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On the teleparallel limit of Poincar´e gauge theory M. Leclerc Section of Astrophysics and Astronomy, Department of Physics, University of Athens, Greece (Dated: 11 January 2005) We will address the question of the consistency of teleparallel theories in presence of spinning matter which has been a controversial subject of discussion over the last twenty years. We argue thattheoriginoftheproblemisnotsimplythesymmetryorasymmetryofthestress-energytensor of the matter fields, which has been recently analyzed by several authors, but arises at a more fundamentallevel,namelyfromtheinvarianceofthefieldequationsunderaframechange,aproblem thathasbeendiscussedlongtimeagobyKopczynskiintheframeworkoftheteleparallelequivalent ofgeneralrelativity. Moreimportantly,weshowthattheproblemisnotonlyconfinedtothepurely teleparallel theory but arises actually in every Poincar´e gauge theory that admits a teleparallel geometry in the absence of spinning sources, i.e. in its classical limit. 5 0 I. INTRODUCTION tothemoregeneralcase(seeremarkattheendofsection 0 III). 2 Recently [1, 2] there has been a revival of the discus- The scope of this article is to show that the problem n siononwhetherornottheDiracfieldcanbeconsistently described in [1] and [2] is actually directly related to the a J coupled to gravity in the framework of the teleparallel frame invariance of the teleparallel Lagrangiananalyzed 1 equivalent of general relativity (TEGR). The authors of in[3]andthatitisnotconfinedtotheteleparallelequiv- 1 [2]cametotheconclusionthatthetheory,withtheusual alent of generalrelativity, but is present in any Poincar´e minimal coupling prescription (which we consider exclu- gauge theory that leads to a teleparallel geometry (with 2 sively in this paper), is not consistent. The reason for equations equivalent to those of general relativity) in its v thisissimplythefactthatthetheoryleadstoasymmet- classical limit, i.e. in the absence of spinning matter 9 ric Einstein equation and thus requires the right hand fields. 1 side of this equation, namely the stress-energy tensor of Inordertofixournotationsandconventions,webriefly 1 1 the Dirac particle, to be symmetric too. Clearly, the review the basic concepts of Riemann-Cartan geometry 1 stress-energy tensor of the Dirac particle, as well as of whichisthebasisofPoincar´egaugetheory. Foradetailed 4 any other particle with intrinsic spin (when minimally introduction, consult the standard reference [8]. Latin 0 coupled), is not symmetric by itself. In other words, re- lettersfromthebeginningofthealphabet(a,b,c...)run c/ quiringits symmetry isa constraintonthe fermionfield. from 0 to 3 and are (flat) tangent space indices. Espe- q Especially, the spin tensor would have to be conserved cially,ηab is the Minkowskimetric diag(1,−1,−1,−1)in - (covariantly),aconditionthatisnotevensatisfiedinthe tangent space. Latin letters from the middle of the al- r g absence of gravitational fields. phabet (i,j,k...) are indices in a curvedspacetime with : Onthe otherhand, the inconsistencyofTEGRhasal- metric gik. We introduce the independent gauge fields, iv ready been claimed twenty years ago in [3] (see also [4]- thetetradeam andtheconnectionΓabm (antisymmetricin X [7]), using a different argumentation. It has been noted ab) and the correspondent field strengths, the curvature r thatTEGRLagrangianpossessesasymmetrythatisnot and torsion tensors a inherit of the matter Lagrangian of a spinning particle. Rab = Γab −Γab +Γa Γcb −Γa Γcb (1) Namely, the Lagrangian and the field equations (in the lm m,l l,m cl m cm l absence of spinning matter) are invariant under what is Ta = ea −ea +eb Γa −ebΓa . (2) lm m,l l,m m bl l bm calledaframetransformation,i.e. aLorentztransforma- tionofthetetradfieldwiththeconnectionheldfixed(see The spacetime connection Γilm and the spacetime metric equation(7)below). Asaconsequenceofthis symmetry, gik can now be defined through the torsion tensor is not entirely determined by the field ea +Γa eb =eaΓi and eaebη =g . (3) equations. Since spinning matter fields do not