Finsler spacetimes with a timelike Killing vector field

Transcription

1 UNIVERSITÀ DEGLI STUDI DI BARI ALDO MORO Dipartimento di Matematica Dottorato di Ricerca in Matematica XXVII Ciclo A.A. 2015/2016 Settore Scientifico-Disciplinare: MAT/03 Geometria Tesi di Dottorato Finsler spacetimes with a timelike Killing vector field Candidato: Giuseppe Stancarone Supervisore della tesi: Prof. Erasmo Caponio Coordinatore del Dottorato di Ricerca: Prof. Luciano Lopez

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3 Contents Introduction v 1 Minkowskian norms Pseudo-Minkowskian norms Examples 6 2 Finsler spacetimes Finslerian metrics Finslerian separation Lorentz-Finsler metrics 15 3 Standard static Finsler spacetimes Finsler spacetimes and integrable timelike Killing vector fields Causality 35 4 Stationary splitting Finsler spacetimes Finsler spacetimes with a complete timelike Killing vector field Static auxiliary Finsler spacetime Conclusions and further discussions 55

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5 Introduction Sometimes it is the people who no one imagines anything of, who do the things that no one can imagine. (Alan Turing) In 1905 Albert Einstein revolutionised the physical vision of the Universe introducing the Special Relativity Theory. That new theory was based on two postulates: 1 The laws of Physics are the same for all observers in uniform rectilinear motion relative to one another (principle of relativity); 2 The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source. Special Relativity was described by a flat geometry and it did not include the gravitational interaction. Einstein thought how to integrate the gravity in a new relativistic framework and in 1915, with the birth of General Relativity, he generalized the principle of relativity, i.e. the laws of physics are the same for all observers (not only for inertial ones) and he added the equivalence principle which states that the gravitational interaction is locally eliminable. Mathematically speaking, the generalized relativity principle tells us that the physical laws are covariant for general coordinates trasformations and not only for Lorentz trasformations. To clarify this aspect, we recall that at every point of a sphere we can introduce a tangent space and approximate, in a neighborhood of that point, its geometry with the Euclidean one. In the same way, we can introduce at every point of the spacetime a new flat spacetime. Specifically, in any point of the spacetime, we have a Lorentzian scalar product on the tangent space of that point which can be diagonalized becoming a Minkowskian scalar product. Then the difference between the Newtonian and Einsteinian vision is the curvature: Newton s Universe is like a box where celestial bodies produce gravitational fields similarly to the electric field generated by charged ones. Instead, Einstein s point of view is that the Universe is like a great

6 vi Introduction sheet having some bodies put on it: as bodies curve the sheet with their weight, celestial bodies curve the space and the time with their mass. Seen in this way, we can state that the gravity, generated by the matter, determines the shape of the Universe. The geometrical model of General Relativity, i.e. Lorentzian geometry, can be generalized to a Finsler geometry of which the former is a particular case. But with a Finslerian model the equivalence principle is violated because the fundamental tensor of a Finsler metric depends on directions too. Thus, at every point of the spacetime there are infinitely many scalar products: one for each direction. In this work we have studied some types of Finsler spacetimes and their causality. We must say that relativity endowed with a Finsler metric has not an empirical evidence. However, there are many theoretical works which consider Finslerian modifications of classical spacetimes (see, e.g. [1], [5], [7], [19], [22], [25], [29], [31], [32], [33], [34], [35],[41], [46], [47], [48], [50], [52], [54], [55], [56], [57]). An important concept discussed in this work is causality. Roughly speaking, the word causality denotes the relation between one event (the cause) and another event (the effect). Many scientific fields such as physics, economy, logic, biology management, engineering, treat the impact of causality on the real life. Aristotle was the first who formalized the causality in four kinds: material cause, formal cause, efficient cause and final cause. As centuries have gone by, this topic has been studied by more and more scientists and philosophers, evolving from the original meaning to a modern one. Physics is the most relevant scientific area where the concept of causality, regarded as loss of information, plays an important role. The discovery of the thermodynamic and entropy principles represented a turning point in the study of causality, which finally reached an evident scientific nature. In particular, physicists working on General Relativity, started to care on causality. In this context, causality refers to the question about which events can influence or be influenced by a given event; mathematically speaking, the question is which points in a spacetime can be joined by causal curves. The study of causality is linked to geometrical properties of the spacetime and the most important results about this are due to Roger Penrose and Stephen Hawking ([24],[44]). They proved the existence of singularities in spacetime: black holes and cosmological singularities. Another aspect of this topic is the hierarchy of spacetime, namely a ladder of spacetimes sharing increasingly better causal properties where in every level there are specific results. At the end of 70 s, some mathematicians and physicists introduced the concept of causal ladder (see [9]), whose foundations were overhauled in [43]. The thesis is organized as follows. Chapter 1 contains its grammar. We analyze geometrical structure on which Finsler spacetimes are based: Minkowskian norms.

7 Introduction vii They are a generalization of classical concept of norm on a vector space. They are studied on conic domains recalling the geometrical structure of causal cones in classical relativity. In Chapter 2 we transplant the theory of Chapter 1 to differentiable manifolds. Here, we define the class of Finslerian metrics and their generalized distance called Finsler separation. Once obtained a background in Finsler geometry, we review some Finsler spacetime definitions in literature with their justifications. As observed in [34], the most important feature that distinguishes a Finsler metric from a Lorentzian one is the fact that it breaks spacetime isotropy also at an infinitesimal scale. At the end of the chapter we give our definition of Finsler spacetime (Definition 2.3.4) which allows us to consider some splitting R M, where the base M is endowed with a classical Finsler metric. Chapter 3 is a collections of our original results on which the preprint [18] is based. We extend the definition of standard static spacetime in the Finslerian case, replacing the Riemannian base with a Finsler one in the presence of an integrable timelike Killing vector field for the Finsler spacetime metric. Moreover we prove a Fermat principle for lightlike geodesic (Proposition 3.1.3). Last section is dedicated to causality: we generalize to this Finslerian case many properties valid in the Lorentzian case and we build the causal ladder. Chapter 4 is the natural progression of Chapter 3, because we perturb a standard static Finsler spacetime with a one-form getting a stationary splitting Finsler spacetime. Now the canonical associated vector field is Killing and timelike but it is no more integrable. Fermat principle is proved also in this more general setting. At the end we observe that the causal properties of a standard stationary Finsler spacetime reduce to the ones of a natural ultrastatic spacetime associated. We stress here that this fact allows one to reduce also the causal theory of a Lorentzian standard stationary spacetime (see [15]) to the one of a ultrastatic Finsler spacetime, as developed in Chapter 3. Acknowledgements I would like to mention some people that contributed in the growth of my scientific and human character and that helped me in the writing of this thesis both in a direct and indirect way. I thank prof. Miguel Angel Javaloyes and prof. Miguel Sanchez for their careful and accurate observations that improved this manuscript. I thank my supervisor prof. Erasmo Caponio for helping and inspiring me during my research activity.

