Toward a Gravitation Theory in Berwald--Finsler Space
aa r X i v : . [ g r- q c ] N ov Toward a Gravitation Theoryin Berwald–Finsler Space
Xin Li and Zhe Chang Institute of High Energy Physics, Chinese Academy of SciencesP. O. Box 918(4), 100049 Beijing, China
Abstract
Finsler geometry is a natural and fundamental generalization of Riemann ge-ometry. The Finsler structure depends on both coordinates and velocities. It isdefined as a function on tangent bundle of a manifold. We use the Bianchi identi-ties satisfied by Chern curvature to set up a gravitation theory in Berwald-Finslerspace. The geometric part of the gravitational field equation is nonsymmetric ingeneral. This indicates that the local Lorentz invariance is violated. Nontrivialsolutions of the gravitational field equation are presented.PACS numbers: 02.40.-k, 04.20.-q [email protected] [email protected] Introduction
The possible violation of Lorentz invariance have been proposed within several modelsof quantum gravity (QG) as well as the Very Special Relativity (VSR)[1]. A succinctlist of QG includes: tensor VEVs originated from sting field theory[2], cosmologicallyvarying moduli scenarios[3], spacetime foam models[4], semiclassical spin–network cal-culations in Loop QG[5, 6], noncommutative geometry gravity[7, 8, 9, 10] and brane–world scenarios[11]. A common feature of these phenomenological studies on Planckscale physics is introducing of modified dispersion relations (MDR) for elementaryparticles. Girelli et al. [12] proposed a possible relation between MDR and Finsler ge-ometry. Gibbons et al. [13] pointed out that VSR is Finsler geometry. In the VSR, CPTsymmetry is preserved. VSR has radical consequences for neutrino mass mechanism.Lepton-number conserving neutrino masses are VSR invariant. The mere observationof ultra-high energy cosmic rays and analysis of neutrino data give an upper bound of10 − on the Lorentz violation[14].The above facts imply that new physics may connected with Finsler geometry.In facts, in 1941 Randers[15] published his work on possible application of Finslergeometry in physics. Properties of Randers space have been investigated exhaustivelyby both mathematicians and physicists[16]-[20].In a recent paper[21], Kostelecky studied the effect of gravitation in the Lorentz-and CPT-violating Standard Model Extension (SME). The incorporation of Lorentzand CPT violation into general relativity based on Riemann-Cartan geometry was dis-cussed. It provided dominant terms in the effective low-energy action for the gravita-tional sector, thereby completing the formulation of the leading-order terms in the SMEwith gravity. It shows that a generalized geometric framework is helpful in constructinga unification theory of gravity and electromagnetism, weak and strong interaction.Finsler geometry is a natural and fundamental generalization of Riemann geometry.The Finsler structure depends on both coordinates and velocities. It is defined as amapping function from tangent bundle of a manifold to R . S. S. Chern[22] provedthat there is a unique connection in the Finsler manifold that is torsion free andalmost g -compatibility. We use the Bianchi identities satisfied by Chern curvatureto set up a gravitation theory in Berwald-Finsler space. The geometric part of thegravitational field equation is nonsymmetric in general. This indicates that the localLorentz invariance is violated. Nontrivial solutions of the gravitational field equationare presented.This paper is organized as follows. In Sec. 2, we briefly review basic concept andnotations of Finsler geometry[23]. The torsion free Chern connection and correspondingcurvature are introduced. The first and second Bianchi identities for curvature arepresented. Sec. 3 is devoted to construct a gravitation theory in Berwald-Finslerspace. Solutions of gravitational field equation are shown in Sec. 4. In the final, wegive conclusion and remarks. 