The problem of motion in gauge theories of gravity
aa r X i v : . [ g r- q c ] O c t The problem of motion in gauge theories of gravity
Serhii Samokhvalov
Dniprovsk State Technical University, Kamianske, UKRAINEE-mail: [email protected]
August 2020
Abstract.
In this article we consider the problem to what extent the motion ofgauge-charged matter that generates the gravitational field can be arbitrary, as well aswhat equations are superimposed on the gauge field due to conditions of compatibilityof gravitational field equations. Considered problem is analyzed from the point of viewsymmetry of the theory with respect to the generalized gauge deformed groups withoutspecification of Lagrangians.In particular it is shown, that the motion of uncharged particles along geodesics ofRiemannian space is inherent in an extremely wide range of theories of gravity and is aconsequence of the gauge translational invariance of these theories under the conditionof fulfilling equations of gravitational field. In the cause of gauge-charged particles, theLorentz force, generalized for gauge-charged matter, appears in equations of motionas a consequence of the gauge symmetry of the theory under the condition of fulfillingequations of gravitational and gauge fields. In addition, we found relationships ofequations for some fields that follow from the assumption about fulfilling of equationsfor other fields, for example, relationships of equations of the gravitational field andthe gauge field of internal symmetry which follow from the assumption about fulfillingof equations of matter fields. In particular, we obtained the identity that generalizesin the case of arbitrary gauge field (and in the presence of gauge-charged matter) theidentity found by Hilbert for the electromagnetic field.At the end of the article there is an Appendix, which briefly describes the mainprovisions and facts from the theory of generalized gauge deformed groups and presentsthe main ideas of a single group-theoretical interpretation of gauge fields of bothexternal (space-time) and internal symmetry, which is an alternative to their geometricinterpretation.
Keywords : gauge theories of gravity, generalized gauge deformed groups, problem ofmotion
1. Introduction
One of the most striking features of the general relativity (GR) is the fact that thematter that generates gravitational field cannot move arbitrarily, but must obey certainequations that follow from equations of the gravitational field as a condition of theircompatibility. This fact was first noted in the fundamental Hilbert’s work [1], in which he problem of motion in gauge theories of gravity G gM = U (1) g × ) T gM [8], which is a semidirect product of the gaugegroup of electrodynamics U (1) g and the generalized gauge group of translations T gM . Inthe aforementioned article, in particular, it was shown that due to the G gM -invarianceof the theory, when fulfilling equations of the gravitational field: a) in the case of dustmatter, particles move along geodesics, and in the presence of an electromagnetic field (ifthe particles are charged) they are additionally affected by the Lorentz force; b) providedthat the electric charge and the total energy-momentum of the system of gravitationaland electromagnetic fields and the field of charged matter are conserved, regardless ofthe method of parameterization of group G gM , the electromagnetic field obeys Maxwell’sequations.The results of article [7] were determined solely by the symmetry of the theoryand did not depend on the specific type of Lagrangians of both gravitational andelectromagnetic fields. The only limitation was that it was assumed that Lagrangiansdepend on derivatives of field variables not higher than the first order. Thus, theresults obtained in [7] are valid not only in GR, but also in a number of othertheories of gravity, in particular in the currently popular f ( T )-theories, which canprovide a theoretical interpretation of the late-time universe acceleration (alternativeto a cosmological constant), avoidance of the Big Bang singularity etc. [9]. he problem of motion in gauge theories of gravity V g are considered. This will allows one to analyze theproblem of motion, in addition to GR and f ( T )-theories, also in a fairly wide class oftheories of gravity, which are now intensively studied, in particular, such as f ( R )-theories[10], Lovelock gravity [11], and others which do not have problems with local-Lorentzinvariance, as f ( T )-theories [12]. In addition, this analysis can be performed in the casewhen the sources of the gravitational field are arbitrary gauge-charged matter fieldsand corresponding gauge fields of internal symmetry (and not only electrically chargedmatter and electromagnetic field). Models where non-Abelian gauge vector fields arethe unique source of inflation and dark energy are now intensively studied [13].This paper is structured as follows.In Section 2 we prove general Theorem about quasilocality of charges associatedwith generalized gauge symmetries G gM for theories with Lagrangians that depend onfields and their derivatives up to an arbitrary order.In Section 3 we study the problem of motion for the abovementioned theories ofgravity in terms of their symmetry with respect to the generalized gauge translationsgroup T gM [8] and show that equations of the gravitational field in an arbitrary gaugetheory of group T gM can be written both in the form similar to the form of Einsteinequation and in a superpotential form, i.e. in the form of the expression of the totalenergy-momentum tensor of the gravitational system through the superpotential (in theform similar to the form of Yang-Mills equations). In addition, Proposition 1 is proved,from which, in particular, follows that in an arbitrary gauge theory of generalized gaugetranslations group T gM for the compatibility of the gravitational field equations dustmatter must move along geodesics of Riemannian space corresponding to the solutionof the gravitational field equations.In Section 4 we assume that matter carries gauge charges and so gauge fields ofinternal symmetry V g are present. The symmetry of the theory in this case extends togroup G gM = V g × ) T gM , which is a semidirect product of the groups V g and T gM . Herewe show that equations of arbitrary gauge field of internal symmetry V g (regardless ofthe specific type of its Lagrangian) in the presence of gravitational field can be writtenboth in the form of Einstein equation and in superpotential form, i.e. as an expression ofthe total current of gauge charges of the system through the superpotential (in the formof Yang-Mills equations). In addition, Proposition 2 is proved, in which relationships ofequations for some fields that follow from the assumption about fulfilling of equationsfor other fields are revealed (for example, relationships of equations of the gravitationalfield and the gauge field of internal symmetry which follow from the assumption aboutfulfilling of equations of matter fields, etc.). In particular, we obtained the identity thatgeneralizes in the case of arbitrary gauge field (and in the presence of gauge-chargedmatter) the identity found by Hilbert for the electromagnetic field [1].At the end of the article there is an Appendix, where we briefly describe the he problem of motion in gauge theories of gravity
2. Quasilocality of gauge charges
Let on the space-time manifold M with coordinates x µ (for which we will use the Greekindices of the middle of the alphabet) the system of fields q I ( x ) (the field index is I )and the action S = Z L ( q, ∂ q ... ∂ ( n ) q ) dx, are given, where L is the Lagrangian, which depends on the fields and their derivatives upto the n -th order, and dx is an element of coordinate volume. Let’s consider infinitesimaltransformations of coordinates and fields: x ′ µ = x µ + δ x µ , q ′ I ( x ) = q I ( x ) + δ q I , in which the action acquires addend ¯ δS = R δ ′ Ldx , where δ ′ L = L ′ ( x ′ ) J − L ( x ) is theintegral variation of the Lagrangian L , and L ′ ( x ′ ) means the Lagrangian calculated forthe fields q ′ at the point x ′ , J is the Jacobian of the transformation x ′ ( x ). By directvariation we obtain: δ ′ L = [ L ] I δ q I + ∂ σ ¯ W σ , ¯ W σ = W σ + Lδ x σ , (1)where [ L ] I := n X k =0 ( − k ∂ ( k ) σ ...σ k ∂ σ ...σ k I L = ∂ I L − ∂ σ ∂ σI L + ∂ (2) σ ρ ∂ σ ρI L − ... (2)are variational derivatives of Lagrangian L with respect to the fields q I , W σ := n − X k =0 [ L ] ν ...ν k σI ∂ ( k ) ν ...ν k δ q I = [ L ] σI δ q I + [ L ] ν σI ∂ ν δ q I + ... where in turn [ L ] ν ...ν l I := n − l X k =0 ( − k ∂ ( k ) σ ...σ k ∂ ν ...ν l σ ...σ k I L = ∂ ν ...ν l I L − ∂ σ ∂ ν ...ν l σI L + ... (3)are variational derivatives of Lagrangian L with respect to the derivations of thefields ∂ ( l ) ν ...ν l q I . Notations are used here ∂ σ := ∂/∂x σ , ∂ ( k ) σ ...σ k := ∂ ( k ) /∂x σ ... ∂x σ k , ∂ σ ...σ k I := ∂/∂ ( k ) σ ...σ k q I . Obviously, from (3) follows (2) if l = 0.Let the infinitesimal action of a generalized gauge group G gM (Appendix) be givenby formulas: δx µ = h µa g a , δ q I = m X p =0 a I µ ...