun-terpart, the possibility to model physical effects is richer. A more practical reasoncomes from a purely mathematical correspondence: the geometry of a concrete classof Finsler manifolds (Randers spaces) is closely related to the conformal structureof a class of spacetimes (standard stationary ones). So, results in one of these twofields can be translated into results on the other one —and, sometimes, this can begeneralized to more general Finsler and Lorentz spaces. The purpose of this paperis to make a brief revision of this topic, emphasizing the applications to Physics.After some preliminary definitions and motivations on Finsler metrics (Section2), we focus on the problem of describing spacetimes by using Finsler elements eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Miguel A. Javaloyes And Miguel S´anchez (Section 3). We introduce and discuss a notion of Lorentz-Finsler metric whichleads to (conic) Finsler spacetimes . Our definition is quite general and, so, severalprevious notions in the literature fit in it. Such a Finsler spacetime provides a conestructure . This notion has been studied systematically very recently [16], and itallows us to introduce classical Causality in the framework of Finsler spacetimes.In particular, some of the key results on this topic can be recovered, Theorem 1.It is worth pointing out: (a) the cone structure of a Finsler spacetime generalizesthe one provided by the chronological futures of a classical spacetime, and definesimplicitly a past cone structure (but no further assumption on reversibility shouldbe necessary if one liked to include in the definition of conic Finsler spacetimethe two cones at each point), and (b) in classical spacetimes, the cone structure isequivalent to the conformal one, but in Finsler spacetimes many non-conformallyrelated Lorentz Finsler metrics will have the same cone structure and, thus, the sameCausality —that is, the information of the Lorentz-Finsler metric not contained inthe cone structure is much richer than in classical spacetimes.In Section 4, we make a brief survey on the recent developments about the geo-metric correspondence between standard stationary spacetimes and Randers spaces.We focus in those questions with more clear applications to relativistic spacetimesand, so, topics such as the causal structure of standard stationary spacetimes orthe gravitational lensing are emphasized. Finally, the conclusions are summarizedin the last section.
2. Finsler metrics and their applications
Let us introduce the very general notion of (conic) pseudo-Finsler metric in a man-ifold M . Let T M denote the tangent bundle of M , π : T M → M , the naturalprojection and A ⊂ T M \ , an open subset satisfying that it is conic (that is, if v ∈ A and λ >
0, then λv ∈ A ), and it projects on all M (i.e., π ( A ) = M ). We saythat a smooth function L : A ⊂ T M \ → R is a (conic) pseudo-Finsler metric if L is positive homogeneous of degree 2, namely, L ( λv ) = λ L ( v ) for any v ∈ A and λ >
0, and the fundamental tensor g , defined as the Hessian of L at every v ∈ A , is nondegenerate. In other words, given v ∈ A , the bilinear symmetric formin T π ( v ) M defined as g v ( u, w ) = 12 ∂ ∂s∂t L ( v + tu + sw ) (cid:12)(cid:12)(cid:12)(cid:12) t = s =0 (1)for any u, w ∈ T π ( v ) M , is non-degenerate.Let us remark that classical Finsler metrics are a particular case of pseudo-Finsler metrics. More specifically, L is a Finsler metric if A = T M \ and thefundamental tensor g is positive definite a for every v ∈ A . In this case, L is alwayspositive, since by homogeneity L ( v ) = g v ( v, v ) for every v ∈ A and we can consider a In this case, positive definiteness follows from non-degeneracy [25, Prop. 2.16 (ii)]. eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2
Finsler metrics and relativistic spacetimes F = √ L , which is positive homogeneous of degree one. The function F is usuallycalled the Finsler metric function, and it is determined by its indicatrix , i.e. the setof its unit vectors.Among the most classical examples of Finsler metrics, we will use Randersmetrics, which are given by F ( v ) = α ( v ) + β ( v ) (2)for every v ∈ T M , where α ( v ) = p h ( v, v ) and h and β are a Riemannian metricand a one-form in M respectively. In fact, the fundamental tensor of F is positivedefinite in v ∈ T M if and only if α ( v ) + β ( v ) > h -norm of β is smaller than 1, so defining aRanders metric. These metrics appear naturally in several contexts as in Zermelonavigation problem, which aims to describe the trajectories that minimize the timein the presence of a mild wind or current modelled by a vector field W . Then, thesetrajectories are given by geodesics of a Randers metric defined as F ( v ) = − g ( v, W ) + p g ( v, W ) + g ( v, v )(1 − g ( W, W ))1 − g ( W, W ) (3)where v ∈ T M , g is a Riemannian metric and g ( W, W ) < M [2]. Ob-serve that Zermelo metrics are always positive for every v ∈ T M \ and then itsfundamental tensor is positive definite. As we will see later, Randers metrics alsoappear naturally associated to stationary spacetimes describing their causal prop-erties (see Section 4). Other remarkable example is given by Matsumoto metric,which describes trajectories minimizing time in the presence of a slope —recallthat going up is slower than going down. This metric is defined as F ( v ) = α ( v ) α ( v ) − β ( v )for every v ∈ T M , and its fundamental tensor is positive definite in v ∈ T M \ if andonly if ( α ( v ) − β ( v ))( α ( v ) − β ( v )) > conic if this inequality is not satisfied by some v = 0. Randers andMatsumoto metrics are particular examples of the class of ( α, β )-metrics, which aredefined as F ( v ) = αφ ( β ( v ) /α ( v )), being φ an arbitrary non-negative real function(see [25] and references therein).
