Clifford Valued Differential Forms, and Some Issues in Gravitation, Electromagnetism and 8 ”Unified” Theories∗ 0 0 2 W. A. Rodrigues Jr. and E. Capelas de Oliveira n a Institute of Mathematics, Statistics and Scientific Computation J IMECC-UNICAMP CP 6065 6 13083-970 Campinas-SP, Brazil 1 [email protected] [email protected] 8 v April 18 2004 5 updated: 16 January 2007 2 0 7 0 4 0 Contents / h p 1 Introduction 2 - h 2 Recall of Some Facts of the Theory of Linear Connections 4 t a 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 m 2.2 Exterior Covariant Differential . . . . . . . . . . . . . . . . . . . 5 : 2.3 Absolute Differential . . . . . . . . . . . . . . . . . . . . . . . . . 7 v i X 3 Clifford Valued Differential Forms 8 r 3.1 Exterior Covariant Differential of Clifford Valued Forms . . . . . 11 a 3.2 Extended CovariantDerivative of Clifford Valued Forms . . . . . 11 3.2.1 Case p=1 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.2 Case p=2 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Cartan Exterior Differential . . . . . . . . . . . . . . . . . . . . . 12 3.4 Torsion and Curvature . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 Some Useful Formulas . . . . . . . . . . . . . . . . . . . . . . . . 15 4 General Relativity as a Sl(2,C) Gauge Theory 16 4.1 The Nonhomogeneous Field Equations . . . . . . . . . . . . . . . 16 ∗Published: Int. J. Mod. Phys. D 13(9), 1879-1915 (2004). This version includes cor- rections of misprints and of some formulas appearing in the original text (mainly is Section 3.5). 1 5 Another Set of Maxwell-Like Nonhomogeneous Equations for Einstein Theory 17 5.1 Sl(2,C) Gauge Theory and Sachs Antisymmetric Equation . . . 19 6 Energy-Momentum “Conservation” in General Relativity 22 6.1 Einstein’s Equations in terms of Superpotentials ⋆Sa . . . . . . . 22 6.2 Is There Any Energy-Momentum Conservation Law in GR? . . . 24 7 Conclusions 27 A Clifford Bundles Cℓ(T∗M) and Cℓ(TM) 28 A.1 Clifford product, scalar contraction and exterior products . . . . 29 A.2 Some useful formulas . . . . . . . . . . . . . . . . . . . . . . . . . 30 A.3 Hodge star operator . . . . . . . . . . . . . . . . . . . . . . . . . 31 A.4 Action of D on Sections of Cℓ(TM) and Cℓ(T∗M) . . . . . . . . 31 ea A.5 Dirac Operator, Differential and Codifferential . . . . . . . . . . 32 A.6 Maxwell Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 33 B Einstein Field Equations for the Tetrad Fields θa 33 Abstract In this paper we show how to describe the general theory of a linear metric compatible connection with thetheory of Clifford valued differen- tialforms. ThisisdonebyrealizingthatforeachspacetimepointtheLie algebraofCliffordbivectorsisisomorphictotheLiealgebraofSl(2,C). In thatwaythepullbackofthelinearconnectionunderalocaltrivialization of the bundle (i.e., a choice of gauge) is represented by a Clifford valued 1-form. That observation makes it possible to realize immediately that Einstein’sgravitationaltheorycanbeformulatedinawaywhichissimilar to a Sl(2,C) gauge theory. Such a theory is compared with other inter- esting mathematical formulations of Einstein’s theory. and