Standard static Finsler spacetimes
SSTANDARD STATIC FINSLER SPACETIMES
ERASMO CAPONIO AND GIUSEPPE STANCARONE
Abstract.
We introduce the notion of a standard static Finsler spacetime R × M where the base M is a Finsler manifold. We prove some results whichconnect causality with the Finslerian geometry of the base extending analogousones for static and stationary Lorentzian spacetimes. Introduction
Recent years have seen a growing interest in Finsler spacetimes. Indeed, Finsle-rian modifications of general relativity have been proposed and Finsler spacetimeshave been considered as possible backgrounds for the Standard Model of particlesand for quantum gravity with Lorentz-symmetry violating fields, see, e.g., [39], [40],[49], [50], [6], [48], [22], [29], [42], [25], [26], [47], [45], [27]. Moreover, several au-thors have started to investigate the geometric and the causal structure of Finslerspacetimes, see, e.g., [38], [28], [39], [20], [24], [34], [35], [1]. Different definitions ofa Finsler spacetime metric have been proposed in the last cited papers but, withthe exception of [28], they do not include a somehow natural generalization to theFinsler realm of the notions of standard static and standard stationary spacetimes.This is related to the fact that such classes of spacetime are defined on productmanifolds ˜ M = R × M where M is identifiable with a spacelike hypersurface of ˜ M .So a Lorentzian Finsler structure on ˜ M inducing a classical Finsler structure on M cannot be smoothly extended (where smooth here means at least C ) to vectorswhich project trivially on T M , due to the lack of regularity of a Finsler functionon the zero section.Motivated by the above problem, we define a Finsler spacetime as follows. Let˜ M be a ( n + 1)-dimensional smooth paracompact connected manifold, n ≥
1. Letus denote by T ˜ M its tangent bundle and by 0 the zero section. Let T ⊂ T ˜ M bea smooth real line vector bundle on ˜ M and T p the fibre of T over p ∈ ˜ M . Let π : T ˜ M \ T → ˜ M be the restriction of the canonical projection, ˜ π : T ˜ M → ˜ M , to T ˜ M \ T and let π ∗ ( T ∗ ˜ M ) the pulled-back cotangent bundle over T ˜ M \ T . Let usconsider the tensor bundle π ∗ ( T ∗ ˜ M ) ⊗ π ∗ ( T ∗ ˜ M ) over T ˜ M \ T and a section ˜ g : v ∈ T ˜ M \ T (cid:55)→ ˜ g v ∈ T ∗ π ( v ) ˜ M ⊗ T ∗ π ( v ) ˜ M . We say that ˜ g is symmetric if ˜ g v is symmetricfor all v ∈ T ˜ M \ T . Analogously, ˜ g is said non-degenerate if ˜ g v is non-degeneratefor each v ∈ T ˜ M \ T and its index will be the common index of the symmetricbilinear forms ˜ g v ; moreover, ˜ g will be said homogeneous if, for all λ > v ∈ T ˜ M \T , ˜ g λv = ˜ g v . Finally, if the above conditions on ˜ g are satisfied, we say that˜ g is the vertical Hessian of a (quadratic) Finsler function if there exists a function Mathematics Subject Classification.
Key words and phrases.
Finsler spacetime, static spacetime, causality. a r X i v : . [ m a t h . DG ] M a r E. CAPONIO AND G. STANCARONE L : T ˜ M → R such that in natural coordinates ( x , x , . . . , x n , v , v , . . . , v n ) of T ˜ M , ∂ L∂v i ∂v j ( v ) = ˜ g v , for all v ∈ T ˜ M \ T . Definition 1.1. A Finsler spacetime ( ˜
M , L ) is a smooth ( n + 1)-dimensional man-ifold ˜ M , n ≥
1, endowed with a smooth, symmetric, homogeneous, non-degenerateof index 1 section ˜ g of the tensor bundle π ∗ ( T ∗ ˜ M ) ⊗ π ∗ ( T ∗ ˜ M ) over T ˜ M \ T , whichis the vertical Hessian of a Finsler function L and such that ˜ g w ( w, w ) < w in a punctured conic neighbourhood of T p in T p ˜ M \ { } and for all p ∈ ˜ M . Remark . Clearly, the Finsler function in the above definition must be fiberwisepositively homogeneous of degree 2 and, so, it is, except for the possible lack oftwice differentiability along T , an indefinite Finsler (also called Lorentz-Finsler orsimply
Finsler ) function as, e.g., in [7, 34, 38]. We could allow more generality bynot prescribing the existence of such a function. This is a quite popular approach toFinsler geometry: see, e.g., [4, 33], and the references therein, where such structuresare called generalized metrics (although in [33] they are sections of the tensor bundle˜ π ∗ ( T ∗ ˜ M ) ⊗ ˜ π ∗ ( T ∗ ˜ M ) with base the whole T ˜ M ). By homogeneity, it is easy toprove that a generalized homogeneous metric is the vertical Hessian of a smoothFinsler function on T ˜ M \ T if and only if its Cartan tensor is totally symmetric,i.e. ∂ ˜ g ij ∂v k ( v ) = ∂ ˜ g ik ∂v j ( v ) for all v ∈ T ˜ M \ T (see, e.g., [4, Theorem 3.4.2.1]).Anyway, there are some arguments against the definition of a Finsler spacetimewithout a Finsler function which will be considered in Remark 2.11. Remark . The requirement about the sign of ˜ g w ( w, w ) for w in a neighbourhoodof T could be weakened by allowing the existence of an open subset ˜ M l ⊂ ˜ M where the reverse inequality holds for any w in a punctured neighbourhood of T p ,for all p ∈ ˜ M l (and then, eventually, the existence of a “critical region” where,in each punctured neighbourhood of T p , there exist vectors w and w such that˜ g w ( w , w ) < g w ( w , w ) > splitting and studied in[15] (see last section). Remark . Whenever T is trivial and ˜ g can be smoothly extended to T \
0, werecover the definition of a time-orientable Finsler spacetime in [34] by choosing ano-where zero continuous section T such that T p ∈ T , for each p ∈ ˜ M . Remark . Definition 1.1 of a Finsler spacetime is a slight generalization of theones appearing in [7, 38, 34] as it takes into account the problem of defining partic-ular Finslerian warped products. It is less general than one given in [28] where L isa.e. smooth on T ˜ M . We point out that all these definitions are purely kinematicalin the sense that they do not take into account the problem of determining physicalreasonable Finslerian field equations (see the discussion of this problem in [49]). Definition 1.6.
A vector w ∈ T ˜ M is called timelike if either w ∈ T \ g w ( w, w ) <
0. Moreover, w ∈ T ˜ M \ T will be said lightlike (resp. causal , spacelike )if ˜ g w ( w, w ) = 0 (resp. ˜ g w ( w, w ) ≤
0, either ˜ g w ( w, w ) > w = 0). Moreover,assuming that a section T as above can be chosen, we say that a causal vector w is future-pointing (resp. past pointing ) if either w is a positive (resp. negative)multiple of T ˜ π ( w ) or ˜ g w ( w, T ) < g w ( w, T ) > γ : I → ˜ M , I ⊆ R , will be called timelike (resp. lightlike, causal,spacelike ) if ˙ γ ± ( s ) are timelike (resp. lightlike, causal, spacelike) for all s ∈ I TANDARD STATIC FINSLER SPACETIMES 3 (where, ˙ γ ± ( s ) denotes the right and the left derivatives of γ at s ). Moreover, acausal curve γ : I → ˜ M will be said future-pointing , (resp. past pointing ) if ˙ γ ± ( s )are future (resp. past) pointing for all s ∈ I . Finally a smooth embedded hyper-surface H ⊂ M which is transversal to T will be called spacelike if for any v ∈ T H ,˜ g v ( v, v ) > Definition 1.7.
Let ( ˜
M , L ) be a Finsler spacetime and γ : [ a, b ) → ˜ M a continuouspiecewise smooth future-pointing causal curve. Then γ is called future extendible ifit has a continuous extension at b ; it is future inextensible otherwise. Analogously γ : ( a, b ] → ˜ M is called past extendible if it can be continuously extended at a and it is past inextensible otherwise. Moreover, a future-pointing causal curve γ : ( a, b ) → ˜ M is inextensible if it is future and past inextensible.Henceforth, we will always consider continuous piecewise smooth curves, so thatwe will often omit to specify it when considering a curve.2. Standard Static Finsler Spacetimes
Let us recall that a Lorentzian spacetime ( ˜
M , g ) is said static if it is endowedwith an irrotational timelike Killing vector field K . This is equivalent to say thatthe orthogonal distribution to K is locally integrable and then for each p ∈ ˜ M thereexists a spacelike hypersurface S , orthogonal to K , p ∈ S , and an open interval I such that the pullback of the metric g by a local flow of K , defined in I × S , isgiven by − Λ dt + g , where t ∈ I , ∂ t is the pullback of K , Λ = − g ( K, K ) and g is the Riemannian metric induced on S by g (see [36, Proposition 12.38]). Thislocal property of static spacetimes justifies the following definition: let M be an n -dimensional Riemannian manifold, Λ : M → (0 , + ∞ ) a smooth, positive functionon M and I ⊆ R an open interval. The warped product I × Λ M , i.e. the manifold I × M endowed with the Lorentzian metric g = − Λ dt + g , where g is the pullbackon I × M of the Riemannian metric on M , is a spacetime called standard static (see[36, Definition 12.36]).The conformal Riemannian metric on M , g / Λ is called optical metric . It playsa fundamental role in the study of light rays of I × Λ M because its pregeodesics arethe projections on M of the light rays in ( I × M, g ) (see [3], [21], [37]). Moreover,many of the causal properties of the spacetime I × Λ M are encoded in the geometryof the conformal manifold ( M, g / Λ). For example, global hyperbolicity of I × Λ M and the fact that the the slices { t } × M , t ∈ I , are Cauchy hypersurfaces areboth equivalent to the completeness of the optical metric (these properties followsby the conformal invariance of causal properties plus Theorem 3.67 and Theorem3.69-(1) with f ≡ Definition 2.1.