3. Finsler-Lorentz metrics and spacetimes
In classical General Relativity, a spacetime is a (connected, Hausdorff) n -manifold M endowed with a Lorentzian metric g and a time orientation, i.e., a continuouschoice of one of the two timelike cones at each point, which will be regarded as future-directed . There are some speculative applications of the replacement of g by a (generalization of a) Finsler metric F , as modeling possible anisotropies ofthe spacetime even at an infinitesimal level, or admitting speeds higher than light.In any case, the possibility to model a general action functional (homogeneous ofeptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Miguel A. Javaloyes And Miguel S´anchez degree two, but not necessarily coming from a quadratic form) justifies the studyof the Lorentz-Finsler approach.There are different possibilities in order to define a Finsler spacetime. Recallthat, for a Riemannian metric g R , the unit vectors constitute the indicatrix of a(standard) Finsler metric; in particular, each unit sphere S p ⊂ T p M is convex (asthe boundary of the unit ball) at each point. Nevertheless, if g is a Lorentzian metric,one has to consider the subsets S + p , S − p ⊂ T p M containing, resp. the spacelike andtimelike unit vectors at p . Notice that S − p always contains two connected parts, eachone concave . For n = 2 these properties also hold for S + p , but for higher dimensions S + p is connected and, at each point p , its second fundamental form (say, with respectto any auxiliary Euclidean product at T p M ) has Lorentzian signature. From thephysical viewpoint, the most important elements of the spacetime are the causalvectors of the metric, since they model the trajectories of massive and masslessparticles. Observe in particular that the length of a causal curve γ : [0 , → M inthe spacetime is computed as ℓ g ( γ ) = Z p − g ( ˙ γ, ˙ γ ) ds and a fundamental property in the spacetime is that causal geodesics locally max-imize this length. If we want to define a more general way of measuring the lengthof curves (but preserving that the length does not depend on the parametrizationof the curve and geodesics are local length-maximizers), then we must consider apositive one-homogeneous function F : A ⊂ T M \ → (0 , + ∞ ) with fundamentaltensor of signature n − L , rather than a one-homogeneous function, —since for a Lorentzian met-ric g , p − g ( v, v ) is not smooth when v is lightlike b . Summing up, we say that a (conic) Finsler spacetime is a manifold endowed with a conic pseudo-Finsler metric L : A → [0 , + ∞ ) which satisfies the following properties:(i) each A p := A ∩ T p M is convex in T p M (i.e., the segment in T p M connectingeach two vectors v p , w p ∈ A p is entirely contained in A p ), in particular, each A p must be strictly included in a half-plane of T p M ,(ii) A has a smooth boundary in T M \ ,(iii) the extension of L as 0 to the closure ¯ A of A is smooth at ˆ A := ¯ A \ (notice that the extension is always continuous at 0 by homogeneity, and b There are some non-trivial issues regarding smoothability. On the one hand, mathematically,it is well known that all Finsler metrics smoothly extendible to the 0-section must come fromRiemannian metrics [45, Prop. 4.1]. On the other, physically, there are situations where it isnatural to consider non-differentiable directions, so that one may allow this possibility explicitly[31]. However, we will not go through these questions in the remainder (essentially, “smooth”might include some residual non-differentiable points with no big harm). eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2
Finsler metrics and relativistic spacetimes this extension cannot be smooth there even for a classical Finsler metric,except if it comes from a Riemannian metric), and(iv) both the fundamental tensor g in (1) and its extension to the points in ∂A \ have signature n − L is extended to ¯ A and g to T v ( T p M ) for all p ∈ M and v ∈ ¯ A p \ . In order to make computations relatedto lightlike geodesics, one can also assume that L is smoothly extended (in a non-unique way) on a neighborhood of ˆ A . We will call F = √ L the Lorentz-Finslermetric (defined on ¯ A ) of the Finsler spacetime. Observe that our notion of Finslerspacetime is based on a “conic” element which is essentially present in most ofprevious literature. In particular:(1) In [5], the function L is defined in all T M and the fundamental tensor is assumedto have signature n −