µ p a ∂ ( p ) µ ...µ p g a = a Ia g a + a I µa ∂ µ g a + ... (4) he problem of motion in gauge theories of gravity h µa and a I µ ...µ s a are functions of x µ and q I ( x ) which specify this action, and g a ( x )are the infinitesimal parameters of the group G gM . In this case, the functions ¯ W σ thatdetermine the integral variation of the Lagrangian (1) are expressed in terms of groupparameters and their derivatives:¯ W σ = − m + n − X p =0 J µ ...µ p σa ∂ ( p ) µ ...µ p g a , (5)where if p = 0 J σa := − n − X k =0 [ L ] ν ...ν k σI ∂ ( k ) ν ...ν k a Ia − Lh σa , (6)and if p > J µ ...µ p σa := − p X l = p − m n − X k = l C lk [ L ] µ ...µ l ν l +1 ...ν k σI ∂ ( k − l ) ν l +1 ...ν k a I µ l +1 ...µ p a . (7)Here C lk is the binomial coefficients. The sequence of indices µ ...µ p in expression (7)for J µ ...µ p σa can be arbitrary, but we will not perform their symmetrization, reservingthe right further to choose their sequence convenient for us.For p = 1 and 2 from (7) follows: J µ σa = − n − X k =0 [ L ] ν ...ν k σI ∂ ( k ) ν ...ν k a I µa − n − X k =1 k [ L ] µ ν ...ν k σI ∂ ( k − ν ...ν k a Ia ,J µρσa = − n − X k =0 [ L ] ν ...ν k σI ∂ ( k ) ν ...ν k a I µρa − n − X k =1 k [ L ] µ ν ...ν k σI ∂ ( k − ν ...ν k a I ρa − n − X k =2 k ( k − L ] µρν ...ν k σI ∂ ( k − ν ...ν k a Ia . Directly from definition (5) follows: ∂ σ ¯ W σ = − m + n X p =0 S µ ...µ p a ∂ ( p ) µ ...µ p g a , (8)where S µ ...µ p a := J µ ...µ p a + ∂ σ J µ ...µ p σa . Suppose now that action S is invariant with respect to transformations (4) of thegroup G gM . The condition of G gM -symmetry of action S is δ ′ L = 0, hence, taking intoaccount (1), it follows [ L ] I δ q I = − ∂ σ ¯ W σ . Therefore, taking into account (4) and (8),we obtain: m X p =0 [ L ] I a I µ ...µ p a ∂ ( p ) µ ...µ p g a = m + n X p =0 S µ ...µ p a ∂ ( p ) µ ...µ p g a , he problem of motion in gauge theories of gravity g a ( x ), which parameterize the group G gM ,gives: [ L ] I a I µ ...µ p a = S { µ ...µ p } a if 0 ≤ p ≤ m , (9) S { µ ...µ p } a = 0 if m < p ≤ m + n , (10)where braces mean symmetrization by the indices contained in them.Identities (9), (10) are strong, i.e. they are fulfilled exclusively due to the G gM -invariance of the theory without assumption about the extremeness of the action.On shell [ L ] I = 0 relation (10) holds for all p (weak identities). For p = 0, 1 and 2it gives: ∂ σ J σa = 0, S µa = J µa + ∂ σ J µσa = 0, S { µν } a = J { µν } a + ∂ σ J { µν } σa = 0 . (11)First of them miens that gauge charges Q a = R V J a dV associated with the currents J σa are conserved ∂ Q a = 0 (here x is the time coordinate, and dV is an element of3-dimensional volume V ). Provided ∂ νσ J µν σa = 0 (12)the rest of identities (11) gives: J µa = − ∂ σ S µσa , S µσa = − S σ µa . (13)So gauge charges in this case are quasilocal Q a = H ∂V S i a ds i and quantities S µνa = J µνa + ∂ σ J µν σa acts as their superpotentials . Here ds i is a vector of elementary areaof the surface ∂V . Using the right to arbitrarily choose the sequence of the first twoindices in J µν σa , index ν with which the condition (12) is satisfied will be located in thepenultimate place. Quasilocality condition (12) is performed, for example, in the case J µ { ν σ } a = 0. Note that if condition (12) is satisfied, we do not need J µ ...µ p a with p > S µνa .The G gM -invariant theory with quasilocality condition (12) we will call the gaugetheory of the generalized gauge group G gM .We concretize now the obtained relations for m = 1. In this case δ q I = a Ia g a + b I µa ∂ µ g a (14)(here the coefficients a I µa from formula (4) are denoted as b I µa ): J µ ...µ p σa = − n − X k = p − C p − k [ L ] µ ...µ p − ν p ...ν k σI ∂ ( k − p +1) ν p ...ν k b I µ p a − n − X k = p C pk [ L ] µ ...µ p ν p +1 ...ν k σI ∂ ( k − p ) ν p +1 ...ν k a Ia (15)in particular J µ σa = − n − X k =0 [ L ] ν ...ν k σI ∂ ( k ) ν ...ν k b I µa − n − X k =1 k [ L ] µ ν ...ν k σI ∂ ( k − ν ...ν k a Ia , (16) he problem of motion in gauge theories of gravity J µρσa = − n − X k =1 k [ L ] µ ν ...ν k σI ∂ ( k − ν ...ν k b I ρa − n − X k =2 k ( k − L ] µρν ...ν k σI ∂ ( k − ν ...ν k a Ia (17)(expression (6) for currents J σa remains unchanged).For n = 1, formulas (6), (16) and (17) give: J σa = − ∂ σI L a Ia − Lh σa , (18) J µ σa = S µ σa = − ∂ σI L b
I µa , J µρσa = 0 (19)and for n = 2 give: J σa = − [ L ] σI a Ia − ∂ ν σI L ∂ ν a Ia − L h σa , (20) J µ σa = − [ L ] σI b I µa − ∂ ν σI L ∂ ν b I µa − ∂ µ σI L a Ia , (21) J µρσa = − ∂ µ σI L b
I ρa . (22)Provided that the quasilocality condition (12) is met, identities (9) in our case arewritten as follows (strong identities):[ L ] I a Ia = ∂ σ J σa , (23)[ L ] I b I µa = J µa + ∂ σ S µσa . (24)In gauge theories, the deformation coefficients h aµ of generalized gauge deformedgroups G gM act as potentials (Appendix), gauge transformations for them (A.15), (A.28)have the form: δh aµ = a aµb g b − ∂ µ g a and transformations all other fields of theory not contain derivatives of groupparameters. So nonzero coefficients b Iσa in this case are only b b σµ a = − δ σµ δ ba (index I now is multiindex bµ ). Thereby identity (24) gives: − [ L ] µa = J µa + ∂ σ S µσa . (25)So the following is true. Theorem . In the gauge theory of the generalized gauge group G gM , the gaugecharges Q a are quasilocal, i.e. their currents J µa have superpotentials S µνa = J µνa + ∂ σ J µν σa = − S νµa : J µa = − ∂ σ S µσa . Moreover, expression of currents throughsuperpotentials is equivalent to equations of the gauge field. This generalizes the corresponding theorem, which was proved in [14], to theLagrangians with higher derivatives of fields.From this theorem, in particular, it follows that the gauge field equations in thearbitrary gauge theory of the generalized gauge group G gM can be written in the formof dynamic Maxwell equations (or Young-Mills equations). he problem of motion in gauge theories of gravity
3. Laws of motion of uncharged matter
Under the theory of gravity we will understand the gauge theory of the translationsgroup, which is the group of Riemannian translations T gM (Appendix). The gravitationalfield potentials are identified with deformation coefficients h mµ , which form inverse matrixto matrix h µm of coefficients of an orthonormal frame (tetrad) X m = h µm ∂ µ (indexed byindices m, n, p, s ) in coordinate frame ∂ µ . Replacement of coordinate indices to frameindices and vice versa will be performed using matrices h mµ and h µm .We assume that in addition to the gravitational field in space-time M there arealso matter fields ψ ξ (these are all fields except the gravitational one). The field indexis ξ . All matter fields are set in the local frame, and therefore are T gM -scalars.The Lagrangian of the gravitational system is defined as the sum L = √ gL G ( h, ∂h, ..., ∂ ( n ) h ) + √ gL ψ ( h, ∂h, ψ, ∂ψ ) , where g = | g µν | , g µν = h mµ h nν η mn . In our case, the field variables q I split into system { h mµ , ψ ξ } which is equivalent to splitting the field index I into the multiindex { mµ , ξ } .With the minimal method of inclusion of the gravitational interaction, L ψ does notdepend on ∂h .In this article, we will not specify either the Lagrangian of gravitational field √ gL G or the Lagrangian of matter √ gL ψ , assuming only gauge translational invariance ( T gM -invariance) of both L G and L ψ , which provides equality to zero of integral variationsof both components of the Lagrangian L at T gM -transformations δ ′ ( √ gL G ) = 0, δ ′ ( √ gL ψ ) = 0.Infinitesimal transformations of coordinates and field variables under the action ofthe group T gM are follows (A.28): δx µ = h µn t n , δ h mµ = − F mµn t n − ∂ µ t m , δψ ξ = − ∂ n ψ ξ t n . (26)Thus in our case a mµn = − F mµn , a ξn = − ∂ n ψ ξ , b mνµ n = − δ mn δ νµ , (27)and all other coefficients b I µm = 0. Here ∂ n := h µn ∂ µ .We introduce now the notation for variational derivatives. Let be[ √ gL G ] µm =: √ gG µm , (28)where G µm is a generalized Einstein tensor, which is calculated by formula (2) with q I → h mµ : √ gG µm = n X k =0 ( − k ∂ ( k ) σ ...σ k ∂ µ, σ ...σ k m ( √ gL G )= ∂ µm ( √ gL G ) − ∂ σ ( √ g ∂ µ,σm L G ) + ∂ (2) σ ρ ( √ g ∂ µ,σ ρm L G ) − ... (29)Here it is accepted that at q I = h mµ : ∂ I = ∂ µm , ∂ ν ...ν l I = ∂ µ,ν ...ν l m , [ L ] I = [ L ] µm ,[ L ] ν ...ν l I = [ L ] µ,ν ...ν l m . In addition[ √ gL ψ ] ξ =: √ g G ξ , (30) he problem of motion in gauge theories of gravity G ξ = f ξ − ∇ σ p σξ , f ξ := ∂ ξ L ψ , p σξ := ∂ σξ L ψ and ∇ σ is a covariant derivativein Riemannian space with metric g µν : ∇ σ p σξ = √ g ∂ σ ( √ g p σξ ). Next, we introduce thenotation [ √ gL ψ ] µm =: −√ g τ µm , (31)so τ µm = σ µm + ∇ σ β µ σm , √ g σ µm := − ∂ µm ( √ gL ψ ), β µ σm := ∂ µ,σm L ψ , (32)where ∇ σ β µσm = √ g ∂ σ ( √ gβ µσm ). This follows from the fact that the tensor density S ν σψ n = √ gβ ν σn is a translational superpotential of the Lagrangian √ gL ψ and thereforeis an antisymmetric quantity (with respect to upper indices) due to the T gM -invarianceof L ψ .It is shown below that the mixed coordinate-frame tensor τ µm can be interpreted asthe energy-momentum tensor of matter. So in the accepted designations[ L ] µm = √ g ( G µm − τ µm ) , and equations of the gravitational field [ L ] µm = 0 in any frame theory of gravity (where h mµ are the gravitational potentials) can be written in the form of Einstein equation G µm = τ µm .Components of currents J µm = J µG m + J µψ m and superpotentials S µνm = S µνG m + S µνψ m are associated with the corresponding components √ gL G and √ gL ψ of the Lagrangian L . To find the components associated with the gravitational Lagrangian, we specifyexpressions (6), (16), (17) for L = √ gL G , taking into account formulas (27): J σG n = n − X k =0 [ √ gL G ] λ,ν ...