Telleparallel Lagrange Geometry and a Unified Field Theory: Linearization of the Field Equations
aa r X i v : . [ g r- q c ] J u l Telleparallel Lagrange Geometry and aUnified Field Theory: Linearization of theField Equations
M. I. Wanas † , Nabil L. Youssef ‡ and A. M. Sid-Ahmed ♮ ∗ † Department of Astronomy, Faculty of Science, Cairo UniversityCTP of the British University in Egypt (BUE)[email protected], [email protected] ‡ Department of Mathematics, Faculty of Science, Cairo [email protected], [email protected] ♮ Department of Mathematics, Faculty of Science, Cairo [email protected], [email protected]
Abstract.
The present paper is a natural continuation of our previous paper: ”Telepar-allel Lagrange geometry and a unified field theory, Class. Quantum Grav., 27 (2010),045005 (29pp)” [14]. In this paper, we apply a linearization scheme on the field equa-tions obtained in [14]. Three important results under the linearization assumption areaccomplished. First, the vertical fundamental geometric objects of the EAP-space loosetheir dependence on the positional argument x . Secondly, our linearized theory in theCartan-type case coincides with the GFT in the first order of approximation. Finally,an approximate solution of the vertical field equations is obtained. Keywords:
Extended Absolute Parallelism geometry, Euler-Lagrange equations, Gen-eralized Field Theory, Extended Teleparallel Unified field theory, Linearization, Cartan-type case.
PACS : 04.50-h, 12.10-g, 45.10.Na, 02.40.Hw, 02.40.Ma.
MSC : 53B40, 53B50, 53Z05, 83C22. ∗ . Motivation and introduction The theory under consideration in the present work is an expansion of the generalizedfield theory (GFT) [3] (formulated in the context of Absolute Parallelism (AP-) geometry(cf. [2], [12], [15]) to the tangent bundle
T M . The theory is formulated in the contextof Extended Absolute Parallelism (EAP-) geometry [16]. The EAP geometry, combineswithin its structure, the geometric richness of the tangent bundle (cf. [7]) and themathematical simplicity of AP-geometry (cf. [9], [12]). The theory, which we refer toas the Extended Teleparallel Unified Field Theory (ETUFT), is constructed in a muchwider and richer context than the GFT. Accordingly, the suggested theory has at leastthe advantages and features of its mother theory (and hopefully more). The GFT has themain properties required for any physical theory unifying gravity and electromagnetism:(1) Its material contents is totally induced by geometry [3].(2) All physical (and geometric) quantities of the theory are derived from one entity,namely, the building blocks of the geometry (the fundamental vector fields formingthe parallelization) [3].(3) The theory shows that the charge of gravity (mass/energy) can generate electro-magnetism [1], [5].(4) The theory shows that there is a direct relation between mass and electric charge.In particular, it shows that a part of mass is electromagnetic in origin and that themass of the electron is totally electromagnetic in origin [13].In addition to the above advantages enjoyed by the GFT, the ETUFT carrieswithin its structure the potentiality of describing interactions other than gravityand electromagnetism [14]. In fact, we conjecture that the vertical field equationsmay express some kind of micro-(or quantum) properties.For the above mentioned reasons and more, we are motivated to study the linearizedform of the suggested field equations in the context of the ETUFT.In the context of geometric field theories, the linearization scheme, although notcovariant, has many advantages. Among these advantages is the determination of theconstants or/and the parameters characterizing a certain theory. Another advantage isto test whether a nonlinear theory covers the domain of a previous, partially successful,linear theory. A third (and important) one is the attribution of some physical meaningto the geometric objects used to construct the theory.The linearization scheme depends mainly on expanding different geometric objects,used in the construction of the theory, in terms of some small parameters and thenneglecting terms of the second and higher order in these parameters. The neglect ofsuch small quantities in the field equations implies the physical assumption that thefield is weak. Also, the neglect of similar quantities in the equations of motion reflectsthe physical assumption that the motion is slow. The two assumptions characterize lowenergy systems. This provides us with a tool to test the theory in low energy regeme.2t should be noted that the production of high energies to test some theories is difficultand sometimes even impossible because of both technological and budgetary reasons.In the present work, we are going to linearize the field equation of the ETUFT[14]. In addition to the advantages of the linearization scheme mentioned above, wehope to throw more light on both the horizontal and vertical geometric objects used inthe construction of the ETUFT. In other words, we hope to illuminate the role of theextra degrees of freedom implied by the ETUFT. A further advantage, which may begained from linearization, is whether one can explore the physical role of the nonlinearconnection characterizing the underlying geometry . Though the mathematical role of thenonlinear connection in the derived field equations is clear , the physical aspect of thisnonlinear connection needs further investigation. The paper is organized in the following manner. In section 2, a short survey of thefield equations obtained in the EAP-context is given. In section 3, we give a brief accounton the process of linearization in the the classical AP-context followed by the processof linearization in the EAP-context. In section 4, we compute the fundamental tensorfields of the EAP-space under our linearization assumption. In section 5, we study theCartan-type case under the linearized condition. In section 6, we give an approximatesolution of the vertical field equations, and finally, we end the paper by some concludingremarks.It should be noted that this paper is a (natural) continuation of [14]. Accordingly,we will use the results of [14] and stick to its notations. In fact, the splitting of the field equations into horizontal and vertical counterparts is made possibledue to the existence of the nonlinear connection. This may be partially achieved if an appropriate physical interpretation of the directional argument y is given. . A short survey of the field equations in the EAP-context We first recall the fundamental tensor fields of the EAP-space. These are given inthe following table [14].