present m,l bl m i ml i k ab ik thesameinvariance(inotherwords,theycoupledirectly Itisunderstoodthatthereexistsaninversetothetetrad, tothetorsion),theirbehavior,whentreatedastestfields, such that eaei = δa. It can now be shown that the i b b cannotbe predictedby the theory. Actually,the authors connection splits in two parts, of [3]-[7] do not confine their analysis to TEGR. Rather, theyconsidertheso-calledone-parameterteleparallelLa- Γab =Γˆab +Kab , (4) m m m grangian,whichleadstothemostgeneralteleparallelge- suchthat Γˆab is torsionfree andthe contortionKa is ometrythatisconsistentwiththeexperimentalsituation. m bm relatedtothetorsionthroughTa =Ka eb−Ka eb. Es- In this article, we confine ourselvesto those Lagrangians ik bi k bk i thatpresentaclassicallimitthatiscompletelyequivalent pecially, the spacetime connection Γˆi constructed from lm to general relativity. The discussion is easily generalized ea +Γˆa eb = eaΓˆi is just the Christoffel connection m,l bl m i ml of general relativity, a function of the metric only. 2 Allquantitiesconstructedwiththetorsionfreeconnec- We consider infinitesimal transformations Λa = δa + b b tion Γˆab or Γˆi will be denoted with a hat. Thus, for εa with εab = −εba. (As tangent space indices, a,b... m lm b instance, Rˆi is the usual Riemann curvature tensor. are and lowered with ηab.) lkm The gauge fields ea and Γab are vector fields with The Poincar´e transformation (5) now takes the form m m respect to the spacetime index m. Under a local gauge δΓab =−εab +εaΓcb +εb Γac , δea =εaec . (8) transformation in tangent space, Λa(xm), they trans- m ,m c m c m m c m b form as The inverse of the tetrad transforms with the inverse transformation, i.e. δem = ε cem. The matter action eam →Λabebm, Γabm →ΛacΛdbΓcdm−Λac,mΛbc. (5) Sm = Lmd4x thereforaeunderagoces the followingchange (up toRa boundary term): The transformation (5) is the basis of Poincar´e gauge theories. Under this transformation, the torsion and the δS = (δLmδem+ δLm δΓab )d4x curvature transform homogeneously. We will refer to it m Z δem a δΓab m a m asPoincar´egaugetransformation,althoughitisactually only the Lorentz part of a Poincar´etransformation after = Z e(2T[ab]+Dmσabm)εabd4x, having fixed the translationalpart to the so called phys- ical gauge. This conception of the Poincar´e transforma- whereD isthecovariantderivativethatactswithΓab m m tion is described in [9]. (For a fundamental treatment in on the tangent space indices and with Γˆi (torsion less) kl a more general framework, see [10].) Every Lagrangian, on the spacetime indices. We conclude that, if the mat- gravitationalor not, should be invariant under (5). ter Lagrangian possesses the symmetry (5), we have the In addition, one can consider the pure Lorentz gauge following (well known) conservation law transformations D σabm+2T[ab] =0. (9) m ea →ea , Γa →ΛaΛd Γc −Λa Λ c, (6) m m bm c b dm c,m b If the matter fields ψ too are subject to a gauge trans- as well as the frame transformations formation (for instance δψ =iεabσabψ in the Dirac case, withtheLorentzgeneratorsσ ),theactionundergoesan ab ea →Λaeb , Γa →Γa . (7) additional change δLmδψ, but this does not contribute, m b m bm bm δψ due to the field equations of the matter fields, which are Clearly, neither (6) nor (7) are symmetries of the Dirac derived from δLm =0. δψ Lagrangian (always speaking of the minimally coupled Clearly, the same argument if applied to the transfor- Lagrangian) nor of the Einstein-Cartan Lagrangian for mation (6) instead of (5) leads to D σabm = 0 and if m instance. Note also that the transformation (5), (6) and applied to the frame change (7) to T[ab] = 0. Since we (7) are not independent. Clearly, a Lorentz transforma- consideronlyLagrangiansthatpossessthePoincar´esym- tion(6)followedbyaframetransformation(7)(withthe metry,thesymmetry(7)willimplythesymmetry(6)and sameparameters)isequivalenttoaPoincar´etransforma- viceversa. Therefore,wecanstatethatiftheLagrangian tion (5). is frame invariant, then we have the conservation laws In the next section, we will investigate under which conditions the