associated with the Finslerian gravitational field

associated with the Finslerian gravitational field

Satoshi Ikeda and Mihai Anastasiei

Dedicated to the memory of Radu Rosca (1908-2005)

Abstract. Some modified connection structures of the Finslerian grav- itational field are considered by modifying the Finslerian independent variables (xⁱ, yⁱ) (i, j = 1,2,3,4) in the following forms: (xⁱ, yⁱ, pj) (pj: co-vector dual toyⁱ); (xⁱ, y^j, z^k) (z^k: another vector chosen at one more microscopic level than they−level); (x, ψ) (ψ: spinor).

Mathematics Subject Classification:53FB40, 83D05.

Key words:Finslerian gravitational fields, connection structures.

1 Introduction

The independent variables of the Finslerian gravitational field are chosen as (xⁱ, y^j), where the vectory(=y^j;j= 1,2,3,4) is attached to each pointx(=xⁱ;i= 1,2,3,4) as the independent internal variable. Therefore, the Finslerian gravitational field is regarded as a unified field between the external (x)−field spanned by points{x}and the internal (y)−field spanned by vectors{y} (cf.[1]).

From another viewpoint, the Finslerian gravitational field is considered the unified field over the total space of the vector bundle whose fibre at each point x is the (y)−field and base manifold is the (x)−field. This vector bundle is not necessarily the tangent bundleT M over the base manifoldM. It can be any vector bundle case in which one applies the geometry of vector bundles as it is described in [2]. But the theory is simpler for the tangent bundle. Thus we consider the tangent bundle as starting point and we will replace it with certain vector bundles in the next sections.

Then, the adapted frame in the total space is set as follows ([2]):

(1.1)

dX^A= (dxⁱ, δyⁱ=dyⁱ+N_jⁱdx^j)

∂

∂X^A = µ δ

δxⁱ = ∂

∂xⁱ −N_i^j ∂

∂y^j, ∂

∂yⁱ

¶ ,

whereX^A= (xⁱ, y^j) (A= 1,2, ...,8), andN_jⁱ denotes the nonlinear connection repre- senting physically the interaction between the (x)−and (y)−fields.

Balkan Journal of Geometry and Its Applications, Vol.11, No.1, 2006, pp. 81-86.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2006.

We say that this interaction is holonomic if d(δyⁱ) = 0 (modulo δyⁱ). A direct calculation gives d(δyⁱ) +ω_jⁱ∧δy^j = Ωⁱ, where ω_jⁱ = ∂N_kⁱ

∂y^j dx^k and 2Ω^k = (δN_j^k δxⁱ − δN_i^k

δx^j )dxⁱ∧dx^j. Thus the said interaction is holonomic if and only if Ω^k = 0. The local vector fields ( ∂

∂yⁱ) span a distribution on T M called vertical and the local vector fields ( δ

δxⁱ) span a distribution that is supplementary to it, called the horizontal distribution. From the equation [ δ

δx^j, δ

δxⁱ] = Ω^k_ij ∂

∂y^k it comes out that the interaction between the (x)−and (y)−fields is holonomic if and only if the horizontal distribution is integrable.

On the basis of (1.1), the Finslerian connection structure is stipulated as

(1.2) ∇ ^∂

∂XC

∂

∂X^B = Γ^A_BC ∂

∂X^A; Γ^A_BC ≡(Lⁱ_jk, C_jkⁱ ).

Namely, two kinds of connection coefficientsLⁱ_jkandC_jkⁱ appear and the covariant derivatives of an arbitrary vectorVⁱcan be defined by

(1.3)

V_|kⁱ =δVⁱ

δx^k +Lⁱ_jkV^j, Vⁱ|k =∂Vⁱ

∂y^k +C_jkⁱ V^j.

This is the most simplified connection structure and geometrically it corresponds to the Finslerian structure. It preserves by parallelism the vertical and the horizontal distributions and makes covariant constant the almost tangent structure onT M.

By the way, the metrical structure is introduced by

(1.4) G≡GABdX^AdX^B =gij(x, y)dxⁱ⊗dx^j+gij(x, y)δyⁱ⊗δy^j.

Here gij(x, y) are the gravitational potentials depending also on the (y)−field.

The condition that ∇ is metrical, that is, ∇G = 0 is equivalent with g_ij|k = 0 and gij|k = 0. In the other words the tensor field is h− covariant constant and v−

covariant constant.One can impose only one of these conditions. The most important is g_ij|k = 0 as it can be seen from [2]. Given the connection ∇ we may consider in particular y_|kⁱ =Lⁱ_jky^j−N_kⁱ called the h− deflection tensor and yⁱ|k =δⁱ_k+C_jkⁱ y^j called the v− deflection tensor. If we set Dij = giky_|j^k and dij = giky^k|j we can associate to gij two tensor fields Fij = 1

2(Dij −Dji), fij = 1

2(dij −dji) that may be regarded ash−andv−electromagnetic tensors associated tog_ij. They verify the identities that generalize the Maxwell equations (see [2]).