8 viii Introduction I thank proff Anna Maria Pastore and Maria Laura Falcitelli for bringing me in the world of mathematical research. I thank my friends Fabio, Ilaria e Sara without whom I cannot either immagine how it would have gone. I thank my friends and colleagues in ADI with whom I shared many battle inside and outside the University. I thank Giulia, Simona and the other friends for the Comitato Giovani UNESCO for noticing how many things I had to say about mathematics and culture. I thank that set of measure zero of people who dedicated even only a little bit of their time supporting me during the PhD studies. Last I thank the lady who taught me how to read, write and count when I was not able to go to school. Ringraziamenti Voglio citare alcune persone che hanno contribuito a sviluppare il mio carattere scientifico e umano e che in maniera diretta e indiretta mi hanno facilitato nella stesura di questa tesi. Ringrazio il professor Miguel Angel Javaloyes e il professor Miguel Sanchez per le loro attente e precise osservazioni che hanno migliorato il tessuto di questo lavoro. Ringrazio il mio supervisor professor Erasmo Caponio per avermi aiutato e ispirato nella mia attivitá di ricerca. Ringrazio le professoresse Anna Maria Pastore e Maria Laura Falcitelli per avermi introdotto nel mondo della ricerca matematica. Ringrazio i miei amici Fabio, Ilaria e Sara senza i quali non immagino come sarebbe andata. Ringrazio i miei amici e colleghi dell ADI, con i quali ho condiviso tante battaglie dentro e fuori l Universitá. Ringrazio Giulia, Simona e gli altri amici del Comitato Giovani UNESCO per essersi accorti di quante cose avevo da dire sulla matematica e la cultura. Ringrazio quell insieme di misura nulla di persone che hanno dedicato anche solo un pó di tempo per sostenermi durante il dottorato di ricerca. Infine ringrazio colei che mi ha insegnato a leggere, scrivere e contare quando non potevo andare a scuola.

9 Chapter 1 Minkowskian norms In this chapter we introduce the definition of a pseudo-minkowski conic norm on a finite dimensional vector space V and we give some examples and results related to it. Most of the material presented here is taken from [28]. 1.1 Pseudo-Minkowskian norms Let V be a vector space with dimension n. Let A V be a non empty, open, connected subset such that if v A then λv A, for every λ > 0. We will call A conic domain. Observe that since A is open, if 0 A then A = V. Definition Let F : A R be a map that satisfies the following properties: (a) F (v) 0, with the equality iff v = 0; (b) F is positive homogeneous of degree 1, that is F (λv) = λf (v), for every λ > 0; (c) F is smooth (away from 0 if A = V ) and the fundamental tensor g defined by 2 g v (u, w) = 1 2 t s F 2 (v + tu + sw) (t,s)=(0,0) (1.1) for each v A and u, w V, is not degenerate. Then F is called a pseudo- Minkowski conic norm on V. If the fundamental tensor g is also positive definitive, then F is called a Minkowski conic norm on V Remark As a difference with [28] we do not allow that pseudo-minkowski conic norm can admit some vector v A such that g v is degenerate. So if F is a pseudo-minkowski conic norm on V which is not a Minkowski conic norm, then,

10 2 1. Chapter 1. Minkowskian norms being F positive on A, necessarily A V. It is worth to point out that, relaxing the condition of being g v non degenerate for each v A, then a positive, positively homogeneous, function F defined on V with F 2 smooth on V \ {0} can exist. For example, a p-norm on R n F (x) := n x i p i=1 1 p, x R n, with p > 2, satisfies the above conditions and g v is positive semi-definite for all v R n. Thus when A = V in Definition 1.1.1, g v must be positive definite on V for every v V \ {0} and F is then called a Minkowski norm. If F is absolutely homogeneous then we will say that F is reversible. Henceforth, we will speak about F as Minkowskian norm, when it is not specified if it is a pseudo- Minkowski conic norm or a Minkowski conic norm. In the following proposition, we will see some simple properties of the fundamental tensor g (see e.g. [27]): Proposition Given a pseudo-minkowski conic norm F and v A, v 0, the fundamental tensor g v is positive homogeneous of degree 0. Moreover for each w V. g v (v, v) = F 2 (v) g v (v, w) = 1 F 2 2 t (v + tw) t=0 (1.2) Proof. For every u, w V and λ > 0 we have 2 g λv (u, w) = 1 2 t s F 2 (λv + tu + sw) (t,s)=(0,0) = = λ2 2 2 t s F 2 (v + t λ u + s λ w) (t,s)=(0,0) = = λ 2 g v ( u λ, w λ ) = g v(u, w). Moreover 2 g v (v, v) = 1 2 t s F 2 (v + tv + sv) (t,s)=(0,0) = 2 = 1 2 t s (1 + t + s)2 F (v) (t,s)=(0,0) = = F 2 (v),

11 1.1. Pseudo-Minkowskian norms 3 and for each w V 2 g v (v, w) = 1 2 t s F 2 (v + tv + sw) (t,s)=(0,0) = t s [(1 + t)2 F 2 (v + s 1 + t w)] (t,s)=(0,0) = 1 2 t (1 + t)2 s (F 2 (v + s 1 + t w)) (t,s)=(0,0) = (1 + t) s F 2 (v + s 1 + t w) (t,s)=(0,0) (1 + t)2 t s F 2 (v + s 1 + t w) (t,s)=(0,0) = h F 2 (v + hw) h= (1 + t)2 t [ h F 2 1 (v + hw) 1 + t ] (h,t)=(0,0) = h F 2 (v + hw) h= (1 + t)2( 1 ) (1 + t) 2 h F 2 (v + hw) (h,t)=(0,0) = h F 2 (v + hw) h=0 1 2 h F 2 (v + hw) h=0 = 1 2 h F 2 (v + hw)) h=0. Now our aim is to show the links between the natural topology on V and the Minkowskian norm F on it. If F is not reversible, we have two notions of balls as the following definition shows: Definition Let F : A [0, ) be a pseudo-minkowskian conic norm on V. The set defined as B + v (r) = {x v + A : F (x v) < r}, (1.3) B v (r) = {x v A : F (v x) < r}, (1.4) are called forward ball and backward ball of center v and radius r, for every v V and r > 0. Since A is open and F is continuous, the forward ball and the backward ball are open in V, while the closed balls B ± v (r) (where the non-strict inequality is used

12 4 1. Chapter 1. Minkowskian norms in the definition) are closed in v ± A but not in V, except when F is a Minkowski norm. By replacing the strict inequality with the equality in the Definition1.1.2, we have the forward sphere and backward sphere S v ± (r), for each v V and r > 0. For convenience, we will work with B = B 0 + (1), B = B+ 0 (1) and S = S 0 + (1), where S is said indicatrix. Remark Observe that, given a Minkowski norm F and the Euclidean norm E on V, the map S E S, v and its inverse are smooth. Hence, in that case, v F (v) S is diffeomorphic to the Euclidean sphere S E. Proposition Let F : A R a pseudo-minkowski conic norm. (i) the indicatrix S is a smooth hypersurface in A and the position vector at each point is transverse to S. (ii) If F is a Minkowski norm, the collection of open forward (backward) balls are a basis for the natural topology of V. Proof. (i) Since F is positive homogeneous df y 0, for every y 0, thus the level set F 1 (1) is a smooth hypersurface. (ii) Let E be a Euclidean norm on V. Since the sphere S E is compact, there exists v S E where F takes the minimum value on S E. If w is a vector such that F (w) F (v), we have F (v) F ( w F (w) E(w) ), namely E(w) F (v) 1 and then BE contains a forward ball. Hence every ball for one norm contains smaller balls for the other one. The backward case follows in analogous way by observing that backward balls are forward for the Minkowski norm F on V defined by F (v) = F ( v) If F is a Minkowski conic norm the intersections between forward and backward balls constitute a topological basis of V (see Proposition 2.8 in [28] ). Now we see a characterization through the unit ball of a pseudo-minkowski conic norm: Proposition Let F : A R be a pseudo-minkowski conic norm, Then (i) B is a closed subset of A and it intersects all directions D v = {λv : λ > 0}, v A; (ii) B is starshaped from the origin i.e. λv B for every v B and λ (0, 1]; (iii) For every v B\{0} there exists a unique λ > 0 such that λv S; (iv) If F is a pseudo-minkowski norm, then it satisfies the strict triangle inequality, that is for every u, v A F (u + v) F (u) + F (v),