ν k σm ∂ ( k ) ν ...ν k F mλ n − √ gL G h σn , (33) J µ σG n = [ √ gL G ] µ, σn + n − X k =1 k [ √ gL G ] λ,µ ν ...ν k σm ∂ ( k − ν ...ν k F mλ n , (34) J µρσG n = [ √ gL G ] µ,ρσn + n − X k =2 k ( k − √ gL G ] λ,µ ρ ν ...ν k σm ∂ ( k − ν ...ν k F mλ n . (35)For n = 1 we have: J σG n = √ g ( ∂ λ, σm L G F mλ n − L G h σn ) ,J µ σG n = √ g ∂ µ, σn L G , J µρσG n = 0 . and for n = 2: J σG n = [ √ gL G ] λ, σm F mλ n + √ g ∂ λ,ν σm L G ∂ ν F mλ n − √ gL G h σn ,J µ σG n = [ √ gL G ] µ, σn + √ g ∂ λ,µσm L G F mλ n , J µρσG n = √ g ∂ ρ,µσn L G . he problem of motion in gauge theories of gravity n = 1, the condition of quasilocality (12) due to J µρσG n = 0 isobviously satisfied. This condition also holds for all f ( ℜ )-theories ( n = 2), where ℜ isthe curvature tensor R nρ σµ . Indeed, in this case J µρσG n = D ρ σµn + D ρ σµn − D σρ µn , where D ρ σµn := 4 ∂f ( ℜ ) /∂R nρ σµ , so J µ { ρσ } G n = 0.We enter the notation: √ g t µm := J µGm , √ gB µνm := S µνGm = J µνGm + ∂ σ J µν σGa . Both Noether’s currents J µm and corresponding superpotentials S µσm in terms ofcoordinate transformations ( T gM -transformations) are tensor densities, which whendivided by √ g become tensors (of appropriate rank), which we will call the tensorcurrents and the tensor superpotentials , respectively. By the frame index in relation tothe global Lorentz transformations, they are vectors, so by all indices in relation to both T gM -transformations and global Lorentz transformations, they are mixed tensors. Thus,the tensor current t µm associated with the gauge translational invariance of Lagrangian L G is mixed energy-momentum tensor of the gravitational field (in the theory of gravitydescribed by the Lagrangian L G ), and B µσm is its tensor superpotential which we willcall tensor of induction of the gravitational field . This tensor in GR is given by B µσm = ω µ σm − h µm R σ + h σm R µ , where ω mµ n is the spin connection and R n := F mmn [15].For the Lagrangian √ gL ψ , the translation current and its superpotential are givenby the formulas: J νψ n = √ g ( β µ νm F mµn + p νξ ∂ n ψ ξ − L ψ h νn ) , S ν σψ n = √ gβ ν σn , which specify for this case formulas (18) and (19), respectively.Consider the identity (24) for both components of the Lagrangian L . For √ gL G wehave: − G µm = t µm + ∇ σ B µσm , (36)and for √ gL ψ : τ νn = J νψ n / √ g + ∇ σ β ν σn . (37)Comparison of the latter identity with expression (32) gives a convenient way to calculatethe translational Noether’s current of the Lagrangian of matter: J νψ n = √ g σ νn = − ∂ νn ( √ gL ψ ) . (38)Therefore, the mixed coordinate-frame tensor σ νn is the energy-momentum tensor ofmatter fields. The tensor τ νn differs from σ νn by the covariant divergence of theantisymmetric tensor β ν σn , so it can also be interpreted as the energy-momentum tensorof matter. At the minimal method of including the gravitational interaction β ν σn = 0,therefore both tensors τ νn and σ νn coincide. he problem of motion in gauge theories of gravity − [ L ] µm / √ g = − ( G µm − τ µm ) = T µm + ∇ σ B µσm , (39)where T µm := t µm + τ µm is the total energy-momentum tensor of the gravitational system.The obtained identity makes it possible to write equations of the gravitational field[ L ] µm = 0 in the superpotential form: ∇ σ B µσm = − T µm , (40)similar to the form of Maxwell’s dynamic equations (and also the form of equations forother gauge fields of internal symmetry). As we can see, this possibility is providedexclusively by the gauge translational invariance of the theory of gravity, and not by aspecific type of gravitational Lagrangian √ gL G .So according to Theorem (Section 2) equations of the gravitational field [ L ] µm = 0in any gauge theory of the generalized gauge group T gM can be written both in the formof Einstein equation G µm = τ µm and in the superpotential form ∇ σ B µσm = − T µm , whichis an expression of the total mixed coordinate-frame energy-momentum tensor of thegravitational system T µm through the covariant divergence of the tensor of induction ofthe gravitational field B µσm , which is its tensor superpotential. The tensor of inductionof the gravitational field B µσm is determined by the gravitational Lagrangian √ gL G andacts as power characteristic of the gravitational field in the theory of gravity based on √ gL G .We turn to the consideration of identity (23). For √ gL G it is reduced to − G µm F mµ n = ∇ σ t σn , (41)or taking into account identity (36) to( t µm + ∇ σ B µσm ) F mµ n = ∇ σ t σn , (42)and for √ gL ψ to τ µm F mµ n − G ξ ∂ n ψ ξ = ∇ σ τ σn . (43)Adding (42) and (43), we obtain:( T µm + ∇ σ B µσm ) F mµ n − G ξ ∂ n ψ ξ = ∇ σ T σn , (44)The identities discussed above are strong. Consider now their weak (weakened)version.From the equations of the gravitational field, due to the antisymmetry of the tensorof induction of the gravitational field B µσm , follows the law of conservation of the totalenergy-momentum of the gravitational system: −∇ µ ∇ σ B µσm = ∇ µ T µm = ∇ µ t µm + ∇ µ τ µm = 0 . (45) he problem of motion in gauge theories of gravity G µm = τ µm , as well as −∇ µ t µm = ∇ µ τ µm , so fromidentity (41) we obtain the identity (weak) ∇ σ τ σn = τ µm F mµ n , (46)which is the equation of transfer of energy-momentum of matter, or the law of motionof matter . Thus, the law of motion of matter (46) is an identity which follows fromequations of the gravitational field due to the gauge translational invariance. From (46)follows that tensor F mµ n describes strength of gravitational field and τ µm its gravitationalcharge . Equality F mµ n = 0 is a sign of flat space.Consider the law of motion of matter (46) in the case of dust matter, when itsenergy-momentum tensor is written as τ µm = π µ u m , where u µ = dx µ /ds is the 4-velosityof particles and π µ = µcu µ is the density of their 4-momentum ( µ is the density ofmass). Substituting this expression in (46), we obtain: π µ ∂ µ u m = π µ u n F nµm − ∇ µ π µ u m . (47)When the mass of matter is conserved ∇ µ π µ = 0, the last term in (47), which has themeaning of the density of reactive force, disappears. So the particles in this case moveaccording to the law u µ ( ∂ µ u m − u n F nµm ) = 0 , (48)which is the geodesic equation of Riemannian space with metric g µν = η mn h mµ h nν , writtenin terms of 4-velocity u m and anholonomic coefficients F nµν .On the gravitational extremal, identity (44) is reduced to equations: G ξ ∂ n ψ ξ = 0 . (49)If the condition rank ( ∂ n ψ ξ ) = f (50)is satisfied, where f is the dimension of the field representation ( ξ = 1 , ...f ), from (49)follows G ξ = 0, or ∇ σ p σξ = f ξ . Therefore, if condition (50) is satisfied, equations ofmatter fields follow from equations of the gravitational field. Condition (50) can besatisfied only in the case f ≤
4, in particular for scalar fields.On the other hand, when equations G ξ = 0 are fulfilled (on the extremal of matterfields) identity (43) also gives the law of motion of matter (46), and from identity (44)it follows that with the additional assumption of the conservation of the total energy-momentum (45), equations( ∇ ν B µνn + T µn ) F nµm = 0 (51)are fulfilled. There are only 4 such equations, and therefore not all equations of thegravitational field follow from them, but only a certain part of them.So the following is true. Proposition 1.
In any gauge theory of the generalized gauge group T gM : he problem of motion in gauge theories of gravity
1) when fulfilling equations of the gravitational field ∇ σ B µσm = − T µm (on thegravitational extremal):a) the law of motion of matter ∇ σ τ σn = τ µm F mµ n is fulfilled, according to which thedust particles move along geodesics of Riemannian space with metric g µν = η mn h mµ h nν ;b) equations G ξ ∂ n ψ ξ = 0 are fulfilled, which are equivalent to equations of matterfields G ξ = 0 if the condition rank ( ∂ n ψ ξ ) = f is satisfied;2) when fulfilling equations of matter fields G ξ = 0 (on the extremal of matterfields):a) the law of motion of matter ∇ σ τ σn = τ µm F mµ n is fulfilled;b) under the additional assumption about conservation of the total energy-momentum ∇ µ T µm = 0 , equations ( ∇ ν B µνn + T µn ) F nµm = 0 are fulfilled, which areequivalent to the part of equations of the gravitational field. As we can see, this result does not depend on a specific expression for Lagrangiansand is a consequence only of the gauge translational invariance of the theory of gravity.