2 Finsler geometry
Denote by T x M the tangent space at x ∈ M , and by T M the tangent bundle of M .Each element of T M has the form ( x, y ), where x ∈ M and y ∈ T x M . The naturalprojection π : T M → M is given by π ( x, y ) ≡ x .A Finsler structure of M is a function F : T M → [0 , ∞ )with the following properties:(i) Regularity: F is C ∞ on the entire slit tangent bundle T M \ F ( x, λy ) = λF ( x, y ) for all λ > n × n Hessian matrix g ij ≡ ( 12 F ) y i y j is positive-definite at every point of T M \
0, where we have used the notation ( ) y i = ∂∂y i ( ).Finsler geometry has its genesis in integrals of the form Z rs F ( x , · · · , x n ; dx dt , · · · , dx n dt ) dt. (1)Throughout the paper, the lowering and raising of indices are carried out by thefundamental tensor g ij defined above, and its matrix inverse g ij . Given a manifold M and a Finsler structure F on T M , the pair (
M, F ) is called as a Finsler manifold. It isobvious that the Finsler structure F is a function of ( x i , y i ). In the case of F dependingon x i only , the Finsler manifold reduces to Riemannian Manifold.The symmetric Cartan tensor can be defined as A ijk ≡ F ∂g ij ∂y k = F F ) y i y j y k , (2)Cartan tensor vanishes if and only if g ij has no y -dependence. So that Cartan tensoris a measurement of deviation from Riemannian Manifold.Using Euler’s theorem on homogenous function, we can get useful properties of thefundamental tensor g ij and Cartan tensor A ijk g ij l i = F y j , (3) g ij l i l j = 1 , (4) y i ∂g ij ∂y k = 0 , y j ∂g ij ∂y k = 0 , y k ∂g ij ∂y k = 0 , (5) y i A ijk = y j A ijk = y k A ijk = 0 , (6)where l i ≡ y i F . 3 .2 Chern Connection The nonlinear connection N ij on T M \ N ij ≡ γ ijk y k − A ijk F γ krs y r y s , (7)where γ ijk is the formal Christoffel symbols of the second kind γ ijk ≡ g is ∂g sj ∂x k + ∂g sk ∂x j − ∂g jk ∂x s ) . (8)The invariant connection under the transform y −→ λy is of the form N ij F ≡ γ ijk l k − A ijk γ krs l r l s . (9)As usually, we define the covariant derivatives ∇ ∂∂x i and ∇ dx i as ∇ ∂∂x i ≡ ω ji ∂∂x j , (10) ∇ dx i ≡ − ω ij dx j , (11)where ω ij is the connection 1-forms. The operator ∇ have the same linear propertywith the covariant derivatives defined on Riemannian manifold.Here, we introduce the Chern connection that is torsion freeness dx j ∧ ω ij = 0 (12)and almost g -compatibility dg ij − g kj ω ki − g ik ω kj = 2 A ijs δy s F . (13)A theorem given by S. S. Chern [22] guarantees the uniqueness of Chern connection.Theorem (Chern): Let (