13 1.1. Pseudo-Minkowskian norms 5 with equality iff u and v are proportional if and only if B is strictly convex (i.e. u, v B, u v, implies λu + (1 λ)v B, with λ (0, 1)). Proof. (i), (ii) and (iii) hold from a straightforward computation. (iv)( ) For every u, v B and λ (0, 1), F (λu + (1 λ)v) < λf (u) + (1 λ)f (v) < 1. Hence B is strictly convex. ( ) The case where u, v are linearly dependent is trivial. So we suppose u, v linearly indipendent. We set ũ = u F (u) and ṽ = v F (v). Thus it holds F (u) F ( F (u) + F (v)ũ + F (v) F (u + v) = F (u) + F (v)ṽ) F (u) + F (v). Since B is strictly convex, the left hand side of previous equation is strictly less than 1. Theorem in [6] shows that if F is a Minkowski norm than the strict triangle inequality holds. Let A be a conic domain of V and B A open, with closed boundary S in A, which satisfies the hypotesis (i),(ii) and (iii) in Proposition Consider the map F : A R defined as F (v) = 1 (1.5) λ(v) where λ(v) is the unique scalar such that λ(v)v S. F is positive, positively homogeneous and B is its unit ball by construction. Moreover, if the boundary S is a smooth (at least C 2 ) hypersurface, then F is smooth away from 0. Thus we have the following proposition: Proposition Let A be a conic domain of V and B A open with smooth boundary S in A. We suppose that the second fundamental form of S is not degenerate. Then F as in (1.5) is a pseudo-minkowski conic norm and its unit ball is equal to B. Moreover, it is a Minkowski conic norm if and only if S is strongly convex (i.e. its second fundamental form, w.r.t. the normal pointing out of B, is negative definite) and it is a Minkowski norm if and only if in addition S is diffeomorphic to a sphere. Proof. F is positive, positive homogeneous of degree 1 and its unit ball is equal B for the previous reasoning. We consider the canonical connection on V which has the Christoffel symbols equal to zero. For every vector fields X, Y on V and v S g v (X, Y ) = HessL(v)(X, Y ) = X(Y (L)) X Y (L)

14 6 1. Chapter 1. Minkowskian norms If X, Y are vector fields on S, decomposing XỸ = ( X Y ) + h(x, Y )N, where X, Ỹ are extension of X and Y on A, N is the normal vector field to S pointing out to B and h the second fundamental form of S, we get, as Z(L) = 0 for any Z T S, g v (X, X) = h v (X, X)N(L). (1.6) Moreover being v transverse to S and g v (v, v) = F 2 (v) > 0, (1.7) we conclude that g v is not degenerate for any v S. The thesis then follows recalling that g v is positively homogeneous of degree 0 in v. Last statement follows by (1.6) and (1.7) observing that N(L) > Examples The simplest example of a Minkowski norm is the Euclidean norm on V where the fundamental tensor for every v V is the Euclidean scalar product. In general, if we have a scalar product, on V, the norm v = v, v defined on A = {v V : v, v > 0} is a pseudo-minkowski conic norm with fundamental tensor,. Let F be a pseudo-minkowski conic norm. We set G = F 2. Observe that it is differentiable and at v = 0 and the derivate there is zero (so G is C 1 on V ). In fact we note that Therefore G(tv) = t 2 G(v) t > 0 G(tv) = t 2 G( v) t < 0, G(te i ) G v i(0) = lim = 0, t 0 t where G v i is the i th -partial derivate, and being G v i smooth on V \ 0 and positively homogeneous, we get lim v 0 G v i(v) = 0 Proposition G is C 2 on V if and only if the fundamental tensor g is a scalar product of some index on V.

15 1.2. Examples 7 Proof. The implication to the left is trivial by the first equality in (1.2). Now we suppose that G is a C 2 on V. For every u, v V and λ > 0, we get g u = g λu = g 0 = g λv = g v. The last equation tells us that the fundamental tensor does not depend on u. Now we see some examples of Minkowskian norms: Example (Example 2.18 in [28]) Let c : [0, 2π] R 2, be a smooth simple periodic curve (i.e. closed and c(0) = c(2π)) that goes around the origin. We write c(θ) = (r(θ)cosθ, r(θ)sinθ). For each θ we consider the radial half line We define S = c([0, 2π]). Observe that l θ = {r(cosθ, sinθ) : r > 0}. c(θ), n(θ) = r2 + 2ṙ 2 r r r 2 + ṙ 2 (1.8) where, is Euclidean product in R 2 and n the unit normal vector to c. Up to 1 factor, (1.8) is the signed curvature of c defined in polar coordinates, which r 2 +ṙ 2 has the same sign of fundamental tensor of F in (1.5) relative to S. Thus if c has positive curvature, by Proposition 1.1.4, the interior region of S will be the unit ball of a Minkowski norm. Now, let θ, θ + be two scalar such that θ + θ < 2π. Then we consider I = (θ, θ + ), the convex conic domain A = θ I l θ and the closed subset B of A delimited by l θ, l θ+ and S. Again B is the unit ball of a Minkowski conic norm that is a Minkowski conic norm if c has positive curvature in [θ, θ + ]. The following example is in 1.3 C. of [6]. Example (Randers norm) Let E an Euclidean norm and β a linear form on V. For v V we define F (v) = E(v) + β(v). (1.9) Observe that F satisfies the condition (b) of Definition We are going to show that F is a Minkowski norm if and only if β < 1. Fix a basis {b i } n i=1 for V and express E(v) = a ij v i v j β = b i v i,

16 8 1. Chapter 1. Minkowskian norms where v = v i b i and (a ij ) is a positive definite symmetric matrix. Then we get β = a ij v i v j, where (a ij ) = (a ij ) 1. One can check that for every v V ; hence 1 F 2 2 v s (v) = F (v)(a sjv j E(v) + b s) v k [F (v)(a sjv j E(v) +b s)] = ( a kjv j E(v) +b k)( a sjv j E(v) +b s)+ F (v) E(v) (a ske 2 (v) a sj v j a sk v k E 2 ). (v) We have just got g ks (v) = ( v k E(v) + b k)( v s E(v) + b s) + F (v) E(v) (a sk v kv s E 2 ). (1.10) (v) We stress that the criterion β < 1 guarantees the positivity of F. By previous computation if we set l i = E = a ijv j v i E(v) and l i = F = l v i i + b i, we can express (1.10) as g ij = F E (a ij l i lj ) + l i l j. (1.11) By an argument in linear algebra discussed in 11.2 of [6], the following equality holds ( ) F n+1 det(g ij ) = det(a ij) (1.12) E Now we consider F ɛ = E + ɛβ and g ɛ its fundamental tensor, where 0 ɛ 1, β 1. By (1.11), detg ɛ is positive and none of the eigenvalues of g ɛ vanish. As ɛ ranges from 0 to 1, none of these eigenvalues can become negative. Hence F is positive definitive. A function F = E + β, with β 1, is called a Randers norm Randers norms belong to a special class of norms called (α, β) norms. In general, we can obtain a Minkowskian norm combining Minkowskian norms with one forms. In fact let (F i ) 1 i n and (β i ) 1 i m be n pseudo-minkowski conic norms and m one-forms on V. Consider a conic domain C R n+m and a continuous function L : C R, L = L(u i, v j ), with 1 i n and 1 j m, such that L is positive and smooth away from 0 and positively homogeneous of degree 2. We define the following function G(v) := L(F 1 (v), F 2 (v),..., F n (v), β 1 (v),..., β m (v)), (1.13)