Table 1: Fundamental second rank tensors of EAP-space
Horizontal VerticalSkew-Symmetric Symmetric Skew-Symmetric Symmetric ξ µν := γ µνα | α ξ ab := γ abd || d γ µν := C α γ µνα γ ab := C d γ abd η µν := C β Λ βµν φ µν := C β Ω βµν η ab := C d T dab φ ab := C d Ω dab χ µν := Λ αµν | α ψ µν := Ω βµν | β χ ab := T dab || d ψ ab := Ω dab || d ǫ µν := C µ | ν − C ν | µ θ µν := C µ | ν + C ν | µ ǫ ab := C a || b − C b || a θ ab := C a || b + C b || a U µν := γ βαµ γ ανβ − γ βµα γ αβν h µν := γ βαµ γ ανβ + γ βµα γ αβν U ab := γ cda γ dbc − γ cad γ dcb h ab := γ cda γ dbc + γ cad γ dcb σ µν := γ βαµ γ αβν σ ab := γ cda γ dcb ω µν := γ βµα γ ανβ ω ab := γ cad γ dbc α µν := C µ C ν α ab := C a C b We now give a short survey of the field equations obtained in [14]. We take for thehorizontal field equations a Lagrangian similar in form (but not in content) to that usedby Mikhail and Wanas in their construction of the GFT [3]. We also assume that thenonlinear connection is independent of the horizontal counterparts of the fundamentialvector fields forming the parallelization. In view of the above, for the horizontal field equations , we start with the followingscalar Lagrangian: Let H = | λ | g µν H µν , This condition is actually satisfied under the Cartan-type condition [14]. H µν := Λ αǫµ Λ ǫαν − C µ C ν . (2.1)The Euler-Lagrange equations [7] for this Lagrangian are given by δ H δλ β := ∂ H ∂λ β − ∂∂x γ (cid:18) ∂ H ∂λ β,γ (cid:19) − ∂∂y a (cid:18) ∂ H ∂λ β ; a (cid:19) = 0 . (2.2)Setting E βσ := 1 | λ | (cid:18) δ H δ j λ β (cid:19) j λ σ , (2.3)the Euler-Lagrange equations (2.2) take the form0 = E βσ = δ βσ H − H βσ − C σ C β − δ βσ C ǫ | ǫ + 2 δ βσ C ǫ C ǫ − C ǫ Λ βǫσ + 2 g αβ C σ | α − g γα Λ βσα | γ − N aα ; a (Λ α σ β − Λ β σ α )+ 2 C ν ( δ βσ N aν ; a − δ βν N aσ ; a ) + 2 g αβ { S σ,α,ǫ C ǫαa R aσǫ } (2.4)Lowering the index β in (2.4) and renaming the indices, we get0 = E µν := g µν H − H µν − C µ C ν − g µν ( C ǫ | ǫ − C ǫ C ǫ ) − C ǫ Λ µǫν + 2 C ν | µ − g ǫα Λ µνα | ǫ − N aǫ ; a (Λ ǫ νµ − Λ µν ǫ ) + 2 g µν C ǫ N aǫ ; a − C µ N aν ; a + 2 S µ,ν,ǫ C ǫµa R aνǫ . (2.5)This is the generalized horizontal field equations in the context of the EAP-geometry.Considering the symmetric part of (2.5), denoting N β := N aβ ; a , it is found that0 = E ( µν ) := ( g µν ◦ R − ◦ R ( µν ) ) + g µν ( σ − h − Q ) − σ µν − h µν − Q ( µν ) )+ N β (Λ µνβ + Λ νµβ ) + 2 g µν C β N β − ( C µ N ν + C ν N µ ) , (2.6)which represents the symmetric part of the generalized horizontal field equations.Setting M µν := N β Λ µνβ , Z µν := C µ N ν , Z := g µν Z µν , (2.7)we conclude, by (2.6), that ◦ R ( µν ) − g µν ◦ R = T ( µν ) ; (2.8) T ( µν ) := 12 g µν ( σ − h − Q + 2 Z ) − ( σ µν − h µν − Q ( µν ) ) + ( 12 N β Ω βµν − Z ( µν ) ) . (2.9) T ( µν ) is interpreted as the generalized energy momentum tensor, constructed from thesymmetric tensors σ µν , h µν , N β Ω βµν , Q ( µν ) and Z ( µν ) .5onsequently, the horizontal Einstein tensor has the form ◦ J µν := ◦ R µν − g µν ◦ R = { g µν ( σ − h ) + ( h µν − σ µν ) } + 12 g µν (2 Z − Q ) + ( 12 N β Ω βµν − Z ( µν ) + Q ( µν ) )+ 12 S µ,ν,α ◦ C αµa R aνα , (2.10)which is subject to the identity ◦ J µσ o | µ = R aσµ ◦ P µa + 12 R aαµ ◦ P αµσa . (2.11)On the other hand, considering the skew-symmetric part of equation (2.5), it is foundthat0 = E [ µν ] = 2 { ( γ µν − ǫ µν − ξ µν + N β Λ βµν ) + ( M [ µν ] − Z [ µν ] ) } + 3 S µ,ν,ǫ C ǫνa R aǫµ . (2.12)Let us define F µν : = ( γ µν − ξ µν + η µν + N β Λ βµν ) + ( M [ µν ] − Z [ µν ] ) + 32 S µ,ν,ǫ C ǫνa R aǫµ = ( γ µν − ξ µν + η µν ) + N β ( γ µνβ + Λ βµν ) + ( 12 N β Λ βµν − Z [ µν ] )+ 32 S µ,ν,ǫ C ǫνa R aǫµ , (2.13)then we get from (2.12) and (2.13) F µν = δ ν C µ − δ µ C ν (2.14)and S µ,ν,σ F µν o | σ = − S µ,ν,σ R aµν ˙ ∂ a C σ . (2.15)Accordingly, if F µν is interpreted as the horizontal electromagnetic field, then (2.15)represents the generalized horizontal Maxwell’s equations and, in view of (2.14), C µ isthe horizontal electromagnetic potential. Again, by (2.13), F µν is constructed from thehorizontal skew-symmetric fundamental tensors of the EAP-space (Table 1) togetherwith the skew-symmetric tensors N β γ µνβ , N β Λ βµν , S µ,ν,ǫ C ǫνa R aǫµ and Z [ µν ] . It is thusconstructed from a purely geometric standpoint.Moreover, if J µ := F µν o | ν , (2.16)then it is deduced that J µ o | µ = 12 { F ǫµ ( ◦ R µǫ − ◦ R ǫµ ) + R aµν F µν o || a } . (2.17)In the case where the nonlinear connection N αµ is integrable [7], (2.17) can be viewed asa generalized conservation law and J µ as the generalized horizontal current density.6or the vertical field equations , we consider a scalar Lagrangian formed of verticalentities, namely, V := || λ || g ab V ab , where V ab := T dea T edb − C a C b . (2.18)The Euler-Lagrange equations reduce to ∂ V ∂λ b − ∂∂y e (cid:18) ∂ V ∂λ b ; e (cid:19) = 0 . (2.19)In this case, we obtain,0 = E ab := g ab V − V ab − g ab ( C e || e − C e C e ) − C a C b − C e T aeb + 2 C b || a − g de T abe || d . (2.20)Considering the symmetric part of (2.20), we get0 = E ( ab ) := ( g ab ◦ S − ◦ S ab ) + g ab (¯ σ − ¯ h ) − σ ab − h ab ) , (2.21)so that ◦ S ab − h ab ◦ S = T ab , (2.22) T ab := 12 g ab (¯ σ − ¯ h ) − ( σ ab − h ab ) , (2.23)where T a b o || a = 0 . (2.24)Consequently, in view of (2.22) and (2.24), T ab could be interpreted as the generalizedvertical energy-momentum tensor, which is, according to (2.23), constructed from thevertical symmetric fundamental tensors of the EAP-space (Table 1).Considering the skew-symmetric part of (2.20), we conclude that if F ab := ( γ ab − ξ ab + η ab ) , (2.25)then F ab = ˙ ∂ b C a − ˙ ∂ a C b . (2.26) F ab is interpretted as the generalized vertical electromagnetic field and C a as the gener-alized vertical electromagnetic potential. Moreover, F ab satisfies the differential identity S a,b,c F ab o || c = 0 . (2.27)It is clear, by (2.25), that F ab is constructed from the vertical skew-symmetric tensorsof Table 1.Finally, if we set J a := F ab o | b (2.28)7hen, similar to (2.17), J a represents the generalized vertical current density and satisfiesthe conservation law J a o | a = 0 . (2.29)
3. Linearization scheme in the EAP-context
We first give a brief account of the process of linearization in the the classical AP-context [4]. The vector fields i λ µ in the Minkowski space of special relativity are givenby i λ µ = i δ µ , (3.1)where i ∈ { , . . . , } , µ ∈ { , . . . , } and i δ µ is the Kronecker delta. To get a space whichdiffers slightly from the flat space, it is assumed that i λ µ = i δ µ + ǫ i h µ (3.2)where i h µ ∈ C ∞ ( M ) represents perturbation terms and the parameter ǫ is assumedto be of small magnitude compared to unity. In this scheme, each geometric object G defined in the