M, F ) be a Finsler manifold. The pulled-back bundle π ∗ T M admits a unique linear connection, called the Chern connection. Its connection formsare characterized by the structural equations (12), (13).We ignore the proof of the theorem, just give some consequence of it directly. Torsionfreeness is equivalent to the absence of dy i terms in ω ij ; namely, ω ij = Γ ijk dx k , (14)together with the symmetry Γ ijk = Γ ikj . (15)4nd almost g -compatibility implies thatΓ ijk = g is (cid:18) δg sj δx k + δg sk δx j − δg jk δx s (cid:19) , (16)where δδx i ≡ ∂∂x i − N ji ∂∂x j . (17)The dual basis of ∂∂y i is δy i ≡ dy i + N ij dx j . (18)As before, we prefer to work with δy i F = 1 F ( dy i + N ij dx j ) , (19)which is invariant under rescaling of y .We will work on two new natural local bases that are dual to each other: { δδx i , F ∂∂y i } for the tangent bundle of T M \ { dx i , δy i F } for the cotangent bundle of T M \ V ≡ V ji ∂∂x j ⊗ dx i be an arbitrary smooth local section of π ∗ T M ⊗ π ∗ T ∗ M . Thedefinition (10), (11) and property of operator ∇ imply that the covariant derivativesof V is ∇ V ≡ ( ∇ V ) ji ∂∂x j ⊗ dx i , (20)where ( ∇ V ) ji ≡ dV i + V ki ω jk − V jk ω ki . (21) ∇ V is a 1-form on T M \
0. Thus, it can be expressed in terms of the natural basis { dx i , δy i F } , ( ∇ V ) ji = V ji | s dx s + V ji ; s δy s F . (22)Using relation between the Chern connection and the connection 1-forms ω ij (14), weobtain the horizontal covariant derivative V ji | s V ji | s = δV ji δx s + V ki Γ ijk − V jk Γ kis , (23)5nd the vertical covariant derivative V ji ; s V ji ; s = F ∂V ji ∂y s . (24)The treatment for tensor fields of higher rank is similar with the methods used onRiemannian manifold. Here, we give results of covariant derivatives of the fundamentaltensor g and the norm 1 vector l : g ij | s = g ij | s = 0 , (25) g ij ; s = 2 A ijs and g ij | s = − ij s , (26) l i | s = l i | s = 0 , (27) l i ; s = δ is − l i l s and l i;s = g is − l i l s . (28) The curvature 2-forms of Chern connection areΩ ij ≡ dω ij − ω kj ∧ ω ik . (29)The expression of Ω ij in terms of the natural basis { dx i , δy i F } is of the formΩ ij ≡ R ij kl dx k ∧ dx l + P ij kl dx k ∧ δy l F + 12 Q ij kl δy k F ∧ δy l F , (30)where R , P and Q are the hh -, hv -, vv -curvature tensors of the Chern connection, re-spectively. The following property is manifest R ij kl = − R ij lk , (31) Q ij kl = − Q ij lk . (32)We are now at the position to demonstrate the Bianchi identities for the curvature.Exterior differential of the structural equation (12) gives dx j ∧ dω ij = 0 . (33)The combination of equations (33) and (12) shows that dx j ∧ Ω ij = 0 . (34)Substituting equation (34) into (30), we get12 R ij kl dx j ∧ dx k ∧ dx l + P ij kl dx j ∧ dx k ∧ δy l F + 12 Q ij kl dx j ∧ δy k F ∧ δy l F = 0 . (35)6he three terms on the left side are completely independent. Thus, all of them shouldvanish. This gives identities R ij kl + R ik lj + R il jk = 0 , (36) P ij kl = P ik jl , (37) Q ij kl = 0 . (38)Then, the curvature 2-forms can be simplified asΩ ij ≡ R ij kl dx k ∧ dx l + P ij kl dx k ∧ δy l F . (39)Tedious but straightforward manipulation of exterior differential on the structuralequation (13) givesΩ ij + Ω ji = − ∇ A ) ijk ∧ δy k F − A ijk (cid:20) d ( δy k F ) + ω kl ∧ δy l F (cid:21) . (40)It can be rewritten into12 ( R ijkl + R jikl ) dx k ∧ dx l + ( P ijkl + P jikl ) dx k ∧ δy l F = − A iju R ukl dx k ∧ dx l − A iju P ukl + A ijl | k ) dx k ∧ δy l F +2( A ijk ; l − A ijk l l ) δy k F ∧ δy l F , (41)where we have used the abbreviations R ikl ≡ l j R ij kl (42) P ikl ≡ l j P ij kl . (43)Equalization of three different types of terms at two sides of equation (41 shows iden-tities R ijkl + R jikl = − A iju R ukl , (44) P ijkl + P jikl = − A iju P ukl + A ijl | k ) , (45) A