17 1.2. Examples 9 for every v A, where A is intersection of the conic domains of (F i ) 1 i n. The following results are Theorem 4.1 and Proposition 4.10 in [28]: Theorem F := G, where G is defined as in (1.13), is a pseudo-minkowski conic norm with conic domain A. The fundamental tensor g of F is positive semidefinite if L u i 0 and Hess(L) is positive semi-definite. Moreover, F is a Minkowski conic norm if, in addition, n i=1 L u i > 0. Proposition Let φ : I R R a positive smooth function. Given a Minkowski conic norm F 0 : A 0 R and a one-form β, we consider the function ( ) β(v) F (v) = F 0 (v)φ, (1.14) F 0 (v) β(v) F 0 (v) for every v A, where A := {v A 0 : I}. F is a Minkowski conic norm if for every s I φ s φ > 0 and φ 0. The (α, β)-norms (see [38]) are a particular case of (1.14), where α denote the square root of a scalar product and β a one form. In [28] the authors generalize (α, β)-norms replacing the square root of a scalar product with a Minkowski norm F 0 and considering the case F 0 + β (Randers type norms), F 0 2 β and F 0 2 F 0 β (Matsumoto type norm). (Kropina type norms)

18 10 1. Chapter 1. Minkowskian norms

19 Chapter 2 Finsler spacetimes In this chapter we introduce the notion of Finslerian manifold and, after briefly analyzing some definition already existing in literature, the one of Finsler spacetime. 2.1 Finslerian metrics Let M be a (n + 1)-dimensional smooth paracompact connected manifold, n 1. Let us denote by T M its tangent bundle and by 0 the zero section. We consider A as a conic sub-bundle of T M, that is π(a) = M, where π : T M M is the natural projection from the tangent bundle to M and A x is a conic domain on the tangent space T x M, for every x M. A conic pseudo-finsler metric on M is a function that assigns smoothly at every point x M a Minkowski conic pseudo-norm F (x, ) on A x. Thus we can see a conic pseudo-finsler metric as a function F : A R such that it is smooth on A, except at most on the zero section, and the functions F (x, ) are Minkowski conic pseudo-norms. When the conic sub-bundle A coincides with T M, we call F a Finsler metric (recall Remark 1.1.1). The pair (M, F ) is said a conic pseudo-finsler manifold and a Finsler manifold if A = T M. We have the following definition: Definition Given a conic pseudo-finsler metric F, the map g that assigns smoothly at each point x M the fundamental tensor g x of the Minkowski conic pseudo-norm F (x, ) is said the fundamental tensor of F. By Definition 1.1 we have that for every (x, v) A \ 0 the fundamental tensor g has the following expression 2 g (x,v) (u, w) = 1 2 t s F 2 (v + tu + sw) (t,s)=(0,0) (2.1)

20 12 2. Chapter 2. Finsler spacetimes where u, w T x M. From now on we will omit the dependence on x, unless necessary, writing g v (u, w). Now if the fundamental tensor g is definite positive, for every x M, F is called a conic Finsler metric (respectively) Finsler metric when A = T M) and the pair (M, F ) a conic Finsler manifold (respectiveley Finsler manifold). All of these definitions determines the class of Finslerian metrics and Finslerian manifolds. Observe that if (x i, y i ) is a local coordinate system of T M in (x, v) T M \ 0, we get g (x,v) = g ij dx i dx j, (2.2) where g ij (x, v) = g v (e i, e j ) = 1 2 F 2 2 y i y j (v). with {e i } basis on T x M. Thus g is the vertical Hessian of 1 2 F 2 and it is homogeneous of degree 0. By (2.2) we have that g (x,v) is (a non-necessarily positive definite) scalar product on T x M, for every v T x M. If we denote by π A : A \ 0 M the restriction of the natural projection from the tangent bundle to M and T M the cotangent bundle of M, we can define πa (T M ) as the pulled-back cotangent bundle over A \ 0. In this way the fundamental tensor g can be understood as a smooth symmetric section of the tensor bundle πa (T M ) πa (T M ). If g is a Riemannian metric on M, setting for every v T x M F (v) = g x (v, v), (2.3) we get that F is a Finsler metric. Moreover, by Proposition 1.2.1, a Finsler metric is C 2 on T M if and only if it is the norm of a Riemannian metric like in (2.3). Of course, it is possible to build Finsler metrics that do not derive from Riemannian metrics. For istance, as in Example 1.2.2, we can consider the Randers metric F (v) = g x (v, v) + β(v) for every (x, v) T M \ 0, where β is a one-form on M such that β x < 1, where β x is the norm of β x with respect to g x. 2.2 Finslerian separation Following [28], we introduce a sort of distance function on a Finslerian manifold that will be called Finslerian separation. The approch is similar to the Riemannian case but as we have the conic set A, we need some constraint on the piecewise smooth curves involved. For this reason we give the following definition:

21 2.2. Finslerian separation 13 Definition Let F : A R be a conic pseudo-finsler metric on M. A piecewise smooth curve γ : [a, b] M is admissible if the right and left derivatives γ ± belongs to A. as Thus let γ : [a, b] M a admissible curve. Its Finslerian length l(σ) is defined l(γ) = b a F ( γ(s))ds. For x 0, x 1 M, denote by I(x 0, x 1 ) the set of all the admissible curves γ joining x 0 to x 1. The map d : M M [0, + ] defined as { inf d(x 0, x 1 ) = I(x0,x 1 ) l(σ) if I(x 0, x 1 ) + if I(x 0, x 1 ) = is called Finslerian separation. Whenever F is a Finsler metric, I(x 0, x 1 ), for all x 0, x 1 M, and the function d : M M [0, + ) is continuous. Moreover it satisfies the triangle inequality but it is not a distance because in general d(x, y) d(y, x) for some x, y M, unless F is absolutely homogeneous of degree one. So the Finslerian separation is, in this case, a generalized distance. As in Definition the lack of symmetry leads to the following definition: Definition Let (M, F ) be a conic pseudo-finsler manifold, the sets defined as B + (x, r) = {y M : d(x, y) < r}, B (x, r) = {y M : d(y, x) < r}, are called respectively forward ball and backward ball. We will denote by B + (x, r) (resp. B (x, r)) the closure of B + (x, r) (resp. B (x, r)). Observe that, in the case where F is a conic Minkowski norm, Definition is a special case where the manifold is a vector space having dimension (n + 1). Proposition 3.9 in [28] ensures us that forward and backward balls are open sets. For every x M, we can choose a compact neighborhood U of x and a Riemannian metric g U on U. If we are able to build a locally finite open covering {U i } such that F (v) g Ui (v, v), for all v A T U i, then by a partition of the unity subordinated to {U i }, we get a Riemannian metric on M that is a lower bound of F. Hence the collections of forward metric balls and backward balls are topological basis of M (see Proposition 3.13 in [28]).