particularly with a supposedly ”unified” field theory of gravitation and electromag- netismproposedbyM.Sachs. WeshowthathisidentificationofMaxwell equations within his formalism is not a valid one. Also, taking profit of the mathematical methods introduced in thepaper we investigate a very polemical issue in Einstein gravitational theory, namely the problem of the ’energy-momentum’ conservation. We show that many statements appearing in the literature are confusing or even wrong. 1 Introduction In this paper we introduce the concept of Clifford valued differential forms1, mathematical entities which are sections of Cℓ(TM)⊗ T∗M. We show how ^ 1Analogous, but non equivalent concepts have been introduced in [13, 64, 66, 63]. In particular [13] introduce clifforms, i.e., forms with values in a abstract (internal) Clifford algebra Rp,q associated with a pair (Rn,g), where n = p+q and g is a bilinear form of signature(p,q)inRn. Theseobjectsdiffer fromtheCliffordvalueddifferentialformsusedin thistext., whithdispensesanyabstract(internal)space. 2 with the aid of this concept we can produce a very beautiful description of the theory of linear connections, where the representative of a given linear connec- tion in a given gauge is given by a bivector valued 1-form. For that objects we introduce the concept of exterior covariant differential and extended covariant derivative operators. Our natural definitions2 are to be compared with other approaches on related subjects (as described, e.g., in [2, 3, 26, 28, 44, 46, 59]) and have been designed in order to parallel in a noticeable way the formalism of the theory of connections in principal bundles and their associatedcovariant derivative operators acting on associated vector bundles. We identify Cartan curvature 2-forms and curvature bivectors. The curvature 2-forms satisfy Car- tan’s second structure equation and the curvature bivectors satisfy equations in complete analogy with equations of gauge theories. This immediately sug- gests to write Einstein’s theory in that formalism, something that has already been done and extensively studied in the past (see e.g., [6, 8]). Our method- ology however, suggests new ways of taking advantage of such a formulation, but this is postponed for a later paper. Here, our investigation of the Sl(2,C) nonhomogeneous gauge equation for the curvature bivector is restricted to the relationshipbetweenthatequationandsomeotherequationswhichusedifferent formalisms,butwhichexpressthesameinformationasthe onecontainedinthe originalEinstein’s equationswritten in classicaltensorformalism. Ouranalysis includes a careful investigation of the relationship of the Sl(2,C) nonhomoge- neousgaugeequationfor the curvature bivectorandsomeinteresting equations appearing in M. Sachs theory [51, 52, 53]. We already showed in [45] that unfortunately M. Sachs identified equivo- cally his basic variables q as being quaternion fields over a Lorentzian space- µ time. Well, they are not. The real mathematical structure of these objects is that they are matrix representations of particular sections of the even Clif- ford bundle of multivectors Cℓ(TM) (called