ijk ; l − A ijk l l = A ijl ; k − A ijl l k . (46)The formula (31) and identities (36),(44) enable us get the fourth property of hh -curvature, R klji − R jikl = ( B klji − B jikl ) + ( B kilj + B ljki ) + ( B iljk − B jkil ) , (47)where, for convenient, we have used the notation B ijkl ≡ − A iju R ukl . On Riemannianmanifold, the Cartan tensor vanish. This means that B ijkl = 0 on Riemannian mani-fold. The familiar properties of Riemannian curvature˜ R ijkl + ˜ R ijlk = 0 , ˜ R ijkl + ˜ R kjli + ˜ R ljik = 0 , ˜ R ijkl + ˜ R jikl = 0 , ˜ R ijkl − ˜ R klij = 0 , hh -curvature (31), (36), (44) and(47). Making use of the identity (45) and equations (6), (27), we may get a constituentrelation for P ijkl , P ijkl = − ( A ijk | l − A jkl | i + A kil | j ) + A uij ˙ A ukl − A ujk ˙ A uil + A uki ˙ A ujl , (48)where ˙ A ijk ≡ A ijk | l l s . (49)Contracting P ijkl with l i in equation (48), we obtain an important relation P jkl ≡ l i P ijkl = − ˙ A jkl . (50)The expression of R and P can be got by substituting the formula (29) into (39), R ij kl = δ Γ ijl δx k − δ Γ ijk δx l + Γ ihk Γ hjl − Γ ihl Γ hjk , (51) P ij kl = − F ∂ Γ ijk ∂y l . (52)These are counterparts of the Riemannian curvature expessed in terms of the Christoffelsymbols ˜Γ ijk ˜ R ij kl = ∂ ˜Γ ijl ∂x k − ∂ ˜Γ ijk ∂x l + ˜Γ ihk ˜Γ hjl − ˜Γ ihl ˜Γ hjk . (53)Before ending the section, we present the second Bianchi identity. Exterior differentialof the Chern connection (29) gives d Ω ij − ω kj ∧ Ω ik + ω ik ∧ Ω kj = 0 . (54)Substituting (39) into the above equation, we obtain12 dR ij kl ∧ dx k ∧ dx l + dP ij kl ∧ dx k ∧ δy l F − P ij kl dx k ∧ d ( δy l F )= 12 R ir kl ω rj ∧ dx k ∧ dx l − R rj kl ω ir ∧ dx k ∧ dx l + P ir kl ω rj ∧ dx k ∧ δy l F − P rj kl ω ir ∧ dx k ∧ δy l F . (55)To evaluate d ( δy l F ), we first rewrite δy l F as δy l F dl l + Γ ljk l k dx j + dFF l l . (56)8hen, one has d ( δy l F ) = dl j ∧ ω lj + l j dω lj + dl l ∧ dFF = l j Ω lj + l j ∧ ω kj ∧ ω lk + ( δy j F − ω jk l k − l j dFF ) ∧ ω lj + ( δy l F − ω lk l k ) ∧ dFF = l j Ω lj + δy j F ∧ ( ω lj − l j δy l F ) , (57)here we have used the identity l i δy i F = dFF (58)to get the third equal.Substituting formula (57) into (54) and noticing the torsion freeness of the Chernconnection, we obtain12 ∇ R ij kl ∧ dx k ∧ dx l + ∇ P ij kl ∧ dx k ∧ δy l F = P ij kl l t dx k ∧ ( 12 R lt rs dx r ∧ dx s + P lt rs dx r ∧ δy s F ) − P ij kl l r dx k ∧ δy r F ∧ δy l F . (59)In natural basis, we can rewrite equation (59) into the form12 ( R ij kl | t − P ij ku R ult ) dx k ∧ dx l ∧ dx t + 12 ( R ij kl ; t − P ij kt | l + 2 P ij ku ˙ A ult ) dx k ∧ dx l ∧ δy t F + ( P ij kl ; t − P ij kl l t ) dx k ∧ δy l F ∧ δy t F = 0 . (60)The three terms in the left side are completely independent. Then, we get the followingidentities R ij kl | t + R ij lt | k + R ij tk | l = P ij ku R ult + P ij lu R utk + P ij tu R ukl , (61) R ij kl ; t = P ij kt | l − P ij lt | k − ( P ij ku ˙ A ult − P ij lu ˙ A ukt ) , (62) P ij kl ; t − P ij kt ; l = P ij kl l t − P ij kt l l . (63) Einstein proposed successfully his general relativity in Riemannian space to describegravity. It is interest to investigate the behaviors of gravitation in a more generalFinsler spaces. Let us briefly recall the setup way of the Einstein field equation onRiemannian manifold. One starts from the second Bianchi identities on Riemannianmanifold ˜ R ij kl | t + ˜ R ij lt | k + ˜ R ij tk | l = 0 . (64)9he metric-compatibility ˜ g ij | k = 0 and ˜ g ij | k = 0 , (65)and contraction of (64) with ˜ g jt gives that˜ R jikl | j + ˜ R il | k − ˜ R ik | l = 0 , (66)where ˜ R il ≡ ˜ R ijjl is the Ricci tensor. Lowering the index i and contracting with ˜ g ik ,we obtain ˜ R jl | j + ˜ R jl | j − ˜ S | l = 0 , (67)where ˜ S = ˜ g ij ˜ R ij is the scalar curvature. An equivalent but more familiar form is( ˜ R jl −