As is understood from (1.2) or (1.3), the connection structure depends essentially on the adapted frame (1.1), which is also constructed by the independent variables (xⁱ, y^j). Therefore, if the independent variables are modified, then the adapted frame

is also modified accordingly and then, the Finslerian connection structure (1.2) is modified correspondingly.

Along this line, in the following, we shall take up the following cases: the indepen- dent variables (xⁱ, y^j) are generalized to (xⁱ, y^j, pj), pj being a covector dual to y^j; (xⁱ, y^j, z^k),z^k being a vector chosen at one more microscopic level than they−level;

(x, ψ),ψbeing a spinor.

2 Modified Connection Structures - I

First, we shall consider the case where the independent variables become (xⁱ, y^j, pj), pj being a covector dual to yj.In fact we replace the tangent bundle T M with the vector bundleT M×T^∗M overM, whereT^∗M is the cotangent bundle ofM. Then, the adapted frame is set as follows (cf. [5])):

(2.1)

dX^A≡(dxⁱ, δyⁱ=dyⁱ+N_jⁱdx^j, δpi =dpi−Mjidx^j),

∂

∂X^A ≡ µ δ

δxⁱ = ∂

∂xⁱ −N_i^j ∂

∂y^j +Mij ∂

∂pj

, ∂

∂yⁱ, ∂

∂pi

¶ , where two kinds of nonlinear connectionsN_jⁱ andMij appear.

The interaction between the (x)−(y)−and (p)−fields is holonomic ifd(δyⁱ= 0 (moduloδyⁱ) andd(δpi) = 0(moduloδpi). We haved(δpi) =−∂Mji

∂p_k δpk−Θjkidx^k∧ dx^j for

2Θjki=∂Mji

∂x^k −∂Mki

∂x^j +Mjh∂Mki

∂ph −Mkh∂Mji

∂ph .

Thus the said interaction is holonomic if and only if Ωⁱ_jk = 0,Θjki = 0. The last equation is equivalent with the integrability of the distribution spanned by (δpi).

Now, the connection structure is given by

(2.2) Γ^A_BC≡(Lⁱ_jk, C_jkⁱ , H_j^ik),

by which the covariant derivatives can be defined as follows: e.g.,

(2.3)

V_|kⁱ = δVⁱ

δx^k +Lⁱ_jkV^j, Vⁱ|k= ∂Vⁱ

∂y^k +C_jkⁱ V^j, Vⁱk^k =∂Vⁱ

∂pk +H_j^ikV^j, In particular, we have

(2.3⁰)

p_i|j =M ij−L^k_ijpk

pi|j =δij−C_ij^k, pik^k=δ^k_i +H_i^jkpj,

Thus new types of electromagnetic tensors could be considered.

The metrical structure is in this case introduced by

(2.4) G≡gijdxⁱ⊗dx^j+gijδyⁱ⊗δy^j+g^ijδpi⊗δpj. A new condition of metrizability of∇appears in the formg^ijk^k= 0.

Next, if the independent variables are chosen as (xⁱ, y^j, z^k), z^k being a vector chosen at one more microscopic level than they−level, then the adapted frame is set as follows:

(2.5)

dX^A≡(dxⁱ, δyⁱ=dyⁱ+N_jⁱdx^j, δz^k =dz^k+A^k_jdy^j+B^k_jdx^j),

∂

∂X^A ≡ µ δ

δxⁱ = ∂

∂xⁱ −N_i^j ∂

∂y^j −B_i^j ∂

∂z^j, δ δyⁱ = ∂

∂yⁱ −A^j_i ∂

∂z^j, ∂

∂zⁱ

¶ , where three kinds of nonlinear connections N_jⁱ, Aⁱ_j and Bⁱ_j appear. In fact here we replace the bundle T M with the product T M ×T M over M. The meaning of the assertion thatz^k is a vector chosen at one more microscopic level than the y−level is provided by the expression δ

δyⁱ = ∂

∂yⁱ −A^j_i ∂

∂z^j. In some sense the variable (yⁱ) become more external then the variables (z^k).

Then, the connection structures is prescribed by (2.6) Γ^A_BC ≡(Lⁱ_jk, C_jkⁱ , Eⁱ_jk),

by which the covariant derivatives can be defined as follows: e.g.,

(2.7)

V_|kⁱ = δVⁱ

δx^k +Lⁱ_jkV^j, Vⁱ|k =δVⁱ

δy^k +C_jkⁱ V^j, Vⁱk^k= ∂Vⁱ

∂z_k +E_jkⁱ V^j, etc. The metrical structure is given by

(2.8) G≡gijdxⁱ⊗dx^j+gijδyⁱ⊗δy^j+gijδzⁱ⊗δz^j.