4. Laws of motion of charged matter
Let us now separate from the matter fields the gauge field A iµ (indices i, j, k ) of internalsymmetry V g (hereinafter simply the gauge field), and by matter we mean all fields ψ ξ ,except the gravitational and the gauge fields. We will consider the matter to be gaugecharged. So the Lagrangian of the theory in this case has three components: L = √ gL G ( h, ∂h, ..., ∂ ( n ) h ) + √ gL A ( h, A, ∂A ) + √ gL ψ ( h, ∂h, A, ψ, ∂ψ )and the field variables q I split into a system { h mµ , A iν , ψ ξ } , which is equivalent to splittingthe field index I into a multiindex { mµ , iν , ξ } .The symmetry of theories under consideration is described by the generalized gaugedeformed group G gM = V g × ) T gM , which has the structure of the semidirect productof groups V g and T gM (Appendix). We will not concretize the theory by specifyingthe components of its Lagrangian and will consider it in general based solely on thesymmetry of the theory. We consider all components of total Lagrangian L to be G gM -invariant. Note only that among theories under consideration, there is a canonicalvariant, which is the general relativity (in the tetrad formalism) together with theMaxwell’s electrodynamics [15]. However, even in the case of Einstein’s theory of gravitythe gauge field may not satisfy the Young-Mills equations, for example, in the case ofnon-quadratic dependence of the Lagrangian √ gL A on the tensor of the gauge field F iµν .Infinitesimal transformations of field variables under the action of the group G gM are described by formulas (A.15), (A.28): δ h mµ = − F mµn t n − ∂ µ t m , δA iµ = F iµn t n − F ijk A jµ υ k − ∂ µ υ i ,δψ ξ = − ( ∂ n + A in Z i ) ψ ξ t n + Z i ψ ξ υ i , (52) he problem of motion in gauge theories of gravity a mµn = − F mµn , a iµn = F iµn , a iµk = − F ij k A jµ ,a ξn = − ( ∂ n + A in Z i ) ψ ξ , a ξi = Z i ψ ξ ,b mνµn = − δ νµ δ mn , b i νµj = − δ νµ δ ij . (53)Here υ i are the parameters of the gauge group of internal symmetry V g , which will bemarked by the indices i, j, k , and Z i are the generators of the field representation of thegroup V g .In addition to the above notations for variational derivatives (28), (30), (31), weintroduce notations for variational derivatives of Lagrangian √ gL A :[ √ gL A ] µi =: √ gG µi , [ √ gL A ] µm =: −√ g θ µm , (54)as well as the variational derivative of the matter Lagrangian √ gL ψ [ √ gL ψ ] µi =: −√ gj µi , (55)which appears due to the fact that matter in our case is considered to be gauge charged.With these notations, the variational derivatives of total Lagrangian L with respectto the potentials h mµ and A iµ of gauge fields of both external (space-time) and internalsymmetry take the form:[ L ] µm = √ g ( G µm − θ µm − τ µm ) , [ L ] µi = √ g ( G µi − j µi ) . The expression for the variational derivatives with respect to matter fields remainsunchanged [ L ] ξ = [ √ gL ψ ] ξ = √ g G ξ , (56)although the functions G ξ = f ξ − ∇ σ p σξ itself in the presence of the gauge field changedue to the dependence of the matter Lagrangian √ gL ψ on the fields A iµ .From the first of definitions (54) follows G µi = − i µi − ∇ σ B µσi , (57)where B µσi := ∂ ν,σi L A , i µi := − ∂ µi L A . When writing expression (57), it is taken into account that tensor B µσi is a tensorsuperpotential associated with transformations of internal symmetry V g , as followsfrom the definition (19) applied to transformations (52). Therefore, tensor B µσi isantisymmetric with respect to upper indices. Tensor B µσi will be called the tensorof induction of the gauge field A iµ .In addition, from the second definition (54) follows θ µm = − ∂ µm ( √ gL A ) / √ g , andfrom definition (55) follows j µi = − ∂ µi L ψ . he problem of motion in gauge theories of gravity √ gL A , new components of currents appear,which will be denoted by the index A . In addition, the expressions for the currents ofthe matter fields change because they become gauge charged.Noether’s currents associated with gauge translations specify formula (18) takinginto account expressions (53) for the corresponding coefficients: J νA n = −√ g ( B µνi F iµn + L A h νn ) , (58) J νψ n = √ g [ β µ νm F mµn + p νξ ( ∂ n + A in Z i ) ψ ξ − L ψ h νn ] . (59)For the gravitational field, the tensor density of energy-momentum remains the same(33). Currents associated with the gauge transformations of the internal symmetry aregiven by expressions: J νA i = √ gB µ νj F jk i A kµ , J νψ i = −√ gp νξ Z i ψ ξ , hence J νψ n = √ g ( β µ νm F mµn + p νξ ∂ n ψ ξ − L ψ h νn ) − A in J νψ i . (60)Identity (24), applied to T gM -transformations of Lagrangian √ gL A , gives √ g θ µm = J µA m and indicates that the quantity θ µm defined by the second relation in (54) is theenergy-momentum tensor of the gauge field. The application of identity (24) for T gM -transformations of total Lagrangian L is reduced to − [ L ] µm / √ g = − ( G µm − θ µm − τ µm ) = T µm + ∇ σ B µσm , (61)where T µm = t µm + θ µm + τ µm is now the total energy-momentum tensor of the gravitational,gauge and matter fields. From this identity it follows that equations of the gravitationalfield [ L ] µm = 0 also in this case can be written both in the form of the Einstein equation G µm = θ µm + τ µm and in the superpotential form ∇ σ B µσm = − T µm .Identity (24) for V g -transformations of Lagrangians √ gL A and √ gL ψ gives: − G νi = J νA i / √ g + ∇ σ B νσi , j νi = J νψ i / √ g . (62)Comparing (62) with (57) we obtain i νi = J νA i / √ g . Therefore, the vectors i νi and j νi arethe currents of the gauge charges of gauge and matter fields, respectively.Due to the V g -symmetry of total Lagrangian L , identity (24) is reduced to − [ L ] νi / √ g = − ( G νi − j νi ) = I νi + ∇ σ B ν σi , (63)where I νi = i νi + j νi is the total current of the gauge charges of the gauge field and thefields of matter. Therefore, according to Theorem (Section 2) equations of the gaugefield [ L ] νi = 0, as well as equations of the gravitational field, can be written both in theform of the Einstein equation G νi = j νi and in the superpotential form ∇ σ B ν σi = − I νi , (64) he problem of motion in gauge theories of gravity I νi through the covariant divergence of the tensor of induction ofthe gauge field B ν σi , which is its tensor superpotential. This is a consequence of the V g -invariance of the theory and does not depend on the specific type of Lagrangians √ gL A and √ gL ψ , which define specific expressions for the quantities B ν σi and I νi . Tensor B ν σi is the power characteristic of the gauge field .Identities (23) corresponding to the T gM -invariance of Lagrangians √ gL G , √ gL A and √ gL ψ have the following form: − G µm F mµ n = ∇ σ t σn , (65) θ µm F mµ n + G µi F iµn = ∇ σ θ σn , (66) τ µm F mµ n − j µi F iµn − G ξ ( ∂ n + A in Z i ) ψ ξ = ∇ σ τ σn (67)and in the sum give an identity corresponding to total Lagrangian L : − ( G µm − θ µm − τ µm ) F mµ n + ( G µi − j µi ) F iµn − G ξ ( ∂ n + A in Z i ) ψ ξ = ∇ σ T σn . (68)We now write identities (23) corresponding to the V g -invariance of Lagrangians √ gL A and √ gL ψ : − G µj F jk i A kµ = ∇ σ i σi , (69) j µj F jk i A kµ + G ξ Z i ψ ξ = ∇ σ j σi , (70)as well as their sum: − ( G µj − j µj ) F jk i A kµ + G ξ Z i ψ ξ = ∇ σ I σi , (71)where it follows G ξ Z i ψ ξ = ∇ σ I σi + ( G µj − j µj ) F jk i A kµ , (72)which allows identity (67) to give another form ∇ σ τ σn = τ µm F mµ n − j µi F iµn − ∇ σ I σi A in − ( G µj − j µj ) F jk i A kµ A in − G ξ ∂ n ψ ξ , (73)and identities (70) to give the form: ∇ σ j σi = j µj F jk i A kµ + ∇ σ I σi + ( G µj − j µj ) F jk i A kµ . (74)Until now, in this section, we have considered exceptionally strong identities. Let’sconsider what these identities on extremals lead to (weakened version of identities).On the gravitational extremal − [ L ] µm / √ g = − ( G µm − θ µm − τ µm ) = T µm + ∇ σ B µσm = 0 , ∇ σ T σn = 0 , (75)so from identity (68) follows( G µi − j µi ) F iµn = G ξ ( ∂ n + A in Z i ) ψ ξ , (76) he problem of motion in gauge theories of gravity ∇ σ τ σn = τ µm F mµ n − j µi F iµn − ( G µi − j µi ) F iµn . (77)On the extremal of gauge field − [ L ] νi / √ g = − ( G νi − j νi ) = I νi + ∇ σ B ν σi = 0 , ∇ σ I σi = 0 , (78)so from (71) follows: G ξ Z i ψ ξ = 0 (79)and from (74) follows ∇ σ j σi = j µj F jk i A kµ , (80)which is the equation of transfer of gauge charges of matter . Identity (66) is reduced toidentity ∇ σ θ σn = θ µm F mµ n + j µi F iµn , (81)which is the equation of transfer of energy-momentum of the gauge field . Identity (67),taking into account (79), is reduced to identity ∇ σ τ σn = τ µm F mµ n − j µi F iµn − G ξ ∂ n ψ ξ , (82)and identity (68) is reduced to identity − ( G µm − θ µm − τ µm ) F mµ n − G ξ ∂ n ψ ξ = ∇ σ T σn . (83)On the extremal of matter fields [ L ] ξ = √ gG ξ = 0, so from (67) follows ∇ σ τ σn = τ µm F mµ n − j µi F iµn , (84)which is the equation of transfer of energy-momentum of gauge charged matter, or thelaw of its motion . The presence of gauge interaction is described here by an additional(in comparison with formula (46)) term − j µi F iµn , which is the Lorentz force densitygeneralized to the case of an arbitrary gauge field of internal symmetry. In addition,from identity (68) follows identity − ( G µm − θ µm − τ µm ) F mµ n + ( G µi − j µi ) F iµn = ∇ σ T σn , (85)and from (70) again follows the equation of transfer of gauge charge of matter (80).From (84) follows that as tensor F mµ n describes strength of gravitational field and τ µm the gravitational charge of matter, tensor F iµn describes strength of the gauge field and j µi the gauge charge of matter .At the simultaneous fulfilling of the equations of gravitational and gauge fields, fromidentity (77) we obtain the law of motion of gauge charged matter (84). At the sametime, from identity (83) follow equations G ξ ∂ n ψ ξ = 0, which generalize equations (49)in the case of gauge interactions (which is taken into account in our case in expression(56) for G ξ ). he problem of motion in gauge theories of gravity G µi − j µi ) F iµn = − ( I µi + ∇ σ B µσi ) F iµn = 0 , which are equivalent to the part of the gauge field equations.When equations of the gauge field and the fields of matter are fulfilledsimultaneously, identity (83) gives − ( G µm − θ µm − τ µm ) F mµ n = ∇ σ T σn , and under the condition of an additional assumption about the conservation of the totalenergy-momentum ∇ σ T σn = 0, taking into account (61), we obtain the identity( ∇ ν B µνn + T µn ) F nµm = 0 , which generalizes equations (51) in the case of the presence of a gauge field, which istaken into account by the term θ µm in the expression for T µm .We summarize the obtained results. Proposition 2.