AP-space can be expressed in the form G = p X r =0 ǫ r G ( r ) ( p ∈ { , , . . . } ) , (3.3)where r denotes the power of ǫ and G ( r ) is the coefficient of ǫ r . For example, using (3.2),the metric tensor g µν is found to be g µν = i λ µ i λ ν = ( i δ µ + ǫ i h µ )( i δ ν + ǫ i h ν )= δ µν + ǫ ( i δ µ i h ν + i δ ν i h µ ) + ǫ ( i h µ i h ν )This linearization proceedure was carried out by Mikhial and Wanas for the GFT([4], [11]). The results obtained showed a perfect agreement of GFT, in the first orderof approximation, with both general relativity and Maxwell’s theories. In addition, thelinearized theory led to the prediction of new features, namely, the existence of a mutualinteraction between both gravitational and electromagnetic fields [4].In the present paper, we carry out a linearization of the field equations obtainedin [14]. We will use the same conventions usually followed by physicists in the currentliterature in which, for example, mesh and world indices are mixed. The treatment hereis, therefore, somewhat less rigorous than our previous paper [14].In analogy to the above linearization process, we assume, in the context of EAP-space, that i λ µ = i δ µ + ǫ i h µ ( x ) + ε i k µ ( y ) , i λ a = i δ a + ǫ i h a ( x ) + ε i k a ( y ) , (3.4) the algebra of smooth functions on M i ∈ { , . . . , } , µ ∈ { , . . . , } , a ∈ { , . . . , } and i h µ , i k µ , i h a , i k a ∈ C ∞ ( T M )represent horizontal and vertical perturbation terms. Here i h µ , i h a are functions of thepositional argument x only whereas i k µ , i k a are functions of the directional argument y only. Moreover, the parameters ǫ and ε are assumed to be of small magnitude comparedto unity: O ( ǫ ) ⋍ O ( ε ) ǫ , ǫε , ε or higher orders can beneglected. This means that we are dealing with a weak field . Finally, e will denoteeither ǫ or ε so that, for example, O ( e ) will mean either O ( ǫ ), O ( ǫε ) or O ( ε ).We interpret x , x and x as space coordinates wheresas x is taken to be the timecoordinate. On the other hand, the vector y a is attatched as an internal variable toeach point x µ . In this sense, y a may be regarded as the spacetime fluctuation (themicro-internal freedom) associated to the point x µ ([7], [8].) We will return to theinterpretation of the directional argument y later on.
4. First order approximation
We now carry out the task of computing the fundamental tensors of the EAP-spaceunder the linearization assumption (3.4). The vertical (resp. horizontal) counterpart isobtained under no condition (resp. in the Cartan type case). Though, we have dealtwith many cases in our previous paper [14], it is the Cartan type case that can lend itselfto the process of linearization . This is because the nonlinear connection and hence allgeometric objects considered are expressed explictly in terms of the fundamental vectorfields. On the other hand, a linearization of the horizontal field equations in the Berwaldtype case gives nothing new, since the derived horizontal field equations in this caseactually coincide with those of the GFT [14]. In preparation to what follows, we set z µν := ( µ h ν + ν h µ ) , w µν := ( µ k ν + ν k µ ); µ h ν := i δ µ i h ν , µ k ν := i δ µ i k ν (4.1) z ab := ( a h b + b h a ) , w ab := ( a k b + b k a ); a h b := i δ a i h b , a k b := i δ a i k b . (4.2)Then, in view of (3.4), we obtain Theorem 4.1.
