arXiv:1505.02935v5 [math-ph] 23 Nov 2016 Preliminaries 2 Introduction 1 Contents nteGoeiso the of Geodesics the On 4 h oetind Sitter de Lorentzian The 3 eeaie nryMmnu osrainLw nd Sitt de in Laws Conservation Energy-Momentum Generalized 5 oetinadTlprle eSte Spacetime Sitter de Teleparallel and Lorentzian . osre urnsAscae oaCvratyConserved in Covariantly Currents a Conserved to Associated Currents 2.3 Conserved 2.2 . h Currents The 2.1 . oa urns t ahmtcladPyia enn 1 . . . . Meaning Physical and Mathematical Its Currents. Komar 2.4 Representation . i ler fted itrGop...... 23 ...... Currents . Conserved . The in . System . Closed 5.3 a for Group Covector Sitter Energy-Momentum de Generalized the of Algebra 5.2 Lie 5.1 Structures Spacetime . uvsOtie rmCntandVrain...... 22 ...... Variations. Constrained From Obtained Curves 4.1 oe nCnevto as qain of Equations Laws, Conservation on Notes oino atradPril ilsin Fields Particle and Matter of Motion .. iie osblt oCntuta Construct to Possibility Limited 2.3.1 .. eemnto fteEpii omof Form Explicit the of Determination 2.4.1 .. h awl ieEquation Like Maxwell The 2.4.2 h eeaalld itrSaeieSrcue...... 25 ...... Structure Spacetime Sitter de Teleparallel the adrA orge r n aulA Wainer A. Samuel and Jr. Rodrigues A. Waldyr ardieuiapb [email protected] [email protected] qain...... 16 ...... Equation J V and IMECC-UNICAMP aur ,2018 1, January J Structures GRT M K J dSL M Π 6 ...... dSℓ α soitdt iln etrFed 9 Fields Vector Killing to Associated 1 = tutr n t Conformal its and Structure g ( Π ∂ α F , 27 ...... ) = CEMC J A J noe Einstein Encodes A 13 ...... in GRT W 9 . . . er 8 . 16 23 17 6 3 0 6 Equation of Motion for a Single-Pole Mass in a GRT Lorentzian Spacetime 29

7 Equation of Motion for a Single-Pole Mass in a de Sitter Lorentzian Spacetime 31

A Clifford Bundle Formalism 32 A.1 CliffordProduct ...... 33 A.1.1 HodgeStarOperator...... 35 A.1.2 Dirac Operator Associated to a Levi-Civita Connection D 36 A.2 Covariant D’ Alembertian, Hodge D’Alembertian and Ricci Op- erators...... 37

B Lie Derivatives and Variations 38

C The Generalized Energy-Momentum Current in (M, g, τg, ) 42 C.1 The Case↑ of a General Lorentzian Spacetime Structure ...... 42 C.2 Introducing D and the Covariant “Conservation” Law for Υ .. 44 C.3 TheCaseofMinkowskiSpacetime ...... 45 C.3.1 The Canonical Energy-Momentum Tensor in Minkowski Spacetime...... 45 C.3.2 The Energy-Momentum 1-Form Fields in Minkowski Space- time...... 45 C.3.3 The Belinfante Energy-Momentum Tensor in Minkowski Spacetime...... 46 C.3.4 The Energy-Momentum Covector in Minkowski Spacetime 46

D TheEnergy-MomentumTensorofMatterinGRT 47

E Relative Tensors and their Covariant Derivatives 49

F Explicit Formulas for Jµν and Jµ4 in Terms of Projective Con- formal Coordinates 50

Abstract In this paper we discuss the physics of interacting tensor fields and particles living in a de Sitter manifold M = S0(1, 4)/S0(1, 3) ≃ R × S3 interpreted as a submanifold of (M˚ = R5,˚g), with ˚g a metric of signature (1, 4). The pair (M, g) where g is the pullback metric of ˚g (g = i∗˚g) is a Lorentzian manifold that is oriented by τg and time oriented by ↑. It is the structure (M, g,τg , ↑) that is primely used to study the energy- momentum conservation law for a system of physical fields (and particles) living in M and to get the equations of motion of the fields and also the equations of motion describing the behavior of free particles. To achieve our objectives we construct two different de Sitter spacetime structures dSL dSTP M = (M, g,D,τg, ↑) and M = (M, g, ∇,τg, ↑), where D is the

