LOCAL SPACE-TIME TRANSLATION SYMMETRY

Transcription

1 GENERAL PHYSICS GRAVITATION LOCAL SPACE-TIME TRANSLATION SYMMETRY G. ZET, C. POPA, D. PARTENIE Department of Physics, Gh. Asachi, Technical University, Iasi, Romania Faculty of Physics, Al. I. Cuza University, Iasi, Romania Received September 6, 005 A theory based on the ravitational aue roup is used in order to obtain a static solution of the field equations for the ravitational potentials. The ravitational aue roup and the correspondin aue covariant derivative are introduced. The strenth tensor of the ravitational aue field is obtained and a aue invariant Laranian is constructed. The field equations of the aue potentials are written and a ravitational conserved current is determined. It is shown that the theory of the ravitational field based on the ravitational aue roup is equivalent to the General Relativity of Einstein. A model with aue potentials dependin only of spatial x coordinate is presented and its aue field equations are written. An analytical solution of these equations is then determined. All the calculations are performed usin a computer alebra packae, runnin on the MapleV platform, alon with several routines that we have written for our model. The interation of the field equation is also performed by this computer alebra packae.. INTRODUCTION In the traditional aue treatment of ravity, the Lorentz roup is localized and the ravitational field is not characterized by aue potentials. It is represented by metric field and the effects of ravity are described by the curvature of the space-time. The aue theory based on the Lorentz roup proves to be correct to the classical level but it is non-renormalizable. However, the aue theories are fundamental in the field theory and, in particular, in the elementary particle physics []. The three non-ravitational interactions (electromanetic, weak, and stron) are completely described by means of aue theory in the framework of the Standard Model (SM). First of all, the aue theory of the unitary roups SU(N) is of fundamental importance in elementary particle physics. The SM of stron and electroweak interactions is based on the aue theory of SU(3) SU() U() roup. In addition, the Grand Unification is described by the auin of SU(5) roup []. Secondly, the Poincaré roup (Lorentz transformations and space-time translations) is also of a fundamental importance in any field theory. After Paper presented at the National Conference of Physics, 3 7 September, 005, Bucharest, Romania. Rom. Journ. Phys., Vol. 5, Nos. 5 6, P , Bucharest, 006

2 56 G. Zet, C. Popa, D. Partenie pioneerin works of Utiyama [], Sciama [3, ], and Kibble [5] it was reconized that ravitation also can be formulated as a aue theory. The aue roups considered in aue theory of ravitation are Poincaré roup [6], de-sitter roup [7], affine roup [8, 9], etc. It is believed that the formulation of ravity as a aue theory on a Minkowski space-time could lead to a consistent quantum theory of ravity. Recently, N. Wu [0] proposed a aue theory of General Relativity (GR) based on the ravitational aue roup (G). The ravitational interaction is considered in this theory as a fundamental interaction in a flat Minkowski space-time, and not as space-time eometry. The ravitation aue roup G consists of eneralized space-time translations, and the ravity is described by aue potentials. If there is ravitational field in space-time, the space-time metric will not be equivalent to Minkowski metric, and space-time will become curved. In other words, in the traditional ravitational aue theory, quantum ravity is formulated in curved space-time. In this paper, we will not follow this way. The underlyin point of view of this new quantum aue eneral relativity is that it is formulated in the framework of traditional quantum field theory, ravity is treated as a kind of physical interactions in flat space-time and the ravitational field is represented by aue potential. In other words, if we put ravity into the structure of space-time, the space-time will become curved and there will be no physical ravity in time, because all ravitational effects are put into space-time metric and ravity is eometry. But, if we study physical ravitational interactions, it is better to rescue ravity from space-time metric and treat ravity as a kind a physical interactions. In this case, space-time is flat and there is physical ravity in Minkowski space-time. For this reason, we will not introduce the concept of curved space-time to study quantum ravity in the most part of this paper. So, the space-time is always flat, the ravitational field is represented by aue potential, and ravitational interactions are always treated as physical interactions. In fact, what ravitational field is represented by aue potential is required by aue principle. In this paper we use the theory based on the ravitational aue roup G to obtain a solution of the field equations for the ravitational potentials on a Minkowski space-time. In Section we define the ravitational aue roup G and then we introduce the aue covariant derivative D μ. The strenth tensor of the ravitational aue field is obtained and a aue invariant Laranian is constructed. The field equations of the aue potentials are written and a ravitational enery-momentum tensor ( T ) μν which is a conserved current is determined. It is shown that the theory of the ravitational field based on the ravitational aue roup G is equivalent to the General Relativity of Einstein.