12 ˜ g jl ˜ S ) | j = 0 . (68)In the weak field limit, gravitation theory should reduce to the Newtonian theory.Einstein suggested his gravitational field equation of the form˜ R jl −
12 ˜ g jl ˜ S = 8 πGT jl , (69)where T jl is the energy–momentum tensor and G is the Newton’s constant.In the paper, we use similar approach to discuss gravitation on Finsler manifold.Let us introduce first two notions for Ricci curvature: the Ricci scalar Ric and theRicci tensor
Ric ij .The Ricci scalar is defined as Ric = g ik R ik , (70)where R ik ≡ l j R jikl l l is symmetric. The Ricci tensor on Finsler manifold was firstintroduced by Akbar-Zadeh[24] Ric ik ≡ ( 12 F Ric ) y i y k , (71)which is manifestly symmetric and covariant. Expanding y derivatives in the definingformula for Ricci tensor Ric ik , we get Ric ik = 14 ( Ric ; i ; k + Ric ; k ; i ) + 34 ( l i Ric ; k + l k Ric ; i ) + g ik Ric. (72)Substituting the defining formula for Ricci scalar
Ric into the above equation, weobtain
Ric ik = 12 ( R sk si + R si sk )+ 14 l j l l ( R sj sl ; k ; i + R sj sl ; i ; k ) − l j l l ( l i R sj sl ; k + l k R sj sl ; i )+ 12 l j ( R si sj ; k + R sj si ; k + R sk sj ; i + R sj sk ; i ) (73)= 12 ( R sk si + R si sk ) + E ik , (74)10here we introduced the notation E ik ≡ l j l l ( R sj sl ; k ; i + R sj sl ; i ; k ) − l j l l ( l i R sj sl ; k + l k R sj sl ; i )+ 12 l j ( R si sj ; k + R sj si ; k + R sk sj ; i + R sj sk ; i ) . (75)Following same setup process for gravitational field equation in Riemannian space,we start from the second Bianchi identities (61). contracting it with g jt , lowering theindex i , and contracting again with g ik , we get R jiil | j + R jilj | i + R jiji | l = g jt g ik ( P jiku R ult + P jilu R utk + P jitu R ukl ) . (76)Using the first Bianchi identity (44) and formula (47), we can divide the left side ofthe above equation into symmetric part labeled by [ ] and nonsymmetric part labeledby { } R jiil | j + R jilj | i + R jiji | l = (cid:18) Ric jl + 12 B kjk l − E jl (cid:19) | j + (cid:18) B jklk + Ric jl + 12 B kjk l − E jl (cid:19) | j − δ jl ( S − E ) | j = [(2 Ric jl − δ jl S ) − (2 E jl − δ jl E )] | j + { B kjk l + 2 B jklk } | j , (77)where E ≡ g ij E ij and S = g ij Ric ij . Using the constituent relation of the hv -curvaturetensor (48), we rewrite the right side of identity (76) as g jt g ik ( P jiku R ult + P jilu R utk + P jitu R ukl )= 2( A jlu | i − A jrl ˙ A riu ) R u ij + 2( A jiu | j − A u | i + A r ˙ A riu − A jri ˙ A rju ) R u il , (78)where A r ≡ g ij A ijr .Finally, we get the equivalent form of the identity (76) (cid:20)(cid:18) Ric jl − g jl S (cid:19) − (cid:18) E jl − g jl E (cid:19)(cid:21) | j + (cid:26) B kjlk + B jklk (cid:27) | j = ( A jl u | i − A jrl ˙ A riu ) R u ij + ( A jiu | j − A u | i + A r ˙ A riu − A jri ˙ A rju ) R uli . (79)A Finsler structure F is said to be of Berwald type if the Chern connection coefficientsΓ ijk in natural coordinates have no y dependence. A direct proposition on Berwaldspace is that