In any gauge theory of the generalized gauge group G gM = V g × ) T gM :
1) when fulfilling equations of the gravitational field ∇ σ B µσm = − T µm and the gaugefield of internal symmetry ∇ σ B µσi = − I µi (on the gravitational and gauge extremals):a) the law of motion of gauge charged matter ∇ σ τ σn = τ µm F mµ n − j µi F iµn is fulfilled;b) equations G ξ ∂ n ψ ξ = 0 are fulfilled, which are equivalent to equations of matterfields G ξ = 0 if the condition rank ( ∂ n ψ ξ ) = f is satisfied;2) when fulfilling equations of the gauge field ∇ σ B µσi = − I µi (on the extremal ofgauge field):a) the equation of transfer of gauge charges of matter ∇ σ j σi = j µj F jk i A kµ is fulfilled;b) the equation of transfer of energy-momentum of the gauge field ∇ σ θ σn = θ µm F mµ n + j µi F iµn is fulfilled;3) when fulfilling equations of matter fields G ξ = 0 (on the extremal of the matterfields):a) the law of motion of gauge charged matter ∇ σ τ σn = τ µm F mµ n − j µi F iµn is fulfilled;b) under the additional assumption of the fulfilling of equations of the gravitationalfield ∇ σ B µσm = − T µm , equations ( I µi + ∇ σ B µσi ) F iµn = 0 are fulfilled, which are equivalentto the part of equations of the gauge field;c) under the additional assumption of the fulfilling of equations of the gauge field ∇ σ B µσi = − I µi , as well as the assumption about conservation of the total energy-momentum ∇ µ T µm = 0 , equations ( ∇ ν B µνn + T µn ) F nµm = 0 are fulfilled, which areequivalent to the part of equations of the gravitational field. We emphasize once again that these results do not depend on specific expressionsfor Lagrangians and are a consequence only of the gauge invariance of the theory.Point 3b) of this Proposition generalizes the result of Hilbert [1] to the arbitrarygauge field and the presence of currents of matter charges. he problem of motion in gauge theories of gravity G gM = U (1) g × ) T gM by changing the method of formation ofthe semidirect product so that at the infinitesimal level it reduced to reparameterizationof the group G gM , namely, instead of υ i we introduce new parameters ¯ υ i according tothe formula υ i = ¯ υ i + C in t n with arbitrary deformation coefficients C in (Appendix). Inthis case, part of transformations (52) of the group G gM changes: δA iµ = ( F iµn − F ij k A jµ C kn − ∂ µ C in ) t n − C in ∂ µ t n − F ij k A jµ ¯ υ k − ∂ µ ¯ υ i ,δψ ξ = − [ ∂ n + ( A in − C in ) Z i ] ψ ξ t n + Z i ψ ξ ¯ υ i , therefore, some of coefficients (53) change, namely: a iµn = F iµn − F ij k A jµ C kn − ∂ µ C in , a ξn = − [ ∂ n + ( A in − C in ) Z i ] ψ ξ ,b iνµn = − δ νµ C in . This results in a new component of the superpotential S ν σA n = √ gB ν σi C in (allothers remain unchanged). In addition, the expressions for the currents associated withthe translations change: there is a renormalization of tensor densities of the energy-momentum of gauge (58) and matter (60) fields (energy-momentum of the gravitationalfield, obviously remains unchanged): J νA n = −√ gB µνi ( F iµn − F ij k A jµ C kn − ∂ µ C in ) − √ gL A h νn = √ g ( θ νn + i νi C in + B µνi ∂ µ C in ) = √ g θ νn − ∂ µ S ν µA n ,J νψ n = √ g [ β µ νm F mµn + p νξ ∂ n ψ ξ − L ψ h νn − ( A in − C in ) j νi ]= √ g ( σ νn + j νi C in ) , which leads to renormalization of the total tensor density of energy-momentum of allfields: J νn = √ g ( T νn − ∇ σ β ν σn + I νi C in + B µνi ∂ µ C in ) , It corresponds the renormalized energy-momentum tensor ¯ T νn := J νn / √ g . Taking thedivergence from this quantity, we obtain: ∇ ν ¯ T νn = ∇ ν T νn + ∇ ν I νi C in + ( I νi + ∇ σ B ν σi ) ∂ ν C in . (86)It follows that when fulfilling equations of the gravitational field which ensure theequality ∇ ν T νn = 0, equation (86) is reduced to equation:( I νi + ∇ σ B ν σi ) ∂ ν C in = 0 , (87)provided that the gauge charges and the total energy-momentum of all fields areconserved (i.e. ∇ ν I νi = 0 and ∇ ν ¯ T νn = 0) for all methods of formation of semidirectproduct of subgroups V g and T gM in the group G gM .The gauge group of electrodynamics V g = U (1) g is one-parameter, so equation(87), in the case of arbitrary functions C n , is reduced to Maxwell’s dynamic equations he problem of motion in gauge theories of gravity ∇ σ B ν σ = − I ν . Therefore, in any G gM = U (1) g × ) T gM -symmetric theory of electrogravity,Maxwell’s dynamic equations ∇ σ B ν σ = − I ν and hence the law of motion of electricallycharged matter ∇ σ τ σn = τ µm F mµ n − j µ F µn are fulfilled when fulfilled equations of thegravitational field ∇ σ B µσm = − T µm , as well as the conservation laws of electric charges ∇ ν I ν = 0 and of total energy-momentum ∇ ν ¯ T νn = 0 for all methods of formation ofsemidirect product of subgroups U (1) g and T gM in group G gM [7].
5. Discussion and conclusion
Why do the laws of such various physical phenomena as gravity, gauge interactionsof internal symmetry and, under certain conditions, the laws of motion of matterfollow from each other? The cause of this phenomenon, as we see, is the infinitegauge symmetry of the theory, but not a specific type of Lagrangians. This symmetrydetermines the structure of field equations, for example, leads to antisymmetry of tensorsof induction of gravitational and gauge fields, allows to represent field equations insuperpotential form (in the form of dynamic Maxwell’s equations), gives expressionof conserved quantity through certain derivatives of Lagrangians, which are in fieldequations. All this allows obtain equations of some fields by requiring equations ofother fields and postulating conservation of certain physical quantities.From a mathematical point of view, with a fairly wide group of infinitetransformations and certain properties of coefficients a Ia , which project the groupparameters on variations of fields, requirement symmetry of theory is sufficient to obtainsome field equations and the principle of least action becomes redundant for it.An important result of this article is that the motion of particles along geodesics ofRiemannian space is inherent in an extremely wide range of theories of gravity and is aconsequence of the gauge translational invariance of these theories under the conditionof fulfilling equations of gravitational field. It is also interesting to note that the originof the Lorentz force, generalized for gauge-charged matter, is a consequence of the gaugesymmetry of the theory under the condition of fulfilling the equations of gravitationaland gauge fields.In the orthonormal frames, the strength of gravitational field in all the abovetheories of gravity is determined by the anholonomic coefficients F mµν , because they is whoforces particles to move along the geodesics. The tensor of induction of the gravitationalfield B µνm , which is determined by a specific type of the theory Lagrangian, is its powercharacteristic, because it is born by the energy-momentum of the gravitational system.The formulas given in this paper make it easy to find tensors of induction of thegravitational field B µνm and corresponding energy-momentum tensors for an arbitrarygauge theory of the generalized gauge translations group T gM . In this paper we showthat such theories include, in particular, f ( T )-theories and f ( ℜ )-theories ( f ( R )-theories,Lovelock gravity, Einstein-Gauss-Bonnet gravity, etc.). Here, the expression of thetensor of induction of the gravitational field B µνm for GR in an orthonormal frame isgiven, and for GR in an arbitrary anholonomic affine frame are presented in [16] (for he problem of motion in gauge theories of gravity the main provisions of this work relating to the problem of motion donot depend on specific expressions for Lagrangians, and hence specific expressions for B µνm , and are determined exclusively by the generalized gauge group of symmetry of thetheory . This, in our opinion, is the main result of this work. Acknowledgments
The author is grateful to Alisa Gryshchenko for her help with the prepare of article.
