Spinning and Spinning Deviation Equations for Special Types of Gauge Theories of Gravity
aa r X i v : . [ g r- q c ] J a n Spinning and Spinning Deviation Equations for Special Types of GaugeTheories of Gravity
Magd E. Kahil Abstract
The problem of spinning and spin deviation equations for particles as defined bytheir microscopic effect has led many authors to revisit non-Riemannian geometryfor being described torsion and its relation with the spin of elementary particles. Weobtain a new method to detect the existence of torsion by deriving the equations ofspin deviations in different classes of non-Riemannian geometries, using a modifiedBazanski method. We find that translational gauge potentials and rotational gaugepotentials regulate the spin deviation equation in the presence of Poincare gaugefield theory of gravity.
Introduction
Einstein’s legacy has bestowed geodesics and null geodesics to examine the trajectorymassive and massless particles respectively, introducing the notation of a test particlewhich ignores the interaction associated with its intrinsic properties. Such a problem maybe assigned to measure the behavior of a certain gravitational field. Yet, the concept of testparticle is counted to be existed relatively rather than absolutely. From this principle, theproblem of spinning objects is necessary to be examined using the Mathissson-Pappetrouequation [1].Due to extending the geometry to become a non-Riemanian, torsion is expressed insome theories of gravity to be interacting with the spin of elementary particle, this isvital to examine the internal symmetry of some gauge theories of gravity with the flavorof Yang-Mills for this issue, one introduces its own corresponding building blocks whichmainly related to the tetrad space , in order to relate it with some properties of elementaryparticles [3-5]. Such theories are developed in different stages from Utyima (1956), Kibble(1961), Sciama (1962), Hehl et al (1976) [3-6]and finally crowned with MAG in 1995.[7]The main theme of these theories is centered on its description in the presence of thetetrad field following same mechanism of Yang-Mills gauge theory of spaces admittingnon vanishing curvature and torsion represented as gauge theories of gravity [8].This approach has led many authors to consider a wide spectrum of theories of grav-ity possessing gauge formulation such as Teleparallel-gravity [9], gauge version of GR intetrad space (torsion-less)[10], and most general one is the Poincare gauge field theory ofgravity [11]. Modern Sciences and Arts University, Giza, EgyptEgyptian Relativity Group, Cairo, EgyptE-mail:[email protected] The Papapertrou Equation in General Relativity:Lagrangian formalism
The Lagrangian formalism of a spinning and precessing object and their correspondingdeviation equation in Riemanian geometry is derived by the following Lagrangian [18] L = g αβ P α D Ψ β Ds + S αβ D Ψ αβ Ds + F α Ψ α + M αβ Ψ αβ (1.1)where P α = mU α + U β DS αβ DS .
Taking the variation with respect to Ψ µ and Ψ µν simultaneously we obtain DP µ DS = F µ , (1.2) DS µν DS = M µν , (1.3)where P µ is the momentum vector, F µ def = R µνρδ S ρδ U ν , and R αβρσ is the Riemann curvature, DDs is the covariant derivative with respect to a parameter S , S αβ is the spin tensor, M µν = P µ U ν − P ν U µ , and U α = dx α ds is the unit tangent vector to the geodesic.Using the following identity on both equations (1) and (2) A µ ; νρ − A µ ; ρν = R µβνρ A β , (1.4)where A µ is an arbitrary vector. Multiplying