Let ( ˜
M , L ) be a Finsler spacetime and K be a vector field on˜ M . Let ψ be the flow of K . We say that K is a Killing vector field if for each v ∈ T ˜ M \ T and for all v , v ∈ T π ( v ) ˜ M , we have:˜ g dψ ¯ t ( v ) ( dψ ¯ t ( v ) , dψ ¯ t ( v )) = ˜ g v ( v , v ) , (1)for any ¯ t ∈ R such that the stage ψ ¯ t is well defined in a neighbourhood U ⊂ ˜ M of π ( v ). E. CAPONIO AND G. STANCARONE
Remark . As proved in [30, Proposition 5.2], this is equivalent to the fact that K is a Killing vector field for ˜ g , in the sense that the Lie derivative L K ˜ g = 0 (see[30, p.136] for the definition of the Lie derivative over the tensor bundle π ∗ ( T ∗ ˜ M ) ⊗ π ∗ ( T ∗ ˜ M ); actually in [30] the base of the tensor bundle is the slit tangent bundle T ˜ M \ T ˜ M \ T ). Definition 2.3.
We say that a Finsler spacetime ( ˜
M , L ) is static if there existsa timelike Killing vector field K such that the distribution of hyperplanes is inte-grable, where ∂ v L ( K ) denotes the one-form on ˜ M given by ∂L∂v i ( K ) dx i . Remark . Observe that the above definition is well posed since L is at least a C function on T ˜ M . In particular, it works also for a smooth global section T of T (in the case when T is trivial) or, more generally, for a vector field K such that K p ∈ T p for some p ∈ ˜ M . Definition 2.5.
We say that a Finsler spacetime is standard static if there exist asmooth non vanishing global section T of T , a Finsler manifold ( M, F ), a positivefunction Λ on M and a smooth diffeomorphism f : R × M → ˜ M , f = f ( t, x ), suchthat ∂ t = f ∗ ( T ) and L ( f ∗ ( τ, v )) = − Λ τ + F ( v ). Remark . The definition of a standard static Finsler spacetime (although calledthere static Finsler spacetime) appeared first in [28, Definition 2]. In [29] thesolution of the vacuum (Finslerian) field equations, introduced in the same paper,is standard static in the region where a certain coefficient B is positive providedthat a constant a is also positive (see [29, Eqs. (16)-(17)]). Remark . Another static future-pointing Killing vector field K will be said stan-dard if the above conditions hold relatively to K , i.e there exist a manifold ( M (cid:48) , G )and a diffeomorphism f (cid:48) : R × M (cid:48) → ˜ M , f (cid:48) = f (cid:48) ( t (cid:48) , x (cid:48) ), such that ∂ t (cid:48) = ( f (cid:48) ) ∗ ( K )and L ( f (cid:48)∗ ( τ (cid:48) , v (cid:48) )) = − Λ K τ (cid:48) + G ( v (cid:48) ).The existence of a standard static vector field is a very rigid condition in com-parison to the Lorentzian case ([44, 2]) where some topological assumptions on thebase M are needed in order to get uniqueness. Indeed we have the following: Proposition 2.8.
If a static Finsler spacetime admits a standard splitting (i.e thestatic vector field is standard) then it is unique up to rescaling t (cid:55)→ t/a , T (cid:55)→ aT , a ∈ (0 , + ∞ ) , of the coordinate t and of the vector field T and up to Finslerianisometries of ( M, F ) .Proof. Assume that another static standard Killing vector field K exists and let f (cid:48) : R × M (cid:48) → ˜ M be a smooth diffeomorphism such that ( f (cid:48) ) ∗ ( K ) = ∂ t (cid:48) and L ( f (cid:48)∗ ( τ (cid:48) , v (cid:48) )) = − Λ K τ (cid:48) + G ( v (cid:48) ), so that G ( v (cid:48) ) = L ( f (cid:48)∗ (0 , v (cid:48) )) for all v (cid:48) ∈ T M (cid:48) .Let L (cid:48) := L ◦ f (cid:48)∗ . Then L = L (cid:48) ◦ ( f (cid:48) ) ∗ , hence L is not twice differentiable along theline bundle K defined by K , because L (cid:48) is not so along the one defined by ∂ t (cid:48) . Thisis possible if and only if K = T , which is equivalent to the fact that K is collinear to T at every point in ˜ M . Since both K and T are Killing vector fields for ˜ g and theybelong to the same timelike cone necessarily they are proportional, i.e. there existsa positive constant a such that K = aT . This follows as in the semi-Riemanniancase, by using the fact that L T ˜ g = L K ˜ g = 0 (see [30, p. 136, Definition 5.1 andProposition 5.2]). Thus the diffeomorphism ( f (cid:48) ) − ◦ f has first component equal to TANDARD STATIC FINSLER SPACETIMES 5 t (cid:55)→ t/a while the second one induces a diffeomorphism φ between M and M (cid:48) suchthat G ( φ ∗ ( v )) = F ( v ) for all v ∈ T M . (cid:3) Remark . Henceforth, we will identify a standard static Finsler spacetime ( ˜
M , L )with the product manifold R × M endowed with the Finsler function L ( τ, v ) = − Λ τ + F ( v ), where Λ and F are, respectively, a positive function and a Finslermetric on M . Remark . Observe that ∂ t is ˜ g ( τ,v ) -orthogonal to { } × T x M for any ( τ, v ) ∈ T ˜ M \ T , x = π M ( v ), where π M : T M → M is the canonical projection. These factsjustify, by analogy with the Lorentzian case, the name “standard static” given to theclass of Finsler spacetime in Definition 2.5. Anyway, as observed in [34, Example 1,Remark 3], differently from the Lorentzian case, a Finsler spacetime can be staticwithout being locally standard static either (in the example of [34], the Killingvector field is not ˜ g ˜ v -orthogonal to ker( ∂ v L ( K π (˜ v ) )), for all ˜ v ∈ ker( ∂ v L ( K ))). Remark . Clearly, in Definition 2.5, we could allow more generality by takinga positive definite, homogeneous, generalized metric g on M (recall Remark 1.2)and/or a function ˜Λ : T M \ → (0 , + ∞ ) which is fiberwise positively homogeneousof degree 0, i.e ˜Λ( λv ) = ˜Λ( v ) for each v ∈ T M \ λ >
0. Let us focus on thelatter case. Observe, first, that the generalized metric ˜ g will not come, in general,from a Finsler function. In such a generalized standard static Finsler spacetimethe set of future-pointing causal vectors J at a point ( t, x ) is given by the non-zerovectors ( τ, v ) satisfying τ ≥ F ( v ) / (cid:112) Λ( v ). Being Λ positively homogeneous of de-gree 0 and positive it satisfies C ( x ) ≤ Λ( v ) ≤ C ( x ), for some positive constants C ( x ) , C ( x ), and for all v ∈ T x M \ { } , so that F/ √ Λ can be extended by conti-nuity in 0. Thus, J is connected. Nevertheless, it is, in general, non-convex (see,e.g., Figure 1). This is in contrast to what happens in Finsler spacetimes definedthrough a Finsler function where the connected components of J are convex (see[34, Theorem 2]) and it should be considered as a serious argument against thedefinition of a Finsler spacetime through a generalized metric which does not comefrom a quadratic Finsler function. In fact, in this case, a reverse Cauchy-Schwarzinequality (see Proposition 2.22 below) cannot hold and there exist causal vectors( τ , v ) , ( τ , w ) which are in the same connected component of the set of causalvectors and such that ˜ g ( τ ,v ) (( τ , v ) , ( τ , w )) > N and two points p, q ∈ N , let Ω pq ( N ) be the set of the continuouspiecewise smooth curve γ on N parametrized on a given interval [ a, b ] ⊂ R andconnecting p to q (i.e. γ ( a ) = p , γ ( b ) = q ). If γ ∈ Ω pq ( N ), we call a (proper)variation of γ a continuous two-parameter map ψ : ( ε, ε ) × [ a, b ] → N such that ψ (0 , s ) = γ ( s ), for all s ∈ [ a, b ], ψ ( w, · ) ∈ Ω pq ( N ) and there exists a subdivision a = s < s < . . . , s k = b of the interval [ a, b ] for which ψ | ( − ε,ε ) × [ s j − ,s j ] is smoothfor all j ∈ { , . . . , k } . Clearly, we can define classes of proper variations of γ as thosesharing the same variational vector field Z . This is, by definition, a continuouspiecewise smooth vector field along γ such that Z ( a ) = 0 = Z ( b ) and Z ( s ) = ∂ψ∂r (0 , s ). By considering any auxiliary Riemannian metric h on N , we see thateach variational vector field Z along γ individuates a variation (and then also aclass of them) by setting ψ ( w, s ) := exp γ ( s ) ( wZ ( s )), for | w | < ε small enough. E. CAPONIO AND G. STANCARONE
Figure 1.