stress-energytensorof the matter fields is Dmσabm =0 and T[ab] =0. (10) symmetric. Then, in section III, we construct the family Until now, we have considered only the matter part of Lagrangians that present a teleparallel limit in the of the Lagrangian. Similar arguments can be applied spinlesscaseanddiscusstheproblemoftheinconsistency to the gravitationalLagrangianL itself, which depends of such theories in the presence of spinning particles in 0 only on ea ,Γab and their first derivatives. If we define relation with their invariance under a frame change (7). C m = −me−1δmL /δΓab and Ea = −(2e)−1δL /δem, ab 0 m m 0 a thegravitationalfieldequationsarisingfromL=L +L 0 m have the form II. FRAME INVARIANCE AND SYMMETRY OF THE STRESS-ENERGY TENSOR Ea =Ta , C m =σ m, (11) m m ab ab where as usual we refer to the first equation as Einstein We now deduce the conservationlaws that follow from equation and to the second one as Cartan equation. the symmetries (5), (6) and (7) of a general matter La- grangian density L , which may depend on ea ,Γab (as Using the same argumentation as before, we can show m m m that every Poincar´e invariant Lagrangian L will satisfy wellas ontheir derivatives)andonmatter fields that we o the Bianchi identity summarize under the notation ψ. Asusual,weusethecanonicaldefinitionsofthe stress- D Cabm+2E[ab] =0. (12) m energy tensor and of the spin density under the form IfL isinadditionframeinvariant,wehavetherelations 0 1 δL 1 δL Tam = 2e δemm, σabm = eδΓamb . DmCabm =0 and E[ab] =0. (13) a m 3 III. POINCARE´ GAUGE THEORY WITH withL from(14)andL somematterLagrangian. The 0 m TELEPARALLEL LIMIT field equations now read A major problem in Poincar´e gauge theory consists in Gˆik = τik+Tik, (16) reducing the 11 parameter Lagrangian (see [11] for in- D Rablm = σabl, (17) m stance) to those Lagrangians which are compatible with the classical experimental situation. Since our experi- with τik =−2a[RabliRablk−(1/4)RablmRablm]. We chose ments until today are confined to the metrical structure (15)asanillustrativeexamplebecauseofitssimplestruc- of spacetime, we can be sure to be in agreement with ture. Its field equations areexactly those ofan Einstein- theexperimentsifthemetricobeystheclassicalEinstein Yang-Mills system. Instead of Rab R ik we can take ik ab equations Gˆ = T . Therefore, we will look for La- any combination of quadratic curvature terms, because ik ik grangians whose Einstein equation Ea = Ta , in the inthefollowing,weareinterestedmainlyintheclassical, m m case of a vanishing spin density of the matter fields, re- teleparallel limit. duces to Gˆ = T . We know at least two such theo- Clearly, if the source is spinless, we get Rab = 0 as ik ik lm ries, namely general relativity (GR) itself (which can be ground state solution. With this solution, we have τ = ik seenastheclassicallimitofEinstein-Cartan(EC)theory, 0, and (16) reduces to the Einstein equation of GR. the zero spin condition leading to zero torsion) and the Wenowcometothediscussionofreferences[1]and[2]. teleparallel equivalent of GR (TEGR) where Rab =0. The main statement in [2] is the fact that TEGR is not lm One goal of Poincar´e gauge theory is to generalize the consistentwhencoupledtotheDiracparticlebecauseits above theories to allow for both dynamical torsion and Einsteinequationhasasymmetriclefthandsidebutthe curvature. This means that we have to include at least stress-energy tensor of the Dirac particle is asymmetric. onetermquadraticinthecurvatureintotheLagrangian. We agree completely with this view, but we will show Ifweseekforaclassicallimitwithzerotorsion,thisterm that the roots of the problem can be traced back to the willcertainlycontributetothe Einsteinequationevenin frame invariance not only of the field equations, but of theclassicallimit,exceptifitisofaveryspecial(andun- their classicallimit (i.e. evenin the absence of the Dirac natural)formlikeRi[klm]R orR[lm]R (here,[ikl] particle as source). Therefore, the discussion should not i[klm] [lm] means total anti-symmetrization of the three indices). be confined to the symmetry properties of T . ik