GAUGE THEORY ON A SPACE-TIME WITH TORSION
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1 GAUGE THEORY ON A SPACE-TIME WITH TORSION C. D. OPRISAN, G. ZET Fculty of Physics, Al. I. Cuz University, Isi, Romni Deprtment of Physics, Gh. Aschi Technicl University, Isi , Romni Received September 6, 005 A guge theory of grvittion bsed on teleprllel grvity (TG) theory is formulted in spce-time with torsion only. The strting point is the Poincré guge theory (PGT). The generl structure of TG nd its connection with generl reltivity (GR) re presented. The field equtions for the grvittionl potentils re written considering the energy-momentum tensor of the electromgnetic field. The Schwrzschild-Reisner-Nordström type solution is obtined by solving the field equtions of TG. Some concluding remrks on the source of torsion re lso given.. INTRODUCTION There re different wys to introduce the grvittionl interction in field theoreticl model. Thus, in generl reltivity (GR), the curvture of the spce-time is used to describe the grvittion. The geometry of the spce-time replces the concept of force. On the other hnd, the teleprllel grvity (TG) ttributes grvittion to torsion [, ] nd it is guge theory for the group of spce-time trnsltions. The grvittionl interction is described in TG by forces, similr to the Lorentz forces in electrodynmics. Therefore, the grvittionl interctions cn be described lterntively in terms of curvture s is usully done in GR, or in terms of torsion s in TG. It is believed tht requesting curved or torsioned spce-time to describe grvity is mtter of convention [3]. In this pper we obtin solution of Reissner-Nordström (RN) type in the cse of TG. The generl structure of TG nd its connection with GR re presented nd the RN solution is obtined by solving the field equtions. These equtions contin the energy-momentum tensor of the electromgnetic field. Section contins review of Poincré guge theory (PGT) on Minkowski spce-time s bse mnifold. In Section 3 non-symmetric connection is constructed strting with guge fields of PGT. The energy-momentum of the electromgnetic field is written in generl form nd it is prticulrized for the Pper presented t the Ntionl Conference of Physics, 3 7 September, 005, Buchrest, Romni. Rom. Journ. Phys., Vol. 5, Nos. 5 6, P , Buchrest, 006
2 53 C. D. Oprisn, G. Zet cse of point-like chrge. The field equtions of TG re then obtined in the Section 4 under generl form. The RN solution of these equtions is given in Section 5. A comprison with GR cse is lso presented. The conclusions nd some remrks re presented in Section 6... POINCARÉ GAUGE THEORY We will denote the genertors of Poincré group P by { P, M b}, b, = 03.,,, Here, P re the genertors of spce-time trnsltions nd Mb = Mb re the genertors of the Lorentz rottions. Suppose now tht P is guge group for grvittion [4]. Correspondingly, we introduce the -form potentil A with vlues in Lie lgebr of the Poincré group, defined by formul: A= ep + ωb M b (.) e = e dx nd ω b =ω b dx re ordinry -forms. The -form defines connection on the spce-time M 4 of our guge model with e nd ω b s guge fields. The -form of curvture F is given by the expression F = da+ [ A, A ] (.) Inserting (.) in (.) nd identifying the result with the definition we obtin the -forms of torsion nd respectively b F = T P + R M b (.3) T nd of curvture b b b R in the form T = de +ω e, (.4) c b b c b R = d ω +ω ω. (.5) We use the Minkowski metric η = (,,, ) b dig on the Poincré group mnifold to rise nd lower the Ltin indices, b, c. Written on the components, the equtions (.4) nd (.5) give: nd respectively b c b c ν ν ν ν ν T = e e + ω e ω e η, (.6) bc b b b c db c db ν ν ν ν ν R = ω ω + ω ω ω ω η. (.7) cd