3 3 Local space-time translation symmetry 57 Section 3 is devoted to the case of a model when the ravitational aue potentials A μ ( x) have a particular symmetry, dependin only on the spatial coordinate x. The correspondin non-null components of the strenth tensor F μν of the ravitational aue field are obtained and then the aue field equations are written. An analytical solution of these equations, which induce the Schwarzschild metric on the aue roup space, is then determined. All the calculations from the Section 3 have been performed by GRTensorII computer alebra packae, runnin on the MapleV platform, alon with several routines that we have written for our model. The interation of the field equation was also performed by this computer alebra packae. The computin proram is listed in the Section and some work instructions are described.. THE TENSOR OF THE GAUGE POTENTIALS The ravitational field is described in GR by the metric tensor of a curved space-time. In the aue theory based on the ravitational aue roup G the ravity is treated as a physical interaction in a Minkowski (flat) space-time and the ravitational field is represented by aue potentials. The infinitesimal transformations of roup G are of the form [0]: U ε = ε P, = 0,,, 3, (.) where ε are the infinitesimal parameters of the roup, and P = i are the enerators of the aue roup. It is known that these enerators commute each other P, P β = 0. (.) However, this property of the enerators does not means that the ravitational aue roup is an Abelian roup, because its elements do not commute [0] U( ε), U ( ε) 0. (.3) There is a difference between the roup T of space-time translations and the ravitational aue roup G. Space-time translations of T are coordinate transformations, that is, the objects or fields (physical systems) in space-time are fixed while the coordinates describin the motion of the physical system undero transformation. But, under the transformations of the ravitational aue roup the space-time coordinate system is fixed, while the objects or fields undero transformation. Therefore, the ravitational aue roup G contains all dynamical information of interactions and it is convenient to use it for studyin the ravitational field.

4 58 G. Zet, C. Popa, D. Partenie Let us suppose now that φ ( x) is a scalar field. Then its ravitational aue transformation under G is: φ( x) φ ( x) = U( ε) φ ( x ). (.) Because U ( ε) 0, the partial derivative of φ ( x) does not transform covariant μ under the aue roup G φx φ x U ε φ x. (.5) μ μ μ In order to construct an action which is invariant under local aue transformations with parameters dependin of coordinates ε =ε ( x ), it is necessary to define a aue covariant derivative [, ] where A ( x) Dμ = μ iaμ( x ), (.6) μ is the aue ravitational field (potential) with values in the Lie alebra of G and is the aue couplin constant of the ravitational interaction. The law of transformation under the aue roup of this potential is: A A U A U i U U μ μ = μ + μ. (.7) The derivative D μ has the property of aue covariance under the aue roup: D D = UD U. (.8) The ravitational A ( x) μ μ μ combination of enerators P where A ( x) μ aue field can be expanded as a linear A x = A x P, (.9) μ μ μ are the ravitational aue potentials, i.e. they are the components of the ravitational aue field. We define now new aue ravitational potentials μ μ μ G x =δ A x, (.0) and introduce their inverses G μ with the properties μ β β μ G G =δ, (.a) μ ν μ ν G G =δ. (.b)