1. The restriction to one connected component of thesubset L − [0 , + ∞ ) \ p ∈ M gives a (conic) Finsler spacetime asdefined here. In [35,31], the definition is essentially as in [5] with the oppositesign of L (and an increasing attention to relax differentiability). The definitionin [36] is somewhat more involved: they introduce an r -homogeneous function L , with r ≥ F = r √ L . This allows one todeal with non-differentiability in cases that extend the lightlike vectors in ourdefinition above. In [1], the author gives a definition analog to ours, but heexcludes the conditions on the boundary (so that one is not worried aboutlightlike vectors).(2) Lorentz-Finsler metrics appear in the context of Lorentz violation. Here, onestarts with a background Lorentz-Minkowski space, but the movement of par-ticles or waves is governed by a Lagrangian with a Lorentz-Finsler behavior(see [37,28,29] and references therein). Under natural hypotheses, the followingpositively homogeneous function is found in [27]: F ( v ) = m p − g ( v, v ) + g ( v, a ) ± p g ( v, b ) − g ( b, b ) g ( v, v ) , where g is the standard metric of Minkowski spacetime R , v is any timelikevector and a, b are two prescribed vector fields in R . The interplay between F and the background metric g becomes important for physical applications (forexample, the group velocity of propagating waves may exceed the light speed in R ). Even though F can be regarded as a Lorentz-Finsler metric, in some casesthe fundamental tensor of F may be degenerate on some timelike directions(when b = 0, it is easy to compute the fundamental tensor of F [25, Proposition4.17], and this becomes degenerate when F ( v ) = p − g ( v, v ) + g ( v, a ) = 0).(3) General physical theories of modified gravity (see [44] or the very recent articles[38,39] and references therein) yield naturally Lorentz-Finsler spacetimes as de-fined above. The exact domain A of L for general expressions in Finsler-Cartangravity (say, as in [39, formula (2)]) may be difficult to compute. However,eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Miguel A. Javaloyes And Miguel S´anchez one can compute A easily in General Very Special Relativity , which possesses asimple Finslerian line element introduced by Bogoslovsky (say, the expression F = ( √− g ) (1 − b ) ω b in [30, formula (I.1)] becomes a Lorentz-Finsler metric withdomain A equal to the intersection at each point of the future time-cone of g and the half-space ω >
0) or in Randers-type cosmologies as [4, formula (3.1)].Notice that our definition of Finsler spacetime is a generalization of the struc-ture obtained in a classical spacetime when one considers only its future causalcones . In the case of Lorentzian metrics, however, the value of the metric on thecausal vectors is enough to determine the metric on all the vectors and, moreover,the lightlike vectors are enough to determine the metric up to a conformal factor.So, the cone structure for Lorentzian metrics is equivalent to the conformal struc-ture. Nevertheless, this does not hold by any means in the case of (conic) Finslerspacetimes: clearly two Lorentz Finsler metrics
F, F ′ on M with the same domain¯ A may not be equal up to a multiplicative function. However, the domain A is a cone structure and, thus, one can reconstruct all the Causality Theory for Finslerspacetimes. Let us review this briefly.Given a Finsler spacetime ( M, L ) a tangent vector v ∈ T M is called (future-directed) timelike if v ∈ A , causal if v ∈ ˆ A , lightlike if v ∈ ˆ A \ A and null if either v is lightlike or the zero vector c . A (piecewise smooth) curve γ on M is also calledtimelike, causal etc. depending on the character of its velocity at all the points. If p, q ∈ M , we say that p lies in the chronological (resp. causal) past of q if thereexists a (future-directed) timelike (resp. causal or null) curve starting at p andending at q ; in this case we write p ≪ q (resp. p ≤ q ) and we also say that q liesin the chronological (resp. causal) future of p . The chronological and causal futuresof p , as well as its corresponding pasts, are then defined formally as in the case ofLorentzian metrics: I + ( p ) = { q ∈ M : p ≪ q } , J + ( p ) = { q ∈ M : p ≤ q } ; I − ( p ) = { q ∈ M : q ≪ q } , J − ( p ) = { q ∈ M : q ≤ p } . One says also that a second Lorentz Finsler metric F ′ has cones wider than F ,denoted F ≺ F ′ when their corresponding domains A, A ′ satisfy ˆ A ⊂ A ′ . Withthese definitions, one can extend directly the causal ladder of classical spacetimes(see for example [33]) to Finsler spacetimes. We recall some of the steps of thisladder. A Finsler spacetime will be called chronological (resp. causal ) when it doesnot admit closed timelike (resp. causal) curves, stably causal when there exists acausal Lorentz-Lorentz metric F ′ with wider cones F ≺ F ′ , and globally hyperbolic c Notice that we have defined the Finsler metric only for “future-directed” causal directions, i.e.,those in ˆ A , and the past directions could be consistently regarded as those in − ˆ A . It would benatural then to extend L to − ˆ A by requiring full homogeneity ( L ( v ) = L ( − v ) for all v ∈ − ˆ V ),but this is not necessary for the issues regarding Causality. However such an extension (as wellas possible further extension of L to non-causal vectors) would be important for the particularphysical theory one may be modelling. eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Finsler metrics and relativistic spacetimes when it is stably causal and the following property holds: J + ( p ) ∩ J − ( q ) is compactfor any p, q ∈ M . Trivially:chronological ⇐ causal ⇐ stably causal ⇐ globally hyperbolic.Taking into account the case of spacetimes, one realizes that there are moreintermediate levels of the ladder as well as many subtle properties and relationsamong them to be considered for Finsler spacetimes. However, we focus here juston a pair of them, with deep implications for the global structure. To this aim, wedefine for a Finsler spacetime ( M, L ):(a) a spacelike hypersurface is a smooth hypersurface S such that no causal vector v is tangent to S ,(b) a (spacelike) Cauchy hypersurface is a spacelike hypersurface S such that anycausal curve which is inextendible in a continuous way, intersects S exactlyonce,(c) a temporal function is a smooth function t such that dt ( v ) > v (thus, its levels t =constant are spacelike hypersur-faces) and,(d) a Cauchy temporal function is a temporal function such that all its levels areCauchy hypersurfaces.In the case of classical spacetimes, results by Geroch [19] and Hawking [21] obtainedat a topological level, plus their improvements to the smooth and metric cases byBernal and S´anchez [7,8], prove the equivalence between being stably causal (resp,globally hyperbolic) and admitting a temporal (resp. Cauchy temporal) function.By using different arguments coming from KAM theory, these results were re-provedand extended to general cone structures by Fathi and Siconolfi [16]. So, as a conse-quence of the latter we get the following result.
Theorem 1.
Let ( M, F ) be a Finsler spacetime.(1) If ( M, F ) is stably causal, then it admits a temporal function (and, thus, itcan be globally foliated by spacelike hypersurfaces).(2) If ( M, F ) is globally hyperbolic, then it admits a Cauchy hypersurface and aCauchy temporal function. Recall, however, that the Finsler spacetime has elements that are not char-acterized by the cone structure. For example, one can define a
Finler separation d ( p, q ) ∈ [0 , ∞ ] between any p, q ∈ M by taking the supremum of the lengths of the(future-directed) causal curves from p to q —in a close analogous to the Lorentzianseparation or distance for spacetimes with Lorentzian metric. The analogies and dif-ferences of d with the Lorentzian case (in particular, its interplay with Causality)deserve to be studied further. Remark 2.
One expects that the converse of parts (1) and (2) of Theorem 1, aswell as most of classical causality theory, hold. To achieve this, typical Lorentzianeptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Miguel A. Javaloyes And Miguel S´anchez techniques, as those regarding limit curves or quasi-limits, must be adapted (recallthe Lorentzian proofs of the converse in [19], [22, Sect. 6.6] or [34, Th. 14.38,Cor. 14.39]). Moreover, as the results in [16] work for very general cone structures,Theorem 1 can be applied to a more general class of spacetimes that includes thosein [35, Example 3.3], which can be non-smooth in some timelike directions, and alsothe cone structure provided by the example (2) above (see [27]).