2 Levi-Civita connection of g and ∇ is a metric compatible parallel connec- tion. Both connections are introduced in our study only as mathematical devices, no special physical meaning is attributed to these objects. In particular M dSL is not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Our approach permit to clarify some misconceptions appearing in the literature, in particular one claiming that free particles in the de Sitter structure (M, g) do not follows timelike geodesics. The paper makes use of the Clifford and spin-Clifford bundles formalism recalled in one of the appendices, something needed for a thoughtful presentation of the concept of a Komar current J A (in GRT) associated to any vector field A generating a one parameter group of diffeomorphisms. The explicit formula for J A in terms of the energy- momentum tensor of the fields and its physical meaning is given. Besides that we show how F = dA (A = g(A, ) satisfy a Maxwell like equation ∂F = J A which encodes the contents of Einstein equation. Our results shows that in GRT there are infinitely many conserved currents, indepen- dently of the fact that the Lorentzian spacetime (representing a gravita- tional field) possess or not Killing vector fields. Moreover our results also show that even when the appropriate timelike and spacelike Killing vec- tor fields exist it is not possible to define a conserved energy-momentum covector (not covector field) as in Special Relativistic Theories.

1 Introduction

In this paper we study some aspects of Physics of fields living and interact- ing in a manifold M = SO(1, 4)/SO(1, 3) R3 S3. We introduce two dif- ferent geometrical spacetime structures that≃ we× can form starting from the manifold M which is supposed to be a vector manifold, i.e., a submanifold of (M,˚ ˚g) with M˚ = R5 and ˚g a metric of signature (1, 4). If i : M M˚ is the inclusion map the structures that will be studied are the Lorentzian→ de dSL Sitter spacetime M = (M, g, D, τg, ) and teleparallel de Sitter spacetime dSTP ↑ M = (M, g, ∇, τg, ) where g = i∗˚g, D is the Levi-Civita connection of g and ∇ is a metric compatible↑ teleparallel connection (see Section 4.1). Our main objective is the following: taking (M, g) as the arena where physical fields live and interact how do we formulate conservation laws of energy-momentum and angular momentum for the system of physical fields. In order to give a meaningful meaning to this question we recall the fact that in Lorentzian space- time structures that are models of gravitational fields in the GRT there are no genuine conservation laws of energy-momentum (and also angular momen- tum) for a closed system of fields and moreover there are no genuine energy- momentum and angular momentum conservation laws for the system consisting of non gravitational plus the gravitational field. We discuss in Section 2.1 a pure mathematical result, namely when there exists some conserved currents V 1J sec T ∗M in a Lorentzian spacetime associated to a tensor field W ∈ sec T1 M and∈ a vector field V sec TM. In Section 2.2 we briefly recall how a conserved energy-momentum tensor∈ for the matter fields is constructed in Special Relativ- ity theories and how in that theory it is possible to construct a conserved energy-

3 momentum covector 1 f or the matter fields. After that we recall that in GRT 1 we have a covariantly “conserved” energy-momentum tensor T ∈ sec T1 M (i.e., D • T = 0) and so, using the results of Section 2.1 we can immediately construct conserved currents when the Lorentzian spacetime modelling the gravitational field generated by T possess Killing vector fields. However, we show that is not possible in general in GRT even when some special conserved currents exist (associated to one timelike and three spacelike Killing vector fields) to build a conserved covector for the system of fields, as it is the case in special relativistic theories. Immediately after showing that we ask the question: Is it necessary to have Killing vector fields in a Lorentzian spacetime mod- elling a given gravitational field in order to be possible to construct conserved currents? Well, we show that the answer is no. In GRT there are an infinite number of conserved currents. This is showed in Section 2.42 were we introduce the so called Komar currents in a Lorentzian spacetime modelling a gravitational field generated by a given (symmetric) energy momentum tensor T and show how any diffeomorphism associated to a one parameter group generated by a vector field A lead to a conserved current. We show moreover using the Clifford 2 bundle formalism recalled in Appendix A that F = dA ∈ sec T ∗M where 1 A = g(A, ) sec T ∗M satisfy, (with ∂ denoting the Dirac operatorV acting ∈ on sections of the CliffordV bundle of differential forms) a Maxwell like equation ∂F = J A (equivalent to dF = 0 and δF = J A). The explicit form of A g − J as a function of the energy-momentum tensor is derived (see Eq.(48)) together 3 with its scalar invariant. We establish that ∂F = J A encode the contents of Einstein equation. We show moreover that even if we can get four conserved currents given one time like and three spacelike vector fields and thus get four scalar invariants these objects cannot be associated to the components of a momentum covector 4 for the system of fields producing the energy-momentum tensor T . We also give the form of J A when A is a Killing vector field and emphasize that even if the Lorentzian spacetime under consideration has one time like and three spacelike Killing vector fields we cannot find a conserved momentum covector for the system of fields. This paper has several appendices necessary for a perfect intelligibility of the results in the main text. Thus it is opportune to describe what is there and where their contents are used in the main text5. To start, in Appendix A we briefly recall the main results of the Clifford bundle formalism used in this paper which permits one to understand how to arrive at the equation ∂F = J A