5 5 Local space-time translation symmetry 59 Usin these components, we can define a metric on the aue roup space as follows: =η GμG ν, (.a) where η = (,,, ) μν β μν β β μν β μ ν =η G G, (.b) dia is the metric on the Minkowski space-time and η μν denotes its inverse. The strenth tensor of the ravitational aue field is iven by the standard expression [3,, 5] Fμν = i Dμ, Dν. (.3) Then, its components, defined by μν μν F = F P, are β β μν μ μ ν μ μ β ν ν β μ F = A A A A + A A. (.) The strenth tensor transforms covariant under the ravitational aue transformations F = UF U. (.5) μν μν In his work [0] N. Wu chooses the Laranian of the ravitational aue potentials in the form L = L 0, (.6) where det ( μν ) = and L = μρ νσ β μρ ν σ β 0 η η β Fμν Fρσ η Gβ G Fμν Fρσ ημρ Gν σ β Gβ Fμν Fρσ. (.7) This expression is quite special for ravitational interactions. Indeed, for ordinary SU(N) aue field theory it is possible to construct only one invariant which is a quadratic form of field strenth. In aue theory of ravitation there are three different aue invariant terms which are quadratic forms of field strenth of ravitational aue field. The interal of action associated to the Laranian L is defined as usually: S = d xl, (.8) It can be verified that this action has ravitational aue symmetry, i.e. it is invariant under the ravitational aue transformations.

6 50 G. Zet, C. Popa, D. Partenie 6 The first order variation of the ravitational aue fields is β μ β μ δ A x = ε A x, (.9) and the correspondin first order variation of the action is Here i T μ μ μ i δ S = d xε T. (.0) is the inertial enery-momentum tensor, whose definition is L Ti β Aν L μaν μ 0 = β +δμ 0. (.) The lobal ravitational aue symmetry of the system ives out the conservation law of inertial enery-momentum tensor μ μ T =0. (.) i This means that the inertial enery-momentum tensor is a conserved current. The Euler-Larane equations for ravitational aue field are L L μ = 0. (.3) A A ν μ ν Introducin (.6) (.7) into the Eqs. (.3) we obtain the followin field A x equations of the ravitational aue fields ( μρ νσ Fσ νρfμ μρfν μ η η β ρσ η ρ + η ρ β F β ν ημρδ ν νρ μ F ) ( T ), ρβ + η δ ρβ = where ( T ) μ of ravitational aue field [0]. The expression of the tensor ( T ) μ (.) is the ravitational enery-momentum tensor, which is the source μ is ν γ γ ( T ) = ηνρηλσ Fβ ( C νρ ) G λg σfβ βγ ρσ λ + η β γ ρσ ( Cλ) G G γ F γ ηνρ λ σ β ( C λρ ) G νg σfβ γ β ρσ λ η β γ ρσ( Cλ) + λρg ν G σ Fβ γ ( C λρ νσ ) G κfβ( D Cγ γ ) β ρσ λ β γ ρσ λ κ ηνρg σg κfμ ( Cγ νρ ) G σg κfβ γ ρσ μ κ + η β γ ρσ( Cκ γ ) + + η + η η