both sides with arbitrary vectors, U ρ Ψ ν aswell as using the following condition [Heydri-Fard et al (2005)] U α ; ρ Ψ ρ = Ψ α ; ρ U ρ , (1.5)and Ψ α is its deviation vector associated to the unit vector tangent U α . Also in a similarway: S αβ ; ρ Ψ ρ = Φ αβ ; ρ U ρ , (1.6)one obtains the corresponding deviation equations [19] D Ψ µ DS = R µνρσ P ν U ρ Ψ σ + F µ ; ρ Ψ ρ , (1.7)and D Ψ µν DS = S ρ [ µ R ν ] ρσǫ U σ Ψ ǫ + M µν ; ρ Ψ ρ . (1.8)Equations (1.7), (1.8) are essentially vital to solve the problem of stability for differentcelestial objects in various gravitational fields. This will examined in detail in our futurework. 2 The Papapertrou Equation in Rieman-Cartan The-ory of Gravity: Lagrangian formalism
The Mathisson-Papapetrou equation in non-Riemanian geometry are generalized formsof both (1.7)and(1.8), as a result of existence of a torsion tensor Λ αβγ , which is consideredas a propagating field defined in the following manner,Λ αβγ def = 12 ( δ αβ φ ,γ − δ αγ φ ,β ) (2.9)where φ is a scalar quantity.Yet, there are two different visions of admitting torsion in path and spinning equations, oneis considering it acting analogously as a Lorentz force, which led some authors to utilizethe concept of torsion force [20]. Others may involve torsion in the affine connection byreplacing the Christoffel symbol with the non-symmetric affine connection [21]. We suggest the following modified Bazanski Lagrangian to obtain the path and pathdeviation equations for non-Riemannian geometry using the notion of torsion force L = g αβ U α D Ψ β DS + Λ αβγ U α U β Ψ γ . (2.10)Thus by taking the variation with respect to Ψ µ , provided that g µν ; ρ = 0 , we obtain, dU µ ds + ( αµν ) U µ U ν = − Λ . .µαβ. U α U β (2.11)Using the following commutation relation A µ ; νρ − A µ ; ρν = R αβρσ A α (2.12)on equations (2.11) provided that DU α dτ = D Ψ α DS , (2.13)we obtain the corresponding deviation equations ∇ Ψ µ ∇ s = R µ.αβγ U α U β Ψ γ + (Λ . .µαβ. ) ; ρ Ψ ρ . (2.14)3 .2 Path and Path Deviation having torsion in a non-symmetricaffine connection Path equations and path deviation equations are obtained its corresponding Bazanskiapproach as follows L def = g µν ˆ U µ ∇ ˆΨ ν ∇ S , (2.15)where, ∇ ˆΨ µν ∇ S = d ˆΨ µν dS + Γ α.βσ ˆΨ σ , (2.16)where Γ α.βσ def = ( αβσ ) + K α.βσ . (2.17)By taking the variation with respect to Ψ α , provided that g µν | ρ = 0 , (2.18)we obtain ∇ ˆ U α ∇ S = 0 , (2.19)i.e. dU µ ds + ( αµν ) U µ U ν = − Λ . .µαβ. U α U β . Using the following commutation relation A µ || νρ − A µ || ρν = ˆ R αβρσ A α + Λ δνρ A µ || δ , (2.20)and ∇ Ψ µ ∇ τ = ∇ U µ ∇ S (2.21)on equation (2.17) we obtain ∇ ˆΨ α ∇ S = ˆ R ανρσ ˆ U ν ˆΨ σ ˆ U ρ + Λ ρµν U α | ρ ˆ U µ ˆΨ ν . (2.22)Thus, we found that equations (2.11) and (2.17) describe the same path equation buttheir corresponding path deviation equations (2.13) and (2.20) are different due to thebuilding blocks of the each type of geometry. For a spinning object, we suggest the following modified Bazanski Lagrangian, to deriveboth spin and spin deviation equations simultaneously. L = g µν P µ D Ψ ν DS + Λ ( µν ) ρ P µ U ν Ψ ρ + S µν D Ψ µν DS + 12 R µνρσ S ρσ U µ Ψ ν + ( P µ U ν − P ν U µ )Ψ µν . (2.23)4aking the variation with respect to Ψ α and Ψ αβ respectively we obtain DP α DS = − Λ .. α ( µν ) P µ U ν + 12 R αρµν S µν U ρ , (2.24)and DS αβ DS = ( P α U β − P β U α ) . (2.25)The associated deviation equations are obtained by considering the following commutationrelation DDS DDτ A α − DDτ DDS A α = R αβρσ A β U ρ Ψ σ (2.26)and DU α DS = D Ψ α Dτ , (2.27)we get D Ψ α DS = R αβρσ U β U ρ Ψ σ + ( 12 R αβµν S µν U β − Λ . . α.