Such terms actually depend only on torsion derivatives Indeed, the Lagrangian (15) is again frame invariant, andvanishinthe zerotorsionlimitvia the Bianchiiden- and thus equation (16) has the same symmetry problem tities in Riemannian space. as the corresponding one considered in [1] and [2]. How- On the other hand, if we are looking for a teleparal- ever, this problem can be cured very easily: We simply lel limit in the zero spin case, we can add all kinds of addatermbR2(withthecurvaturescalarR=eiekRab ) a b ik terms quadratic in the curvature, R Rik, R2..., with- to (15). This term is clearly not frame invariant (al- ik outchangingtheclassicallimitofthetheory. Suchterms thoughPoincar´einvariant)andthusbreakstheunwanted willleadonlytocontributionsthatvanishinthezerocur- symmetry. (Any other quadratic curvature term that is vature limit. These are the Lagrangians we investigate not frame invariant does the same job. Again, the term in this paper. R2 servesasillustrativeexample.) Especially,wewillget Apart from the quadratic curvature terms, we have to anadditionalasymmetriccontribution∼R(4R −g R) ik ik modify the TEGR Lagrangian such that it is suitable to (16), allowing therefore for an asymmetric T . Fur- ik for a first order variation without the use of Lagrange ther, we get a contribution to the Cartan equation (17) multipliers (see [12]). The suitable Lagrangian can be of the form ∼ D (ei ekR). Therefore, from the point of i [a b] found (in a more general framework) in [13]. It consists view of the discussion in [1] and [2], which focuses on of the sum of the teleparallel and the EC Lagrangian the symmetry properties of the Einstein equation, the (eL0 =L0), problem has been solved. 1 1 1 However,in the absence of spinning sources, we get as L0 =R− 4TiklTikl− 2TiklTlki+ 2TkikTmmi. (14) before the groundstate solutionRabik =0,andtherefore the(teleparallel)EinsteinequationGˆ =T . Notethat This Lagrangian, apart from a divergence term, is es- ik ik T is now supposed to be symmetric, since the source is sentially the Einstein-Hilbert Lagrangian (expressed in ik classical. Theseequationsareonceagainframeinvariant. terms ofthe tetrad)(see [13]or[14]). Itleads,inthe ab- What does that mean? Well, let us fix the Poincar´e sence of spinning matter, to the GR equation Gˆ =T ik ik gauge by imposing Γab = 0. Then, from the Ein- andthe Cartanequationisidenticallyfulfilled. (Inother m stein equation, we can determine the metric g . But words, δL /δΓab = 0.) This means that Γab remains ik 0 m m the tetrad field will be determined only up to a Lorentz completely undetermined. transformation ea → Λaeb . This is the problem that Note that (14) is frame invariant and consistently, the m b m has been discussed in [3] twenty years ago in the frame- Einstein tensor is symmetric. Let us now look at the work of the teleparallel equivalent of general relativity. Lagrangian For classical matter, this is not a problem, because it L=L +aRab R lm+L , (15) couples to the metric alone. Especially, the geodesics 0 lm ab m 4 of a classical test particle will not depend on the gauge firmed and analyzed in greater detail in the follow-up choice. However,spinningparticlescoupledirectlytothe articles [4]-[7]. In order to solve the problem completely, tetrad(ortothetorsion,whichisnotatensorunder(7)) the torsionhasto be fixed (determined) completely even andthe(semiclassical)trajectoryofatestparticleenter- in the classical limit (and especially in the ground state ingourfields, aswellasits spinprecessionequation,will of the theory). Therefore, if we want a Poincar´e theory depend on the specific frame we choose. We can there- tohaveageneralrelativitylimitinthe spinlesscase,this forenottakethe pointofview thatallthe solutionsthat limit cannot correspond to a teleparallel geometry, but differ only by a frame change are equivalent. should be described by a fixed torsion, most probably We can even reduce the whole discussion to the com- Ta =0, i.e. a Riemannian geometry. ik pletegroundstateofthefieldequations. Thegroundstate solution of the Einstein equation is g = η and that