To the first order of approximation, we have (a) i λ µ ⋍ i δ µ − ǫ µ h i ( x ) − ε µ k i ( y ) ; i λ a ⋍ i δ a − ǫ a h i ( x ) − ε a k i ( y ) . (b) g µν ⋍ δ µν + ǫz µν ( x ) + εw µν ( y ); g µν ⋍ δ µν − ǫz µν ( x ) − εw µν ( y ) . (c) g ab ⋍ δ ab + ǫz ab ( x ) + εw ab ( y ); g ab ⋍ δ ab − ǫz ab ( x ) − εw ab ( y ) . (d) C abc ⋍ ε ( a k b ; c )( y ) . (e) T abc ⋍ ε ( a k b ; c − a k c ; b )( y ) . (f ) C b ⋍ ε ( a k b ; a − a k a ; b )( y ) . This implies, in particular, that both Maxwell’s and general relativity theories are an outcome ofour field equations under the Berwald condition. g) ◦ C abc ⋍ ε ( w ab ; c + w ac ; b − w bc ; a )( y ) . (h) γ abc ⋍ ε { a k b ; c − ( w ab ; c + w ac ; b − w bc ; a ) } ( y ) . (j) Ω abc ⋍ ε { ( a k b ; c + a k c ; b ) − ( w ab ; c + w ac ; b − w bc ; a ) } ( y ) . Proof.
We prove (a) , (e) and (g) only. The rest can be proved in a similar manner. (a) Assume, to the first order of e , that i λ µ = i D µ + ǫ i H µ + ε i K µ . Then, in view of i λ µ i λ ν = δ µν , we have i D µ = i δ µ , ǫ { i H µ i δ ν + i h ν i δ µ } = 0 and ε { i K µ i δ ν + i k ν i δ µ } = 0 . Bythe second relation above, we get i H µ i δ ν = − i h ν i δ µ . Multiplying by j δ ν , we obtain j H µ = − ( j δ ν i h ν ) i δ µ = − i h j i δ µ := − µ h j . Similarily, by ε { i K µ i δ ν + i k ν i δ µ } = 0 , weconclude that j K µ = − µ k j . The expression of i λ a is obtained in a similar manner,using the relation i λ a i λ b = δ ab . (e) By (a) , we have C abc = i λ a ˙ ∂ c i λ b = { i δ a − ǫ a h i − ε a k i + O ( e ) }{ ε i k b ; c } = ε a k b ; c + O ( e ) . (g) By (c) , we have g ad = δ ad − ǫz ad ( x ) − εw ad ( y ) + O ( e ). Moreover, g cd ; b = εw cd ; b ( y ) + O ( e ) , g bd ; c = εw bd ; c ( y ) + O ( e ) , g bc ; d = εw bc ; d ( y ) + O ( e ) . Consequently, ◦ C abc = 12 g ad { g cd ; b + g bd ; c − g bc ; d } ⋍ εδ ad { w dc ; b + w db ; c − w bc ; d } ( y ) = 12 ε { w ab ; c + w ac ; b − w bc ; a } ( y ) . In the light of the above theorem, using the relevant definitions, we obtain
Proposition 4.2.
To the first order of ε , the following hold: (a) ǫ ab ⋍ ε ( d k a ; bd − d k b ; ad )( y ) , (b) θ ab ⋍ ε ( d k a ; bd + d k b ; ad − d k d ; ab )( y ) , (c) ξ ab ⋍ ε { a k b ; dd − ( w ab ; dd + w ad ; bd − w bd ; ad ) } ( y ) , (d) ψ ab ⋍ ε { ( d k a ; bd + d k b ; ad ) − ( w ad ; bd + w bd ; ad − w ab ; dd ) } ( y ) , (e) χ ab ⋍ ε ( d k a ; bd − d k b ; ad )( y ) . Let W abc := ( w ab ; c + w ac ; b − w bc ; a ) . Then the following tensors contain no terms offirst order:
Proposition 4.3.
To the second order of ε , the following hold: (a) γ ab ⋍ ε { ( c k d ; c − c k c ; d )( a k b ; d − W abd ) } ( y )10 b) φ ab ⋍ ε { ( c k d ; c − c k c ; d )( d k a ; b + d k b ; a − W dab ) } ( y ) . (c) η ab ⋍ ε { ( c k d ; c − c k c ; d )( d k a ; b − d h b ; a ) } ( y ) . (d) ω ab ⋍ ε { ( d k a : c − W dac )( c k b ; d − W cbd ) } ( y ) . (e) σ ab ⋍ ε { ( d k c ; a − W dca )( c k d ; b − W cdb ) } ( y ) . (f ) α ab ⋍ ε { ( c k a ; c − c k c ; a )( c k b ; c − c k c ; b ) } ( y ) . (g) h ab ⋍ ε { ( d k c ; a − W dca )( c k b ; d − W cbd ) + ( d k a ; c − W dac )( c k d ; b − W cdb ) } ( y ) . (h) U ab ⋍ ε { ( d k c ; a − W dca )( c k b ; d − W cbd ) − ( d k a ; c − W dac )( c k d ; b − W cdb ) } ( y ) . Corollary 4.4.
Assume that (3.4) holds. Then, up to the first order of approxima-tion, the vertical fundamental geometric objects of the EAP-space are functions of thedirectional argument y only.