7 7 Local space-time translation symmetry 5 + ηλρg ν G σ G κ Fβ D Cγ λρg ν G σ G κ Fβ D Cγ β γ ρσ λ κ η β γ ρσ λ κ μ ηλρηνσ ( C Fβ νρ ) ( CσG λfβ μ β λ ρσ η β λ ρσ) + μ μ + ηνρ ( CσG λfβ λρ ) ( G νg σg ν σ ) Fβ λ β ρσ + η μ ρσ β λ δβδ δλ λρ μ μ η ( G νg σg ν σ ) Fβ μ ρσ β λ δ δβδλ γ γ η κρg νg λg σfβf κρg νg λg σfβf β γ ρσ κλ + η γ β ρσ κλ G γ F F G γ ημρηλσ ν β μρ λσ νfβ γ F 8 β ρσ μλ η η βγ ρσ 6 μλ μρg ν G λ G σ γ FβF μρg ν G λ G σ γ η Fβ γ ρσ γ ρσf. 8 β μλ + η β μλ This new tensor is different from that previously introduced into the Eq. (.). The inertial enery-momentum tensor i T μ is iven by the conservation law associated to the lobal ravitational aue symmetry. It defines out an enery momentum -vector whose time component ( Pi ) 0 ravitational enery-momentum tensor ( T ) ( P) = 3 d 0 i x T i, (.5) = H ives out the Hamiltonian of the system. The is iven by the field equations of the ravitational aue field. It define also a conserved enery-momentum -tensor μ ( P ) = 3 d 0 x T, (.6) whose time component ( P ) 0 μ = E ives out the ravitational enery of the system, which is the source of the ravitational aue fields. It is easy to verify T is also a conserved current that the ravitational enery-momentum tensor ( ) μ μ The two enery-momentum tensors i μ T =0. (.7) T μ and ( T ) μ are different and this has as consequence the result that the inertial mass is different from ravitational mass. In the first order of approximation their difference is proportional to A μ and it is too small to be detected in experiments developed in weak ravitational

8 5 G. Zet, C. Popa, D. Partenie 8 field like those of the Earth. But, in the environment of stron ravitational field the difference will become relatively larer and will be easier to detect. If we use the metric β defined in the space of the aue roup [see Eq. (.)], then we can calculate the quantities [6] ( β δ βδ δ β) γ Γ γδ β = +. (.8) γ From this definition, we can see that Γ β looks like the affine connection in eneral relativity. But now, it has no eometric meanin; it is not the affine connection in the curved space-time of the eometric picture of the ravity. The quantities define a connection in the aue roup space of the physical model for ravity. The correspondin Riemann tensor is defined by δ δ δ η δ η δ βγ γ β β γ β γη γ βη R = Γ Γ +Γ Γ Γ Γ. (.9) The Ricci tensor is defined, as usually, by the equation and the scalar curvature by γ β βγ R = R, (.30) γ R= R. (.3) γ If we consider the ravitational interactions of a matter field φ ( x) havin the action S M, then the total action of the system is S = + π d xr SM. (.3) 6 G The variational principle δ S = 0 with respect to aue ravitational A μ x ives the followin field equations potentials Rβ Rβ = 8 πgt β, (.33) where T β is the symmetric enery-momentum tensor of the mater field which is defined by [7, 8] T β = δ δ S M β. (.3) Therefore, we obtained in (.33) just the Einstein's field equations. They are equivalent with the aue field equations (.) and this proves that the aue model implies the eneral relativity of Einstein.

9 9 Local space-time translation symmetry MODEL OF GRAVITATIONAL GAUGE FIELD We apply the previous results to the case when the ravitational aue potentials A μ ( x) have a dependence only of the coordinate x in the aue roup space. We choose these potentials in the form A μ ( x) A x = , (3.) A( x) A( x) where ( x, y, z, t ) are Cartesian coordinates on the aue roup space, and A( x) is a function dependin only of the variable x. Then the new ravitational aue G x defined by the Eqs. (.0), become potentials μ, and their inverse are G G μ μ A x = 0 0 0, A x 00 0 A( x) = A x (3.) (3.3) The components β of the metric tensor defined by the Eq. (.) are iven then by β 00 0 ( A( x) ) = ( ) A x (3.)