1The energy-momentum covector is an element of a vector space and is not a covector field. 2This section is an improvement of results first presented in [36]. 3The symbol ∂ denotes the the Dirac operator acting on sections of the Clifford bundle Cℓ(M, g). See Appendix A. 4Not a covector field. 5Some of the material of the Appendices is well known, but we think that despite this fact theri presentation here will be useful for most of our readers.

4 in Section 2.2.6. Lie derivatives and variations of tensor fields is discussed in Appendix B. In Section C1 we derive from the Lagrangian formalism conserved currents for fields living in a general Lorentzian spacetime structure and the corresponding generalized covariant energy-momentum “conservation” law. We compare these results in Section C2 with the analogues ones for field theo- ries in special relativistic theories where the Lorentzian spacetime structure is Minkowski spacetime. We show that despite the fact that we can derive con- served quantities for fields living and interacting in M ℓDS we cannot define in this structure a genuine energy-momentum conserved covector for the system of fields as it is the case in Minkowski spacetime. A legitimate energy-momentum covector for the system of fields living in (M = SO(1, 4)/SO(1, 3), g) exist only in the teleparallel structure M dSTP . This is discussed in Section 5.2 after re- calling the Lie algebra and the Casimir invariants of the Lie algebra of de Sitter group in Section 5.1. In Appendix E we derive for completeness and to insert the Remark 32 the so called covariant energy-momentum conservation law in GRT. Appendix D recalls the intrinsic definition of relative tensors and their covariant derivatives. Appendix E present proofs of some identities used in the main text. As we already said the main objective of this paper is to discuss the Physics of interacting fields in de Sitter spacetime structures M dSL and M dSTP . In particular we want also to clarify some misunderstandings concerning the roles of geodesics in the M dSℓ. So, in section 3 we briefly recall the conformal rep- resentation of the de Sitter spacetime structure M dSL and prove that the one timelike and the three spacelike “translation” Killing vector fields of (M, g) de- fines a basis for almost all M. With this result we show in Section 4 that the method using in [32] to obtain the curves which extremizes the length function of timelike curves in de Sitter spacetime with the result that these curves are not geodesics is equivocated, since those authors use constrained variations in- stead of arbitrary variations of the length function. Even more, the equation obtained from the constrained variation in [32] is according to our view wrongly interpreted in its mathematical (and physical) contents. Indeed, using some of the results of Section 5.2 and the results of Section 6 which briefly recall Papapetrou’s classical results [29] deriving the equation of motion of a probe single-pole particle in GRT, we show in Section 7 that contrary to the authors statement in [32] it is not true that the equation of motion of a single-pole obtained from a method similar to Papapetrou’s one in [29] but using the gen- eralized energy-momentum tensor of matter fields in M dSL gives an equation of motion different from the geodesic equation in M dSL and in agreement with the one they derived from his constrained variation method. Indeed, we prove that

6The Clifford bundle formalism permits the representation of a covariant Dirac spinor field as certain equivalence classes of even sections of the Clifford bundle, called Dirac-Hestenes spinor field (DHSF). These objects are a key ingredient to clarify the concept of Lie derivative of spinor fields of and give meaningful definition for such an object , something necessary to study conservation laws in Lorentzian spacetime structures when spinor fields are present. Our approach to the subject is descrbed in [20] and athoughtful derivation of Dirac equation in de Sitter structure (M, g) using DHSFs is given in [39].