ik ik of the Cartan equation is Rab = 0. Without physi- lm cal consequences, we can fix the Poincar´e gauge by the IV. CONCLUSION requirement Γab = 0. Obviously, this state is invari- m ant under (7). We can therefore determine neither the As a result, we conclude that the teleparallel equiva- tetrad, nor the torsion(which is not a tensor under (7)). lent of general relativity is not consistent in presence of These fields however are measurable since they couple minimally coupled spinning matter. We showedthat the to spinning particles. Clearly, this problem arises in any argument given in [2], i.e. that the Einstein equation theory whose field equations reduce in the classical limit to Rab =0 and Gˆ =T . hasa symmetricleft handside whereasthe stress-energy lm ik ik tensorofspinningmatterfieldisnotsymmetric,actually Finally, there is the possibility of adding the term has its roots in the frame invariance of the teleparallel λT[ikl]T (the square of the totally antisymmetric tor- [ikl] Lagrangiandiscussed in [3]. sion part) to the Lagrangian. This changes the classical limitslightlybutinawayconsistentwiththeexperimen- Furthermore,wecouldshowthateveryPoincar´egauge tal situation, for an arbitrary constant λ. (The so-called theory that leads, in the absence of spinning matter, to one-parameter teleparallel theory, see [15]). This breaks a teleparallel geometry with an Einstein equation equiv- the frame invariance of the classical limit (and even of alent to GR suffers from the same inconsistency. Even the groundstate),but it has beenshownin [3] that there if the Lagrangian itself is not frame invariant, the field is a remaining invariance of the form ea → Λaeb with equations in their classical limit will be frame invariant m b m Λa aspecialLorentztransformationthatleavestheaxial again. A spinning test particle entering these fields how- b torsion part unaffected. Therefore, taking into account ever will couple directly to the torsion (which is not a this new termwouldsolvethe problemforthe Diractest tensor under the frame change), and its behavior (spin particle, which couples only to the axial torsion, but if precession, trajectory...) will depend on the arbitrary we consider higher spin fields or macroscopic spin po- choice of a specific frame. larized bodies, the problem reappears, since the latter The problem with such theories has also been ana- couplealsototheothertorsionparts(vectorandtensor) lyzedin[17], basedona completelydifferentargumenta- which remain undetermined (see [16] for semi-classical tion (3+1-decomposition). The conclusions are similar, equations of momentum propagation and of precession however, our argumentation is much simpler and shows for general spinning test bodies). The complete discus- clearly which class of theories suffers from the inconsis- sion, in the framework of the purely teleparallel theory, tency andwhy there isa relationtothe symmetryofthe can be found in [3]. The results of [3] have been con- Einstein equation. [1] E. W. Mielke, Phys. Rev.D 69, 128501 (2004) 044004 (2000) [2] Y. N. Obukhov and J.G. Pereira, Phys. Rev. D 69, [11] M.S. Gladchenkoand V.V.Zhytnikov,Phys.Rev.D50, 128502 (2004) 5060 (1994) [3] W. Kopczynski,J. Phys. A 15, 493 (1982) [12] Y.N.ObukhovandJ.G.Pereira,Phys.Rev.D67,044016 [4] J.M. Nester, Class. QuantumGrav. 5, 1003 (1988) (2003) [5] F.M.Mu¨ller-HoissenandJ.Nitsch,Phys.Rev.D28,718 [13] Y.N. Obukhov, E.J. Vlachynsky, W. Esser and F.W. (1983) Hehl, Phys. Rev.D 56, 7769 (1997) [6] W.H.Cheng,D.C.ChernandJ.M.Nester,Phys.Rev.D [14] M.O. Katanaev, Gen. Rel. Grav. 25, 349 (1993) 38, 2656 (1988) [15] K. Hayashi and T. Shirafuji, Phys. Rev. D 19, 3524 [7] M.Blagojevi´candI.A.Nikoli´c,Phys.Rev.D62,024021 (1979) (2000) [16] K. Hayashi, K. Nomura and T. Shirafuji, Prog. Theor. [8] F.W. Hehlet al., Rev.Mod. Phys. 48, 393 (1976) Phys 86, 1239 (1991) [9] G. Grignani and G. Nardelli, Phys. Rev. D 45, 2719 [17] R.D. Hecht, J. Lemke and R.P. Wallner, Phys. Rev. D (1992) 44, 2442, (1991) [10] R. Tresguerres and E.W. Mielke, Phys. Rev. D 62,

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