10 5 G. Zet, C. Popa, D. Partenie 0 The non-null components Fμν Eq. (.3) are t tx F of the strenth tensor field defined in the = t A x Fxt =, A x where A ( x) denotes the derivative of (3.5) A x with respect to the spatial variable x. Introducin (3.) and (3.5) into the Eq. (.), we obtain only one independent aue field equation ( ) A x A x A x = 0. (3.6) Therefore, the aue field equations (.) for the previous considered model reduce to only one independent equation for the unknown function (aue potential) A( x). Its solution can be easy obtained if we separate the variables and then interate it. We have ± + ax + b A( x) = (3.7) where a and b are two arbitrary constants of interation. In particular, if we chose a = and b = 0, then the solution (3.7) becomes A x ± + x = (3.8) Now, introducin the solution (3.8) into the Eq. (3.), we obtain the non-null components β of the metric tensor on the aue roup space β x = ( + ) x The correspondin line element is ds = dx + dy + dz ( + x ) d t. + x (3.9) (3.0) and it describe the ravitational field of our model. On the other hand, if we write the Eq. (.33) for the spherical model above considered and suppose that the enery-momentum tensor of the mater field

11 Local space-time translation symmetry 55 vanishes T μν = 0, then we obtain the same independent field equations as in Eq. (3.6). Therefore, the aue theory based on the ravitational aue roup G ives out the result of General Relativity. The aue theory has the advantae that the space-time is Minkowski (flat) and it could lead to a consistent quantum theory of ravity. All the calculations from the Section 3 have been performed by GRTensorII computer alebra packae, runnin on the MapleV platform, alon with several routines that we have written for our model. The interation of the field equation (3.6) was also performed by this computer alebra packae. The computin proram is listed in the followin Section.. COMPUTING PROGRAM In GRTensor, when the oal is the calculation of components of indexed objects (in particular tensors) or definin new tensors, first of all, we must to specify the space-time eometry [7, 8]. We loaded the metric of the space-time usin the qload(mink) command. The Minkowski metric of the eta miu niu and its inverse by space-time is denoted in our proram by { } eta inv{ miu niu}. The command rdef is included to facilitate the specification of new tensors in a simple and natural manner. It allows tensors to be defined either as an equation in terms of previously defined tensors, or by manual entry of their components. Inner and outer products of tensors, symmetrization, and derivatives can all be specified as part of the tensor definitions. Furthermore, index symmetries of the newly defined tensors can be included. The interation of the field equations has been done by command with(detools, odeadvisor). A ^ alphamiu, were introduced The aue potentials Aμ ( x ), denoted by { } by manual entry of their components, and the quantities Eq. (.0), have been denoted by Gb{ ^ } Gbinv{ ^ alpha miu}. G x defined in μ, alpha miu and their inverses by The analytical proram allows calculatin: the components of the strenth tensor field F ^ alphamiuniu, the components of the metric μν, F denoted by { } β, denoted by b{ miu niu }, the components of the ravitational enery-momentum tensor, T ^ niu alpha, and the field T ν denoted by { } equations, denoted in our proram by EQ{ ^ niu alpha }. The expression under the derivative μ on the left-hand side of the aue field equations (.) has