The set of the f. p. lightlike vectors (in blue) in R × R with the (flat) static metric ˜ g = − e v v v dt + dx + dy . In cyan,it is represented the plane of vectors ( τ, v , v ) which are ˜ g (1 , , -orthogonal to the lightlike vector (1 , , E : Ω pq ( ˜ M ) → R , E ( γ ) = 12 (cid:90) ba (cid:0) − Λ( σ ) ˙ ζ + F ( ˙ σ ) (cid:1) ds. As ˜ M splits as R × M , the path space Ω pq ( ˜ M ) is identifiable with the productΩ t p t q ( R ) × Ω x p x q ( M ), where ( t p , x p ) = p and ( t q , x q ) = q and any curve γ ∈ Ω pq ( ˜ M )has two components γ ( s ) = ( θ ( s ) , σ ( s )). Definition 2.12.
A continuous piecewise smooth curve γ : [ a, b ] → ˜ M is a (affinelyparametrized) geodesic of ( ˜ M , L ) if it is a critical point of the energy functional,i.e. if ddr ( E ( ψ ( r, · )) | r =0 = 0, for all proper variations ψ of γ . Theorem 2.13.
A curve γ : [ a, b ] → ˜ M , γ ( s ) = ( θ ( s ) , σ ( s )) , is a geodesic of ( ˜ M , L ) if and only if the following equations are satisfied in local natural coordinates ( t, x , . . . , x n , τ, v , . . . , v n ) on T ˜ M : − ∂ Λ ∂x i ( σ ) ˙ θ + ∂F ∂x i ( ˙ σ ) − dds (cid:18) ∂F ∂v i ( ˙ σ ) (cid:19) = 0 , i = 1 , . . . , n (2)Λ( σ ) ˙ θ = const . (3) σ and θ are, respectively, C and C on [ a, b ] and there exists a constant C ∈ R such that L ( ˙ γ ( s )) = C , for all s ∈ [ a, b ] .Moreover, if γ is a non-constant geodesic then: ( a ) if it is spacelike or lightlike(i.e. C ≥ ) then ˙ σ never vanishes and γ is smooth; ( b ) if σ is constant equal to x ∈ M on the whole interval [ a, b ] then C < , d Λ( x ) = 0 and ˙ θ is constant too;vice versa, if d Λ( x ) = 0 then, for each θ ∈ R and m (cid:54) = 0 , the curve s ∈ [ a, b ] (cid:55)→ ( θ + m ( s − a ) , x ) ∈ ˜ M is a timelike geodesic. TANDARD STATIC FINSLER SPACETIMES 7
Remark . In particular the function s ∈ [ a, b ] (cid:55)→ ∂F ∂v i ( ˙ σ ( s )) is C on [ a, b ], infact it is differentiable also at the instants s where ˙ σ ( s ) = 0. Proof.
By considering variational vector fields Z which are, respectively, of the type( Y,
0) and (0 , W ) and have compact support in any open maximal interval I where σ is smooth and ˙ σ ( s ) (cid:54) = 0, we deduce by standard arguments that in local coordinates( t, x , . . . , x n , τ, v , . . . , v n ) of T ˜ M , any critical point γ = ( θ, σ ) of E satisfies, insuch interval I , equations (2) and (3). Actually, it can be easily seen that (3) holdson all [ a, b ] and then, as σ is continuous, θ is C on [ a, b ]. If ¯ s ∈ [ a, b ] is an isolatedzero of ˙ σ ( s ), and σ is smooth in a neighbourhood J of ¯ s then, as the functions − ∂ Λ ∂x i ( σ ) ˙ θ + ∂F ∂x i ( ˙ σ ) and ∂F ∂v i ( ˙ σ ( s )) are continuous in J , we deduce, from the factthat (2) is satisfied in a left and in a right neighbourhood of ¯ s , that s (cid:55)→ ∂F ∂v i ( ˙ σ ( s ))is differentiable with continuous derivative at ¯ s and (2) is satisfied also at ¯ s . Bythis information at isolated zeroes of ˙ σ , we see that the same argument applies ata zero where σ is smooth and which is an accumulation point of isolated zeroes.On the other hand, at the instants s j , j ∈ { , . . . , k } , where ˙ σ has a break, bytaking any vector w j ∈ T σ ( s j ) M , j ∈ { , . . . , k − } , and a variational vector field(0 , W j ), such that W j ( s j ) = w j and W ≡ s j ,we get, using that (2) is satisfied both in [ s j − , s j ] and [ s j , s j +1 ], ∂F ∂v i ( ˙ σ − ( s j )) w ij = ∂F ∂v i ( ˙ σ + ( s j )) w ij , hence ∂F ∂v ( ˙ σ − ( s j )) = ∂F ∂v ( ˙ σ + ( s j )). Being the map v ∈ T x M (cid:55)→ ∂F ∂v ( v ) ∈ T ∗ x M ahomeomorphism (see e.g. [13, p.373]) we deduce that ˙ σ − ( s j ) = ˙ σ + ( s j ). Thus, σ isa C curve on [ a, b ] and, from (3), θ is a C function. Then, from s -independenceof the Lagrangian L and the fact that it is positively homogeneous of degree 2, weknow that L ( ˙ γ ( s )) = ∂L∂v ( ˙ γ ( s ))[ ˙ γ ( s )] − L ( ˙ γ ( s )) = const . := C j ∈ R in each interval[ s j , s j +1 ] where γ is smooth. As γ is a C curve on [ a, b ], the constants C j mustagree, i.e. L ( ˙ γ ( s )) = const . := C ∈ R on [ a, b ].The converse clearly follows by observing that each C curve γ which solves (2)and (3) must necessarily be a critical point of the energy functional.Last part of the theorem follows by observing that, being Λ >
0, if
C > σ (cid:54) = 0 everywhere in [ a, b ] and if γ is non-constant and C = 0 then at each instant¯ s ∈ [ a, b ] where ˙ σ (¯ s ) = 0, necessarily, also ˙ θ (¯ s ) = 0 and, from (3), ˙ θ ≡ F ( ˙ σ ) ≡
0, i.e. ˙ σ ≡
0, a contradiction since γ was not constant. As the verticalHessian of F is invertible at each non-zero vector, from (2) written in normal form,we deduce that if C ≥ σ is smooth (i.e. it is at least twice differentiable alsoat the instants s j , j ∈ { , . . . , k } . Finally, if σ is constant and equal to x ∈ M then necessarily C < θ is a non-zero constant while, from (2), ∂ Λ ∂x i ( x ) = 0. Analogously, one can check that last statement holds true. (cid:3) Definition 2.15. A pregeodesic of ( ˜ M , L ) is any C curve γ : [ c, d ] → ˜ M admittinga reparametrization ϕ : [ a, b ] → [ c, d ] which is C , regular and orientation preserving(i.e. ˙ ϕ > ) such that γ ◦ ϕ is a geodesic. As lightlike vectors w = ( v, τ ) of ( R × M, ˜ g ) are defined by the equation g v ( v, v ) − Λ τ = 0, we give, in analogy to the Lorentzian case, the following definition: Definition 2.16.
The optical “metric” of a standard static Finsler spacetime isthe positive definite homogeneous section v ∈ T M \ (cid:55)→ g v . E. CAPONIO AND G. STANCARONE
Remark . Notice that the optical metric is the fundamental tensor of the Finslermetric ˜ F = F/ √ Λ on M .Definition 2.16 is justified by the following result: Proposition 2.18.
Let ( ˜
M , L ) be a standard static Finsler spacetime. A curve γ : [ a, b ] → ˜ M , γ ( s ) = ( θ ( s ) , σ ( s )) is a future-pointing lightlike geodesic if and onlyif σ is a (non-constant) pregeodesic of the Finsler metric F/ √ Λ on M parametrizedwith (cid:112) Λ( σ ) F ( ˙ σ ) = const . and θ ( s ) = θ ( a ) + (cid:82) sa F ( ˙ σ ) √ Λ( σ ) dτ .Proof. Let us assume that γ is a future-pointing lightlike geodesic of ( ˜ M , L ). By(3) and the fact that γ is lightlike we get that (cid:112) Λ( σ ) F ( ˙ σ ) must be constant on[ a, b ]. Hence, Eq. (2) is equivalent to2 (cid:112) Λ( σ ) F ( ˙ σ ) (cid:32) − ∂ Λ ∂x i ( σ ) F ( ˙ σ )(Λ( σ )) / + 1 (cid:112) Λ( σ ) ∂F∂x i ( ˙ σ ) − dds (cid:32) (cid:112) Λ( σ ) ∂F∂v i ( ˙ σ ) (cid:33)(cid:33) = 0 . As (cid:112) Λ( σ ) F ( ˙ σ ) is positive, the above equation is satisfied if and only if − ∂ Λ ∂x i ( σ ) F ( ˙ σ )(Λ( σ )) / + 1 (cid:112) Λ( σ ) ∂F∂x i ( ˙ σ ) − dds (cid:32) (cid:112) Λ( σ ) ∂F∂v i ( ˙ σ ) (cid:33) = 0 . (4)This is the Euler-Lagrange equation (in natural local coordinates of T M ) of thelength functional associated to F/ √ Λ, hence σ is a pregeodesic of the Finsler man-ifold ( M, F/ √ Λ). The converse immediately follows by using invariance under C ,regular, orientation preserving reparametrizations of the solutions of (4). (cid:3) Remark . Notice that the length functional of the Finsler metric F/ √ Λ on thepath space Ω x p x q ( M ) coincides, up to a constant, with the arrival time functional T pl xq of the standard static Finsler spacetime ( ˜ M , L ); this is the functional definedon the set of the future-pointing lightlike curves γ connecting p with the line l x q = s (cid:55)→ ( s, x q ) and defined as T pl xq ( γ ) = t ( γ ( b )). Hence, Proposition 2.18 can beinterpreted as a Fermat’s principle for light rays in the Finsler spacetime ( ˜ M , L ),namely the critical point of T pl xq are all and only the lightlike pregeodesic of ( ˜ M , L ).For a general version of the Fermat’s principle in a Finsler spacetime defined througha quadratic Finsler function which is smooth on T ˜ M \ Remark . Let ˜ L : T ˜ M → R be the function given by ˜ L ( τ, v ) = F ( v ) − τ .We call the pair ( ˜ M , ˜ L ) the ultrastatic Finsler spacetime associated to ( ˜ M , L ) andwe denote by G the square of ˜ F : G = F and by ˜ d the distance associated to ˜ F .From Proposition 2.18 we immediately deduce that the Finsler spacetime ( ˜ M , L )and the ultrastatic one ( ˜
M , ˜ L ) share the same lightlike pregeodesics. In other words,invariance of lightlike geodesics under conformal changes of the metric (which is afundamental property of Lorentzian spacetime) also holds in the class of standardstatic Finsler spacetime under conformal factors depending only on x ∈ M .It has been known at least since [7] that the causal structure of a tangent spacein a Finsler spacetime can be weird. In fact, [7] contains some two-dimensional TANDARD STATIC FINSLER SPACETIMES 9 examples of Finsler spacetime where the causal cone at a point x (i.e. the theset of the vectors v ∈ T x ˜ M \ { } which are causal) has more than two connectedcomponent. Recently, in [34], it has been proved that such pathologies are confinedin dimension two if the Finsler spacetime is time oriented and the Finsler spacetimemetric is reversible. For a standard static spacetime ( ˜ M , L ) things are simpler asthe following proposition shows:
Lemma 2.21.