5. Linearization in the Cartan-type case
We now consider the Cartan-type case. In this case, the nonlinear connection coeffi-cients N aµ are given by N aµ = y b ( i λ a ∂ µ i λ b )[16]. Consequently, all geometric objects of theEAP-space are expressed explicitely in terms of the fundamental vector fields formingthe parallelization.Similar to Theorem 4.1, we have the following Theorem 5.1.
Assume that the canonical d -connection is of Cartan type. Then, up tothe first order approximation, the following hold: (a) N aµ ⋍ ǫ { y b ( a h b,µ ( x )) } . (b) Γ αµν ⋍ ǫ ( α h µ,ν )( x ) , Γ abν ⋍ ǫ ( a h b,ν )( x ) . Consequently, N ν := N aν ; a ⋍ ǫ a h a,ν ( x ) . (c) Λ αµν ⋍ ǫ ( α h µ,ν − α h ν,µ )( x ) . (d) C µ ⋍ ǫ ( α h µ,α − α h α,µ )( x ) . (e) ◦ Γ αµν ⋍ ǫ ( z µα,ν + z να,µ − z µν,α )( x ) . (f ) γ αµν ⋍ ǫ { α h µ,ν − ( z µα,ν + z να,µ − z µν,α ) } ( x ) . (g) Ω αµν ⋍ ǫ { ( α h µ,ν + α h ν,µ ) − ( z µα,ν + z να,µ − z µν,α ) } ( x ) . Proof.
We prove (a) only. The rest is similar. We have N aµ = y b ( i λ a ∂ µ i λ b )= y b { i δ a − ǫ a h i − ε a k i + O ( e ) }{ ǫ i h b,µ } = ǫ ( y b a h b,µ ) + O ( e ) .
11n the next two propositions, we assume that the canonical d -connection is of Cartantype . Then, using the relevant definitions, taking into account Theorem 5.1 and setting Z βµν := ( z βµ,ν + z βν,µ − z νµ,β ) , we obtain Proposition 5.2.
To the first order of ǫ , we have (a) ǫ µν ⋍ ǫ ( α h µ,αν − α h ν,αµ )( x ) , (b) θ µν ⋍ ǫ ( α h µ,αν + α h ν,αµ − α h α,µν )( x ) , (c) ξ µν ⋍ ǫ { µ h ν,αα − Z µνα,α } ( x ) , (d) ψ µν ⋍ ǫ { ( α h µ,να + α h ν,µα ) − Z αµν,α } ( x ) , (e) χ µν ⋍ ǫ ( α h µ,να − α h ν,µα )( x ) . The following tensors are of order ǫ . Proposition 5.3.
To the second order of ǫ , we have (a) γ µν ⋍ ǫ { ( β h α,β − β h β,α )( µ h ν,α − Z µνα ) } ( x ) (b) φ µν ⋍ ǫ { ( β h α,β − β h β,α )( α h µ,ν + α h ν,µ − Z αµν ) } ( x ) . (c) η µν ⋍ ǫ { ( β h α,β − β h β,α )( α h µ,ν − α h ν,µ ) } ( x ) . (d) ω µν ⋍ ǫ { ( α h µ,β − Z αµβ )( β h ν,α − Z βνα ) } ( x ) . (e) σ µν ⋍ ǫ { ( α h β,µ − Z αβµ )( β h α,ν − Z βαν ) } ( x ) . (f ) α µν ⋍ ǫ { ( β h µ,β − β h β,µ )( σ h ν,σ − σ h σ,ν ) } ( x ) . (g) h µν ⋍ ǫ { ( β h α,µ − Z βαµ )( α h ν,β − Z ανβ ) + ( β h µ,α − Z βµα )( α h β,ν − Z αβν ) } ( x ) . (h) U µν ⋍ ǫ { ( β h α,µ − Z βαµ )( α h ν,β − Z ανβ ) − ( β h µ,α − Z βµα )( α h β,ν − Z αβν ) } ( x ) . The relation equivalent to (2.5) in the Cartan type case is given by0 = E µν := g µν H − H µν − C µ C ν − g µν ( C ǫ | ǫ − C ǫ C ǫ ) − C ǫ Λ µǫν + 2 C ν | µ − g ǫα Λ µνα | ǫ − N ǫ (Λ ǫ νµ − Λ µν ǫ ) + 2 g µν C ǫ N ǫ − C µ N ν . (5.1)Consequently, C µ N ν , C µ N µ , N β Λ µνβ and N β Λ βµν are the additional terms appearingin the horizontal field equations under the Cartan condition that have no counterpartsin the field equations obtained in the context of the GFT. Proposition 5.4.