µν ) ρ Ψ ρ (2.28)and D Ψ αβ DS = S ρ [ β R α ] ργδ U γ Ψ δ ; δ U µ Ψ ν + ( P α U β − P β U α ) ; ρ Ψ ρ . (2.29) If we replace the covariant derivative in Riemanian geometry by the absolute derivative inEinstein-Cartan geometry, we suggest the following Lagrangian of spinning and spinningdeviation objects with precession L = g µν ˆ P µ ∇ ˆΨ ν ∇ S + S µν ∇ ˆΨ µν ∇ S + 12 ˆ R µνρσ ˆ S ρσ U ν Ψ µ + ( ˆ P µ U ν − ˆ P ν U µ )Ψ µν , (2.30)such that ˆ P µ def = m ˆ U α + ˆ U β ∇ S αβ ∇ S , (2.31)regarding that, ˆ S µν ˆ U ν = 0 (2.32), then taking the variation with respect to ˆΨ α and ˆΨ αβ .Thus, we obtain ∇ ˆ P α ∇ S = 12 ˆ R ανρσ ˆ S ρσ U ν , (2.33)and ∇ ˆ S αβ ∇ S = ( ˆ P α U β − ˆ P β U α ) . (2.34)5sing the commutation relation (2.22) and equation (2.27) on (2.33) and (2.34), we obtainthe following set of deviation equations viz, ∇ Ψ α ∇ S = ˆ R αβγδ P β U γ Ψ δ + Λ ρσδ U σ Ψ δ P σ | ρ + 12 ( ˆ R αρσǫ S σǫ U ρ ) | ρ Ψ ρ , (2.35)and ∇ Ψ αβ ∇ S = ˆ S ρ [ β ˆ R α ] ργδ U γ Ψ δ + Λ δµν S αβ | δ U µ Ψ ν + ( ˆ P α U β − ˆ P β U α ) | ρ Ψ ρ . (2.36)We can find that there is a link between spin deviation tensor and torsion of space time.While, the spinning motion has no explicit relation with the same torsion tensor whichconfirms Hehl’s point of view [22]. This result has led to find out its detailed descriptionin case of taking into account the microstructure of any system. Such a trend can reachto define its contents with in the context of tetrad space. The concept of torsion of space time may give rise to revisit its existence in a tetradspace, which may give rise to express space-time as a system of two different coordinatesystems. At each point of space-time is defined by the vector x µ , µ = 0 , , , g µν . each point is associated with tangent space becoming a fiber of itscorresponding tangent bundle given by Minkowski space whose metric tensor is definedby η ab def = dig (1 , − , − , −
1) .Accordingly, this type of description may be analogous to explain the underlyinggeometry associated with some gauge theories e.g. it is analogous to internal gaugetheories, in which gravity becomes as a special gauge theory [12]. Thus, as a result ofsimilarity between the above mentioned space-time and gauge theory, it is of interest toderive the gauge approach of equations of motion for different particles.From this perspective, the problem of invariance of any quantities must be a covariantderivative invariant under general coordinate transformation (GCT) and Local Lorentztransformation (LLT) that are expressible in terms of gauge potential of translation androtation in the following way .The building blocks of the space is two quantities, one represents the tetrad vector ( e aµ ,and the other is the generalized spin connection Ω ijµ ). The tangent space is raised andlowered by the Minkowski space, while the space-time indices are raised and lowered bythe Riemannain metric g µν def = e aµ e bν η ab , (3.37) g µν def = e aµ e bν η ab . (3.38)This type of geometry defined its curvature tensor R ijµν [25] is defined as follows R ijµν def = Ω ijν,µ − Ω ijµ,ν + Ω kjνµ Ω ikµ − Ω kjµµ Ω ikν (3.39)6nd its corresponding torsion tensor Λ aµν is becoming asΛ aµν def = ( e aµ,ν − e aν,µ + Ω abµ e bν − Ω abν e bµ ) (3.40), provided that ∇ ν e mµ = 0 , (3.41)i.e. e mµ || ν ≡ . Due to the following condition [23], g µν || σ = 0 , which becomes e aµ,ν − Γ λµν e aλ + Ω a.bν e bµ = 0 , and Γ ανµ e mα = e αm ( e mν,µ + Ω .m .