Let ( ˜
M , L ) be a standard static Finsler spacetime. Then the set offuture-pointing causal vectors at a point ( t, x ) ∈ ˜ M has only one connected convexcomponent. Moreover, for each c > , the set J ( c ) of the future-pointing timelikevectors in T ( t,x ) ˜ M such that L (˜ v ) ≤ − c } is also connected and strictly convex.Proof. Observe that, by definition, future-pointing causal vectors ˜ v = ( τ, v ) ∈ T ( t,x ) ˜ M are all and only the non-zero vectors satisfying τ ≥ ˜ F ( v ) (recall that,for all ˜ v ∈ T ˜ M \ T , as ˜ g ˜ v (˜ v, ∂∂t ) ≤ τ has non-negative sign). Being ˜ F continuouson T x M and fiberwise convex (see, e.g [46, Lemma 1.2.2]) its epigraph in T x M isconnected and convex. For the last part of the proposition, observe that ˜ v ∈ J ( c ) ifand only if τ ≥ (cid:112) G ( v ) + α where α = c/ Λ( x ). A simple computation shows thatfor each v ∈ T x M \ { } , the fiberwise Hessian of √ G + α at v is given by ∂ ˜ F∂v i ( v ) ∂ ˜ F∂v j ( v ) (cid:112) G ( v ) + α (cid:18) − G ( v ) G ( v ) + α (cid:19) + ˜ F ( v ) (cid:112) G ( v ) + α ∂ ˜ F∂v i ∂v j ( v ) . Since ∂ ˜ F∂v i ∂v j ( v ) w i w j ≥ w = λv , for some λ ∈ R and, moreover, ∂ ˜ F∂v i ( v ) v i = F ( v ) >
0, we get that the Hessian of √ G + α is positivedefinite for any v ∈ T x M \
0. At v = 0, observe that the differential of √ G + α is 0 and (cid:112) G ( v ) + α > √ α = (cid:112) G (0) + α . Thus, being √ G + α a C function on T x M we conclude that it is strictly convex and its epigraph is also (connected) andstrictly convex. (cid:3) By Lemma 2.21 we get the following reverse Cauchy-Schwarz inequality:
Proposition 2.22.
Let ( ˜
M , L ) be a standard static spacetime and ˜ v, ˜ w ∈ T ( t,x ) ˜ M ,future-pointing causal vectors. Then − ∂L∂ ˜ v i (˜ v ) ˜ w i ≥ (cid:112) − L (˜ v ) (cid:112) − L ( ˜ w ) , (5) with equality if and only if ˜ v and ˜ w are proportional.Proof. Recalling that L is C in T ˜ M and smooth outside T , the same proof of [34,Theorem 3] gives − ˜ g ˜ v (˜ v, ˜ w ) ≥ (cid:112) − L (˜ v ) (cid:112) − L ( ˜ w ) , for any ˜ v ∈ T ( t,x ) ˜ M \ T ( t,x ) and ˜ w ∈ T ( t,x ) ˜ M , with equality if only if ˜ v and ˜ w areproportional. Then (5) extends to any ˜ v ∈ T ( t,x ) ˜ M by continuity, recalling that, byhomogeneity, − ∂L∂ ˜ v i (˜ v ) ˜ w i = − ˜ g ˜ v (˜ v, ˜ w ). Moreover if ˜ v ∈ T , so ˜ v = ( τ ,
0) for some τ > w ∈ T . In fact if ˜ w = ( τ , w ),with w (cid:54) = 0, then the right-hand side of (5) is equal to τ (cid:112) Λ( x ) (cid:112) Λ( x ) τ − F ( w )which is strictly less than the left-hand side equal to Λ( x ) τ τ . (cid:3) Remark . In particular, the above proposition implies that if ˜ v, ˜ w ∈ T ( t,x ) ˜ M \T ( t,x ) , are future-pointing causal vectors then ˜ g ˜ w ( ˜ w, ˜ v ) ≤ g ˜ v (˜ v, ˜ w ) ≤
0; more-over ˜ g ˜ v (˜ v, ˜ w ) = 0 if and only ˜ v and ˜ w are proportional and lightlike. Causality
In a Lorentzian spacetime (
M, g ), two events p, q ∈ M are said chronologically (resp. causally ) related and denoted with p (cid:28) q (resp. p ≤ q ), if there existsa future-pointing timelike (resp. causal) curve γ from p to q ; p is said strictlycausally related to q , denoted with p < q , if p ≤ q and p (cid:54) = q . The chronological (resp. causal ) future of p ∈ M is defined as I + ( p ) := { q ∈ M : p (cid:28) q } ( J + ( p ) = { q ∈ M : p ≤ q } ). Analogous notions appear reversing the binary relation (cid:28) ,namely we have the chronological (resp. causal ) past I − ( p ) = { q ∈ M : q (cid:28) p } ( J − ( p ) = { q ∈ M : q ≤ p } ). For further details see [8], [36].The above definitions can be trivially extended to a Finsler spacetime ( ˜ M , ˜ g )but notice that in the Lorentzian setting the chronological and the causal past ofan event p can be equivalently defined by considering past-pointing timelike andcausal curves starting at p . Clearly, in a Finsler spacetime, this is true only when thegeneralized metric is absolutely homogeneous, i.e. ˜ g v = ˜ g − v for any v ∈ T ˜ M \ T .In the general case, given a future-pointing causal vector v or a future-pointingcausal curve, − v and the curve parametrized with the opposite orientation can bespacelike. So the chronological (resp. causal) past of an event p will be defined byconsidering only future-pointing timelike (resp. causal) curves arriving at p .One immediate consequence of the definitions of the chronological and the causalfuture of a point is that, for all p ∈ ˜ M : I + ( p ) ⊂ J + ( p ) , I − ( p ) ⊂ J − ( p ) . (6) Remark . After [13] and mostly [14, 12], it is clear that Finsler geometry playsa prominent role in the description of the causal structure of a class of stationaryspacetime which generalizes the standard static one in the following sense: underthe same notation as in Section 2, let us also consider a one-form on M ; then definea Lorentzian metric g on the product R × M as g = g + ω ⊗ dt + dt ⊗ ω − Λ dt , where ω is the pullback on R × M of the one form on M . A Finsler metric R of Randerstype emerges then as the optical metric of ( R × M, g ), R = ( ω + √ Λ g + ω ⊗ ω ).Analogously, the metric structure of the Finsler manifold ( M, F/ √ Λ) associated toa static Finsler spacetime ( ˜
M , L ) can be related to its causal structure. As a firstexample, the following proposition analogous to [14, Prop. 4.2] holds:
Proposition 3.2.