The following holds: (a) C µ N ν = ǫ { ( α h µ,α − α h α,µ ) c h c,ν } ( x ) . b) N β Λ µνβ = ǫ { c h c,β ( µ h ν,β − ( µ h β,ν ) } ( x ) . (c) N β Λ βµν = ǫ { c h c,β ( β h µ,ν − β h ν,µ ) } ( x ) . In view of Propositions (5.2), (5.3) and (5.4), we have
Corollary 5.5.
Assume that (3.4) holds. Then, up to the first order of approximation,the purely horizontal fundamental geometric objects of the EAP-space in the Cartan typecase are identical to their corresponding counterparts in the context of classical AP-space[11] and are functions of the positional argument x only. This is because all extra termsappearing in the horizontal field equations in ETUFT are of order ǫ . Consequently, thehorizontal field equations under the Cartan condition coincides with the GFT up to thefirst order of approximation. One reading of Corollary 5.5 is that our constructed unified field theory seems tobe a plausable generalization of the GFT. Not only do the horizontal field equations inthe Berwald-type case coincide with those of the GFT [14], but also the horizontal fieldequations in the Cartan-type case coincide with the GFT in the first order of approxi-mation . This also means that our theory (under the Cartan type condition) differs fromthe GFT only when dealing with strong fields, that is, in the second (and higher orders)of ǫ . In other words, the two theories actually coincide when dealing with weak fields .
6. Approximate solutions of the vertical field equations
In this final section, we will examine the solutions of the vertical field equationscorresponding to the first order of approximation.By (2.22) and (2.23), the vertical Einstein tensor is expressed in terms of the funda-mental tensors in the form: ◦ S ab − h ab ◦ S = 12 g ab (¯ σ − ¯ h ) − ( σ ab − h ab ) . (6.1)As easily checked, in view of Theorem 4.1 (c) and Proposition 4.3 (e) and (g) , (6.1)reduces in the first order of approximation to( ◦ S ab ) (1) − δ ab ◦ S (1) = 0 . (6.2)Contracting, we get ◦ S (1) = 0, so that (6.2) reduces to( ◦ S ab ) (1) = 0 . (6.3)Now, we have [14] ◦ S ab = −
12 ( θ ab − ψ ab + φ ab ) + ω ab . (6.4)Consequently, in view of Proposition 4.3 (( φ ab ) (1) = ( ω ab ) (1) = 0), (6.4) reduces in thefirst order of approximation to( ◦ S ab ) (1) = 12 { ( ψ ab ) (1) − ( θ ab ) (1) } , (6.5)13o that, by Proposition 4.2 (b) and (d) , noting that w dd = 2 d k d , we obtain( ◦ S ab ) (1) = 12 { ( d k a ; bd + d k b ; ad ) − w da ; bd − w db ; ad + w ab ; dd − ( d k a ; db + d k b ; ad ) + 2 d k d ; ab } = 12 { w ab ; dd − w ad ; bd − w bd ; ad + w dd ; ab } . The above equation, together with (6.3), gives w ab ; dd − w ad ; bd − w bd ; ad − w dd ; ab = 0 . (6.6)Consequently, w ab ; dd = w ad ; bd + w bd ; ad − w dd ; ab = ( w ad ; bd − w dd ; ab ) + ( w bd ; ad − w dd ; ab ) , which can be expressed in the form w ab ; dd = (cid:8) ( w ad ; d − w dd ; a ) ; b + ( w bd ; d − w dd ; b ) ; a (cid:9) . (6.7)Hence, if ( ◦ C add ) (1) = 0, that is, w ad ; d = w dd ; a , then the symmetric part of the verticalfield equations in this case reduces to w ab ; dd = (cid:8) ∂ ∂ ( y ) + ∂ ∂ ( y ) + ∂ ∂ ( y ) + ∂ ∂ ( y ) (cid:9) w ab = 0 . (6.8)In view of the interpretation of the variable y a , (6.8) represents, `a la Kaluza , a waveequation in the micro-internal dimension .On the other hand, by (2.25) and (2.26), the vertical generalized electromagneticfield is given by F ab := ( γ ab − ξ ab + η ab ) , (6.9) F ab = ˙ ∂ b C a − ˙ ∂ a C b . (6.10)By Proposition 4.3, (6.9) and (6.10) reduce in the first approximation to( C a ; b ) (1) − ( C b ; a ) (1) = − ( ξ ba ) (1) (6.11)Now, in the first order of ε , C a ⋍ ε ( d k a ; d − d k d ; a ) ⋍ ε ( C a ) (1) ; C a ; b ⋍ ε ( d k a ; db − d k d ; ab ) ⋍ ε ( C a ; b ) (1) . (6.12)Consequently, ( C a ; b ) (1) = ( C a ) (1); b = d k a ; db − d k d ; ab . (6.13)14ontraction of relation (6.6) yields w dd ; aa = w ad ; ad , (6.14)equivalently, d k d ; aa = d k a ; da . (6.15)Hence, in view of (6.13) and (6.15), we conclude that( C d ; d ) (1) = ( C d ) (1); d = 0 (6.16)Differentiating (6.11), taking into account (6.16), we obtain( C a ; dd ) (1) = ( C a ) (1); dd = − ( ξ da ) (1); d = − ( ξ da ; d ) (1) (6.17)Consequently, by J a := F ab o | b , where J a is the vertical current density, we obtain (cid:8) ∂ ∂ ( y ) + ∂ ∂ ( y ) + ∂ ∂ ( y ) + ∂ ∂ ( y ) (cid:9) ( C a ) (1) = − ( J a ) (1) . (6.18)Hence, if ( J a ) (1) = 0, then ( C a ) (1) again satisfies a wave equation in the micro-internaldimension .The vertical part of the field equations gives rise to two different 4-dimensionalLaplace equations which would be wave equations if the metric was not positive def-inite, but had Lorentz signature. The meaning of