Appendix A. Group-theoretic description of gauge fields
An infinite (local gauge) symmetry lies in a basis of modern theories of fundamentalinteractions. Theory of gravity (general relativity) is based on the idea of covariance withrespect to the group of space-time diffeomorphisms. Theories of strong and electroweakinteractions are gauge theories of internal symmetry. Moreover, the existence of theseinteractions is considered to be necessary for ensuring the local gauge symmetries.But any physical theory can be written in covariant form without introduction of agravitational field. Similarly, as first had been emphasized in [17], for any theory withglobal internal symmetry G corresponding gauge symmetry G g can be ensured withoutintroduction of nontrivial gauge fields by pure gauging. Presence of the gravitational orthe gauge field of internal symmetry is manifested in presence of deformation – curvatureof Riemannian space, or fiber bundles with connection accordingly.Formally, nontrivial gauge fields are entered by continuation of derivatives up tocovariant derivatives ∂ µ → ∇ µ . Their commutators characterize strength of a field,which is considered, and from the geometrical point of view - curvature of correspondingspace. On the other hand, covariant derivatives set infinitesimal space-time translationsin the gauge field (curved spaces). That is why it is possible to suppose, that forintroduction of nontrivial arbitrary gauge fields it is necessary to consider groups (widerthan gauge groups G g ) which would generalize gauge groups G g to a case of nontrivialaction on space-time manifold and contain the information about arbitrary gauge fields in which motion occurs. Consequently, such groups must contain the informationabout appropriate geometrical structures with arbitrary variable curvature and set thesegeometrical structures on manifolds where they act. Hence from the mathematicalpoint of view such groups should realize the Klein’s Erlangen Program [18] for thesegeometrical structures.For a long time it was considered that such groups do not exist.
E.Cartan [19] hasnamed the situation in question as
Riemann-Klein’s antagonism – antagonism between
Riemann’s and
Klein’s approaches to geometry. There are attempts of modifying ofthe Klein’s Program for geometrical structures with arbitrary variable curvature bymeans of refusal of group structure of used transformations with usage of categories [20]quasigroups [21] and so on. One can encounter widespread opinion that nonassociativityhe problem of motion in gauge theories of gravity realization of the Klein’s Program for thegeometrical structures with arbitrary variable curvature (Riemannian space and fiberbundles with connection) can be fulfilled within the framework of the so called infinitedeformed groups which generalize gauge groups to the case of nontrivial action on thebase space of bundles with use of idea of groups deformations.
Such groups have been constructed in [8]. Klein’s Erlangen Program was realizedfor fiber bundles with connection in [22] and for Riemannian space in [23].This is important for physics because the widely known gauge approaches to gravity(see, for example, [24]) in fact gives gauge interpretation neither to metric fields nor toframe (tetrad) ones. An interpretation of these as connections in appropriate fibrings hasbeen achieved in way of introduction (explicitly or implicitly) of the assumption aboutexistence of the background flat space (see, for example, [25]). That is unnatural forgravity. The reason for these difficulties lies in the fact that the fiber bundles formalismis appropriate only for the internal symmetry Lie group, which do not act on the space-time manifold. But for the gravity this restriction is obviously meaningless because itdoes not permit considering gravity as the gauge theory of the translation group.In this Appendix we also show that generalized gauge deformed groups give a group-theoretic description of gauge fields (gravitational field with its metric or frame partsimilarly to gauge fields of internal symmetry) which is alternative for their geometricalinterpretation [26].This approach allow to overcome the well known Coleman-Mandula no-go theoremwithin the framework of generalized gauge deformed groups and gives new possibilitiesto unification gravity with gauge theory of internal symmetry [27].
Appendix A.1.
Generalized gauge groups
Gauge groups of internal symmetry G g are a special case of infinite groups and havesimple group structure – the infinite direct product of the finite-parameter Lie groups G g = Q x ∈ M G where product takes on all points x of the space-time manifold M . Groups G g act on M trivially: x ′ µ = x µ .For the aim of a generalization of groups G g to the case of nontrivial action onthe space-time manifold M , let’s now consider a Lie group G M with the multiplicationlaw (˜ g · ˜ g ′ ) α = ˜ ϕ α (˜ g, ˜ g ′ ) which act on the space-time manifold M (perhaps inefficiently)according to the formula x ′ µ = ˜ f µ ( x, ˜ g ). The infinite Lie group ˜ G gM is parameterized bysmooth functions ˜ g α ( x ) which meet the condition (cid:12)(cid:12)(cid:12) d ν ˜ f µ ( x, ˜ g ( x )) (cid:12)(cid:12)(cid:12) = 0 ∀ x ∈ M (where d ν := d/dx ν ). For parameters of group ˜ G gM we will use the Greek indices from thebeginning of the alphabet. The multiplication law in ˜ G gM is determined by the formulas:(˜ g × ˜ g ′ ) α ( x ) = ˜ ϕ α (˜ g ( x ) , ˜ g ′ ( x ′ )) , (A.1) x ′ µ = ˜ f µ ( x, ˜ g ( x )) . (A.2) he problem of motion in gauge theories of gravity G gM a group [8]. Formula(A.2) sets the action of ˜ G gM on M . In the case of trivial action of the group G M on M , x ′ µ = x µ and ˜ G gM becomes the ordinary gauge group G g = Q x ∈ M G . We name thegroups ˜ G gM generalized gauge groups. For the clearing of the groups deformations idea we will consider spheres of differentradius R . All of them have isomorphic isometry groups - groups of rotations O (3).The information about radius of the spheres is in structural constants of groups O (3),which in the certain coordinates may be written as: F = 1 /R , F = − F = 1.Isomorphisms of groups O (3), which change R , correspond to deformations.For gauge groups ˜ G gM some isomorphisms also play a role of deformations of spaceof groups representations, but unlike deformations of finite-parametrical Lie groups suchdeformations are more substantial, as these allow to independently deform space in itsdifferent points.Let us pass from the group ˜ G gM to the group G gM isomorphic to it by the formula g a ( x ) = H a ( x, ˜ g ( x )) (Latin indices assume the same values as the corresponding Greekones). The smooth functions H a ( x, ˜ g ) have the properties:1) H a ( x,
0) = 0 ∀ x ∈ M ;2) ∃ K α ( x, g ) : K α ( x, H ( x, ˜ g )) = ˜ g α ∀ ˜ g ∈ G, x ∈ M .The group G gM multiplication law is determined by its isomorphism to the group˜ G gM and formulas (A.1), (A.2):( g ∗ g ′ ) a ( x ) = ϕ a ( x, g ( x ) , g ′ ( x ′ )) := H a ( x, ˜ ϕ ( K ( x, g ( x )) , K ( x ′ , g ′ ( x ′ )))) , (A.3) x ′ µ = f µ ( x, g ( x )) := ˜ f µ ( x, K ( x, g ( x ))) . (A.4)Formula (A.4) sets the action of G gM on M .We name such transformations between the groups ˜ G gM and G gM as deformations of infinite Lie groups, since (together with changing of the multiplication law) thecorresponding deformations of geometric structures of manifolds subjected to groupaction are directly associated with them. We name the functions H a ( x, ˜ g ) deformationfunctions , functions h ( x ) aα := ∂H a ( x, ˜ g ) /∂ ˜ g α | ˜ g =0 deformation coefficients , and thegroups G gM infinite (generalized gauge) deformed groups. Let us consider expansion ϕ a ( x, g, g ′ ) = g a + g ′ a + γ ( x ) abc g b g ′ c + 12 ρ ( x ) abcd g d g ′ b g ′ c + ... (A.5)The functions ϕ a , setting the multiplication law (A.3) in the group G gM , are explicitly x dependent, so the coefficients of expansion (A.5) are x dependent as well. So x dependent became structure coefficients of group G gM ( structure functions versusstructure constants for ordinary Lie groups): F ( x ) abc := γ ( x ) abc − γ ( x ) acb (A.6)and coefficients R ( x ) adbc := ρ ( x ) adbc − ρ ( x ) adcb , (A.7) he problem of motion in gauge theories of gravity curvature coefficients of the deformed group G gM .Since ξ ( x ) µa := ∂ a f µH ( x, g ) | g =0 = h ( x ) αa ˜ ξ ( x ) µα , where ∂ b := ∂/∂g b and h ( x ) αa isinverse to the h ( x ) aα matrix, the generators X a := ξ ( x ) µa ∂ µ ( ∂ µ := ∂/∂x µ ) of thedeformed group G gM are expressed through the generators ˜ X α := ˜ ξ ( x ) µα ∂ µ of the group˜ G gM by the deformation coefficients: X a = h ( x ) αa ˜ X α . So in infinitesimal (algebraic)level, deformation is reduced to independent in every point x ∈ M nondegenerate linertransformations of generators of the initial Lie group. Theorem A1.