4. Stationary to Randers correspondence and beyond
From a classical viewpoint, a correspondence between some purely geometric el-ements of Lorentzian and Finslerian manifolds has been developed recently. Sucha correspondence provides a precise description of certain objects in (classical)spacetimes in terms of an associated Finsler space. The precise correspondenceis developed between a particular class of spacetimes, the standard complete-conformastationary ones, or just stationary , for short, and a precise class of Finslermanifolds, the Randers ones (see (2)). But some consequences and techniques canbe extrapolated to general Lorentzian and Finslerian manifolds.We start with a trivial observation. Probably the simplest examples ofLorentzian manifolds are the products ( R × M, g L = − dt + π ⋆ g ), where π : R × M → M and t : R × M → R are the natural projections. The geometricproperties of ( R × M, g L ) depend on those of ( M, g ). In particular, each curve( R ⊃ ) I ∋ t c ( t ) ∈ M parametrized with unit speed yields naturally two lightlikecurves t → ( ± t, c ( t )) (future-directed with the sign “ + ” and past-directed with“ − ”, for the natural time orientation of the spacetime), which are geodesics iff c is a geodesic in ( M, g ). A less trivial spacetime is obtained if we admit crossterms between the time and space parts (independent of the time t ). This can bedescribed by using a 1-form ω on M , namely, we consider the spacetime: V = ( R × M, g L ) , g L = − dt + π ⋆ ω ⊗ dt + dt ⊗ π ⋆ ω + π ⋆ g . (4)Now, we introduce the following Finslerian Fermat metrics associated to (4): F ± = p g + ω ± ω, notice that these are metrics of Randers type, and F − is the reverse Finsler metricof F + , so, we will write simply F for the latter and ˜ F for the former. If we considercurves c + and c − in M which are unit for F and ˜ F resp., the curves in the spacetime t → ( ± t, c ± ( t )) are again (future or past directed) lightlike curves in V , and eachone is a geodesic up to parametrization iff so is the corresponding original curve c + or c − (see the details in [12] or [13]). This suggests the possibility of describing theproperties related to the Causality and conformal structure in (4) in terms of thegeometry of the corresponding Randers space ( M, F ).It is worth pointing out that, as only conformally invariant properties will betaken into account, the class of spacetimes to be considered includes those conformalto (4). This class of spacetimes can be characterized intrinsically as those whicheptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2
Finsler metrics and relativistic spacetimes are distinguishing (a causality condition less restrictive than strong causality) andadmit a complete timelike conformal vector field. In fact, the conformal change ofthe metric g by − g/g ( K, K ), plus the result in [25], allows one to find the expression(4), which defines a normalized standard stationary spacetime . So, one will find acorrespondence between the conformal properties of the elements in this class ofmetrics (
V, g L ) and the geometric properties of Randers spaces ( M, F ). This hasbeen carried out at different levels (see [9,10,11,12,13,18,20,24] or [23] for a review),and we will focus here in three of them, with clear physical applications.
Causal structure
Nicely, Fermat metrics allow one to determine the chronological and causal futureand past of any point in the stationary spacetime (4). For example, if we take(0 , p ) ∈ { } × M then the intersection of I + (0 , p ) with the slice t = t of thespacetime is equal to { t } × B + ( p, t ), where B + ( p, t ) is the open ball of center p and radius t for the Fermat metric F (= F + ). From these considerations, one candescribe in a precise way the causal structure of the spacetime. Concretely, one has[13]: Theorem 3.
A stationary spacetime (4) is always causally continuous and it is • Causally simple (i.e., it is causal, and the causal futures and pasts J ± ( p ) areclosed) if and only if ( M, F ) is convex, i.e. any pair ( p, q ) in M can be connectedby means of an F + -geodesic of length equal to the Finslerian distance d F ( p, q ) . • Globally hyperbolic if and only if the closed balls for the symmetrized distanceof d F are compact. • Globally hyperbolic with slices t = constant that are Cauchy hypersurfaces if andonly if d F is forward and backward complete. Other causal elements which are described naturally with the Finslerian ele-ments are the Cauchy horizons and developments. For example, given a subset A ⊂ { } × M , its future Cauchy horizon H + ( A ) is the graph, in R × M of thefunction which maps each y ∈ ¯ A to d F ( M \ A, y ), i.e., the d F -distance from thecomplement of A to y . Remark 4.