22 14 2. Chapter 2. Finsler spacetimes Given a Finslerian metric F on M, we have recalled above that the Finslerian manifold (M, F ) can be endowed with a generalized distance. Hence we have the following definition: Definition A sequence {x i } in M is called a forward (resp, backward) Cauchy sequence if, for ɛ < 0, there exists a positive integer N (depending on ɛ) such that d(x i, x j ) < ɛ (resp. d(x j, x i ) < ɛ), for every N i < j. So the next definition ratifies definitely the analogy between the classic metric spaces and the Finslerian manifold. Definition A Finslerian manifold (M, F ) is said to be forward complete with respect to its distance function d if every forward Cauchy sequence converges in M. It is backward complete with respect to d if every backward Cauchy sequence converges. A Finsler manifold (M, F ) is said to be forward geodesically complete if every geodesic γ : [a, b) M, parametrized to have constant Finslerian speed, can be extended to a geodesic defined on [a, + ). In similar way A Finsler manifold (M, F ) is said to be backward geodesically complete if every geodesic γ : (a, b] M parametrized to have constant Finslerian speed, can be extended to a geodesic defined on (, b]. If a geodesic γ, parametrized to have constant Finslerian speed, is both forward and backward geodesically complete then it is called simply geodesically complete. These two notions are not equivalent ( unlike Riemannian geometry) as showed in In 12.6 of [6]. Theorem in [6] is the Hopf-Rinow Theorem in Finslerian case: Theorem Let (M, F ) be any Finsler manifold. Then the following statements are equivalent: (a) (M, F ) is forward complete. (b) (M, F ) is forward geodesically complete. (c) At every point x M, the exponential map ( see 5.3 in [6]) is defined on all of T x M (d) At some point x M, the exponential map is defined on all of T x M. (e) Every closed and forward bounded subset of (M, F ) is compact. Moreover, if any of the above holds, then every pair of points in M can be joined by a minimizing geodesic. Remark The same Theorem holds analogously (but independently) for the backward completeness. In this case we want to stress that the exponential map is relative to the reverse Finsler metric F (v) = F ( v).

23 2.3. Lorentz-Finsler metrics 15 The following Lemma will be used in the next sections: Lemma 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 Finslerian length σ : [a, b) M (resp. σ : (a, b] M) there exists a point x 0 M such that σ(s) x 0, as s b (resp. as s a). Proof. ( ) Let l(σ) (0, + ) be the length of σ. Set K = {x M : d(σ(a), x) l(σ)}. Since K is forward bounded and closed, we have that it is compact by Theorem 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 0 K such that σ(s k ) converges to x 0, up to subsequences. If lim s b σ(s) x 0, there would exist an ε > 0 such that σ leaves the set {x M : d(x, x 0 ) ε} infinite times while, definitively, d(σ(s k ), x 0 ) < ε/2. This is a contradiction because σ has a finite length. ( ) We suppose that (M, F ) is not forward complete manifold. Therefore, there exists a geodesic σ : [a, b) M, hence with finite length, which is not extendible to s = b. 2.3 Lorentz-Finsler metrics In this section, we start commenting some definitions about Finsler spacetimes and their motivations. The idea is to replace the Lorentzian metric of General Relativity with a Finslerian metric. The most important difference is that in a Finsler spacetime the spatial isotropy is broken because the fundamental tensor has a dipendence from direction too. Thus we have different metrics for every direction in a tangent space to M. In classical General Relativity a spacetime is an n-lorentzian manifold (M, g) endowed with a time orientation, that is a continuous function that assigns every point x M a choice of timecone in x. If this function is smooth, then the time orientation is equivalent to existence of smooth timelike vector field on M (see [44] Lemma 32 p. 145). A causal curve γ : [a, b] M represents a trajectory of a massive or a massless particle and its length is l(γ) = b a g( γ, γ)ds. (2.4) The causal curves that maximize locally the previous functional are the timelike or lightlike geodesics. So to generalize the classical spacetime, we could consider a conic pseudo-finsler metric F : A T M (0, + ) with fundamental tensor g v

24 16 2. Chapter 2. Finsler spacetimes having index 1, for all v A. This F is called a Lorentz-Finsler metric. In analogy with General Relativity, a Lorentz-Finsler metric could be enough to describe the trajectories of massive particles. But if we consider a massless one then its trajectory should be a curve γ such that F ( γ) = g γ ( γ, γ) = 0 (i.e. it is a lightlike curve). In this case we should consider an "energy fuctional" rather than a length functional because the latter does not admit first variation if γ is lightlike. Given a conic subset A T M, the previous reasoning justifies the following definition: Definition A quadratic conic pseudo-finsler metric is a function L : A R with the following properties: (i) L is C k on A, k 2; (ii) L(x, λv) = λ 2 L(x, v), λ > 0 (iii)the fundamental tensor g π A (T M) π A (T M), locally defined by g = g ij (x, v)dx i dx j = 2 L v i v j (x, v)dxi dx j, is not degenerate. Thus the Finslerian case (2.4) is replaced by for every admissible curve γ. E(γ) = b a L( γ)ds, (2.5) Remark If F is a conic pseudo-finsler metric on M we get that L = F 2 is a quadratic conic pseudo-finsler metric. If L : A 1 R is a quadratic conic pseudo- Finsler metric, set A 2 = {v A 1 : L(v) 0}, then F = L A2 is a conic pseudo-finsler metric defined on A 2 with fundamental tensor sign(l)g. Notice that F can be extended continuously to the boundary of A 2. The pair (M, L) will be called a quadratic conic pseudo-finsler manifold (Lorentz- Finsler manifold if g has index 1 for all v A). Thus by Proposition and Remark we get: L(v) = 1 2 g ij(v)v i v j, L y i (v) = g ij(v)v j, (2.6) where v i are the components v in T x M relative to a local coordinate system (x i, y i ) in T M Now we are able to define the first type of Finsler spacetime as in [29]:

25 2.3. Lorentz-Finsler metrics 17 Definition (Javaloyes-Sanchez [29]) A Finsler spacetime is a Lorentz-Finsler manifold (M, L) that satisfies the following properties: (i) L is positive on A, (ii) A x = A T x M is convex for every x M, (iii) A has smooth boundary in T M \ 0, (iv) L can be extended on a neighborhood of A \ 0 in T M \ 0, it is equal to zero on A and it is smooth on such a neighborhood, (v) The extension of fundamental tensor to the points in A \ 0 has index 1. Remark Very recently, a more general definition of a Finsler spacetime appared in [1]. That definition assumes regularity of L only an a neighborhood of the null cone A. This would be enough also for our purpose of defining a Lorentz- Finsler metric on a warped product (see Definition and the discussion above it). Anyway, as we will consider some concrete examples of Lorentz-Finsler metrics which are, a priori, defined on a larger subbundle of T M \ 0 than the only future causal bundle Ā \ 0, we prefer here to not follow the point of view of [1]. Remark A possible alternative and more general way to define a Finsler spacetime is starting from a smooth symmetric section g of the tensor bundle π A (T M ) π A (T M ) over A. However in next chapter we will clarify because it is not convenient to define spacetimes with fundamental tensors which do not derive from a quadratic Finslerian function. Given two Lorentzian metrics g, g on M, we suppose that they have the same causal cone for every x M. In particular, g, g have the same lightlike vectors. If we suppose that M has dimension 2 (the n-dimensional case is analogous), let B = {e 1, e 2 } be an orthonormal basis for g x, at x M. Set v 1 = 1 2 (e 1 + e 2 ) and v 2 = 1 2 (e 1 e 2 ), then B = {v 1, v 2 } is another basis of lightlike vectors. Observe that M B (g ) = cm B (g) where c = g x(v 1, v 2 ) and M B ( ) is the matrix rapresentation w.r.t. the basis B. Hence g and g are conformal metrics, that is there exists a smooth function c(x) on M such that g x = c(x)g x. So a Lorentzian metric is determined up to a conformal factor by its causal cone. It is not possible to apply the same reasoning to Finslerian metrics. In fact, given a spacetime (M, L), a lightlike vector u T x M, with x M, is a vector of A x \ {0 x }. Hence g u (u, u) = 0, (2.7) where g is the fundametal tensor of L. Since there is the u-dependence of g, we do not know if the vectors v 1, v 2 above are lightlike in the Finslerian case. Since in