µ . n e nν ) . Consequently, the relationship between the generalized spin connection and contortionof space time can be obtained asΩ .m .n. µ = − e νn ( e mν,µ + Γ mνµ )comparing with (2.17) one obtains that, one getsΩ .m .n. µ = ω .m .n. µ + K .m .n. µ where ω .m .n. µ spin connection associated with Christoffel symbol, and K .m .n. µ = e νn e mα K ανµ . Special cases: i Tele-parallelism: Ω ijµ = 0 → ω ijµ = − K ijµ gives that the spin connection is equivalent toRicci coefficient of rotation .(ii) General Relativity in a gauge form: K ijµ = 0 → Ω ijµ = ω ijµ gives that the genealizedspin connection is equivalent to Ricci coefficient of rotation. The study of microscopic structure of particles gives rise to utilize the richness Yang-Millsguage theory to express gravity as a gauge theory having internal gauge invariance. Thiscan be done by involving two different types of coordinate system one for GCT and theother for internal gauge invariance such as LLT. Consequently, it is worth mentioningthat local Poincare gauge theory is an appropriate theory to explain gravity at this level.This approach was achieved by Hehl after a long process of versions by, Utimama, Kibble, Ricci coefficient of rotation A ρµν def = e ρi e iµ ; ν f aα is equivalentto e aµ and the rotational gauge Γ abβ is equivalent to Ω abµ . This may give rise to find anothersimilarity between the gauge field strength F abµν and curvature of space time admitting theanholonomic coordinates R abµν [24]. to be added for Poincare gauge theory Translationalgauge potential e nµ and rotational gauge potential Γ abµ in which the commutation derivativeoperators[11]. As in gauge theories the commutation relation is defined as with respectto the gauge field strength[ ˜ D a , ˜ D b ] = f αa f βb ( F . . .µναβ s αβ − F . . nαβ ˜ D n ) , (3.42)which becomes is equivalent to[ ∇ a , ∇ b ] = e αa e βb ( R . . .µναβ s ab − Λ . . nαβ ∇ n ) . (3.43) In this case, Ω abµ = 0 the conventional absolute parallelism geometry: a pure gauge theoryfor translations [9],[12].Using Acros and Pirra method[12], one can find out thatˆ R αβγδ def = Γ αβγ,δ − Γ αβδ,γ + Γ ǫβγ Γ αǫδ − Γ ǫβδ Γ αǫγ ≡ , Γ ρµν = e ρc ( e cµ,ν − e cν,µ ) , Λ αβγ = Γ αβρ − Γ αρρ . As, in GR the spin connection is equal to Ricci coefficient of rotation ω abµ def = e aρ e ρa ; µ thus, ω abµ def = A abµ − γ abµ , where K abµ its contortion defined as K abµ = 12 e cµ (Λ ac.b + Λ ab.c − Λ a.bc ) . Consequently, one obtains, R cdµν = ˆ R cdµν − K cdµν , where K cdµν def = γ cdν ; µ − γ cdµ ; ν + γ caµ γ adν − γ caν γ adµ . 8 .3 General Relativity: A Tetrad Version of Gravitational GaugeTheory Collins et al (1989)[10] described GR as a gauge theory of gravity subject to the follow-ing gauge potential vectors e aµ and ω ij. .µ to represent translational and rotational gaugepotentials respectively.The equations of physics will contain derivatives of tensor fields and it is therefore nec-essary to define the covariant derivatives of tensor fields under the transformations GCTand LLT, one must need to define two types of connection fields to be associated witheach of them. Accordingly the Christoffel symbol n αµν o is referred to GCT while the spinconnection ω abµ as related to LLT.which is considered a torsion less condition of Poincare gauge theory D µ e mν def = e mν,µ − ( ανµ ) e mα + ω .m .