3 3 Guge theory on spce-time with torsion 533 The quntity T ν is the torsion tensor nd the quntity R b ν is the curvture tensor of the connection A defined by the eqution (). The connection A defines structure Einstein-Crtn (EC) on the spce-time nd we will denote the corresponding spce by U 4. This spce hve both torsion nd curvture. In the next Section, we will consider the cse of spce-time with non-null torsion nd vnishing curvture to construct the teleprllel theory of grvity (TG).. TELEPARALLEL GRAVITY We interpret e ν s tetrd fields nd ω b ν s spin connection. A model of guge theory bsed on Poincré group nd implying only torsion cn be obtined choosing ω b ν =0. Then the curvture tensor R b ν vnishes nd the torsion in eqution (.6) becomes: T = e e (.) ν ν ν Expressed in coordinte bsis, this tensor hs the components: e is the inverse of. ν ν ν T = e e e e, (.) e Now, we define the Crtn connection Γ on the spce-time M 4 with nonsymmetric coefficients ν ν Γ = e e. (.3) This definition is suggested by the expression (.) of the torsion components. Therefore, the connection Γ hs the torsion given by the usully formul ν ν ν T =Γ Γ. (.4) With respect to the connection Γ, the tetrd field is prllel, tht is: ν ν e = e Γ e = 0. (.5) Curvture nd torsion hve to be considered s properties of the connections nd therefore mny different connections re llowed on the sme spce-time [5]. For exmple, strting with the tetrd field e we cn define the Riemnnin metric: ν b ν g =η e e. (.6)
4 534 C. D. Oprisn, G. Zet 4 Then, we cn introduce the Levi-Civit connection ν ν ν ν Γ = +. g g g g (.7) This connection is metric preserving: ν ν ν g = g+γ g +Γ g = 0. (.8) The reltion between the two connections Γ nd Γ ν ν ν Γ =Γ + K, (.9) K ν= T ν+ Tν T ν (.0) is the contortion tensor. The curvture tensor of the Levi-Civit Connection Γ is: τ ν ν τ ν ν R = Γ +Γ Γ ν (.) Becuse the connection coefficients Γ re symmetric in the indices nd ν, its torsion is vnishing. Therefore, the Levi-Civit connection Γ hve non-null curvture, but no torsion. Contrrily, the Crtn connection Γ presents torsion, but no curvture. Indeed, using the definition (.3), we cn verify tht the curvture of the connection Γ vnishes identiclly: τ ν ν τ ν R = Γ +Γ Γ ν 0. (.) Then, substituting (.9) into the expression (.), we obtin: is ν ν ν R = R + Q 0, (.3) τ ν ν τν Q = D K +Γ K ν (.4) is the non-metricity tensor. Here τ τ ν= ν+γ τ ν Γ τν DK K K K (.5) is the teleprllel covrint derivtive.
5 5 Guge theory on spce-time with torsion 535 ν The eqution (.3) hs n interesting interprettion [6]: the contribution R coming from the Levi-Civit connection Γ compenstes exctly the contribution Q ν coming from the Crtn connection Γ, yielding n identiclly zero Crtn curvture tensor R ν. Now, ccording to GR theory, the dynmics of the grvittionl field is determined by the Lgrngin [7]: ν ν L GR 4 gc = R, 6πG (.6) R= g R is the sclr curvture of the Levi-Civit connection Γ, g= det g. Then, substituting R s obtined from (.3), one obtins up to divergences [6]: G is the grvittionl constnt nd ( ν ) e= det( e ) =, 4 L = ec ν TG S T ν, 6πG (.7) g nd ( ) ν ν ν ν ν S = S = K g T + g T (.8) is tensor written in terms of the Crtn connection only. The eqution (.7) gives the Lgrngin of the TG s guge theory of grvittion for the trnsltion group. It is proven [8] tht the trnsltionl guge theory of grvittion TG with the Lgrngin L TG qudrtic in torsion is completely equivlent to generl reltivity GR with usul Lgrngin L GR liner in the sclr curvture. Therefore, the grvittion presents two equivlent descriptions: one GR in terms of metric geometry nd nother one TG in which the underlying geometry is provided by teleprllel structure. In the next Section we will obtin the field equtions of grvittion within TG theory. 3. FIELD EQUATIONS Tking the vrition of the Lgrngin L TG in Eq. (.7) with respect to the guge field e, one obtins the teleprllel version of the grvittionl field equtions:
6 536 C. D. Oprisn, G. Zet 6 ν ν = S e S nd ( es ν 4πG ) ( ej ) ν = 0, c4 S ν g Sτν T ν Tν T ν T ν νt = τ = + δ δ. 4 (3.) The quntity j in Eq.(3.) is the guge grvittionl current, defined nlogous to the Yng-Mills theory: L j e S T e L = 4 TG = c ν ν+ TG e e 4πG e. (3.) The current j represents the energy-momentum of the grvittionl field. The term es is clled superpotentil in the sense tht its derivtive yields the guge current ej. Due to the nti-symmetry of S in the indices nd the quntity ej is conserved s consequence of the field equtions, i.e. ej = 0. (3.3) Mking use of Eq. (.3) to express e, the field equtions (3.) cn be written in purely spce-time form: ( es ) 4πG ( t 4 e c ) = 0, (3.4) t is the cnonicl energy-momentum pseudo-tensor of the grvittionl field [5], defined by the expression: c4 ν ν TG 4πG e t = Γ S + δ L. (3.5) It is importnt to notice tht the cnonicl energy-momentum pseudo-tensor t is not simply the guge current j with the Lorentz index chnged to the spce-time index. It incorportes lso n extr term coming from the derivtive term of Eq. (3.) 4 c ν ν t = e j + Γ S. 4πG (3.6) Like the guge current, consequence of the field eqution: ej the pseudo-tensor et is conserved s