26 18 2. Chapter 2. Finsler spacetimes Definition the classical causal cones are replaced pointwise by domains A x, (in other words the classical definition of Lorentzian spacetime is a particular case of Definition when one considers only its future causal cone) we might have two Lorentz-Finsler metrics F and F on M with the same domain which are not equal up to a conformal factor. Definition takes into account the problem of defining null vectors on a Finsler spacetime preserving some regularity for the Lorentz-Finsler function L. We point out that several authors had already considered Finslerian extensions of spacetimes; we want to cite here Asanov [5], who considers a strictly positive, positively homogeneous of degree one, C 5 function, F on an conic subset of T M \ 0, with non degenerate fundamental tensor (i.e. a conic Lorentz-Finsler metric in our terminology). The idea of considering a quadratic Finsler function rather than a positively homogeneous one was introduce by Beem in [8] Now we would like to present another recent definition of the Finsler spacetime (see Definition below), which includes the one in [8] and in [46]. Here the author considers a positively homogeneous of degree 2 function L : T M \ 0 R, with locally Lipschitz 3 th derivates such that the fundamental tensor g (which is, then, locally C 1,1 ) as in (2.6) has index 1 for all v T M \ 0. Since L is defined on all T M \ 0, there is not a choice of a conic sub-bundle as in Definition 2.3.2, hence in [41] Minguzzi defines, for every x M, the following subsets of T x M \ 0, I x = {v : g v (v, v) < 0} J x = {v : g v (v, v) 0} (2.8) E x = {v : g v (v, v) = 0} resembling the sets of timelike, causal and lightlike vectors at x M, as they extend in a canonical way the definition of timelike, causal and lightlike vectors given in the Lorentzian case. It s well know that the set I x could have more than two connected components. Proposition 3.1 in [46] states that the connected components of I x are convex and Proposition 2 in [41] shows that its number is independent of x. Moreover, a reverse Cauchy-Schwartz inequality holds (see Theorem 3 in [41] or Proposition 2.4 in [1]), i.e. for every u, v belonging to a connected component of J x it holds: g u (u, v) L(u) L(v), (2.9) therefore we have that any couple of vectors u, v in the same of J x satisfies g u (u, v) 0 and g v (v, u) 0. Moreover fixed a timelike vector field K, if any vector u in a connected component of J x satisfies g u (u, K x ) 0 then K x lies in the same component (see Proposition 5 in [8]). Thus we can use a timelike continuous vector field K to time-orienting

27 2.3. Lorentz-Finsler metrics 19 (M, L) and we can call u T x M such that g u (u, K x ) < 0, a causal future-pointing vector. Given the previous properties, the following definition of Finsler spacetime is given in [41] Definition (Minguzzi [41]) A Lorentz-Finsler manifold (M, L), with L : T M R, is time orientable if it admits some continuous global timelike vector field. A Finsler spacetime is a time orientable Lorentz-Finsler manifold where a continuous global timelike vector field has been chosen. In Chapter 1 we introduced a notion of the reversible Minkowskian norm. In a similar way a (quadratic) conic pseudo-finsler metric L is reversible if for each tangent vector v it holds L(v) = L( v), (hence L must be defined in a cone bundle such that A x = A x, for every x M). Proposition (Theorem 7 in [41]) Let (M, L) be a spacetime as in Definition with dim M > 2. If L is reversible then I x has two connected components for every x M. Proof. Let (x, v) T M \ 0. If v and v were in the same connected component we would get g v ( v, v) < 0 by (2.9). Since L is reversible we have g v ( v, v) = g v (v, v) 0. Thus v and v do not lie in the same connected components. Let I 1 x a connected component of I x and set I 1 x = { v : v I 1 x}. Of course also I 1 x is connected. Thus the number of connected components are even. By Proposition 4 in [41], since we have that for every connected component I k x, I k x have non-empty intersection with others connected components, we can conclude the thesis. In next chapters we will study the notions of standard static and stationary splitting Finsler spacetimes. Such classes of spacetime are defined on product manifolds M = R M where M is identifiable with a spacelike hypersurface of M. So a Lorentzian-Finsler metric on M inducing a classical Finsler metric on M cannot be smoothly extended (where smooth here means at least C 2 ) to vectors which project trivially on T M, due to the lack of regularity of a Finsler function on the zero section. Motivated by this problem, we define a Finsler spacetime as follows. Let M be a (n+1)-dimensional smooth paracompact connected manifold, n 1 Let T T M be a 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

28 20 2. Chapter 2. Finsler spacetimes T M \T. Let us consider the tensor bundle π (T M) π (T M) over T M \T and a section g : v T M \ T g v T π(v) M T π(v) M. We say that g is symmetric if g v is symmetric for all v T M \ T. Analogously, g is said non-degenerate if g v is non-degenerate for each v T M \T and its index will be the common index of the symmetric bilinear forms g v ; moreover, g will be said homogeneous if, for all λ > 0 and v T M \ T, g λv = g v. Definition A Finsler spacetime ( M, L) is a smooth (n+1)-dimensional manifold M, n 1, endowed with a smooth, symmetric, homogeneous, non-degenerate of index 1 section g of the tensor bundle π (T M) π (T M) over T M \ T, which is the fundamental tensor of a quadratic Finsler function L : T M \ T R and such that g w (w, w) < 0 for each w in a punctured conic neighbourhood of T p in T p M \ {0} and for all p M. Clearly, the quadratic Finsler function in the above definition must be fiberwise positively homogeneous of degree 2 and, so, it is, except for the possible lack of twice differentiability along T, a quadratic Lorentz-Finsler metric defined on T M. We could allow more generality by not prescribing the existence of such a function. This is a quite popular approach to Finsler geometry: see, e.g., [4, 40], and the references therein, where such structures are called generalized metrics (although in [40] they are sections of the tensor bundle π (T M) π (T M) with base the whole T M). By homogeneity, it is easy to prove that a generalized homogeneous metric is the vertical Hessian of a smooth Finsler function on T M \ T if and only if its Cartan tensor is totally symmetric, i.e. g ij (v) = g v k ik (v) for all v T M \ T (see, e.g., [4, v j Theorem ]). Remark The requirement about the sign of g w (w, w) for w in a neighborhood of T could be weakened by allowing the existence of an open subset Ml M where the reverse inequality holds for any w in a punctured neighborhood of T x, x M l (and then the existence of a critical region where, in each punctured neighborhood of T, there exist vectors w 1 and w 2 such that g w1 (w 1, w 1 ) < 0 and g w2 (w 2, w 2 ) > 0). This, for example, would be the case for a Finslerian modification of a class of spacetime called SSTK splitting, studied in [16]. Remark Whenever T is trivial and g can be smoothly extended to T \ 0, we get Definition by choosing a no-where zero continuous section K such that K x T, for each x M. In our definition of Finsler spacetime, causal vectors, curves, etc. are defined as follows: Definition A vector w T M \ 0 is called

29 2.3. Lorentz-Finsler metrics 21 -timelike if either w T or g w (w, w) < 0, -lightlike if g w (w, w) = 0, -causal, if g w (w, w) 0, -spacelike, if g w (w, w) > 0. Moreover, assuming that a section K as above can be chosen, we say that a causal vector w is -future pointing if either w is a positive (resp. negative) multiple of K π(w) or g w (w, K) < 0, -past pointing if g w (w, K) > 0). Coherently with previous definitions, continuous piecewise smooth curve 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. Moreover, a causal 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 hypersurface H M which is transversal to T will be called spacelike if for any v T H, g v (v, v) > 0.