µ . n e nν , (3.44)provided that D ρ D µ e mν = D µ D ρ e mν . (3.45)Using this concept it turns out that GR may be expressed in terms of connecting e µa , ω abµ and n αµν o together g µν def = e aµ e bν η ab , (3.46)such that, ( αµν ) def = 12 g ασ ( g νσ,α + g σα,ν − g αν,σ ) (3.47). Thus, the curvature tensor may be defined ,due to gauge approach, in terms of spinconnection ω abµ R c.dµν def = ω cdν,µ − ω cdµ,ν + ω caµ ω adν − ω caν ω adµ . Using this concept it turns out that GR may be expressed in terms of connecting e µa , ω abµ and n αµν o together. Accordingly where R α.µdc the curvature tensor may be defined ,due togauge approach, in terms of spin connection ω abµ R c.dµν def = ω cdν,µ − ω cdµ,ν + ω caµ ω adν − ω caν ω adµ . We suggest the following Lagrangian to derive both path and path deviation equation forgauge theories having a torsion force. L = e µa e νb U a D Ψ b DS + e αa Λ αβγ P a U β Ψ γ , (4.48)9o obtain its corresponding path equations by applying the Bazanski approach tobecome, DU a DS = − e aα Λ . . α ( βγ ) . U β U γ . (4.49)Applying the following commutation relations of equation (2.20) and following (2.21),we obtain after some manipulation its corresponding path deviation equations D Φ a DS = e aµ K µαβσ U α U β Φ σ + Λ . . a .βγ U γ U β ) ; ρ Φ ρ . (4.50) Thus spinning and spinning equations are obtained from taking the variation with respectto Ψ µ and Ψ µν simultaneously, for the following Lagrangian L = e µa e νb P a D Ψ b DS +Λ aβγ P a U β Ψ γ − K abµν S µν U b Ψ a + S ab D Ψ ab DS +( P a U b − P b U a )Ψ ab (4.51)to obtain DP a DS = − K aνρδ S ρδ U ν + Λ . . a .βγ P α U β . (4.52)and De aα e bβ S ab DS = ( P α U β − P β U α ) , (4.53)and DS cd DS = ( P c U d − P d U c ) . (4.54)Using the commutation relations as mentioned above, we obtain; D Φ a DS = K abβσ P b U β Φ σ + ( R abµν S µν U α + Λ . . a .βγ P γ U β ) ; ρ Φ ρ , (4.55)and D Ψ cd DS = S ρ [ d ˆ R c ] ργδ U γ Ψ δ + ( P c U d − P d U c ) ; ρ Ψ ρ . (4.56) The Lagrangian formalism of path and path deviation equation of gauge theories oftorsion-less is given as L def = e µa e νb U a D Ψ b DS . (4.57)10pplying the Bazanski approach , taking the variation with respect to Ψ α , we obtain De αc ˆ U c DS = 0 , (4.58)provided that De aα DS ≡ , we get DU c DS = 0 . (4.59)Applying the following commutation relation A a || νρ − A a || ρν def = ˆ R cbρσ A b , (4.60)on equation (4.59) and using (2.13) we obtain D Ψ c DS = R cbρσ ˆ U b ˆΨ σ ˆ U ρ (4.61)Substituting with the path equation in the deviation equation we get D Ψ c DS = R cbρσ U b ˆΨ σ U ρ . (4.62) Thus spinning and spinning equations are obtained from taking the variation with respectto Ψ µ and Ψ µν simultaneously, for the following Lagrangian L = e µa e νb P a D Ψ b DS + 12 ( R abµν ) S µν U b Ψ a + e aµ e bν S ab D Ψ µν DS + e µa e νb ( P a U b − P a U b )Ψ µν , (4.63)to obtain its corresponding path equation using the Bazanski approach to get DP a DS = 12 ( R abµν ) S µν U ν , (4.64)and DS ab DS = ( P α U β − P β U α ) , (4.65) e aα e bβ DS ab DS + S ab D ( e aα e bβ ) DS = ( P α U β − P β U α ) . (4.66)Multiplying both sides by e cα e dβ e cα e dβ e aα e bβ DS ab DS + e cα e dβ S ab D ( e aα e bβ ) DS = e cα e dβ ( P α U β − P β U α ) , (4.67)11uch that D ( e aα e bβ ) DS = 0 . Consequently, one obtains DS cd DS = ( P c U d − P d U c ) . (4.68)Accordingly, the deviation equation is obtained by applying the commutation relationand (2.27) on (4.52) and (4.54); D Φ µ DS = R µαβσ P α U β Φ σ + ( R µαab S ab U α + Λ . . α.