The results in stationary spacetimes can also be translated to resultsin Randers metrics, sometimes generalizable to any Finsler manifold. Among them,we point out (see [13]): • The theorem above suggests that the compactness of the closed symmetrizedballs of d F , which is a weaker hypothesis than the commonly used (forwardor backward) completeness of d F , can substitute the last hypothesis in manyresults. And, in fact, this is the case in classical theorems of Finsler Geometry,such as Myers’ theorem or the sphere theorem. • From the known fact that any globally hyperbolic spacetime admits a smoothspacelike Cauchy hypersurface, one can deduce that any Randers metric R witheptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Miguel A. Javaloyes And Miguel S´anchez compact symmetrized balls admits a trivial projective change ( R → R + df )such that the corresponding new Randers metric has the same pregeodesics as R , and it is forward and backward complete . This result has been extended byMatveev [32] for any Finsler metric. • The results on horizons (a substantial topic in Lorentzian Geometry) in sta-tionary spacetimes, yield directly results on Randers spaces. For example, thetranslation of a well-known result by Beem and Krolak [6] yields the followingproperty for any subset A of a Randers manifold ( M, R ): p ∈ M is a differen-tiable point of the distance from p if and only if it can be crossed by exactly oneminimizing segment. This result has been extended to any Finslerian manifoldby Sabau and Tanaka [42].
Visibility and gravitational lensing
Assume that in our spacetime (4), a point w represents an event and a line l ,obtained as an integral curve of ∂ t , represents the trajectory of a stellar object. Wewonder if there exist lightlike geodesics from l to w (i.e., whether an observer at w can see the object l ) and, in this case, if there are many of such geodesics (i.e., theexistence of a lens effect such that l is seen in two different directions). This situationis applicable to cosmological models such as Friedmann-Lemaitre-Robertson-Walkerones, as they are conformally stationary ( l would represent a “comoving observer”of the model). The problem becomes more realistic if we choose some (open) region R × D ⊂ R × M which contains w and l , and search for geodesics contained in thisregion.These problems are related to the convexity of D for the Fermat metric (inthe sense of the last subsection) which turns to be related to the convexity of itsboundary ∂D . There are several different notions for the latter, as the local and infinitesimal convexity and, as shown in [3], they are equivalent to its geometricconvexity . The latter means that, given any pair of points of D , any geodesic con-necting them and contained in the closure ¯ D , must be entirely contained in D . The(geometric) convexity of D turns out equivalent to the light (geometric) convexity of the boundary of R × D ( ⊂ R × M ) (i.e., the property of convexity holds whenrestricted to lightlike geodesics) and, finally, this is equivalent to the question ofexistence of connecting geodesics, yielding [10]: Theorem 5.
Assume that the closed balls of ¯ D computed for the restriction of thesymmetrized distance d sF are compact (which happens for example, if the intersec-tions with ¯ D of the closed symmetrized balls in M are compact or, simply, if d F iscomplete on all M ). Then, the following assertions are equivalent:(i) ( D × R , g L ) is causally simple (i.e. ( D, F ) is convex, Theorem 3).(ii) ∂D is convex for the Fermat metric F .(iii) R × ∂D is light-convex for the Lorentzian metric g L .(iv) Any point w = ( t p , p ) ∈ R × D and any line l q := { ( τ, q ) ∈ R × D : τ ∈ R } , eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Finsler metrics and relativistic spacetimes with p = q , can be joined in R × D by means of a future-directed lightlikegeodesic z ( s ) = ( t ( s ) , x ( s )) , s ∈ [ a, b ] , which minimizes the (future) arrivaltime T = t ( b ) − t ( a ) (i.e., such that x minimizes the F -distance in D from p to q ).(v) Idem to the previous property but replacing “future” by “past” and “ F -distance” by (the reverse) “ ˜ F -distance”. Remark 6.