30 22 2. Chapter 2. Finsler spacetimes

31 Chapter 3 Standard static Finsler spacetimes In this chapter and in the next we will consider a class of Finsler spacetimes which satisfy Definition Finsler spacetimes and integrable timelike Killing vector fields Let us recall that a Lorentzian spacetime ( M, g) is said static if it is endowed with an irrotational timelike Killing vector field K. This is equivalent to say that the orthogonal distribution to K is locally integrable and then for each p M there exists 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, is given by Λdt 2 + g 0, where t I, t is the pullback of K, Λ = g(k, K) and g 0 is the Riemannian metric induced on S by g (see [44, Proposition 12.38]). This local property of static spacetimes justifies the following definition: let M be an n- dimensional Riemannian manifold, Λ: M (0, + ) a smooth, positive function on M and I R an open interval. The warped product, i.e. the manifold I M endowed with the Lorentzian metric g = Λdt 2 + g 0, where g 0 is the pullback on I M of the Riemannian metric on M, is a spacetime called standard static (see [44, Definition 12.36]). Let us extend the above picture to Finsler spacetimes. First we need the notion of a Killing vector field.

32 24 3. Chapter 3. Standard static Finsler spacetimes Definition 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 1, v 2 T π(v) M, we have: g dψ t(v)(dψ t(v 1 ), dψ t(v 2 )) = g v (v 1, v 2 ), (3.1) for any t R such that the stage ψ t is well defined in a neighbourhood U M of π(v). As observed in (2.2), the fundamental tensor g is a section of the tensor bundle π (T M) π (T M) over T M \ T, but generally it is not a scalar product on T M \ T because it is degenerate on vertical fields. In [36] it is showed how to define a Lie derivative L K relative to a vector field K such that Definition is equivalent to the condition L K g = 0, similarly to the semi-riemannian case. Actually in [36] the base of the tensor bundle is the slit tangent bundle T M \ 0 but the reader can check the validity of the equivalence when considering T M \ T. Let K be a Killing vector field on M and set K c the complete lift of K. Then K c is completely determined by its action on complete lifts on T M f c, for every smooth function f on M: f c (v) := v(f) K c (f c ) := (K(f)) c for all v T M. We define a bundle map i : π (T M) T T M such that, for every (y, z) π (T M), i(y, z) is the initial velocity of parametrized straight line s y + sz; in local coordinate (x i, y i ), if z = z i x i (x) for some x M, it holds i ( y, z i ) = z i x i y i. Observe that the map i induces also a injective homomorphism between X(π) and X(T M \ T ), denotated always by i, where X(π) and X(T M \ T ) are the sets of smooth sections of π (T M) over T M \ T and T (T M \ T ). In analogous way, we define a map j : T T M π (T M) such that j(w) = (z, π (w)), for every w T z T M. Observe that Ii = ker j and we have the following exact sequence 0 π (T M) i T T M j π (T M) 0. Thus another homomorphism between X(T M \ T ) and X(π), denoted always by j, is defined and it holds the following exact sequence 0 X(π) i X(T M \ T ) j X(π) 0.

33 3.1. Finsler spacetimes and integrable timelike Killing vector fields 25 The vertical vector fields are the elements of V (T M \ T ) := Ii = ker j. We can think K as a section of pull-back bundle π (T M) over T M \ T, i.e. K X(π), and consider a section Ỹ X(π). We define the Lie derivative relative to K on the tensor algebra of the pull-back bundle π (T M) over T M \T such that: L K (f) := K c (f), L K (Ỹ ) := i 1 (L K ci(ỹ )), (3.2) where f is a smooth function on T M \ T and L K c is the Lie derivative relative to K c on T M. Observe that the second equation in 3.2 is well posed. In fact, if local expressions are Ỹ = U k and K = K h, where U k depend on y i too, then we x k x h get [K c, i(ỹ )] = [ K h x h, U k ] [ K h + x k = K h U k x h y k + Kh x = (K h U k x i yi y h i yi U k y h, U k ] y k y k U h Kk x h x h + Kh U k yi xi y h U h Kk x h Since [K c, i(ỹ )] = L K ciỹ, then L Kci(Ỹ ) is vertical. Moreover, we have for every X, Y X(π), ) y k. y k (L K g)(x, Y ) = K c (g(x, Y )) g(l K X, Y ) g(x, L K Y ). (3.3) Thus, repeating the proof of Proposition 5.2 in [36] we get Proposition K is a Killing vector if and only if L K g = 0. Definition We say that a Finsler spacetime ( M, L) is static if there exists a timelike Killing vector field K such that the distribution of hyperplanes ker( v L(K)) is integrable, 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 1 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 We say that a Finsler spacetime is standard static if there exist a smooth non vanishing global section T of T, a Finsler manifold (M, F ), a positive function Λ on M and a smooth diffeomorphism f : R M M, f = f(t, x), such that t = f (T ) and L(f (τ, v)) = Λτ 2 + F 2 (v).

34 26 3. Chapter 3. Standard static Finsler spacetimes Remark The definition of a standard static Finsler spacetime (although called there static Finsler spacetime) appeared first in [34, Definition 2]. In [35] the solution of the vacuum (Finslerian) field equations, introduced in the same paper, is standard static in the region where a certain coefficient B is positive provided that a constant a is also positive (see [35, Eqs. (16)-(17)]). Remark Another static future-pointing Killing vector field K will be said standard if the above conditions hold relatively to K, i.e there exist a manifold (M, G) and a diffeomorphism f : R M M, f = f (t, x ), such that t = (f ) (K) and L(f (τ, v )) = Λ K τ 2 + G 2 (v ). The existence of a standard static vector field is a very rigid condition in comparison to the Lorentzian case ([51, 3]) where some topological assumptions on the base M are needed in order to get uniqueness. Indeed we have the following: Proposition If a static Finsler spacetime admits a standard splitting (i.e the static vector field is standard) then it is unique up to rescaling t t/a, T at, a (0, + ), of the coordinate t and of the vector field T and up to Finslerian isometries of (M, F ). Proof. Assume that another static standard Killing vector field K exists and let f : R M M be a smooth diffeomorphism such that (f ) (K) = t and L(f (τ, v )) = Λ K τ 2 + G 2 (v ), so that G 2 (v ) = L(f (0, v )) for all v T M. Let L := L f. Then L = L (f ), hence L is not twice differentiable along the line bundle K defined by K, because L is not so along the one defined by t. This is 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 they belong to the same timelike cone necessarily they are proportional, i.e. there exists a positive constant a such that K = at. This follows as in the semi-riemannian case, by using the fact that L T g = L K g = 0. Thus the diffeomorphism (f ) 1 f has first component equal to t t/a while the second one induces a diffeomorphism φ between M and M such that G 2 (φ (v)) = F 2 (v) for all v T M. 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) = Λτ 2 + F 2 (v), where Λ and F are, respectively, a positive function and a Finsler metric on M. Definition A Finsler spacetime is said locally standard static if for any point p M there exists a neighborhood U of p and a diffeomorphism φ : I p S p U, where I p is an interval in R and S p a open manifold, such that, named t the natural coordinate of I p, φ ( t ) is a section of T and for all (τ, v) T (t,x)(i p