βγ P α U β ) ; ρ Φ ρ , (4.69)and D Ψ cd DS = S ρ [ d ˆ R c ] ργδ U γ Ψ δ + ( P c U d − P d U c ) ; ρ Ψ ρ . (4.70) Path equations and path deviation equations are obtained its corresponding Bazanskiapproach as follows L def = e µa e νb ˆ U a ∇ ˆΨ b ∇ S . (4.71)By taking the variation with respect to Ψ α , we obtain ∇ e αc ˆ U c ∇ S = 0 , (4.72)provided that ∇ e aα ∇ S ≡ , we get ∇ ˆ U c ∇ S = 0 . (4.73)Thus, the spin deviation equations are obtained by applying the commutation relationand (2.17) on both ∇ ˆΨ c ∇ S = ˆ R cbρσ ˆ U b ˆΨ σ ˆ U ρ + e δb Λ bνρ U a || δ (4.74)Substituting with the path equation in the deviation equation we get ∇ ˆΨ c ∇ S = ˆ R cbρσ ˆ U b ˆΨ σ ˆ U ρ (4.75)12 .6 Spin and Spin Deviation Equation Poincare Gauge Theory We suggest the following Lagrangian of spinning and spinning deviation objects withprecession by replacing the covariant derivative in (1.1) by the absolute derivative asdescribed in (2.15) to get L = e µa e νb ˆ P a ∇ ˆΨ b ∇ S + S ab ∇ ˆΨ ab ∇ S + 12 ˆ R abµν ˆ S ab U ν Ψ µ + ( ˆ P a U b − ˆ P b U a )Ψ ab , (4.76)such that ˆ P µ def = m ˆ U α + ˆ U β ∇ S αβ ∇ S , (4.77)Taking the variation with respect to ˆΨ α and ˆΨ αβ and after some manipulations,we obtain ∇ ˆ P a ∇ S = 12 ˆ R abρσ ˆ S ρσ U b , (4.78)and ∇ ˆ S ab ∇ S = ( ˆ P a U b − ˆ P b U a ) . (4.79)And their corresponding deviation equations are obtained by applying the commuta-tion relation (2.20) and (2.21) on both (4.79) and (4.79) to get ∇ Ψ a ∇ S = ˆ R abβγ P b U β Ψ γ + e ρc Λ cβγ U β Ψ γ P a | ρ + 12 ( ˆ R abρσ S ρσ U b ) | δ Ψ δ , (4.80)and ∇ Ψ ab ∇ S = e bβ e cδ ˆ S δ [ β ˆ R a ] cγδ U γ Ψ δ + e δc Λ cµν S αβ | δ U µ Ψ ν + ( ˆ P a U b − ˆ P b U a ) | ρ Ψ ρ . (4.81)Thus, we see clearly the interaction between spin deviation tensor and torsion of spacetime is expressed in terms of curvature (rotational) and torsion (translational) strengthfields. In our present work, we have developed he Bazanski approach to obtain spin and spindeviation equations in non-Riemanian geometry. This approach was applied for obtain-ing path equations for some geometries admitting non vanishing curvature and torsionsimultaneously. [26-28]Due to the resultant equations (2.35),(2.36), we have figured out that torsion is explicitlymentioned in spin deviation equations ,even if one puts P µ = mU µ ,. Such a result isin favor of Hehl’s argument falsifying the measurement of torsion from identifying thespin tensor, for a spinning object in an orbit. This result is an alternative approach tomeasure torsion from spinning equations using non-minimal coupling without introducing13icro-structure[29].Accordingly, we have obtained (2.35) and (2.36) of Poincare gauge theory of gravity asdescribed within a tetrad space. Consequently, using the analogy between tetrads andgauge theories, we obtained equations (4.82), (4.83) that show how a tetrad vector e µi isequivalent to gauge translational potential explicitly mentioned in the equation while thegeneralized spin tensor Ω ijµ , which is equivalent to gauge rotational potentials as men-tioned implicitly in the presence of curvature and torsion of space-time simultaneously.These equations are considered to be the generalized cases for some special classes ofgauge theory of gravity of path and spin deviation equations (4.55) and (4.56) having atorsion force, as well as their counterpart of (4.69) and (4.70) for a gauge theory of gravityhaving a torsion free.Thus, we conclude that torsion of space time can be tested for any spinning object in anytype of gravitational fields by examining its spin deviation equations. Acknowledgement
The author would like to thank Professor T.Harko his remarks and comments.