Commutators of generators of the deformed group G gM are linercombinations of generators with structure functions as coefficients: [ X a , X b ] = F ( x ) cab X c . (A.8)For generalized gauge nondeformed group ˜ G gM we have [ ˜ X α , ˜ X β ] = ˜ F γαβ ˜ X γ , where˜ F γαβ is structure constants of the initial Lie group G M . So for ˜ G gM F γαβ = ˜ F γαβ . Appendix A.2.
Group-theoretic description of connections in fiber bundlesand gauge fields of internal symmetry
Let P = M × V be a principal bundle with the base M (space-time) and a structuregroup V with coordinates ˜ υ i and the multiplication law (˜ υ · ˜ υ ′ ) i = ˜ ϕ i (˜ υ, ˜ υ ′ ) (indices i, j, k ). As usually, we define the left l ˜ υ : P = M × V → P ′ = M × ˜ υ − · V and the right r ˜ υ : P = M × V → P ′ = M × V · ˜ υ action V on P .Let’s consider a group G M = T M ⊗ V where T M is the group of space-timetranslations. The group G M parameterized by the pair ˜ t µ , ˜ υ i , has the multiplication law(˜ g · ˜ g ′ ) µ = ˜ t µ + ˜ t ′ µ , (˜ g · ˜ g ′ ) i = ˜ ϕ i (˜ υ, ˜ υ ′ ) and act on M inefficiently: x ′ µ = x µ + ˜ t µ . Onecan define the left action of the group G M on the principal bundle P : x ′ µ = x µ + ˜ t µ , υ ′ i = l i ˜ υ ( υ ).The group ˜ G gM is parameterized by the functions ˜ t µ ( x ), ˜ υ i ( x ) which meet theconditions (cid:12)(cid:12)(cid:12) δ µν + ∂ ν ˜ t µ ( x ) (cid:12)(cid:12)(cid:12) = 0 ∀ x ∈ M . The multiplication law in ˜ G gM is(˜ g × ˜ g ′ ) µ ( x ) = ˜ t µ ( x ) + ˜ t ′ µ ( x ′ ), (˜ g × ˜ g ′ ) i ( x ) = ˜ ϕ i (˜ υ ( x ) , ˜ υ ′ ( x ′ )) , (A.9) x ′ µ = x µ + ˜ t µ ( x ) , (A.10)where (A.10) determines the inefficient action of ˜ G gM on M with the kern of inefficiency- gauge group V g . The group ˜ G gM has the structure V g × ) T gM , which is a semidirectproduct of the groups V g and T gM , act on P as: x ′ µ = x µ + ˜ t µ ( x ) , υ ′ i = l i ˜ υ ( x ) ( υ )and is the group aut P of automorphisms of the principal bundle P .Let us deform the group ˜ G gM → G gM by means of deformation functions withadditional properties:3) H µ ( x, ˜ t, ˜ υ ) = ˜ t µ ∀ ˜ t ∈ T, ˜ υ ∈ V, x ∈ M ;4) H i ( x, , ˜ υ ) = ˜ υ i ∀ ˜ υ ∈ V, x ∈ M . he problem of motion in gauge theories of gravity G gM is parameterized by the functions t µ ( x ) = ˜ t µ ( x ) and υ i ( x ) = H i ( x, ˜ t ( x ) , ˜ υ ( x )) . Obviously, the group G gM , as well as the group ˜ G gM , has thestructure V g × ) T gM and acts on P as: x ′ µ = x µ + t µ ( x ) , υ ′ i = l iK ( x,t ( x ) ,υ ( x )) ( υ ) , (A.11)where functions K i ( x, t ( x ) , υ ( x )) are determined by equation: K i ( x, t ( x ) , υ ( x )) = ˜ υ i ( x ).Properties 3), 4) result in the fact that among deformation coefficients of the group G gM , x -dependent is only h ( x ) iµ = ∂ ˜ µ H i ( x, ˜ t, (cid:12)(cid:12)(cid:12) ˜ t =0 =: − A ( x ) iµ (where ∂ ˜ µ := ∂/∂ ˜ t µ ).Generators of the G gM -action on P (A.11) are split in the pair X µ = ∂ µ + A ( x ) iµ ˜ X i , X i = ˜ X i , where ˜ X i are generators of the left action of the group V on P . This result in the naturalsplitting of tangent spaces T u in any point u ∈ P to the direct sum T u = T τ u ⊕ T υ u subspaces: T τ u = { t µ X µ } , T υ u = { υ i X i } . The distribution
T τ u is invariant with respect to the right action of the group V on P , and T υ u is tangent to the fiber. So T τ u one can treated as horizontal subspaces ofthe T u and generators X µ - as covariant derivatives. This set in the principal bundle P connection and deformation coefficients A ( x ) iµ are the coordinates of the connectionform, which on submanifold M ⊂ P may be written as ω i = A ( x ) iµ dx µ . Necessarycondition of existence of group G gM (A.8) for generators X µ gives[ X µ , X ν ] = F ( x ) iµν X i (A.12)where F ( x ) iµν = F ijk A ( x ) jµ A ( x ) kν + ∂ µ A ( x ) iν − ∂ ν A ( x ) iµ (A.13)are the structure functions of the group G gM and F ijk = ˜ F ijk are the structure constantsof the Lie group V . Relationship (A.13) one can write in the form: dω i = − F ij k ω j ∧ ω k + Ω i , (A.14)where Ω i = F ( x ) iµν dx µ ∧ dx ν play the role of the curvature form on submanifold M . Soequation (A.14) is a structural equation for connection, which has been set on principalbundle P by action of group G gM . Theorem A2.
Acting on the principal bundle P = M × V deformed group V g × ) T gM sets on P structure of connection. Any connection on the principal bundle P = M × V may be set in such a way. This theorem realizes Klein’s Erlangen Program for fiber bundles with connection[22]. We should emphasize that for the setting of a geometrical structure in P it is enoughto consider the infinitesimal action (A.11) of the group G gM . he problem of motion in gauge theories of gravity A ( x ) iµ , a strength tensor – with structure functions F ( x ) iµν ofthe group G gM [8]. All groups G gM obtained one from another by internal automorphisms,which are generated by the elements υ ( x ) ∈ V g , describe the same gauge field. Theseautomorphisms lead to gauge transformations for fields A ( x ) iµ and for infinitesimal υ i ( x )give: A ′ ( x ) iµ = A ( x ) iµ − F ij k A ( x ) jµ υ k ( x ) − ∂ µ υ i ( x ) . (A.15) Appendix A.3.
Group-theoretic description of Riemannian spaces andgravitational fields
The structure of Riemannian space is a special case of structure of affine connectionin frame bundle and consequently it can be set by the way described above with theapplication of the deformed group SO ( n ) g × ) Dif f M . If we force the metricity andnon-torsionity conditions, generators of translations X µ = ∂ µ + Γ( x ) ( mn ) µ ˜ S ( mn ) , where˜ S ( mn ) is generators of group SO ( n ), become covariant derivatives in Riemannian space.For the setting of Riemannian structure by such means, it is enough to consider thegroup SO ( n ) g × ) Dif f M on algebraic level – on level its generators.Potentials of a gravitational field in the given approach are represented by theconnection coefficients Γ( x ) ( mn ) µ instead of the metrics or verbein fields that wouldcorrespond to sense of a gravitational field as a gauge field of translation group whichis born by energy-momentum tensor, instead of spin.Now we will show that the Riemannian structure on M is naturally set alsoby a narrower group than SO ( n ) g × ) Dif f M , namely, the deformed group ofdiffeomorphisms T gM , though it demands consideration of its action on M up to thesecond order on parameters.Let G M = T M where T M is the group of space-time translations. In this case(˜ t · ˜ t ′ ) µ = ˜ t µ + ˜ t ′ µ and x ′ µ = x µ + t µ . The group ˜ T gM is parameterized by the functions˜ t µ ( x ), which meet the condition (cid:12)(cid:12)(cid:12) δ µν + ∂ ν ˜ t µ ( x ) (cid:12)(cid:12)(cid:12) = 0 ∀ x ∈ M . The multiplication law in˜ T gM is (˜ t × ˜ t ′ ) µ ( x ) = ˜ t µ ( x ) + ˜ t ′ µ ( x ′ ) , (A.16) x ′ µ = x µ + ˜ t µ ( x ) , (A.17)where (A.17) determines the action of ˜ T gM on M . The multiplication law indicatesthat ˜ T gM is the group of space-time diffeomorphisms Diff M in additive parametrization.Thus, in the approach considered, the Diff M group becomes the gauge group of localtranslations. The generators of the ˜ T gM action on M are simply derivatives ˜ X µ = ∂ µ andthis fact corresponds to the case of the flat M and the absence of gravitational field.Let us deform the group ˜ T gM → T gM : t m ( x ) = H m ( x, ˜ t ( x )). The multiplication lawin T gM is determined by the formulas:( t ∗ t ′ ) m ( x ) = ϕ m ( x, t ( x ) , t ′ ( x ′ )) := H m ( x, K ( x, t ( x ))+ K ( x ′ , t ′ ( x ′ ))) , (A.18) he problem of motion in gauge theories of gravity x ′ µ = f µ ( x, t ( x )) := x µ + K µ ( x, t ( x )) . (A.19)Formula (A.19) sets the action of T gM on M .Let us consider expansion H m ( x, ˜ t ) = h ( x ) mµ [˜ t µ + 12 Γ( x ) µνρ ˜ t ν ˜ t ρ + 16 ∆( x ) µνρσ ˜ t ν ˜ t ρ ˜ t σ ] . (A.20)With usage of formula (A.18), for coefficients of expansion (A.5) we can obtain γ mpn = h mµ (Γ µp n + h νp ∂ ν h µn ) , (A.21) ρ mspn = h mµ (∆ µspn − Γ µnσ Γ σps − h νn ∂ ν Γ µσρ h σp h ρs ) . (A.22)So formulas (A.6), (A.7) for structure coefficients and curvature coefficients of deformedgroup T gM give F nµν = − ( ∂ µ h nν − ∂ ν h nµ ) , (A.23) R µν κρ = ∂ κ Γ µρν − ∂ ρ Γ µκν + Γ µκσ Γ σρν − Γ µρσ Γ σκν . (A.24)In these formulas deformation coefficients h mµ and h µm we use for changing Greek indexto Latin (and vice versa).Formulas (A.23) and (A.24) show that groups T gM contain information aboutgeometrical structure of space M where they act. The generators X m = h νm ∂ ν of the T gM -action (A.19) on M can be treated as frames (tetrads). Structure functions F nµν are anholonomic coefficients of frames X m .Let us write the multiplication law of the group T gM (A.18) for infinitesimal secondfactor: ( t ∗ τ ) m ( x ) = t m ( x ) + λ ( x, t ( x )) mn τ n ( x ′ ) , (A.25)where λ ( x, t ) mn := ∂ n ′ ϕ m ( x, t, t ′ ) | t ′ =0 . Formula (A.25) gives the rule for the addition ofvectors, which set in different points x and x ′ or a rule of the parallel transport of avector field τ from point x ′ to point x : τ m ( x ) = λ ( x, t ( x )) mn τ n ( x ′ ), or in coordinate basis τ µ ( x ) = ∂ ˜ ν H µ ( x, ˜ t ) τ ν ( x + ˜ t ). This formula determines the covariant derivative ∇ ν τ µ ( x ) = ∂ ν τ µ ( x ) + Γ( x ) µσν τ σ ( x ) , (A.26)where functions Γ( x ) µσν set the second order of expansion (A.20) and play the role ofcoefficients of an affine connection. They are symmetric on the bottom indexes so torsionequals zero. If η mn is a metric of a flat space, in the manifold M we can determine metrics g µν = h mµ h nν η mn . In this case generators X m = h νm ∂ ν of group T gM form an orthonormalframe (tetrad).If we force γ .ksl + γ .lsk = 0 (lowering indices we perform by metric η mn ), we can showthat coefficients Γ ρµν of expansion (A.20) may be written asΓ ρµν = 12 g ρσ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ) . (A.27) he problem of motion in gauge theories of gravity Christoffel symbols { µσν } and curvature coefficients R µν κρ of group T gM coincide with the Riemann curvature tensor . In this case group T gM is called the group of Riemannian translations [28]. Theorem A3.