35 3.1. Finsler spacetimes and integrable timelike Killing vector fields 27 S p ), L(dφ (t,x) (τ, v)) = Λτ 2 + F 2 (v), where Λ and F are respectively a positive function and a Finsler metric on S p. We call the manifolds S p, p M, the local spacelike leaves of ( M, L). Remark As observed in [41, Example 1, Remark 3], differently from the Lorentzian case, a Finsler spacetime can be static without being locally standard static (in the example of [41], the Killing vector field is not gṽ-orthogonal to D π(ṽ) for all ṽ D π(v) = ker( v L(K π(v) )). We observe that, although L is not twice differentiable along vectors y T, its fiberwise second Gateaux derivative 2 L(u) := 2 y 2 s t L(u + sy + ty) (t,s)=(0,0) exists for all u, y T with u = ay for some a R \ 0. Indeed, by homogeneity, we have L(u + sy + ty) = L((a + s + t)y) = (±a + s + t) 2 L(±y), for small s, t R, where in the last equation we have the positive sign if a > 0 and the negative one otherwise. This fact will be used in the next theorem when we characterize static Finsler spacetimes which are locally standard. Theorem Let ( M, L) be a static Finsler spacetime with timelike Killing vector field and D = ker( L v (K)) be its integrable distribution. Then, (M, L) is locally standard static with local spacelike leaves the integral manifolds of D, if and only if both the following conditions are satisfied: (a)l(k) = L( K), (b) L(y ± K π(y) ) = L(y) + L(±K π(y) ), for all y D. Proof. ( ) Let p M and D p = ker( L v (K p)) T p M. Let Sp be an integral manifold of D at p. Let (0, v) T (0,x) (I p S p ) and dφ (t,x) (0, v) = y. As φ ( t ) = φ ((0, 1)) = K, we have Moreover, L(K) = L(φ ( t )) = L(φ ( t )) = L( φ ( t )) = L( K). L(y ± K) = L ( dφ (t,x) (0, v) ± dφ (t,x) ((1, 0)) = L(dφ (t,x) (0, v) ± (1, 0) ) = L ( dφ (t,x) (0, v)) + L(dφ (t,x) (±(1, 0)) ) = L(y) + L(±K π(y) ). ( ) Let p M and S p be an integral manifold of D at p and I p be an interval s.t. the map φ : I p S p M, φ(t, q) = ψ t (q), is a diffeomorphism onto a neighborhood U of p in M, where ψ is the flow of K. Consider a non vanishing smooth section V : S p T M, where S p is an open neighborhood U of p in S p. Set Y q = (dψ t ) x (V x ),

36 28 3. Chapter 3. Standard static Finsler spacetimes with q = φ(t, x). So Y is a non vanishing smooth vector field in a neighborhood U U M of p. The evalutation g Y of the fundamental tensor of L in Y becomes, then, a Lorentzian metric on U (and, by definition of Y, K is a Killing vector field for g Y ). In particular g Yq (K q, K q ) = g Vx (K x, K x ), for all q U, q = φ(t, x). Let V x = v x + τ(x)k x where v x D x and τ(x) R (recall that being L v (K)[K] = L(K) < 0, K x is transversal to D x and then T x M = Dx [K x ]) Let us assume that τ(x) < 0 (the case τ(x) > 0 is analogous): g Vx (K x, K x ) = 2 s t L( v x + τ(x)k x + sk x + tk x ) (t,s)=(0,0) = 2 s t L(v x) + s t[ 2 ( τ(x) s t) 2 L( K x ) ] (t,s)=(0,0) = 0 + L( K x ) < 0, for all x S p. Thus K is a timelike vector field for g Y in U. Analogously, for all u D x g Vx (K x, u) = = 2 s t L( v x + τ(x)k x + sk x + tu ) (t,s)=(0,0) s t[ 2 ( τ(x) s) 2 L( K x ) ] (s,t)=(0,0) + 2 s t L(v x + tu) (s,t)=(0,0) = 0 Hence D is the orthogonal distribution to K w.r.t. g Y and in particular, being K timelike, g Vx (u, u) > 0 for all u D x and x S p. Finally g Vx (u, u) = 2 s t L(v x + τ(x)k x + (s + t)u) (s,t)=(0,0) = 2 s t L(v x + (s + t)u) (s,t)=(0,0) + 2 s t L(τ(x)K x) (s,t)=(0,0) = g vx (u, u). Thus g Vx D x D x depends only the component of V x on D x and then it is actually the fundamental tensor of a (classical) Finsler metric F on S p such that F 2 (v) := L(v) = 1 2 g v(v, v) = 1 2 g V (v, v), for all V T x M such that V = v + τ(x)k x, v D x, x S p.

37 3.1. Finsler spacetimes and integrable timelike Killing vector fields 29 Putting toghether the above facts, we get L ( dφ (t,x) (τ, v) ) = Λ(x)τ 2 + F 2 (v), for all (τ, v) R T x S p and x S p, where Λ(x) = L(K x )(= L( K x )). By Proposition applied to (U, L), we also get that necessarily K x T for every x U. Remark Clearly, in Definition 3.1.3, we could allow more generality by taking a positive definite, homogeneous, generalized metric g on M (recall the comment after Definition 2.3.4) and/or a function Λ: T M \ 0 (0, + ) which is fiberwise positively homogeneous of degree 0, i.e Λ(λv) = Λ(v) for each v T M \ 0 and λ > 0. Let us focus on the latter case. Observe, first, that the generalized metric g will not come, in general, from a Finsler function. In such a generalized standard static Finsler spacetime the set of future pointing causal vectors J at a point (t, x) is given by the non-zero vectors (τ, v) satisfying τ F (v)/ Λ(v). Being Λ positively homogeneous of degree 0 and positive it satisfies C 1 Λ(v) C 2, for some positive constants C 1, C 2, and for all v T x M, so that F/ Λ can be extended by continuity in 0. Thus, J is connected. Nevertheless, it is, in general, non-convex (see, e.g., Figure 3.1). This is in contrast to what happens in Finsler spacetimes defined through a Finsler function where the connected components of J are convex (see [41, Theorem 2]) and it should be considered as a serious argument against the definition of a Finsler spacetime through a metric which does not come from a quadratic Finsler function. In fact, in this case, a reverse Cauchy-Schwartz inequality (see Proposition below) cannot hold and there exist causal vectors (τ 1, v), (τ 2, w) which are in the same connected component of the set of causal vectors such that g (τ1,v 1 )((τ 1, v 1 ), (τ 2, w)) > 0 (see Figure 3.1). Let us determine the geodesic equations in a standard static Finsler spacetime. Given a manifold N and two points p, q N, let Ω pq (N) be the set of the continuous piecewise smooth curve γ on N parametrized on a given interval [a, b] R and connecting 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 : (ɛ, ɛ) [a, b] N such that ψ(0, s) = γ(s), for all s [a, b], ψ(r, ) Ω pq (N) and there exists a subdivision a = s 0 < s 1 <..., s k = b of the interval [a, b] for which ψ ( ɛ,ɛ) [sj 1,s j ] is smooth for all j {1,..., k}. Clearly, we can define classes of proper variations of γ as those sharing the same variational vector field Z. This is, by definition, a continuous piecewise 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

38 30 3. Chapter 3. Standard static Finsler spacetimes Figure 3.1: The set of the f. p. lightlike vectors (in blue) in R R 2 with the (flat) 4 v 2 2 v static metric g = e 1 2+v2 2 dt 2 + dx 2 + dy 2. In cyan, it is represented the plane of vectors (τ, v 1, v 2 ) which are g (1,1,0) -orthogonal to the lightlike vector (1, 1, 0). that each variational vector field Z along γ individuates a variation (and then also a class of them) by setting ψ(r, s) := exp γ(s) (rz(s)), for r < ɛ small enough. Let us consider the energy functional E : Ω pq ( M) R, E(γ) = b a ( Λ(σ) θ2 + F 2 ( σ) ) ds. As M splits as R M, the path space Ω pq ( M) is identifiable with the product Ω tpt q (R) Ω xpx q (M), where (t p, x p ) = p and (t q, x q ) = q. Definition A continuous piecewise smooth curve γ : [a, b] M is a (affinely parametrized) geodesic of ( M, L) if it is a critical point of the energy functional, i.e. if d dr (E(ψ(r, )) r=0 = 0, for all proper variations ψ of γ. Theorem 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 coordinate (t, x i, τ, v i ) i {1,...,n} on T M: Λ x i (σ) θ 2 + F 2 x i ( σ) d ( ) F 2 ds v i ( σ) = 0, i = 1,..., n (3.4) Λ(σ) θ = const. (3.5)