12 56 G. Zet, C. Popa, D. Partenie been denoted by EX{ ^miu ^ niu alpha }. On the other hand, we used the analytical proram to calculate the connection coefficients Γ γ β, denoted by GAMMA{ ^ amma alpha beta }, the components of the curvature Riemann tensor R δ βγ, denoted in proram by RIM{ ^ delta alpha beta amma }, the components of the Ricci R β, denoted by RIC{ alpha beta }, and the curvature scalar R, denoted by RSC. The correspondin Einstein's equations have been denoted by EQU{ alpha beta }. The differential equation (3.6) has been denoted by ode and then it was interated by the commands: odeadvisor(ode) and dsolve(ode). The comments are inserted as instructions noted by symbol #. Below we list the part of proram which allows definin and calculatin the quantities previously specified. PROGRAM GAUGE THEORY.mws > restart; >rtw; #Metric of the Minkowski space-time; >qload(mink); #Gaue potentials; >rdef(`a{^alpha miu}`); rcalc(a(up,dn)); #New aue potentials; >rdef(`gb{^alpha miu}:=kdelta{^ alpha miu}-*a{^ alpha miu}`); >rcalc(gb(up,dn)); #Inverse of new aue potentials; >rdef(`gbinv{^miu alpha}`); rcalc(gbinv(up,dn)); #Strenth tensor of the aue potentials; >rdef(`f{^alpha miu niu}:=a{^ alpha niu,miu}-a{^alpha miu,niu} -*A{^ beta miu}*a{^alpha niu, beta}+*a{^beta niu} *A{^alpha miu, beta}`); rcalc(f(up,dn,dn)); #Metric on the aue roup space; >rdef(`b{miu niu}:=eta{alpha beta}*gbinv{miu ^alpha}*gbinv{niu ^beta}`); rcalc(b(dn,dn)); #Expression under the derivative in Eq. (.); >rdef(`ex{^miu ^niu alpha}:=(/)*etainv{^miu ^rho}*etainv{^niu ^sima}*b{alpha beta}*f{^beta rho sima}-(/)*etainv{^niu ^rho} *F{^miu rho alpha}+(/)*etainv{^miu ^rho}*f{^niu rho alpha}-(/) *etainv{^miu ^rho}*kdelta{^niu alpha}*f{^beta rho beta}+(/) *etainv{^niu ^rho}*kdelta{^miu alpha}*f{^beta rho beta}`); >rcalc(ex(up,up,dn));

13 3 Local space-time translation symmetry 57 #Component terms in expression of ravitational enery-momentum tensor; >rdef(`t{^niu alpha}:=(/)*etainv{^niu ^rho}*etainv{^lambda ^sima} *b{beta amma}*f{^beta rho sima}*a{^amma lambda,alpha}`); >rcalc(t(up,dn)); >rdef(`t{^niu alpha}:=(/)*etainv{^niu ^rho}*gbinv{^lambda beta}*gbinv{^sima amma}*f{^beta rho sima}*a{^amma lambda,alpha}`); >rcalc(t(up,dn)); >rdef(`t3{^niu alpha}:=(/)*etainv{^niu ^rho}*gbinv{^lambda amma}*gbinv{^sima beta}*f{^beta rho sima}*a{^amma lambda,alpha}`); >rcalc(t(up,dn)); >rdef(`t{^niu alpha}:=(/)*etainv{^lambda ^rho}*gbinv{^niu beta}*gbinv{^sima amma}*f{^beta rho sima}*a{^amma lambda,alpha}`); >rcalc(`t(up,dn)`); >rdef(`t5{^niu alpha}:=(/)*etainv{^lambda ^rho}*gbinv{^niu amma}*gbinv{^sima beta}*f{^beta rho sima}*a{^amma lambda,alpha}`); >rcalc(t5(up,dn)); >rdef(`t6{^niu alpha}:=(/)*etainv{^lambda ^rho}*etainv{^niu ^sima}*b{alpha beta}*gbinv{^kapa amma}*f{^beta rho sima}*gb{^tau lambda}*a{^amma kapa,tau}`); rcalc(t6(up,dn)); >rdef(`t7{^niu alpha}:=(/)*etainv{^niu ^rho}*gbinv{^sima alpha}*gbinv{^kapa amma}*f{^miu rho sima}*c{^amma kapa,miu}`); >rcalc(t7(up,dn)); >rdef(`t8{^niu alpha}:=(/)*etainv{^niu ^rho}*gbinv{^sima beta}*gbinv{^kappa amma}*f{^beta rho sima}*a{^amma kapa,alpha}`);>rcalc(t8(up,dn)); >rdef(`t9{^niu alpha}:=(/)*etainv{^lambda ^rho}*gbinv{^niu beta}*gbinv{^sima alpha}*gbinv{^kapa amma}*f{^beta rho sima}*gb{^tau lambda}*a{^amma kapa,tau}`); rcalc(t9(up,dn)); >rdef(`t0{^niu alpha}:=(/)*etainv{^lambda ^rho}*gbinv{^niu alpha}*gbinv{^sima beta}*gbinv{^kapa amma}*f{^beta rho sima}*gb{^tau lambda}*c{^amma kapa,tau}`); rcalc(t0(up,dn)); >rdef(`s{^miu alpha lambda rho sima}:=b{alpha beta}*c{^miu lambda}*f{^beta rho sima}`); rcalc(s(up,dn,dn,dn,dn)); >rdef(`t{^niu alpha}:=(/)*etainv{^lambda ^rho}*etainv{^niu ^sima}*(s{^miu alpha lambda rho sima,miu})`); rcalc(t(up,dn)); >rdef(`r{^beta alpha rho}:=a{^sima lambda}*gbinv{^lambda alpha}*f{^beta rho sima}`); rcalc(r(up,dn,dn)); >rdef(`t{^niu alpha}:=(/)*etainv{^niu ^rho}*(r{^beta alpha rho,beta})`); rcalc(t(up,dn)); >rdef(`q{rho}:=a{^sima lambda}*gbinv{^lambda beta}*f{^beta rho sima}; rcalc(q(dn)); >rdef(`t3{^niu alpha}:=(/)*etainv{^niu ^rho}*(q{rho,alpha})`);