5 from the equation describing the motion of a single-pole the geodesic equation follows automatically. Finally, in Section 8 we present our conclusions.

2 Preliminaries

Let (M, g,D,τg, ) be a general Lorentzian spacetime. Let M be an open ↑ µ U ⊆ µ set covered by coordinates x . Let eµ = ∂µ be a basis of T and ϑ = µ { } { } µ µU { dx the basis of T ∗ dual to the basis ∂µ , i.e., ϑ (∂ν ) = δν . We denote } U { } µ ν by g a metric of the cotangent bundle such that if g = gµν ϑ ϑ then g = µν µρ µ µ ⊗ g ∂µ ∂ν with g gρν = δν . We introduce also ∂ and ϑµ respectively as the reciprocal⊗ bases of e and ϑµ , i.e., we have{ } { } { µ} { } µ µ µ µ g(∂ν , ∂ )= δν , g(ϑ , ϑν )= δν (1) µ Next we introduce in T the tetrad basis eα = hα∂µ and in T ∗ the α α µ U { } U cotetrad basis γ = hµγ which are dual basis. We introduce moreover the basis eα and {γ as the} reciprocal bases of e and γα satisfying { } { α} { α} { } β β β β g(eα, e )= δα, g(γ , γα)= δα. (2) Moreover recall that it is

g = η γα γβ = ηαβγ γ , αβ ⊗ α ⊗ β g = ηαβe e = η eα eβ. (3) α ⊗ β αβ ⊗

2.1 The Currents V and K J J αβ 0 αβ βα ˇ α Let W = W eα eβ sec T2 M with W = W and W = Wαβγ γβ sec T 2M, W ⊗= η ∈η W ςτ and W = W αγβ e sec T 1M. For the⊗ ∈ 0 αβ ας βτ β ⊗ α ∈ 1 applications we have in mind we will say that W, Wˇ and W are physically equivalent. Note that W (an example of an extensor field7) is such that

1 1 W : sec T ∗M sec T ∗M, → ^ α ^β W (V )= VαWβ γ . (4) Define the divergence of W as the 1-form field

D W := (D W α)γβ (5) • α β where α α α α ι ι α D W := (De W ) = e (W )+Γ W Γ W . (6) α β α β α β αι β − αβ ι Moreover, introduce the 1-form fields

1 β αβ := W γα sec T ∗M. (7) W ∈ ^ 7See Chapter 4 of [34].

6 Remark 1 Take notice for the developments that follows that the Hodge coderiva- tive of the 1-form fields βis (see Appendix): W β κ αβ δ = γ yDe (W γ ) gW − κ α = e (W αβ)γκyγ W αβΓι γκyγ − κ α − κα ι = e (W αβ) W αβΓκ . − α − κα So, D W =0 does not implies that δ β =0. • gW

α Now, given a vector field V = V eα and the physically equivalent covector α field V = V γα define the current

1 α (V = V α sec T ∗M ֒ sec ℓ(M, g) (8 J W ∈ ^ → C Of course, writing = J γβ (9) JV β we have · = V αW . (10) Jβ αβ Recalling (see Appendix A) that ⋆1= τg define

4 αβ αβ (T = W ⋆ 1= W τg sec T ∗M ֒ ℓ(M, g). (11 g ∈ ^ →C Then, we have, with ∂ denoting the Dirac operator,

α d⋆ V = ∂ ∧ ⋆ V = γ (De ⋆ V ) g J gJ ∧ α g J α α = De (γ ⋆ V ) De γ ⋆ V . (12) α ∧ gJ − α ∧ gJ

α α α Taking into account that γ ⋆ V = ⋆(γ y V ) and De γ ⋆ V = ∧ gJ g J α ∧ gJ α ⋆(De γ y V ) we can write Eq.(12) as g α J