7 7 Guge theory on spce-time with torsion 537 But, due to the pseudo-tensor chrcter of et = 0. (3.7), t this conservtion lw cn not be expressed with covrint derivtive, in contrst with j cse. Using the previous results, we will prove in the next Section tht the RN solution cn be obtined from the field equtions (3.4) of the teleprllel theory of grvity with the energy-momentum tensor on the right-hnd side, i.e. es k t = k T, (3.8) e we denoted k = 4πG for simplicity. The energy-momentum tensor will c4 correspond to the electromgnetic field creted by point-like electricl chrge. 3.. REISSNER-NORDSTRÖM SOLUTION Becuse we re looking for sphericlly symmetric solution of the field equtions, we will choose the Minkowski metric ( sin ) ds = c dt dr r dθ + θd ϕ (4.) on the spce-time mnifold. The coordintes x 0, x, x, x 3 correspond to ct, r,θ,ϕ respectively. Then we will consider the guge theory bsed on Poincré group with ω b ν = 0 described in Section 3. The tetrd field e will be chosen under the form [9]: ( 000) ( 0 00) 0 e = A,,,, e =,,,, A ( 00 0) ( 000 sin ) 3 (4.) e =,, r,, e =,,, r θ. (4.b) A is n unknown function depending only of the 3D rdius r. The metric g ν =η eb ν will hve then the following mtrix form b e ( gν ) A = A, 0 0 r 0 θ r sin (4.3)
8 538 C. D. Oprisn, G. Zet 8 with its inverse ( gν ) A 0 A 0 0 = r r sinθ (4.4) We use the bove expressions to compute the coefficients Γ ν of the Crtn connection, the components T ν of the torsion tensor, of the tensor ν S nd of the cnonicl energy-momentum pseudo-tensor components of the tensor T ν re: ctgθ T = r A 0, T = T3 = A, T3 =, A r r. t The non-null (4.5) A = da/ dr denotes the derivtive of the function Ar with respect to the vrible r. We list lso the non-null components of the tensor cnonicl energy-momentum pseudo-tensor for S we hve: S nd of the t of the grvittionl field. Thus, ( + ) 0 θ = 0 ctg 3 ArA A S A 0, S0 = S =, S = S3 =, r r r (4.6) nd for, t 0 3 A ra + A t0 = t = t = t3 =, kr (4.7) ArA + Actgθ =. kr t (4.7b) The energy-momentum tensor of the electromgnetic field creted by point-like electric chrge Q hs the following mixed components written in mtrix form:
9 9 Guge theory on spce-time with torsion 539 T Q kr 4 Q kr 4 =. Q kr 4 Q kr (4.8) Now, using these components we obtin from (3.8) the following equtions of grvittionl field in TG theory: + Q raa A =, r (4.9) + Q AA ra + raa =. r3 (4.9b) A = da/ dr is the second derivtive of Ar with respect to r. It is esy to verify tht the eqution (4.9b) cn be obtined by the derivtion of (4.9) with respect to r. Therefore, we hve only one independent field eqution (4.9) for single unknown function Ar. The solution of the eqution (4.9) is A Q = + +, r r ( r) α (4.0) α is n rbitrry constnt of integrtion. If we choose this constnt so tht α= m, then the solution (4.0) correspond to the RN metric. Indeed, using the metric components from the eqution (4.3) nd the form (4.0) of the solution, we cn write the corresponding line element s (with c = units) = Q ds m + dt dr r ( dθ + sinθdϕ ). r r m Q + r r (4.) It corresponds to the grvittionl field creted by point-like mss m hving lso n electricl chrge Q. The components of the metric g ν cn be used therefore s guge invrint vribles of our model.
10 540 C. D. Oprisn, G. Zet 0 REFERENCES. M. Clcd nd J. G. Pereir, Int. J. Theor. Phys. 4, p. 79, 00.. V. C. Andrde nd J. G. Pereir, Torsion nd electromgnetic field, rxiv:qr-qc/970805, v, Jn S. Cpozziello nd G. Lmbise Storniolo, Ann. Phys. (Leipzig) 0, p. 8, G. Zet nd V. Mnt, Int. J. Modern Phys. C3, p. 509, M. Blgojevic, Three lectures on Poincré guge theory, rxiv:qr-qc/030040, v, Feb V. C. Andrde, L. C. T. Guillen nd J. G. Pereir, Teleprllel grvity: n overview, rxiv:qr-qc/ 00087, V. C. Andrde nd J. G. Pereir, Phys. Rev. D56, p. 4689, V. C. Andrde, L. C. T. Guillen nd J. G. Pereir, Teleprllel spin connection, rxiv:gr-qc/0040, 30 Apr G. Zet, Schwrzschild solution on spce-time with torsion, rxiv:gr-qc/ v, 4 Aug L. Lndu nd E. Lifchitz, Théorie du chmp, Ed. Mir, Moscou, G. Zet, Unified theory of fundmentl interctions in spce-time with torsion (in preprtion).
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