Let ( ˜
M , L ) be a standard static Finsler spacetime. For all p =( t , x ) ∈ ˜ M we have: I + ( p ) = (cid:91) r> (cid:0) { t + r } × B + ( x , r ) (cid:1) , (7) I − ( p ) = (cid:91) r> (cid:0) { t − r } × B − ( x , r ) (cid:1) , where B + ( x , r ) and B − ( x , r ) denote, respectively, the forward and the backwardopen ball (see [5, § ) of centre x and radius r of the Finsler metric ˜ F = F/ √ Λ .Moreover, I ± ( p ) are open subsets of ˜ M .Proof. Let us reasoning only for I + as the statements for I − can be proved anal-ogously. Let ( t, x ) ∈ I + ( t , x ) and γ ( s ) = ( θ ( s ) , σ ( s )) be a timelike curve joining( t , x ) to ( t, x ), so we have that θ is an increasing function and ˜ F ( ˙ σ ) < ˙ θ . Inte-grating this inequality, we get ˜ d ( x , x ) < t − t . Hence, x ∈ B + ( x , t − t ) and weconclude ( t, x ) ∈ { t + r } × B + ( x , r ), r = t − t . Conversely, let x ∈ B + ( x , r ), for TANDARD STATIC FINSLER SPACETIMES 11 some r >
0, and σ : [0 , a ] → M be a unit ( F/ √ Λ)-speed curve joining x to x , suchthat a < r . The curve γ ( s ) = ( t + ra s, σ ( s )) is then future-pointing , timelike andconnects the points ( t , x ) and ( t + r, x ).For the last statement observe that if p = ( t, x ) ∈ I + ( p ) then x ∈ B + ( x , t − t ).Let ε = ( t − t − ˜ d ( x , x )) / t − ε, + ∞ ) × B + ( x, ε ) is contained in I + ( p ). (cid:3) As claimed by Barrett O’Neill [36, p. 293], a fundamental problem in a Loren-tzian manifold is to determine which pairs of points can be joined by a timelikecurve. Our aim, next, is to extend to standard static Finsler spacetimes [36, Prop.10.46] stating that there are timelike curves from p to q arbitrarily near to everycausal curve γ which is not a lightlike pregeodesic. This properties has been alreadyproved in [1, Prop. 7.6], anyway we give here an elementary proof that exploits thesplitting R × M .Let γ : [ a, b ] → ˜ M , γ ( s ) = ( θ ( s ) , σ ( s )) be a curve and ψ : ( − ε, ε ) × [ a, b ] → ˜ M , ψ ( w, s ) = ( ζ ( w, s ) , η ( w, s )), be a variation of γ with variational vector field W =( Y, Z ). Let us denote by ψ w , w ∈ ( − ε, ε ), the (longitudinal) curve in the variationgiven by ψ w : [ a, b ] → ˜ M , ψ w ( s ) = ψ ( w, s ) and by ˙ ψ w its velocity, ˙ ψ w = ∂∂s ψ ( w, s ). Lemma 3.3.
Let γ be a continuous piecewise smooth causal curve and ψ a variationof γ such that ∂∂w L ( ˙ ψ w ) | w =0 < then, for sufficiently small w > , the associatedlongitudinal curve ψ w is timelike.Proof. As ∂∂w ˜ g ( ˙ ψ w , ˙ ψ w ) | w =0 = ∂∂w L ( ˙ ψ w ) | w =0 < . and ˜ g ( ˙ γ, ˙ γ ) ≤ (cid:3) In any chart (cid:0) ( R × U ) × ( R × R n ) , ( t, x , . . . , x n , τ, v , . . . , v n ) (cid:1) of T ˜ M , such that γ ([ a, b ]) ∩ ( R × U ) (cid:54) = ∅ , we have ∂∂w L ( ˙ ψ w ) = ∂∂w (cid:16) − Λ( η w ) ˙ ζ w + F ( ˙ η w ) (cid:17) = (cid:18) − ∂ Λ ∂x i ( η w ) ˙ ζ w + ∂F ∂x i ( ˙ η w ) (cid:19) ∂η iw ∂w − η w ) ˙ ζ w ∂ ˙ ζ w ∂w + ∂F ∂v i ( ˙ η w ) ∂ ˙ η iw ∂w , up to the finite number of instants s j where, eventually, ˙ γ has breaks (clearly, theabove equation is satisfied, separately, in some intervals of the type ( s j − ε j , s j ],[ s j , s j + ε j ), ε j , ε j > γ (cid:0) ( s j − ε j , s j + ε j ) (cid:1) ⊂ R × U ).For w = 0, the above equation yields ∂∂w L ( ˙ ψ w ) | w =0 = (cid:18) − ∂ Λ ∂x i ( σ ) ˙ θ + ∂F ∂x i ( ˙ σ ) (cid:19) Z i − σ ) ˙ θ ˙ Y + ∂F ∂v i ( ˙ σ ) ˙ Z i . (8)Now we consider the following equation in each coordinate system as above: ∂F ∂v i ( ˙ σ ) ˙ Z i + (cid:18) ∂F ∂x i ( ˙ σ ) − ∂ Λ ∂x i ( σ ) ˙ θ (cid:19) Z i − σ ) ˙ θ ˙ Y = − α (9)where α is a positive constant. Proposition 3.4.
Let ρ : [ a, b ] → ˜ M , ρ ( s ) = ( ρ ( s ) , ρ ( s )) , be a future-pointingcausal curve of ( ˜ M , L ) that is not a lightlike pregeodesic, then there exists a propervariation of ρ by timelike curves. Proof. If ρ is a timelike curve then the thesis follows by a simple continuity argu-ment. Then assume that ρ is not timelike. Being ρ causal and future-pointing,Λ( ρ ) ˙ ρ > a, b ], thus we can reparametrize ρ on the same interval [ a, b ] to ob-tain a curve γ = γ ( s ) = ( θ ( s ) , σ ( s )) such that Λ( σ ) ˙ θ = C for some positive constant C . Let us consider a covering { R × U i } i ∈{ ,...,k } of γ by k charts of ˜ M and a subdi-vision a = s < s < . . . < s k of the interval [ a, b ] such that γ ([ s j − , s j ]) ⊂ R × U j for all j ∈ { , . . . , k } . As ρ was not a lightlike pregeodesic, necessarily it must exista piecewise smooth vector field Z along σ , with Z ( a ) = Z ( b ) = 0 such that k (cid:88) j =1 (cid:90) s j s j − (cid:18) ∂F ∂v i ( ˙ σ ) ˙ Z i + (cid:18) ∂F ∂x i ( ˙ σ ) − ∂ Λ ∂x i ( σ ) ˙ θ (cid:19) Z i (cid:19) ds (cid:54) = 0 , otherwise, from (2), γ would be a geodesic of ( ˜ M , L ) and then (being causal andnot timelike) necessarily a lightlike one. Clearly, up to consider the opposite vector − Z , we can assume that the above summation is negative. Let us define on thecoordinate system associated to R × U j h j = ∂F ∂v i ( ˙ σ ) ˙ Z i + (cid:18) ∂F ∂x i ( ˙ σ ) − ∂ Λ ∂x i ( σ ) ˙ θ (cid:19) Z i , j ∈ { , . . . , k } , then Y ( s ) = 12 C m ( s ) (cid:88) j =1 (cid:90) s j s j − h j ( µ ) dµ + (cid:90) ss m ( s ) h m ( s )+1 dµ + α ( s − a ) , where m ( s ) ∈ { , . . . , k − } , such that s ∈ ( s m ( s ) , s m ( s )+1 ] (with the conventionthat if m ( s ) = 0 then the first term in the right-hand side is equal to 0), solves (9)and for α = − b − a (cid:80) kj =1 (cid:82) s j s j − h j ( µ ) dµ we get that α > Y ( b ) = 0. From (8),this implies that ∂∂w L ( ˙ ψ w ) | w =0 = − α < , and then, by Lemma 3.3, we conclude. (cid:3) As a consequence of Proposition 3.4 we immediately get the following fundamen-tal properties of the relations (cid:28) and ≤ (compare also with [35, Corollary1]): Corollary 3.5.
Let p, q, z ∈ ˜ M . If p ≤ q and q (cid:28) z (or vice versa) then p (cid:28) z . Moreover, the equality between the closures of the chronological and the causalfuture of a point also easily follows:
Corollary 3.6.
For all p ∈ ˜ M , J + ( p ) = I + ( p ) (and J − ( p ) = I − ( p ) ).Proof. From (6), it is enough to show that J + ( p ) ⊂ I + ( p ). Let q = ( t, x ) ∈ J + ( p ).Clearly, if q = p then q ∈ I + ( p ). So, let us assume that p < q ; let γ be acausal future-pointing curve between p and q and and consider a sequence of points q k = ( t k , x ), with t k (cid:38) t . As the line l x is timelike, from Corollary 3.5, we get p (cid:28) q k , hence q ∈ I + ( p ). (cid:3) Let us now give a first definition in the causal hierarchy of Finsler spacetimeswhich, as the following ones in Definitions 3.9 and 3.14 are formally the same as inLorentzian spacetimes.
TANDARD STATIC FINSLER SPACETIMES 13
Definition 3.7.
A Finsler spacetime ( ˜
M , ˜ g ) is causally simple if for all p ∈ ˜ M , J ± ( p ) are closed and no closed future-pointing causal curve exists.The following result extends to standard static Finsler spacetimes [14, Th. 4.3-(a)]. Theorem 3.8.