hv –part of the Chern curvature vanishes identically P ij kl = 0 , (80)and the hh –part of the Chern connection reduce to R ij kl = ∂ Γ ijl ∂x k − ∂ Γ ijk ∂x l + Γ ihk Γ hjl − Γ ihl Γ hjk . (81)11o that, in Berwald space the identity (79) reduces as (cid:20) Ric jl − g jl S (cid:21) | j + (cid:26) B kjlk + B jklk (cid:27) | j = 0 . (82)Thus, the counterpart of the Einstein’s field equation on Berwald space takes the form (cid:20) Ric jl − g jl S (cid:21) + (cid:26) B kk jl + B kj lk (cid:27) = 8 πGT jl . (83)The gravitational field equation on Berwald space is obvious different from the Ein-stein’s field equation. The geometric part contains nonsymmetric term. Thus, in gen-eral, the energy–momentum tensor T jl is not symmetric. It means that local Lorentzinvariance is violated in general. At this section, we present examples of Berwald-Finsler space. Kikuchi[25] proved thatin a Randers space of Berwald type, one has˜ b i | j ≡ ˜ b i,j − ˜ b k ˜ γ kij = 0 , (84)where ˜ γ kij is the Christoffel symbols of Riemannian metric ˜ a ≡ ˜ a ij dx i ⊗ dx j . In Randersspace, one can derive straightforwardly the expression of the geodesic spray coefficientsas G i ≡ γ ijk y j y k = (˜ γ ijk + l i ˜ b j | k ) y j y k + (˜ a ij − l i ˜ b j )(˜ b j | k − ˜ b k | j ) αy k , (85)and the Chern connection asΓ ijk = ( N ij ) y k + 12 g it y s ( N st ) y j y k . (86)It is not difficult to check that the geodesic spray coefficients satisfy that12 ∂G i ∂y j = N ij . (87)Thus in Randers spaces of Berwald type, the geodesic spray coefficients reduce to G i = ˜ γ ijk y j y k . (88)The Chern connection reduces to Γ ijk = ˜ γ ijk . (89)Then, the hh –curvature takes the form R ij kl = ∂ ˜ γ ijl ∂x k − ∂ ˜ γ ijk ∂x l + ˜ γ ihk ˜ γ hjl − ˜ γ ihl ˜ γ hjk . (90)12n 4-dimensional Randers space, the Robertson-Walker metric˜ a ij = diag { , − a ( t )1 − kr , − a ( t ) r , − a ( t ) r sin θ } (91)and the constraint ˙ a + k = 0 (92)gives nontrivial solution of the gravitation in the Berwald-Finsler space.A possible solution of (83) for Berwald-Finsler space with one extra dimension isof the form ˜ a ij = diag { , − a ( t )1 − kr , − a ( t ) r , − a ( t ) r sin θ, } , (93)˜ b i = { , , , , c } , (94)where c is constant. In this paper, we have setup a gravitation theory in a torsion freeness Berwald-Finslerspace. The geometric part of the gravitational field equation is , in general, nonsym-metric. This fact indicates that the local Lorentz invariance is violated in the Finslermanifold. This is in good agreement with discussions on special relativity in Finslerspace[13, 12, 20]. Nontrivial solutions of gravitation in Berwald-Finsler space werepresented.However, problems still remain. How to construct a gravitation in general Finslerspace is still a open question. It is well-known that in Riemannian space the signof section curvature K ( x ) determine the type of geometry near x (hyperbolic, flat orspherical). In the landscape of Finslerian, the sign of K ( x, y ) depend on the direction y of our line of sight. This make it possible to encounter all three types of geometryduring a survey. In such a cosmology model, one may wish to find a natural explanationfor why the early universe is asymptotic flat. Acknowledgements
We would like to thank Prof. C.-G. Huang for helpful discussion. The work wassupported by the NSF of China under Grant No. 10575106.
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