paravector fields in mathematical literature). Here we show that the identification proposed M. Sachs of a new antisymmetricfield[51,52,53]inhis‘unified’theoryasanelectromagneticfield is,accordingtoouropinion,anequivocatedone. Indeed,aswillbeprovedinde- tail, M. Sachs ‘electromagnetic fields’ F (whose precise mathematical nature ab is disclosed below) are nothing more than some combinations of the curvature bivectors3, objects that appear naturally when we try to formulate Einstein’s gravitational theory as a Sl(2,C) gauge theory. The equations found by M. Sachs, which are satisfied by his antisymmetric fields F looks like Maxwell ab equations in components, but they are not Maxwell equations. However, we cansaythattheydorevealonemoreofthemanyfacesofEinstein’sequations4. Taking profitofthe mathematicalmethods introducedin thispaper we also discuss some controversial, but conceptually important issues concerning the lawofenergy-momentumconservationinGeneralRelativity,showingthatmany 2Ourdefintions,forthebestofourknowledge, appearshereforthefirsttime. 3ThecurvaturebivectorsarephysicallyandmathematicallyequivalenttotheCartancur- vature 2-forms, since they carry the same information. This statement will become obvious fromourstudyinsection 3.4. 4Someotherfacesofthatequations areshownintheAppendices. 3 statements appearing in the literature are confusing and even wrong. The paper contains two Appendices. Appendix A recalls some results of the Clifford calculus necessary for the calculations presented in the main text. Appendix B recalls the correct intrinsic presentation of Einstein’s equations in terms of tetrad fields {θa}, when these fields are sections of the Clifford bundleandcomparethatequations,whichsomeotherequationsforthatobjects recently presented in the literature. 2 Recall of Some Facts of the Theory of Linear Connections 2.1 Preliminaries In the general theory of connections [9, 31] a connection is a 1-form in the cotangentspace of a principalbundle, with values in the Lie algebra of a gauge group. In order to develop a theory of a linear connection5 N ω ∈secT∗PSOe (M)⊗sl(2,C), (1) 1,3 with an exterior covariant derivative operator acting on sections of associated vector bundles to the principal bundle PSOe (M) which reproduces moreover 1,3 the well known results obtained with the usual covariant derivative of tensor fields in the base manifold, we need to introduce the concept of a soldering form N θ ∈secT∗PSOe (M)⊗R1,3. (2) 1,3 Let be U ⊂ M and let ς : U → ς(U) ⊂ PSOe (M).We are interested in the 1,3 N N pullbacks ς∗ω and ς∗θ once we give a local trivialization of the respective bun- N dles. As it is well known [9, 31], in a localchart hxµi coveringU, ς∗ θ uniquely determines 1 θ =e ⊗dxµ ≡e dxµ ∈secTM ⊗ T∗M. (3) µ µ ^ Now, we givethe Cliffordalgebrastructure to the tangent bundle, thus gen- erating the Clifford bundle Cℓ(TM) = Cℓ (M), with Cℓ (M) ≃ R intro- x x 1,3 x duced in Appendix A. [ We recallmoreover,a well knownresult [34], namely, that for each x∈U ⊂ M the bivectors of Cℓ(T M) generate under the product defined by the com- x mutator,the Lie algebrasl(2,C). We thusareleadto definethe representatives in Cℓ(TM)⊗ T∗M for θ and for the the pullback ω of the connection in a ^ 5In words, ωN is a 1-form inthe cotangent space of the bundle of ortonornal frames with valuesintheLiealgebrasoe1,3≃sl(2,C)ofthegroupSOe1,3. 