3 Modified Connection Structures - II

In this section, we shall consider the case where the independent variables become (x, ψ),ψbeing a spinor. Then, if we put the adapted frame in the form (cf. [4])

(3.1)

dX^A≡(dxⁱ, δψ=dψ+ Θiψdxⁱ+ Ξψdψ≡P δ),

∂

∂X^A ≡ µ δ

δxⁱ = ∂

∂xⁱ −Ni ∂

∂ψ, ∂

∂ψ =P⁻¹ ∂

∂ψ

¶ ,

where Θ_i and Ξ denote the spin gauge fields andδψ=dψ+N_idxⁱ,P =I+ Ξψand P Ni ≡Θiψ, then the connection structure can be introduced by

(3.2) Γ^A_BC≡(Lⁱ_jk, Lk, C_jⁱ, C),

by which the following covariant derivatives can be defined: e.g.,

(3.3)

V_|kⁱ = δVⁱ

δx^k +Lⁱ_jkV^j, Vⁱ|=∂Vⁱ

∂ψ +C_jⁱV^j,

etc. Further, for a spinorial field Φ(x, ψ), its covariant derivatives are given by

(3.4)

DΦ =dΦ−ΘiΘdxⁱ−ΞΦdψ= (Φ|k)dx^k+ (Φ|)δψ, Φ|k = δΦ

δx^k −ΘkΦ, Φ|= ∂Φ

∂ψ −ΞΦ, where Θk= Θk−ΞNk and Ξ =P⁻¹Ξ.

4 Other Comments

In this Section, we shall show some other modified connection structures.

First, if the adapted frame (1.1) is changed to

(4.1)

dX^A≡(dxⁱ, δyⁱ=P_jⁱdy^j+Qⁱ_jdx^j),

∂

∂X^A ≡ µ δ

δxⁱ = ∂

∂xⁱ −N_i^j ∂

∂y^j, δ

δyⁱ = (P⁻¹)^j_i ∂

∂y^j

¶ , whereN_jⁱ= (P⁻¹)ⁱ_lQ^l_j, then the connection structure is given by (4.2) Γ^A_BC ≡(Lⁱ_jk−N_k^lC_jlⁱ, (P⁻¹)^l_jC_lkⁱ ).

Second, if the independent variables are chosen as (xⁱ, pj) and the adapted frame in this case is given by (cf. [5])

(4.3)

dX^A≡(δxⁱ=dxⁱ+ Π^jiδpj, δpi=dpi−Mjidx^j),

∂

∂X^A ≡ µ δ

δxⁱ = ∂

∂xⁱ +Mij ∂

∂pj, δ δpi = ∂

∂pj −Π^ij δ δx^j

¶ , then the connection structure is given by (see (2.2))

(4.4) Γ^A_BC ≡(Lⁱ_jk, H_j^ik−Π^klLⁱ_jl).

Finally, if the nonlinear connectionN_jⁱin (1.1) is changed asNⁱ_j =N_jⁱ−F_jⁱ, then the connection structure becomes (cf.[2])

(4.5) Γ^A_BC ≡(Lⁱ_jk=Lⁱ_jk+C_jlⁱF_k^l, Cⁱ_jk=C_jkⁱ ).

Thus, we can consider many interesting modified connection structures by modi- fying the independent variables or the adapted frame in various ways.

References

[1] S. Ikeda,Advanced Studies in Applied Geometry, Seizansha, Sagamihara, 1995.

[2] R. Miron, M. Anastasiei,Vector Bundles and Lagrange Space with Applications to Relativity, Geometry Balkan Press, Bucharest, 1997.

[3] R. Miron,The Geometry of Higher Order Hamilton Spaces, Kluwer Acad. Publ., Dordrecht, 2003.

[4] S. Ikeda,Some structural considerations on the Finslerian gravitational field, Ten- sor, N.S., 59 (1998), 1-5.

[5] R. Miron, D. Hrimiuc, H. Shimada, S.V. Sabau,The Geometry of Hamilton and Lagrange Spaces, Kluwer Acad. Publ., Dordrecht, 2001.

Authors’ addresses:

Satoshi Ikeda

Department of Mechanical Engineering, Faculty of Science and Technology Tokyo University of Science, Noda, Chiba 278-8510, Japan.

Mihai Anastasiei

Faculty of Mathematics, University ”AL.I.Cuza” Ia¸si, 700506, Ia¸si, Romania

Mathematics Institute ”O.Mayer” Ia¸si, Ia¸si Branch of the Romanian Academy, Ia¸si, Romania.

email: [email protected]