The previous result solves the question of existence of connectinglightlike geodesics. The question of multiplicity has some possibilities. First, theexistence of a conjugate point of the lightlike geodesic z at w . This is equivalent tothe existence of a conjugate point at p for its projection x [11, Theorem 13], andit is regarded as trivial. Second, the non-triviality of the topology of D may yielda topological lensing . In fact, one can prove that, whenever D is not contractible,infinitely many connecting lightlike geodesics (with diverging arrival times) willexist.The results can be also extended to the case of timelike geodesics, prescribingits length (i.e., the lifetime of the massive particles represented by such geodesics).The idea relies on a reduction of the problem to the lightlike one, by consideringan extra spacelike dimension, see [13, Section 4.3] and [10, Section 5.2]. Causal boundaries and further questions
The studied stationary to Randers correspondence can be also applied to the studyof the causal boundary of the spacetime in terms of Finsler elements. We recall that,in Mathematical Relativity, the Penrose conformal boundary is commonly used, inspite of the fact that it is not an intrinsic construction, and there are problems toensure its existence or uniqueness. The causal boundary is a conformally invariantalternative, which is intrinsic and can be constructed systematically in any stronglycausal spacetime (see [17] for a comprehensive study of this boundary). The com-putation of the causal boundary and completion of a stationary spacetime (4) hasbeen carried out in full generality in [18]. It must be emphasized that this boundaryhas motivated the definition of a new
Busemann boundary in any Finslerian mani-fold. Even more, this has stimulated the study of further properties of the Gromovboundary in both, Riemannian and Finlerian manifolds.It is also worth mentioning two further topics of current interest. The first oneis the correspondence at the level of curvatures between the Weyl tensor of thestationary spacetime and the flag curvature of the Randers space, see [20]. Thesecond one is the possibility to extend the studied correspondence to the case ofspacetimes with a nonvasnishing complete Killing vector field, whose causal typemay change from timelike to lightlike and spacelike. Such metrics are related toa generalization of the Zermelo metrics which includes the possibility of a strongwind (i.e., with a speed higher than the one which can be reached by the engine ofthe ship), see [14].eptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2 Miguel A. Javaloyes And Miguel S´anchez
5. Conclusions
As a first goal, we have introduced a very general notion of Finsler spacetime whichincludes many previous ones. This notion emphasizes the role of the cone structure,showing that many properties of (classical, relativistic) spacetimes are potentiallytransplantable to this setting —in particular, this holds for a major Lorentzianresult such as Theorem 1. Remarkably, in relativistic spacetimes the cone structureis equivalent to the conformal structure and, thus, it is governed by the Weyl tensor.However, the metrics F , F of two Finsler spacetimes with the same cone structureare not by any means necessarily conformal, and one is lead to deal directly withthe cone structure. This cone structure would remain then at a more basic levelthan any possible Weyl-type tensor and, thus, also at a much basic level than theCartan or Chern connections (of course, such connections will be relevant in orderto describe concrete physical effects).The importance on the choice of the connection is stressed because a Finslermetric admits several associated connections. Among them, one has the classicalCartan connection, the Berwald connection, the Chern-Rund connection and theHashiguchi connection (see [40] or [41, § convexity wth theproperties related to relativistic observability and lensing, (c) accurate relationsbetween ideal Finslerian boundary and the causal boundary (and, implicitly then,the conformal one) of spacetimes, or (d) interesting relations between Finsleriancurvature and Weyl curvature [20]. This equivalence has suggested new results inboth, Finslerian and Lorentzian geometries. Moreover, a forthcoming work [14] willshow its extendability and applicability to new fields —namely, an extension ofclassical Finsler metrics which allows one to model strong winds in Zermelo navi-gation problem is introduced, and its geometrical properties are shown equivalentto the conformal ones of a class of spacetimes wider than standard stationary ones,which allow the possibility of horizons and black holes. Acknowledgments
Comments by Kostelecki, Laemmerzahl, Perlick, Szilasi and the referee are warmlyacknowledged. Both authors are partially supported by the Grant P09-FQM-4496 (J. Andaluc´ıa) with FEDER funds. The first-named author is also partiallyeptember 6, 2018 1:59 WSPC/INSTRUCTION FILESanchez˙Javaloyes˙Revised2
Finsler metrics and relativistic spacetimes supported by MINECO-FEDER project MTM2012-34037 and Fundaci´on S´enecaproject 04540/GERM/06, Spain. This research is a result of the activity developedwithin the framework of the Programme in Support of Excellence Groups of theRegi´on de Murcia, Spain, by Fundaci´on S´eneca, Regional Agency for Science andTechnology (Regional Plan for Science and Technology 2007-2010). The second-named author acknowledges the support of IHES, Bures sur-Yvette, France, for athree months stay in 2013 (which included the period of celebration of the meetingIFWGP); he is also partially supported by MTM2010–18099 (MICINN-FEDER). References [1] G. S. Asanov,
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