4 given gauge (that we represent with the same symbols): 1 1 1 θ =e dxµ =e θa ∈sec TM ⊗ T∗M ֒→Cℓ(TM)⊗ T∗M, µ a ^ ^ ^ 1 ω = ωbce e θa 2 a b c 1 2 1 1 = ωbc(e ∧e )⊗θa ∈sec TM ⊗ T∗M ֒→Cℓ(TM)⊗ T∗M. (4) 2 a b c ^ ^ ^ Before we continue we must recall that whereas θ is a true tensor, ω is not a true tensor, since as it is well known, its ‘components’ do not have the tensor transformation properties. Note that the ωbc are the ‘components’ of a the connection defined by D eb =−ωb ec, ω =−ω =η ωd , (5) ea ac abc cba ad bc where D is a metric compatible covariantderivative operator6 defined on the ea tensor bundle, that naturally acts on Cℓ(TM) (see, e.g., [11]). Objects like θ andω will be called Cliffordvalued differential forms (or Clifford valued forms, forshort),andinsections3and4 wegiveadetailedaccountofthe algebraand calculusofthatobjects. But,beforewestartthisprojectweneedtorecallsome concepts of the theory of linear connections. 2.2 Exterior Covariant Differential Oneofourobjectivesistoshowhowtodescribe,withourformalismanexterior covariant differential (EXCD) which acts naturally on sections of Clifford val- ueddifferentialforms(i.e.,sectionsofsecCℓ(TM)⊗ T∗M )andwhichmimics theactionofthepullbackoftheexteriorcovariantd^erivativeoperatoractingon sectionsofavectorbundleassociatedtotheprincipalbundlePSOe (M),oncea 1,3 linear metric compatible connection is given. Our motivation for the definition of the EXCD is that with it, the calculations of curvature bivectors, Bianchi identities, etc., use always the same formula. Of course, we compare our defi- nition, with other definitions of analogous, but distinct concepts, already used in the literature, showing where they differ from ours, and why we think that ours seems more appropriate. In particular, with the EXCD and its associated extended covariantderivative (ECD) we canwrite Einstein’s equations in such a way that the resulting equation looks like an equation for a gauge theory of the group Sl(2,C). To achieve our goal, we recall below the well known defini- tion of the exterior covariant differential dE acting on arbitrary sections of a vector bundle E(M) (associated to PSOe (M) and having as typical fiber a l- 1,3 dimensionalrealvectorspace)andonendE(M)=E(M)⊗E∗(M),the bundle of endomorphisms of E(M). We recall also the concept of absolute differential acting on sections of the tensor bundle, for the particular case of lTM. V 6Aftersection3.4,Dea referstotheLevi-Civitacovariantderivativeoperator. 5 Definition 1 The exterior covariant differential operator (ECDO) dE acting on sections of E(M) and endE(M) is the mapping 1 dE:secE(M)→secE(M)⊗ T∗M, (6) ^ such that for any differentiable function f : M → R, A ∈ secE(M) and any p q F ∈sec(endE(M)⊗ T∗M), G∈sec(endE(M)⊗ T∗M) we have: ^ ^ dE(fA)=df ⊗A+fdEA, dE(F ⊗ A)=dEF ⊗ A+(−1)pF ⊗ dEA, ∧ ∧ ∧ dE(F ⊗ G)=dEF ⊗ G+(−1)pF ⊗ dEG. (7) ∧ ∧ ∧ InEq.