14 58 G. Zet, C. Popa, D. Partenie >rcalc(t3(up,dn)); >rdef(`l{^miu ^niu alpha lambda rho}:=((gbinv{^niu beta}*gbinv{^sima alpha}*gb{^miu lambda}-kdelta{^niu beta}*kdelta{^sima alpha}*kdelta{^miu lambda})*f{^beta rho sima})`); >rcalc(l(up,up,dn,dn,dn)); >rdef(`t{^niu alpha}:=(/(*))*etainv{^lambda ^rho}*l{^miu ^niu alpha lambda rho,miu}`); rcalc(t(up,dn)); >rdef(`y{^miu ^niu alpha lambda rho}:=((gbinv{^niu alpha}*gbinv{^sima beta}*gb{^miu lambda}-kdelta{^niu alpha}*kdelta{^sima beta}*kdelta{^miu lambda})*f{^beta rho sima})`); >rcalc(y(up,up,dn,dn,dn)); >rdef(`t5{^niu alpha}:=(/(*))*etainv{^lambda ^rho}*y{^miu ^niu alpha lambda rho,miu}`); rcalc(t5(up,dn)); >rdef(`t6{^niu alpha}:=(/)*etainv{^kapa ^rho}*gbinv{^niu beta}*gbinv{^lambda alpha}*gbinv{^sima amma}*f{^beta rho sima}*f{^amma kapa lambda}`); rcalc(t6(up,dn)); >rdef(`t7{^niu alpha}:=(/)*etainv{^kapa ^rho}*gbinv{^niu amma}*gbinv{^lambda alpha}*gbinv{^sima beta}*f{^beta rho sima}*f{^amma kapa lambda}`); rcalc(t7(up,dn)); >rdef(`t8{^niu alpha}:=(/8)*etainv{^miu ^rho}*etainv{^lambda ^sima}*b{alpha amma}*gbinv{^niu beta}*f{^beta rho sima}*f{^amma miu lambda}`); rcalc(t8(up,dn)); >rdef(`t9{^niu alpha}:=(/6)*etainv{^miu ^rho}*etainv{^lambda ^sima}*b{beta amma}*gbinv{^niu alpha}*f{^beta rho sima}*f{^amma miu lambda}`); rcalc(t9(up,dn)); >rdef(`t0{^niu alpha}:=(/8)*etainv{^miu ^rho}*gbinv{^niu alpha}*gbinv{^lambda beta}*gbinv{^sima amma}*f{^beta rho sima}*f{^amma miu lambda}`); rcalc(t0(up,dn)); >rdef(`t{^niu alpha}:=(/)*etainv{^miu ^rho}*gbinv{^niu alpha}*gbinv{^lambda amma}*gbinv{^sima beta}*f{^beta rho sima}*f{^amma miu lambda}`); rcalc(t(up,dn)); #Gravitational enery-momentum tensor; >rdef(`t{^niu alpha}:=t{^niu alpha}+t{^niu alpha}-t3{^niu alpha} T{^niu alpha}+t5{^niu alpha}+t6{^niu alpha}-t7{^niu alpha}+t8{^niu alpha}+t9{^niualpha}-t0{^niu alpha}-t{^niu alpha}-t{^niu alpha}+t3{^niu alpha}+t{^niu alpha}-t5{^niu alpha}-t6{^niu alpha}+t7{^niu alpha} T8{^niu alpha}-t9{^niu alpha}-t0{^niu alpha}+t{^niu alpha}`); rcalc(t(up,dn)); #Gravitational aue field equations ; >rdef(`eq{^niu alpha}:=ex{^niu alpha}+*t{^niu alpha}`); rcalc(eq(up,dn)); rdisplay(_); #Christoffel coefficients ;