κ α κ α α κ β d⋆ V = eα(V )Wκ + V eα(Wκ )+Γ αβ·· V Wκ τg (13) g J ·
Also, we can easily verifiy from Eq.(5) that

⋆[(D W )(V)] = [(D W )(V)]τg g • • κ α α ι κ ι α κ = (V eα(Wκ )+Γ αι·· WκV Γ ακ·· Wι V ) τg. (14) · − · Now, let £ be the (standard) Lie derivative operator. Let us evaluate the produuct of £Vg (eα, eβ) by T, i.e.,

4 (£Vg (eα, eβ)) T sec T ∗M. (15) ∈ ^ 7 From Cartan magical formula we get

α α α £Vγ = d(V )+ V y(∂ γ ) α ι ς α ι ∧ς α ι = eι(V )γ V Γ ςι··γ + V Γ ις··γ . (16) − · · Then,

ι κ ι κ [£Vg (eα, eβ)] T = [(ηικ£Vγ γ + ηικγ £Vγ ) (eα, eβ)]T κι⊗ ς κ ⊗ι = [2eκ(Vι)W +2V Γ ις··Wκ]T (17) · and we get from Eqs.(13), (14) and (17) the important identity [5]

(£Vg (eα, eβ)) T =2d⋆ V 2 ⋆ [(D W )(V)]. (18) J − g •

From Eq.(18) we see that if V is a conformal Killing vector field, i.e., £Vg (eα, eβ)= 2ληαβ we have ⋆ λtrW = d⋆ V ⋆[(D W )(V)] (19) g g J − • α where trW is the trace of the matrix with entries Wβ .

2.2 Conserved Currents Associated to a Covariantly Con- served W Definition 2 We say that W is “ covariantly conserved” if D W = 0 (20) • In this case, if V = K is a Killing vector field then £Kg = 0 and we have

d⋆ K = ⋆[(D W )(K)] (21) g J g • and the current 3-form field ⋆ K is closed, i.e., d⋆ K = 0, or equivalently gJ g J (taking into account the definition of the Hodge coderivative operator δ) g

δ K =0. (22) gJ 8 In resume, when we have Killing vector fields Ki, i =1, 2, ..., n “covariant conservation” of the tensor field W , i.e., D W = 0 implies in genuine con- • servation laws for the currents K , i.e., from δ K = 0 we can using Stokes J i gJ I theorem build the scalar conserved quantities 1 (Ki) := ′ ⋆ K (23) E 8π Σ g J i R where N is the region where K has support and ∂N =Σ+Σ′ + Ξ where Σ, Σ′ are spacelike surfaces and J is null at Ξ (spatial infinity). JKi 8The maximum number is 10 when dim M = 4 and that maximum number occurs only for spacetimes of constant curvature.

8 2.3 Conserved Currents in GRT Associated to Killing Vec- tor Fields Before studying the conditions for the existence or not of genuine energy- momentum conservation laws in GRT, let us recall from Appendix C.3.4 that 9 4 in Minkowski spacetime (M R , η, D, τη, ) we can introduce global coor- µ ≃ ↑ dinates x (in Einstein-Lorentz-Poincar´egauge) such that η(eµ,eν ) = ηµν , { } µ and Deµ eν = 0, where the eµ = ∂/∂x is simultaneously a global tetrad and a coordinate basis. Also the{ ϑµ = dxµ} is a global cotetrad and a coordinate cobasis. { } µ 4 Moreover, the eµ = ∂/∂x are also Killing vector fields in (M R , η) and thus we have for a closed physical system (consisting of particles≃ and fields in interaction living in Minkowski spacetime and whose equations of motion are derived from a variational principle with a Lagrangian density invariant under spacetime translations) that the currents10

ν 1 4 (α = T ϑ sec T M ֒ ℓ(M, η),M R (24 =: Tα J∂/∂x αν ∈ →C ≃ V are the conserved energy-momentum 1-form fields of the physical system under consideration, for which we know that the quantity (recall Eq.(238))

P = P ϑα = P Eα, (25) α |o α with

Pα = ⋆ α (26) Z g T are the components of the conserved energy-momentum covector (CEMC) P of the system.