References [1] A.Papapetrou , Proceedings of Royal Society London A , 248(1951).[2]E. Corinaldesi and A. Papapetrou Proceedings of Royal Society London A , 259(1951)[3]Utyiama, R.(1956) Phys Rev. 101 1597[4]Kibble, T.W. (1960) J. Math Phys., 2, 212[5]Hehl,F. W., von der Heyde, P. Kerlik, G.D. and Nester, J.M. (1976) Rev Mod Phys 48,393-416[6] Hehl, F.W. (1979),1, Proceedings of the 6th Course of the International School ofCosmology and Gravitation on ”Spin, Torsion and Supergravity” de ed. P.G. Bergamannand V. de Sabatta , held at Erice .[7] Hehl,F.W., McCrea, J.D. Mielke, E.W. and Ne’eman, Y.(1995)Phys Reports 1-171[8]Ali, S.A., Carfao, C. , Capozziello, S. and Corda, Ch. (2009) arXiv:0907.0934[9]Hayashi, K. and Shirifuji, T. (1979) Phys. Rev. D, , 3524.[10] Collins, P., Martin, A. and Squires, E. ” Particle Physics and Cosmology ”, John Wileyand Sons, New York.(1989)[11]Hammond, R. (2002) Rep. Prog. Phys, 65, 599-449[12]Acros, H. I. and Pereira, J.G. (2004) International Journal of Modern Physics D ,2193[13]Mao, Y, Tegmark, M., Guth, A. and Cabi, S. (2007) Phys. Rev. D , 104029 ;arXiv:gr-qc/060812[14] Hehl, F.W. (1971) Phys. Lett.36A, 225.[15] Hojman, S.(1978) Physical Rev. D , 2741.([16])Yasskin, P.H. and Stoeger, W.R. (1980) Phys. Rev. D 21, 2081[17]Bazanski, S.L. (1989) J. Math. Phys., , 1018 ;14ahil, M.E. (2006) , J. Math. Physics ,052501.[18) Kahil, M.E. (2015) Odessa Astronomical Publications, vol 28/2, 126. [19] Mohseni, M. (2010), Gen. Rel. Grav., , 2477 .20 Acros, H. I., Andrade, V.C. and Pereira, J.G. (2004) arXiv;gr-qc/0403074[21] Hojman, S., Rosenbaum, M. and Ryan, M.P.(1979) Physical Rev. D , 430[22]Hehl, F.W., Obukhov, Yu, N. and Puetzfeld, D. (2013) A , 99 .[24] Cianfani, F., Montani, G. and Scopelliiti, V. (2015) arXiv: 1505.00943[25]Fabbri, L. and Vignolo, S. (2011) arXiv:1201.286[26] Wanas, M.I., Melek, M. and Kahil, M.E. (2000) Grav. Cosmol., , 319.[27] Wanas, M.I. and Kahil, M.E.(1999) Gen. Rel. Grav., , 1921. ;Wanas, M.I., Melek, M. and Kahil, M.E. (2002) Proc. MG IX, part B, p.1100, Eds.V.G. Gurzadyan et al. (World Scientific Pub.); gr-qc/0306086.[28] Wanas, M.I., Kahil, M.E. and Kamal, Mona. (2016) Grav. Cosmol.,22