A standard static Finsler spacetime ( ˜
M , L ) is causally simple ifand only if for any couple ( x, y ) ∈ M × M there exists a geodesic of the metric ˜ F = F/ √ Λ joining x to y with length equal to the their ˜ F -distance, ˜ d ( x, y ) .Proof. ( ⇒ ) Let x, y ∈ M and let σ : [0 , → M be a piecewise smooth curvecurve, say, from x to y . The curve γ ( s ) = (cid:16)(cid:82) s ˜ F ( ˙ σ ) dµ, σ ( s ) (cid:17) is then lightlikeand future-pointing. This shows that J + (0 , x ) ∩ l y (cid:54) = ∅ . Let t y := inf { t ∈ R :( t, y ) ∈ J + (0 , x ) } ; clearly, ( t y , y ) ∈ J + (0 , x ) = J + (0 , x ). Moreover ( t y , y ) (cid:54)∈ I + (0 , x )otherwise, being I + (0 , x ) open there would exist ( t, y ) ∈ I + (0 , x ) with t < t y . Thus,from Proposition 3.4, any causal future-pointing curve connecting (0 , x ) to ( t y , y )must be a lightlike geodesic and, from Proposition 2.18, its component on M isa pregeodesic of ( M, ˜ F ). Moreover, the ( ˜ F )-length of this component must beequal to ˜ d ( x, y ) otherwise it would be possible to consider, as above, a lightlikefuture-pointing curve whose future endpoint would have t -coordinate less than t y .( ⇐ ) Let ( t , x ) ∈ ˜ M and ( t, x ) ∈ J + ( t , x ). Consider a geodesic σ : [0 , → M connecting x to x and such that (cid:96) ( σ ) = ˜ d ( x , x ). Then, the curve γ ( s ) = (cid:16) t + (cid:82) s ˜ F ( ˙ σ ) dµ, σ ( s ) (cid:17) is lightlike and future-pointing. Let { ( t k , x k ) } be a sequenceof points such that ( t k , x k ) ∈ J + ( t , x ) and ( t k , x k ) → ( t, x ), as k → + ∞ . Con-sider a sequence of future-pointing causal curves γ k , each one connecting ( t , x )to ( t k , x k ), so that ˜ d ( x , x k ) ≤ t k − t . Then, by the continuity of ˜ d , we get˜ d ( x , x ) ≤ t − t . Thus, if ˜ d ( x , x ) < t − t then, by (7), ( t , x ) (cid:28) ( t, x ) and, if˜ d ( x , x ) = t − t the same lightlike curve γ gives ( t , x ) ≤ ( t, x ). (cid:3) Since the Finslerian distance ˜ d of ˜ F is not symmetric, the lack of symmetry givesrise to two notions of completeness: the forward completeness and the backward one(see [5, § M, F ) is a sequence { x k } k ∈ N with the property that for all ε > ν ∈ N such that for all k > j ≥ ν , d ( x j , x k ) < ε (resp. d ( x k , x j ) < ε ), where d is thedistance associated to F . Then ( M, F ) is said forward (resp. backward ) complete ifany forward (resp. backward) Cauchy sequence is convergent.Forward and backward completeness of the metric ˜ F are related to the existenceof some particular Cauchy hypersurfaces in ( ˜ M , L ). Definition 3.9. A Cauchy hypersurface in a Finsler spacetime ( ˜
M , ˜ g ) is a topo-logical hypersurface that is met exactly once by every inextensible future-pointingcausal curve.The following theorem extends to standard static Finsler spacetime a well knownresult, already cited at the beginning of Section 2, valid for standard static Loren-tzian spacetimes, and [14, Theorem 4.4] for standard stationary ones. Theorem 3.10.
A slice (and then any slice) S t = { t }× M is a Cauchy hypersurfaceof the standard static Finsler spacetime ( ˜ M , L ) if and only the Finsler manifold ( M, ˜ F ) is forward and backward complete. Before proving Theorem 3.10 we need the following well known result that wereport for the reader convenience.
Lemma 3.11.
Let ( M, F ) be a Finsler manifold. Then ( M, F ) is forward (resp.backward) complete if and only if for any piecewise smooth curve of finite Finslerianlength σ : [ a, b ) → M (resp. σ : ( a, b ] → M ) there exists a point x ∈ M such that σ ( s ) → x , as s → b (resp. as s → a ).Proof. Let us show the equivalence only for forward completeness since the back-ward case is completely analogous.( ⇒ ) Let (cid:96) ( σ ) ∈ (0 , + ∞ ) be the length of σ . Set K = { x ∈ M : d ( σ ( a ) , x ) ≤ (cid:96) ( σ ) } .Since K is forward bounded and closed, we have that it is compact by Hopf-Rinowtheorem (see [5, Th.6.6.1]). Fix a sequence { s k } in [ a, b ) such that s k → b . Since σ ([ a, b )) ⊆ K , the compactness implies that there exists a point x ∈ K such that σ ( s k ) converges to x , up to subsequences. If lim s → b σ ( s ) (cid:54) = x , there would existan ε > σ leaves the set { x ∈ M : d ( x, x ) ≤ ε } infinite times while,definitively, d ( σ ( s k ) , x ) < ε/
2. This is a contradiction because σ has a finite length.( ⇐ ) We suppose that ( M, F ) is not forward complete manifold. Therefore, thereexists a geodesic σ : [ a, b ) → M , hence with finite length, which is not extendibleto s = b . (cid:3) Proof of Theorem 3.10. ( ⇒ ) By contradiction, assume that ( M, ˜ F ) is not, say, for-ward complete. Then, from Lemma 3.11, there exists σ : [ a, b ) → M having finitelength, ˜ (cid:96) ( σ ) = (cid:82) ba ˜ F ( ˙ σ ) ds , and such that it does not converge as s → b . Take t ∈ R such that t + (cid:96) ( σ ) < t and define θ : [ a, b ) → R , θ ( s ) = t + ˜ (cid:96) ( σ | [ a,s ] )and γ : [ a, b ) → R × M , γ ( s ) = ( θ ( s ) , σ ( s )). Observe that γ is a future-pointinglightlike curve, by definition, and it is future-inextensible because σ is inextensible.Hence, any extension of γ as a past inextensible causal curve does not intersect S t ,a contradiction.( ⇐ ) Let ( t , x ) ∈ R × M , with t < t . We consider a future-inextensible causalcurve γ : [ a, b ) → R × M , γ ( s ) = ( θ ( s ) , σ ( s )) that starts from ( t , x ). As θ is an increasing function, γ cannot intersect S t more than once. So assume, bycontradiction, that γ does not meet S t then, as θ is also continuous, lim s → b θ ( s ) ∈ ( t , t ]. Moreover, being γ causal, (cid:82) sa ˜ F ( ˙ σ ) dµ ≤ θ ( s ) − t , for all s ∈ [ a, b ). Hence,there exists lim s → b − (cid:82) sa ˜ F ( ˙ σ ) dµ ∈ [0 , t − t ], i.e σ has a finite length and thereforefrom Lemma 3.11 it is extendible. Hence, γ is future extendible. The proof for apast-inextensible curve is analogous. (cid:3) Notice that, as L and ˜ L share the same causal curves, S t = { t } × M is aCauchy hypersurface also for ( ˜ M , ˜ L ). Anyway, the completeness assumptions inTheorem 3.10 involve the optical metric ˜ F and not F . We can give some conditionson Λ ensuring forward and backward completeness of the conformal metric ˜ F = F/ √ Λ, provided that F is forward and backward complete. To this end, let usintroduce the following notions.Given a Finsler manifold ( M, F ) and a function Λ : M → R , we say that Λ is forward (resp. backward ) subquadratic , if there exist constants c , c > x ∈ M , | Λ( x ) | ≤ c d (¯ x, x ) + c ( | Λ( x ) | ≤ c d ( x, ¯ x ) + c ) for some ¯ x ∈ M .Clearly, by the triangle inequality, one immediately sees that previous definitionsare independent from the point ¯ x (up to change the constants c and c ). Moreover,we say that Λ is subquadratic if it is both forward and backward subquadratic. TANDARD STATIC FINSLER SPACETIMES 15
Proposition 3.12.
Let ( M, F ) be a forward (resp. backward) complete Finslermanifold. If Λ is a positive smooth forward (resp. backward) subquadratic functionthen ( M, ˜ F ) is forward (resp. backward) complete.Proof. Let { x k } k ∈ N be a forward Cauchy sequence for ( M, ˜ F ) and let us denote by˜ d the distance induced by ˜ F . We can find two indexes ν and ¯ m such that, for all k > ¯ m > ν , ˜ d ( x ¯ m , x k ) < √ c . Let σ k : [0 , → M be a curve joining x ¯ m to x k andsuch that (cid:90) ˜ F ( ˙ σ k ) ds < √ c . Now we evaluate the F -distance d ( x ¯ m , x k ): (cid:90) F ( ˙ σ k ) ds = (cid:90) (cid:112) Λ( σ k ) F ( ˙ σ k ) (cid:112) Λ( σ k ) ds ≤ (cid:90) ˜ F ( ˙ σ k )( c d (¯ x, σ k ) + c ) ds ≤ (cid:90) ˜ F ( ˙ σ k )( c (2 d ( x ¯ m , σ k ) + 2 d (¯ x, x ¯ m ) ) + c ) ds ≤ (cid:90) ˜ F ( ˙ σ k ) (cid:32) c (cid:18)(cid:90) s F ( ˙ σ k ) dµ (cid:19) + 2 c d (¯ x, x ¯ m ) + c (cid:33) ds< √ (cid:90) F ( ˙ σ k ) ds (cid:32) c d (¯ x, x ¯ m ) + c c (cid:82) F ( ˙ σ k ) ds (cid:33) ds. This shows that the sequence { (cid:82) F ( ˙ σ k ) ds } m ∈ N is bounded. Thus { x k } is forwardbounded with respect to d and therefore it admits a converging subsequence. Theproof in the backward case is analogous. (cid:3) From the above proposition and Theorem 3.10 we get:
Corollary 3.13. If ( M, F ) is forward and backward complete and Λ is subquadratic,then S t = { t } × M is a Cauchy hypersurface in ( ˜ M , L ) . The strongest property in the causal hierarchy of spacetimes is global hyperbol-icity:
Definition 3.14.