2.3.1 Limited Possibility to Construct a CEMC in GRT Now, recall that in GRT a gravitational field generated by an energy-momentum 11 T is modelled by a Lorentzian spacetime (M, g, D, τg, ) where the relation between g and T is given by Einstein equation which using↑ the orthonormal bases e and γα introduced above reads { α} { } 1 Gα = Rα δαR = T α. (27) β β − 2 β − β Moreover, defining G = Gαγβ e and recalling that T = T αγβ e it is β ⊗ α β ⊗ α D G =0, D T = 0 (28) • • 9See [34] for the remaning of the notation. 10 Keep in mind that in Eq.(24) the Tαν are the ν-component of the current Tα = J∂/∂xα and moreover are here taken as symmetric. See Appendix C.3.3. 11 In fact, by an equivalence classes of pentuples (M, g, D, τg , ↑) modulo diffeomorphisms.

9 Based only on the contents of Section 2.1, given that D T = 0, it may seems at first sigh12 that the only possibility to construct conserved• energy- momentum currents α in GRT are for models of the theory where appropriate Killing vector fieldsT (such that one is timelike and the other three spacelike) exist. However, an arbitrarily given Lorentzian manifold (M, g,D) in general does not have such Killing vector fields.

Remark 3 Moreover, even if it is the case that if a particular model (M, g,D) of GRT there exist one timelike and three spacelike Killing vector fields we can construct the scalar invariants quantities Pα, α = 0, 1, 2, 3 given by Eq.(26) we cannot define an energy-momentum covector P analogous to the one given by Eq.(25) This is so because in this case to have a conserved covector like P it is necessary to select a γα at a fixed point of the manifold. But in general there is no physically meaningful way to do that, except if M is asymptoti- cally flat13 in which case we can choose a chart such that at spatial infinity α 0 α α lim ~x γ (x , ~x)= ϑ = dx and lim ~x γ gµν = ηµν | |→∞ | |→ Thus, paroding Sachs and Wu [43] we must say that non existence of genuine conservation laws for energy-momentum (and also angular momentum) in GRT is a shame.

Remark 4 Despite what has been said above and the results of Section 2.1 we next show that there exists trivially an infinity of conserved currents (the Komar currents) in any Lorentzian spacetime modelling a gravitational field in GRT. We discuss the meaning and disclose the form of these currents, a result possible due to a notable decomposition of the square of the Dirac operator acting on sections of the Clifford bundle ℓ(M, g). C Remark 5 We end this subsection recallng that in order to produce genuine conservation laws in a field theory of gravitation with the gravitational field equations equivalent (in a precise sense) to Einstein equation it is necessary to formulate the theory in a parallelizable manifold and to dispense the Lorentzian spacetime structure of GRT. Details of such a theory may be found in [37].

2.4 Komar Currents. Its Mathematical and Physical Mean- ing Let A sec TM be the generator of a one parameter group of diffeomorphisms ∈ of M in the spacetime structure M, g, D, τg, which is a model of a gravi- h 1 ↑i tational field generated by T sec T1 M (the matter fields energy-momentum momentum tensor) in GRT. It∈ is quite obvious that if we define F = dA, where 1 A = g(A, ) sec T ∗M ֒ ℓ(M, g), then the current ∈ →C V J A = δF (29) −g 12See however Section 2.4 to learn that this naive expectation is incorrect. 13The concept of asymptotically flat Lorentzian manifold can be rigorously formulated with- out the use of coordinates, as e.g., in [48]. However we will not need to enter in details here.

10 is conserved, i.e., δJ A =0. (30) g Surprisingly such a trivial mathematical result seems to be very important by 14 15 people working in GRT who call J A the Komar current [18]. Komar called