(7),writingF =Fa⊗f(p),G=Gb⊗g(q)whereFa,Gb ∈sec(endE(M)), a b f(p) ∈sec pT∗M and g(q) ∈sec qT∗M we have a b ^ ^ F ⊗ A= Fa⊗f(p) ⊗ A, ∧ a ∧ (cid:16) (cid:17) F ⊗ G= Fa⊗f(p) ⊗ Gb⊗g(q). (8) ∧ a ∧ b (cid:16) (cid:17) In what follows, in order to simplify the notation we eventually use when there is no possibility of confusion, the simplified (sloppy) notation (FaA)⊗f(p) ≡(FaA)f(p), a a Fa⊗f(p) ⊗ Gb⊗g(q) = FaGb f(p)∧g(q), (9) a ∧ b a b (cid:16) (cid:17) (cid:0) (cid:1) where FaA ∈ secE(M) and FaGbmeans the composition of the respective endomorphisms. LetU ⊂M beanopensubsetofM,hxµiacoordinatefunctionsofamaximal atlasofM,{e }acoordinatebasisofTU ⊂TM and{s },K=1,2,...l abasis µ K 1 foranysecE(U)⊂secE(M). Then,abasisforanysectionofE(M)⊗ T∗M is given by {s ⊗dxµ}. ^ K Definition 2 The covariant derivative operator D : secE(M) → secE(M) eµ is given by . dEA= D A ⊗dxµ, (10) eµ where, writing A=AK⊗s we hav(cid:0)e (cid:1) K D A=∂ AK⊗s +AK⊗D s . (11) eµ µ K eµ K 1 Now, let examine the case where E(M) = TM ≡ (TM) ֒→ Cℓ(TM). Let {e }, be an orthonormalbasis of TM. Then,using E^q.(11) and Eq.(5) j dEe =(D e )⊗θk ≡e ⊗ωk j ek j k j ωk =ωkθr, (12) j rj 6 1 where the ωk ∈sec T∗M are the so-called connection 1-forms. j Also, for v=vie^∈secTM, we have i dEv=D v⊗θi =e ⊗dEvi, ei i dEvi =dvi+ωivk. (13) k 2.3 Absolute Differential l Now, let E(M)=TM ≡ (TM)֒→Cℓ(TM).Recallthattheusualabsolute ^l differential D of A∈sec TM ֒→secCℓ(TM) is a mapping (see, e.g., [9]) ^ l l 1 D:sec TM →sec TM ⊗ T∗M, (14) ^ ^ ^ l such that for any differentiable A∈sec TM we have ^ DA=(D A)⊗θi, (15) ei l whereD AisthestandardcovariantderivativeofA∈sec TM ֒→secCℓ(TM). ei ^ l Also,foranydifferentiablefunctionf :M →R,anddifferentiableA∈sec TM we have ^ D(fA)=df ⊗A+fDA. (16) Now,ifwesupposethattheorthonormalbasis{e }ofTM issuchthateach j 1 e ∈ sec TM ֒→ secCℓ(TM), we can find easily using the Clifford algebra j structure^of the space of multivectors that Eq.(12) can be written as: 1 De =(D e )θk = [ω,e ]=−e yω j ek j 2 j j 1 ω = ωabe ∧e ⊗θk 2 k a b 1 2 1 1 ≡ ωabe e ⊗θk ∈sec TM ⊗ T∗M ֒→secCℓ(TM)⊗ T∗M, 2 k a b ^ ^ ^ (17) where ω is the representative of the connection in a given gauge. The general case is given by the following proposition. l Proposition 3 For A∈sec TM ֒→secCℓ(TM) we have ^ 1 DA=dA+ [ω,A]. (18) 2 7 Proof. The proof is a simple calculation, left to the reader. Eq.(18) can now be extended by linearity for an arbitrary nonhomogeneous multivector A∈secCℓ(TM). l Remark 4 We see that when E(M) = TM ֒→ secCℓ(TM) the absolute differential D can be identified with the ex^terior covariant derivative dE. We proceed now to find anappropriate exterior covariantdifferentialwhich actsnaturallyonCliffordvalueddifferentialforms,i.e.,objectsthataresections of Cℓ(TM)⊗ T∗M (≡ T∗M ⊗Cℓ(TM)) (see next section). Note that we cannot simp^ly use the ^above definition by using E(M) = Cℓ(TM) and endE(M) = endCℓ(TM), because endCℓ(TM) 6= Cℓ(TM)⊗ T∗M. Instead, we must use the above theory and possible applications as a^guide in order to find an appropriate definition. Let us see how this can be done. 3 Clifford Valued Differential Forms Definition 5 A homogeneous multivector valued differential form of type (l,p) l p is a section of TM ⊗ T∗M ֒→ Cℓ(TM)⊗ T∗M, for 0 ≤ l ≤ 4, 0 ≤ ^ ^ ^ p ≤ 4. A section of Cℓ(TM)⊗ T∗M such that the multivector part is non homogeneous is called a Clifford ^valued differential form. l p p We recall, that any A ∈ sec TM ⊗ T∗M ֒→ secCℓ(TM)⊗ T∗M can always be written as ^ ^ ^ 1 A=m ⊗ψ(p) ≡ mi1...ile ...e ⊗ψ(p) (l) l! (l) i1 il 1 = m ⊗ψ(p) θj1 ∧...