Acting on the manifold M group of Riemannian translations T gM sets on M structure of Riemannian space. Any Riemannian structure on the manifold M may be set in such a way. This theorem realizes Klein’s Erlangen Program for Riemannian space [23].Information about Christoffel symbols is contained in the second order of expansion(A.20) of deformation functions, and about curvature in functions ρ mspn , which determine third order of expansion (A.5) in the multiplication law of the group T gM . So in thisapproach we need consider not only infinitesimal (algebraic) level in the group T gM (asin previous section), but higher levels, too.The gravitational field potentials are identified with deformation coefficients h ( x ) mµ ,strength tensor of the gravitational field – with structure functions F ( x ) msn of the group T gM [8]. All groups T gM obtained one from another by internal automorphisms describethe same gravitational field. These automorphisms, which can always be connected withthe coordinate transformations on M , lead to a general covariance transformation lawfor fields h ( x ) mµ and for infinitesimal t m ( x ) yield: h ′ ( x ) mµ = h ( x ) mµ − F ( x ) msn h ( x ) sµ t n ( x ) − ∂ µ t m ( x ) . (A.28)The transformation law (A.28) is similar to the transformation law (A.15) forgauge fields of internal symmetry and the only difference consists in the replacement ofstructure constants of finite Lie group by structure functions of the infinite deformedgroup T gM . This fact permits us to interpret the group T gM as the gauge translationgroup and the vector fields h mµ as the gauge fields of the translation group.So we show that infinite deformed (generalized gauge) groups:a) set on manifolds where they act geometrical structures of Riemannian spaces orfiber bundles with connection with arbitrary variable curvature;b) give a group-theoretic description of gauge fields – gravitational field with itsframe part and gauge fields of internal symmetry. References [1] Hilbert D. Die Grundlagen der Physik,
Math.-phys. Klasse (1915) 395–408[2] Noether E. Invariant Variation Problems, Transport theory and statistical physics (1971) 183–207 (English translation), arXiv:physics/0503066 [physics.hist-ph][3] Einstein A. and Grommer J. Allgemeine Relativit¨atstheorie und Bewegungsgesetz, Phys.-math.Kl. (1927) 2–14[4] Fock V. On the Motion of Finite Mass in the General Relativity,
Journal of Experimental andTheoretical Physics (1939) 375–428 (in Russian)[5] Einstein A. and Grossmann M. Entwurf Einer Verallgemeinerten Relativit¨atstheorie und EinerTheorie der Gravitation, Teubner, Leipzig 1913. Reprinted as , Doc.13 CPAE[6] Oltean M., Epp R., Sopuerta C., Spallicci A. and Mann R. Motion of Localized Sources inGeneral Relativity: Gravitational Self-force from Quasilocal Conservation Laws, Phys. Rev. D (2020) 064060, arXiv:1907.03012 [gr-qc] he problem of motion in gauge theories of gravity [7] Samokhvalov S. The Laws of Electrodynamics in the Gauge Theory of Gravity, Math. mod. (2001) 5–10 (in Ukrainian)[8] Samokhvalov S. Group-theoretical Description of Gauge Fields, Theor. Math. Phys. (1988)709–717[9] Cai Y., Capozziello S., De Laurentis M. and Saridakis E. f(T) Teleparallel Gravity andCosmology, Rept. Prog. Phys. (2016) 106901, arXiv:1511.07586 [gr-qc][10] Pinto P., Del Vecchio L., Fatibene L. and Ferraris M. Extended Cosmology in Palatini f(R)-theories, J. Cosmology and Astroparticle Physics (2018) 044, arXiv:1807.00397 [gr-qc][11] Petrov A. Field-theoretical Construction of Currents and Superpotentials in Lovelock Gravity,
Class. Quantum Grav. (2019) 235021, arXiv:1903.05500 [gr-qc][12] Maluf J., Ulhoa S., da Rocha-Neto J. and Carneiro F. Difficulties of Teleparallel Theoriesof Gravity with Local Lorentz Symmetry, Class. Quantum Grav. (2020) 067003,arXiv:1811.06876 [gr-qc][13] Guarnizo A., Orjuela-Quintana J. and Valenzuela-Toledo C. Dynamical Analysis ofCosmological Models with non-Abelian Guge Vector Fields, Phys. Rev. D (2020) 083507,arXiv:2007.12964 [gr-qc][14] Samokhvalov S. Group-theoretical Basis of the Holographic Principle, Math. mod. (2010) 7–11(in Ukrainian)[15] Samokhvalov S. and Vanyashin V. Group Theory Approach to Unification of Gravity withInternal Symmetry Gauge Interactions. I. Canonical Electrogravity, Class. Quantum Grav. (1991) 2277–2282, arXiv:1802.08958 [gr-qc][16] Samokhvalov S. About the Symmetry of General Relativity, J. Geom. Symm. Phys. (2020)75–103, arXiv:2001.07081 [gr-qc][17] Ogievetski V. and Polubarinov I. On the Meaning of the Gauge Invariance, Nuovo Cimento (1962) 173–180[18] Klein F. Vergleichende Betrachtungen Uber Neuere Geometrische Forschungen (ErlangenProgram), in: On Foundations of Geometry , pp. 399–434. Gostekhteorizdat, Moscow (1956)(Russian translation)[19] Cartan E. Group Theory and Geometry, in:
On Foundations of Geometry , pp. 438–507.Gostekhteorizdat, Moscow (1956) (Russian translation)[20] Sulanke R. and Wintgen P. Differentialgeometrie und Faserbundel, Veb Deutscher Verlag derWissenschaften, Berlin 1972[21] Sabinin L. Methods of Nonassociative Algebra in Differential Geometry, in: Koboyashi S. andNomizu K. (eds.)
Foundations of Differential Geometry , pp. 293–334. Nauka, Moscow (1981)(in Russian)[22] Samokhvalov S. On Setting of Connections in Fiber Bundles by the Action of Infinite Lie Groups, Ukrainian Math. J. (1991) 1599–1603[23] Samokhvalov S. Group-theoretic Description of Riemannian Spaces, Ukr. Math. J. (2003)1238–1248, arXiv:0704.2967 [math.DG][24] Hehl F., Heyde P., Kerlich G. and Nester J. General Relativity with Spin and Torsion: Fondationsand Prospects, Rev. Mod. Phys. (1976) 393–416[25] Cho Y. Einstein Lagrangian as the Translational Yang-Mills Lagrangian, Phys. Rev. D (1976)2521–2525[26] Konopleva N. and Popov V. Gauge Fields, Atomizdat, Moscow 1980 (in Russian)[27] Samokhvalov S. Group Theory Approach to Unification of Gravity with Internal SymmetryGauge Interactions. II. Relativity of Charges and Masses, Probl. Nucl. Phys. Cosm. Rays (1991) 50–58, arXiv:1802.09842 [gr-qc][28] Samokhvalov S. and Balakireva E. Group-theoretic Matching of the Length and the EqualityPrinciples in Geometry, Rus. Math. (Iz. VUZ)59