A Finsler spacetime ( ˜
M , ˜ g ) is globally hyperbolic if it is causal(i.e. no future-pointing closed causal curve exists) and J + ( p ) ∩ J − ( q ) is a compactsubset, for every p, q ∈ ˜ M .Classically, a spacetime is globally hyperbolic if it is strongly causal (namely, noalmost closed future-pointing causal curve exists). As shown recently in [11], theweaker property of being causal can be used instead. Anyway, in a standard staticFinsler spacetime both conditions hold because every future-pointing causal curvehas increasing t -component. The following theorem is the analogous of [14, Th.4.3-(b)] which concerns standard stationary Lorentzian spacetimes. Theorem 3.15.
A standard static Finsler spacetime ( ˜
M , L ) is globally hyperbolicif and only if ¯ B + ( x, r ) ∩ ¯ B − ( y, s ) is compact, for every x, y ∈ M and r, s > ,where ¯ B ± are the the closure of the forward and backward balls on M w.r.t. to thedistance ˜ d associated to F/ √ Λ . Proof. ( ⇒ ) Let us first show that J + ( t , x ) is closed for all ( t , x ) ∈ ˜ M . Let( t, x ) ∈ J + ( t , x ). Clearly we can find ( t , x ) such that ( t, x ) ∈ J + ( t , x ) ∩ I − ( t , x ). Then, being I − ( t , x ) open, any sequence { ( t k , x k ) } ⊂ J + ( t , x ) andconverging to ( t, x ) is definitively contained in I − ( t , x ) and admits, by globalhyperbolicity, a converging subsequence to a point (˜ t, ˜ x ) ∈ J + ( t , x ) ∩ J − ( t , x ).As (˜ t, ˜ x ) must be equal to ( t, x ), it follows that J + ( t , x ) is closed. Analogously,the same holds for J − ( t , x ). Thus, for any ( r, x ) ∈ ˜ M , by Proposition 3.2 andCorollary 3.6, we get { } × (cid:0) ¯ B + ( x, r ) ∩ ¯ B − ( x, r ) (cid:1) = I + ( − r, x ) ∩ I − ( r, x ) = J + ( − r, x ) ∩ J − ( r, x ) = J + ( − r, x ) ∩ J − ( r, x ) . Hence, the left-hand side is a compact set and this can be easily seen to be equivalentto the thesis (see, e.g. [14, Prop. 2.2]).( ⇐ ) For every ( t , x ) , ( t , x ) ∈ ˜ M , by Proposition 3.2 and Corollary 3.6, we get: J + ( t , x ) ∩ J − ( t , x ) = (cid:91) s ∈ [0 ,t − t ] { t + s } × (cid:0) ¯ B + ( x , s ) ∩ ¯ B − ( x , t − t − s ) (cid:1) . As in [14, p. 936], one obtains that the right-hand side is a compact subset and,then, J + ( t , x ) ∩ J − ( t , x ) is compact. From [14, Prop. 2.2 and Th. 5.2], weknow that the compactness of ¯ B + ( x, r ) ∩ ¯ B − ( y, s ), for any x, y ∈ M and r, s > x, y ∈ M can be joined by a geodesic from x to y with length ˜ d ( x, y ). Hence, from Theorem 3.8, both J + ( t , x ) and J − ( t , x ) areclosed and this concludes the proof. (cid:3) Remark . Notice that, in the proof of Theorem 3.15 we have also shown thatglobal hyperbolicity implies causal simplicity as in the causal ladder for Lorentzianspacetime. From the Finslerian Hopf-Rinow theorem (see [5, Th.6.6.1]), it is clearthat forward and backward completeness of the metric ˜ F implies compactness ofthe intersections of the closed balls (in other words, the fact that a slices S t isa Cauchy hypersurface implies that ( ˜ M , L ) is globally hyperbolic). Anyway, thelatter condition is weaker than forward or backward completeness (see [14, Ex-ample 4.6]), so a standard static Finsler spacetime might be globally hyperbolicbut with slices S t which are not Cauchy hypersurface. It is well known that in aLorentzian spacetime ˜ M , global hyperbolicity is equivalent to the existence of a smooth , spacelike Cauchy hypersurface [9] and a
Cauchy temporal function [10], i.e.a smooth function f : ˜ M → R which is strictly increasing on any future-pointingcausal curve and whose level sets are (spacelike, smooth) Cauchy hypersurfaces. Asobserved in [24, Theorem 1], the existence of a Cauchy temporal function followsalso in a Finsler spacetime by using weak KAM theory as shown in [19]. Indeed,[19, Th. 1.3] extends the result in [10] to a manifold ˜ M endowed with a conestructure C , i.e. a continuous map (w.r.t. the Hausdorff metric in local coordi-nates) p ∈ ˜ M (cid:55)→ C p ⊂ T x ˜ M , where C p is a closed convex cone with vertex at0, with non-empty interior and not containing any complete affine line. Clearly,from Lemma 2.21, the set of future-pointing causal vectors plus the zero section ina standard static Finsler spacetime defines a cone structure. It is however worthpointing out that, when applied to Finsler spacetimes, [19, Th. 1.3] does not auto-matically gives a spacelike hypersurface (in the sense of Definition 1.6). This is dueto the fact that a cone structure takes into account only future-pointing directions TANDARD STATIC FINSLER SPACETIMES 17 and the cones might flatten “at infinity”, i.e. along some lines corresponding tofuture-pointing lightlike vectors at the points outside an arbitrarily large compactsubset of ˜ M . Thus, in principle, the tangent bundle of a level set of a temporalfunction might contain vectors which are causal and past-pointing.For this reason, we introduce the following: Definition 3.17.
Let ( ˜
M , L ) be a Finsler spacetime. A hypersurface
H ⊂ ˜ M issaid future spacelike if T H does not contain any future-pointing causal vector.We can now show that a future Cauchy hypersurface can be always constructedas the graph of a smooth function on M (as for a standard stationary Lorentzianspacetime, see the proof of [14, Th. 5.10]). Proposition 3.18.
Let ( ˜
M , L ) be a globally hyperbolic standard static Finslerspacetime such that ˜ F is not forward or backward complete. Then there existsa smooth function f : M → R such that S f = { ( f ( x ) , x ) : x ∈ M } is a futurespacelike, smooth, Cauchy hypersurface.Proof. As ( ˜
M , L ) is globally hyperbolic, from Theorem 3.15, the sets ¯ B + ( x, r ) ∩ ¯ B − ( x, r ) are compact for any x ∈ M and r >
0. Then from [32, Th. 1], there existsa smooth function f : M → R such that ˜ F − df is a backward and forward completeFinsler metric on M (in particular, ˜ F ( v ) − df ( v ) > v ∈ T M \ v ∈ T S f , ˜ v = ( df ( v ) , v ), v ∈ T M , we have L (˜ v ) = − Λ( x ) (cid:0) df ( v ) (cid:1) + F ( v ),where x = ˜ π (˜ v ). Being ˜ F ( v ) − df ( v ) > v ∈ T M \
0, we conclude that L (( df ( v ) , v )) > v such that df ( v ) > S f is a futurespacelike hypersurface. Let γ : ( a, b ) → ˜ M , γ ( s ) = ( θ ( s ) , σ ( s )), be an inextensiblecausal curve and assume that γ intersects S f at least twice at the instants s , s ∈ ( a, b ), s < s . Thus, we have0 = (cid:90) s s (cid:0) ˙ θ − df ( ˙ σ ) (cid:1) ds ≥ (cid:90) s s (cid:0) ˜ F ( ˙ σ ) − df ( ˙ σ ) (cid:1) ds, which is possible if and only if σ is constant on [ s , s ] but this is a contradictionbecause S f is the graph of f . Let now assume that γ does not intersect S f . Then,being θ and σ continuous, it must be either θ − f ◦ σ < θ − f ◦ σ > a, b ).Assume that the former inequality holds and take s ∈ ( a, b ). We have, for any s > s , (cid:90) ss ˜ F ( ˙ σ ) − df ( ˙ σ ) dµ ≤ θ ( s ) − θ ( s ) − f ( σ ( s )) + f ( σ ( s )) < f ( σ ( s )) − θ ( s ) . Therefore σ | [ s ,b ) has finite length w.r.t. ˜ F − df . As this metric is forward complete,from Lemma 3.11, σ is extendible in b , i.e. there exist x b ∈ M such that x b =lim s → b σ ( s ). So, by continuity of f and monotonicity of θ also θ is extendible in b ,which contradicts the fact that γ was future inextensible. By a similar reasoning,using that ˜ F − df is also backward complete and γ is past inextensible, we obtainthat also the second inequality cannot hold. (cid:3) The following proposition shows that, under the condition of finite reversibility (10) (reversibility is a measure of how much a Finsler metric is far from beingreversible; it was introduced in [41]), it is possible to modify f to get a smooth,spacelike (not only future spacelike), Cauchy hypersurface. Proposition 3.19.