15 5 Local space-time translation symmetry 59 >rdef(`gamma{^amma alpha beta}:=(/)*binv{^amma ^delta}*(b{alpha delta,beta}+b{beta delta,alpha}-b{alpha beta,delta})`); >rcalc(gamma(up,dn,dn)); #Riemann tensor; >rdef(`rim{^delta alpha beta amma}:=gamma{^delta alpha beta, amma}-gamma{^deltaalpha amma,beta}+gamma{^eta alpha beta}*gammm{^delta amma eta}-gm{^eta alpha amma}*gamma{^delta beta eta}`); rcalc(rim(up,dn,dn,dn)); #Ricci tensor; >rdef(`ric{alpha amma}:=kdelta{beta ^delta}*rim{^beta alpha delta amma}`); rcalc(ric(dn,dn)); #Scalar curvature; >rdef(`rsc:=binv{^alpha ^amma}*ric{alpha amma}`); rcalc(rsc); #Einstein field equations; >rdef(`equ{alpha beta}:=ric{alpha beta}-/*b{alpha beta}*rsc`); >rcalc(equ(dn,dn)); rdisplay(_); We list below the ravitational field equations EQ obtained after runnin the proram. They are written as a -dimensional matrix, each line bein iven successively in a quadratic bracket. Of course, we have to put each matrix element equal to zero to obtain the field equations. EQU[a]*``[b] = matrix([[0, 0, 0, 0], [0, -*(diff(a(x),x)^*-diff(a(x),`$ `(x,))+diff(a(x),`$`(x,))**a(x)), 0, 0], [0, 0, -*(diff(a(x),x)^*-diff(a(x),` $`(x,))+diff(a(x),`$`(x,))**a(x)), 0], [0, 0, 0, 0]]); REFERENCES. T. P. Chen and L. F. Li, Gaue Theory of Elementary Particle Physics, Clarendon Press, Oxford, 98.. R. Utiyama, Phys. Rev. 0, 597 (956). 3. D. W. Sciama, On the analoy between chare and spin in eneral relativity, in: Recent Developments in General Relativity, Peramon Press, Oxford 96.. D. W. Sciama, Rev. Mod. Phys. 36, 63 (96). 5. D. W. Sciama, Rev. Mod. Phys. 36, 03 (96). 6. T. W. B. Kibble, J. Math. Phys., (96). 7. M. Blaojević, Three lectures on Poincaré aue theory, arxiv:r qc/03000, Feb F. Gronwald, Metric-affine aue theory of ravity. I. Fundamental structure and field equations, Int. J. Modern, Physics D6, 63 (997). 9. G. Zet, V. Manta and S. Babeti, De-Sitter aue theory of ravitation, Int. J. Mod. Phys. C, (003). 0. N. Wu, Renormalizable Quantum Gaue General Relativity, arxiv:r qc/03090, September 8, G. Zet, I. Gottlieb and V. Manta, Nuovo Cimento B, 607 (996).

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