∧θjp p! (l) j1...jp 1 = l!p!m(i1l)...ilei1...eil ⊗ψj(1p.)...jpθj1 ∧...∧θjp (19) 1 = l!p!Aij11......ijlpei1...eil ⊗θi1 ∧...∧θip. m p Definition 6 The ⊗ product of A = A⊗ψ(p) ∈ secCℓ(TM)⊗ T∗M and ∧ ^ 8 m q B = B⊗χ(p) ∈secCℓ(TM)⊗ T∗M is the mapping7: ^ l p ⊗ :secCℓ(TM)⊗ T∗M ×secCℓ(TM)⊗ T∗M ∧ ^ ^ l+p →secCℓ(TM)⊗ T∗M, ^ mm A⊗ B = AB⊗ψ(p)∧χ(q). (20) ∧ l p Definition 7 Thecommutator[A,B]of A∈sec TM⊗ T∗M ֒→secCℓ(TM)⊗ p m q ^ ^q T∗M and B ∈ TM ⊗ T∗M ֒→secCℓ(TM)⊗ T∗M is the map- p^ing: ^ ^ ^ l p m q [ , ]:sec TM ⊗ T∗M ×sec TM ⊗ T∗M ^ ^ ^ ^ |l+m| k p+q →sec(( T∗M)⊗ T∗M) k=X|l−m|^ ^ [A,B]=A⊗ B−(−1)pqB⊗ A (21) ∧ ∧ Writing A = l1!Aj1...jlej1...ejlψ(p), B = m1!Bi1...imei1...eimχ(q), with ψ(p) ∈ p q sec T∗M and χ(q) ∈sec T∗M, we have ^ ^ 1 [A,B]= l!m!Aj1...jlBi1...im ej1...ejl,ei1...eim ψ(p)∧χ(q). (22) (cid:2) (cid:3) The definition of the commutator is extended by linearity to arbitrary sections of Cℓ(TM)⊗ T∗M. ^ Now, we have the proposition. p q Proposition 8 Let A∈secCℓ(TM)⊗ T∗M, B ∈secCℓ(TM)⊗ T∗M, r ^ ^ C ∈A∈secCℓ(TM)⊗ T∗M. Then, ^ [A,B]=(−1)1+pq[B,A], (23) and (−1)pr[[A,B],C]+(−1)qp[[B,C],A]+(−)rq[[C,A],B]=0. (24) Proof. It follows directly from a simple calculation, left to the reader. Eq.(24) may be called the graded Jacobi identity [4]. m m 7AandB aregeneralnonhomogeous multivectorfields. 9 2 p r Corollary 9 Let be A(2) ∈ sec (TM)⊗ T∗M and B ∈ sec (TM)⊗ q ^ ^ ^ T∗M. Then, ^ [A(2),B]=C, (25) r p+q where C ∈sec (TM)⊗ T∗M. ^ ^ Proof. It follows from a direct calculation, left to the reader.(cid:4) 2 1 l Proposition 10 Let ω ∈ sec (TM) ⊗ T∗M , A ∈ sec (TM) ⊗ p m ^q ^ ^ T∗M.B ∈sec (TM)⊗ T∗M. Then, we have ^ ^ ^ [ω,A⊗ B]=[ω,A]⊗ B+(−1)pA⊗ [ω,B]. (26) ∧ ∧ ∧ Proof. Using the definition of the commutator we can write [ω,A]⊗ B =(ω⊗ A−(−1)pA⊗ ω)⊗ B ∧ ∧ ∧ ∧ =(ω⊗ A⊗ B−(−1)p+qA⊗ B⊗ ω) ∧ ∧ ∧ ∧ +(−1)p+qA⊗ B⊗ ω−(−1)pA⊗ ω⊗ B ∧ ∧ ∧ ∧ =[ω,A⊗ B]−(−1)pA⊗ [ω,B], ∧ ∧ from where the desired result follows. From Eq.(??) we have also8 (p+q)[ω,A⊗ B] ∧ =p[ω,A]⊗ B+(−1)pqA⊗ [ω,B] ∧ ∧ +q[ω,A]⊗ B+(−1)ppA⊗ [ω,B]. ∧ ∧ Definition 11 The action of the differential operator d acting on l p p A∈sec TM ⊗ T∗M ֒→secCℓ(TM)⊗ T∗M, ^ ^ ^ is given by: dA⊜ej1...ejl ⊗dAj1...jl (27) 1 =ej1...ejl ⊗dp!Aij11......ijplθi1 ∧...∧θip. We have the important proposition. p q Proposition 12 LetA∈secCℓ(TM)⊗ T∗M andB ∈secCℓ(TM)⊗ T∗M. Then, ^ ^ d[A,B]=[dA,B]+(−1)p[A,dB]. (28) Proof. The proof is a simple calculation, left to the reader. We now define the exterior covariant differential operator (EXCD) D and the extended covariant derivative (ECD) D acting on a Clifford valued form er l p p A∈sec TM ⊗ T∗M ֒→secCℓ(TM) ⊗ T∗M, as follows. ^ ^ ^ 8The result printed in the original printed version is (unfortunately) wrong. However (fortunately)except fordetails,itdoesnotchange anyoftheconclusions. 10