this result, however, will be clarifiedif a clear physical interpretation of the vector y a is given .We end the paper with the following remarks and comments: • As previously mentioned, a possible interpretation of the coordinates would to take x , x and x as space coordinates and x as time coordinates. On the other hand,the vector y a is attached as an internal variable to each x µ . According to Miron [8], y a may be regarded as spacetime fluctuations (micro internal freedom) associated tothe point x µ . It could be argued that this interpretation, however, seems somewhatincompatible with (3.4). This is because we are actually combining dimensions ofdifferent natures, namely, a macro dimension x and a micro dimension y . Toavoid this (apparent) incompatibility, we shall take x µ as stated above and leavethe meaning of the directional argument y a unspecified, at least for the present. • In any geometric field theory, authors try to attribute physical meaning to thegeometric objects present in the theory. One way to accomplish this is to comparethe new theory with previously existing field theories. The GFT is a field theoryunifying electromagnetism and gravitation. It is thus a theory that generalizesboth Maxwell’s electromagnetic theory and Einstein’s general theory of relativ-ity. The comparison of the GFT with these two theories, resulted in attributingsome physical meaning to certain geometric objects occuring in the GFT (usingcertain systems of units). A new scheme, called
Type Analysis [13], has also15een suggested in the context of the GFT. Its aim is to test the AP-geometry(the underlying geometry of the GFT) for representing certain physical fields. Theprocedure of Type Analysis is usually applied before solving the field equations.It is a covariant procedure formulated in the language of tensors. Some tensorsare used to characterize the AP-space under consideration. Roughly, the typesof spaces considered are as follows: spaces with or without electromagnetic field,spaces with or without gravitational fields, and, finally, spaces in which both fieldsare present. The strengths of the fields involved are also taken into account. Moreprecisely, some tensors are defined indicating the presence or absence of electomag-netic and gravitational fields. Not all possible combinations, however, are allowed.There are certain constraints. For example, a space with a non-vanishing electro-magnetic field necessarily contains a non-vanishing gravitational field. • One of our future aims is to generalize the procedure of Type Analysis to ourconstructed field theory. It should be noted that this is not an easy task. This isbecause both the physical and the mathematical scope of the ETUFT are muchwider and richer in content than the GFT. Therefore, in the context of the ETUFT,an “extension” of the procedure of Type Analysis (applied to the (horizontal)field equations in the Cartan type case and the vertical field equations in thegeneral case) is expected to be more complicated and much more involved. Someof the reasons for such complications are the following: First, the (horizontal)field equations in the Cartan type are more complex than their correspondingcounterparts in the GFT (they coincide only in the first order of approximation).Secondly, we have two types of “e” in our linearized field theory. Last, but notleast, our geometric objects, unlike the classical AP-geometry (in which the GFTis formulated), depend, in general, on both the directional argument x and thepositional argument y . • On the other hand, an application of (some method similar to) the procedureof Type analysis to the ETUFT may help us illuminate the physical role of thenonlinear connection and give a plausible physical interpretation to the directionalargument y . This, in turn, may shed more light on the possibility that the ETUFT- in addition to unifying gravity and electromagnetism - could also describe somemicro physical phenomena. This is easy to see. An electromagnetic field carries energy, that is mass (since energy and mass areequivalent). But mass is the source of gravitational field. In conclusion, we note that both Einstein’s and Maxwell’s equations in the ETUFTare doubled . This is due to the splitting of the field equations induced by thenonlinear connection. An interpretation of the new equations is needed. Perhapsit is possible to connect one of Maxwell’s fields with SU (2) gauge fields. Such astep seems necessary, given the spectacular success of Weinberg-Salam theory. Thispoint will be the subject of future research. References [1] R. S. De Souza and R. Opher,
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