Under the assumptions and notations of Proposition 3.18, as-sume also that α := sup v ∈ T M \ F ( v ) F ( − v ) < + ∞ , (10) then S f/α is a spacelike Cauchy hypersurface.Proof. We show that, under (10), both ˜ F − df /α and ˜ F − − df /α , are Finsler metricwith the former which is also forward and backward complete, where ˜ F − denotesthe reverse Finsler metric associated to ˜ F , ˜ F − ( v ) := ˜ F ( − v ), and f is a functionsuch that ˜ F − df is a forward and backward complete Finsler metric, [32, Th. 1].Observe that α ≥ F is a reversible Finslermetric, i.e. F ( v ) = F ( − v ) for all v ∈ T M . Clearly, we have ˜ F ( v ) − df ( v ) α > v ∈ T M \
0. Moreover, for all v ∈ T M \ df ( v ) ≥
0, we have df ( v ) α ≤ df ( v ) F ( − v ) F ( v ) = df ( v ) ˜ F ( − v )˜ F ( v ) < ˜ F ( − v ) = ˜ F − ( v ). On the other hand, if df ( v ) < F − ( v ) − df ( v ) α >
0. Hence, both ˜ F − df /α and ˜ F − − df /α , are Finsler metric (see,e.g., [23, Cor. 4.17]). Clearly, ˜ F − df /α remains backward and forward completebecause it is 1 /α -homothetic, to α ˜ F − df . Thus the same proof of Proposition 3.18shows that S f/α is a future spacelike Cauchy hypersurface. Actually, for df ( v ) < F − ( − v ) > − df ( v ) α , hence − ˜ F ( v ) < df ( v ) α which is equivalent, togetherwith ˜ F ( v ) > df ( v ) α , to F ( v ) − Λ( x ) ( df ( v )) α >
0, for all v ∈ T x M and x ∈ M . Since, T S f/α = { ( df ( v ) α , v ) : v ∈ T M } and L (( df ( v ) α , v )) = F ( v ) − Λ( x ) ( df ( v )) α , we havedone. (cid:3) We have seen in Theorem 3.8 that causal simplicity is equivalent to the existenceof a ˜ F -length minimizing geodesic between any couple of points ( x, y ) ∈ M × M andthen from Proposition 2.18 and Remark 2.19 to the existence of a future-pointinglightlike geodesic between, ( t , x ) and l y (for any t ∈ R ) which minimizes thearrival time functional T ( t ,x ) l y . Under non-trivial topology of the manifold M wecan obtain also a multiplicity result. Corollary 3.20.
Let ( ˜
M , L ) be a globally hyperbolic standard static Finsler space-time such that M is not contractible. Then for any p, q ∈ ˜ M there exist infinitelymany lightlike future-pointing geodesics { γ k } k ∈ N connecting p to l x q and such that T pl xq ( γ k ) → + ∞ , as k → + ∞ .Proof. From global hyperbolicity, ¯ B + ( x, r ) ∩ ¯ B − ( y, s ) is compact for any x, y ∈ M and r, s >
0. Then from [14, Th. 5.2] we know that there exists a sequence { σ k } ofgeodesics for ˜ F from x p to x q such that ˜ (cid:96) ( σ k ) → + ∞ . Thus, it is enough to applyProposition 2.18 and Remark 2.19. (cid:3) Remark . The above corollary is based on L¨usternik-Schnirelmann theory forFinsler geodesics as developed in [13]. Indeed the fact that M is not contractibleimplies that the L¨usternik-Schnirelmann category of the based loop space is equalto + ∞ and compact subsets of it, with arbitrarily large category, do exist [18].Similar results concerning existence and multiplicity of geodesics of ( ˜ M , L ) betweentwo given events can be obtained by matching the results in [31] and the ones in[13].
TANDARD STATIC FINSLER SPACETIMES 19 Conclusions and further discussions
We have extended the class of standard static Lorentzian spacetimes R × M byconsidering Finslerian optical metrics. A spherical symmetric Finsler optical metrichas been proposed in [28] to get a perturbation of the Schwarzschild metric. Asobserved in [28], the most important feature that distinguishes a Finsler metricfrom a Lorentzian one is the fact that it breaks spacetime isotropy also at aninfinitesimal scale. In the case of a standard static Finsler spacetime, the anisotropyis confined to the restspaces { t } × M (where t is the natural coordinate on R ),relative to the observer field ∂ t , through the optical metric F/ Λ. We also observethat the map ( t, x ) (cid:55)→ ( − t, ϕ ( x )) of ˜ M , where ϕ is a diffeomorphism of M , flipsthe future causal cone in the past one at T ( − t,ϕ ( x )) ˜ M . If we assume that Λ isconstant and F is a locally Minkowski, non-reversible, metric on M (i.e. thereexists a covering of M by coordinate systems such that, in the corresponding naturalcoordinates ( x , . . . , x n , v , . . . , v n ) of T M , F depends only on ( v i )) then the causalstructure of ˜ M is not invariant under the P T -transformation ( τ, v , . . . , v n ) (cid:55)→ ( − τ, − v , . . . , − v n ). So such a particular class of standard static Finsler spacetimescould be interesting for some extensions of the Standard Model of particles when CP T symmetry and Lorentz-invariance violation are considered [43, 17].A wider class of splitting Finsler spacetimes ˜ M = R × M can be obtained, takinga one-form ω on M and a function Λ : M → R which is non-necessarily positive,by considering the quadratic Finsler function L : T ˜ M → R , L (( τ, v )) = − Λ τ + 2 τ ω ( v ) + F ( v ) . It can be easily seen (compare with [15, Prop. 3.3]) that the fundamental tensor˜ g = − Λ dt + ω ⊗ dt + dt ⊗ ω + g of L , where g is the fundamental tensor of F , is asmooth, symmetric section of index 1 of the tensor bundle π ∗ ( T ˜ M ) ⊗ π ∗ ( T ˜ M ) over T ˜ M \T , where T is again the line bundle defined by ∂ t , if and only if Λ( x )+ (cid:107) ω (cid:107) x > x ∈ M , where (cid:107) ω (cid:107) x = min v ∈ T x M \{ } max w ∈ T x M \{ } | ω x ( w ) | (cid:112) g v ( w, w ) , (so, in particular, if Λ is a positive function). The vector field ∂ t is still a Killingvector field but it is no more timelike (it is spacelike at the points ( t, x ) whereΛ( x ) < x ) = 0) and no more ˜ g ( τ,v ) -orthogonal to the vectorsin { } × T x M , x = π M ( v ) (recall Remark 2.10). On the other hand, the function t : ˜ M → R is a temporal function in the sense that it is smooth and dt (˜ v ) (cid:54) = 0 forany causal vector ˜ v ∈ T ˜ M . This allows us to define future-pointing causal vectorsas the ones such that dt (˜ v ) >
0. So, for a function Λ which is not positive ourdefinition of a Finsler spacetime (Definition 1.1) should be now intended in theweaker sense of Remark 1.3: ˜ g ˜ v (˜ v, ˜ v ) is positive in a punctured neighbourhood of T p , for any p = ( t, x ) ∈ ˜ M such that Λ( x ) >
0, while there always exist vectors˜ v , ˜ v , in any punctured neighbourhood of T ( t,x ) , such that ˜ g ˜ v (˜ v , ˜ v ) > g ˜ v (˜ v , ˜ v ) <
0, iff Λ( x ) = 0. In the case where Λ is positive, such Finsler spacetimescome into play: (a) already when one consider a different splitting of the type R × S f , in standard static Finsler spacetime, after a coordinate change of the type( t, x ) (cid:55)→ ( t + f ( x ) , x ) (recall Proposition 3.19); (b) as a Finslerian generalizationof a standard stationary Lorentzian spacetime where the Riemannian metric g is replaced by the fundamental tensor g of the Finsler metric F . In the general case where Λ is not positive, they represent the spacetime counterpart of windFinslerian structures morally in the same way as SSTK splittings correspond towind Riemannian structures (see [15]) and they certainly deserve further study. Inparticular, a challenging task is the study of the relations between their causalityproperties and the geometry of their optical structure (associated to future-pointinglightlike vectors). The latter is described, in the general case of a non-positive Λ,by two conic pseudo-Finsler metrics (in the sense of [23]): F o ( v ) = F ( v ) − ω ( v ) + (cid:112) Λ F ( v ) + ω ( v ) , (11) F ol ( v ) = − F ( v ) ω ( v ) + (cid:112) Λ F ( v ) + ω ( v ) , (12)The cone structure A ⊂
T M , where the first metric is defined, is given by thewhole tangent space T x M , for all x ∈ M with Λ( x ) >
0, by the vectors v suchthat ω x ( v ) < x ) = 0 and by the vectors { v ∈ T M l : ω ( v ) < , Λ F ( v ) + ω ( v ) ≥ } on M l . This last cone structure is also theset where F l is defined and we have, there, 0 < F o ( v ) ≤ F ol ( v ), with equality ifand only if Λ F ( v ) + ω ( v ) = 0. This is the condition that the projections of thelightlike vectors of ( ˜ M , L ) on
T M l must satisfy. The break of continuity (see [19,Def. 2.25]) of the map x ∈ M (cid:55)→ A x at the critical region { x ∈ M : Λ( x ) = 0 } iscertainly a difficulty in dealing with such cone structures but the spacetime pointof view, as shown in [15], is of help since the map that to p ∈ ˜ M associates the setof future-pointing causal vectors at p is instead continuous. Acknowledgements
We would like to thank the referee for his stimulating comments and for pointingout some references. We also thank some interesting comments and remarks by M.A. Javaloyes and M. S´anchez.EC is a member of and has been partially supported during this research bythe “Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Appli-cazioni” (GNAMPA) of the “Istituto Nazionale di Alta Matematica (INdAM)”,by “FRA2011, Politecnico di Bari”, and by the project MTM2013- 47828-C2-1-P(Spanish MINECO with FEDER funds).GS is a member of the “Gruppo Nazionale per le Strutture Algebriche Geometrichee loro Applicazioni” (GNSAGA).
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Gen. Relativity Gravitation , 1015-1042 (2012) Dipartimento di Meccanica, Matematica e Management,Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy
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