E = ⋆ J A = ⋆ F (31) V g ∂V g R R the generalized energy. To understand why J A is considered important in GRT write the action for the gravitational plus matter and non gravitational fields as 1 = + = Rτ g + . (32) A Lg Lm −2 Lm R R R R Now, the equations of motion for g can be obtained considering its variation under an (infinitesimal) diffeormorphism h : M M generated by A. We have 0 16 → 0 that g g′ = h∗g = g + δ g where the variation δ g = £Ag. Taking 7→ − into account Cartan’s magical formula (£AM = AydM + d(AyM), for any g g M sec T ∗M) we have ∈ V δ0 = δ0 + δ0 A Lg Lm = R £A R £A − Lg − Lm = R d(Ay ) R d(Ay ) − Lg − Lm := Rd(⋆ )R (33) gC R where ⋆ = Ay g Ay m + K (34) g C − L − L with dK =0. To proceed introduce a coordinate chart with coordinates xµ for the region µ µ 1 µ { } of interest U M. Recall that = 2 Rϑ are the Einstein 1-form 17 ⊂µ µ µ G R − fields , with = Rν ϑ the Ricci 1-forms and R the curvature scalar. Einstein equation obtainedR form the variation principle δ0 = 0 is µ := µ + µ = 0, µ µ ν A E G T µν with = Tν ϑ the energy-momentum 1-form fields and moreover DµG = T µν 0= DµT . Next write explicitly the action as [19] 1 = R det gdx0dx1dx2dx3 + L det gdx0dx1dx2dx3. (35) A −2 − m − R p R p 14Komar called a related quantity the generalized flux. 15V denotes a spacelike hypersurface and S = ∂V its boundary. Usualy the integral E is calculated at a constant x0 time hypersurface and the limit is taken for S being the boundary at infinity. 16Please do not confuse δ0 with δ. g 17 µ µ ν µ µ 1 µ G = Gν ϑ where Gν = Rν − 2 δν are the components of the Einstein tensor. Moreover, µ µ ν we write E = Eν ϑ .

11 We have immediately

0 1 µν 0 1 2 3 δ = E (£Ag) det gdx dx dx dx 2 µν A − p− R µν 0 1 2 3 = E DµAν det gdx dx dx dx − p− = R D (Eµν A ) det gdx0dx1dx2dx3 − µ ν − ν p = R (∂y Aν )τ g − gE R ν = ⋆ δ( Aν ) g g E R ν = d(⋆ Aν ). (36) − gE R From Eqs.(33) and (36) we have

ν d(⋆ Aν )+ d(⋆ )=0, (37) gE gC R and thus ν δ( Aν )+ δ = 0 (38) g E g C 1 ν Thus, the current sec T ∗M is conserved if the field equations = 0 C ∈ E are satisfied. An equationV (in component form) equivalent to Eq.(38) already ν appears in [18] (and also previously in [4] ) who took = Kν +N with δN = 0. C E g 0 Here, to continue we prefer to write an identity involving only δ g = 0 µ A δ g. Proceeding exactly as before we get putting (A)= Aµ that there L 1 G G existsR N sec T ∗M such that ∈ V δ (A) + δN =0. (39) gG g and we see that we can identify

N := µA + L (40) −G µ where δL = 0. Now, we claim that g

1 Proposition 6 There exists a L sec T ∗M such that ∈ V µ N = Aµ + L = δdA = J A, (41) −G g − where J A was defined in Eq.(29).

18 . (Proof. To prove our claim we suppose from now one that T ∗M ֒ ℓ(M, g →C 18 V Cℓ(M, g) is the Clifford bundle of differential forms, see Appendix and if more details are necessary, cosnult, e.g., [34].

12 Then it is possible to write 1 µA = µA RA G µ R µ − 2 1 = ∂ ∂A RA ∧ − 2 1 = ∂ ∂A + ∂ ∂A RA ∂ ∂A ∧ · − 2 − · 1 = ∂2A RA ∂ ∂A − 2 − · 1 = δdA dδA RA ∂ ∂A (42) −g − g − 2 − · where ∂ ∂ is the Ricci operator and ∂ ∂ =  is the D’Alembertian operator. Then we∧ take · µ 1 Aµ dδA RA ∂ ∂A = δdA (43) − G − g − 2 − · g and of course it is19 1 L = dδA RA ∂ ∂A (44) − g − 2 − · proving the proposition. Now that we found a L satisfying Eq.(41) we investigate if we can give some 1 nontrivial physical meaning to such N = J sec T ∗M. − A ∈ V 2.4.1 Determination of the Explicit Form of J A We recall that the extensor field T acts on A as

T (A)= µA T µ Thus, since µA = T (A) (45) G µ − we have from Eq.(43) 1 δdA = T (A) dδA RA ∂ ∂A. (46) g − g − 2 − · µ We can write Eq.(46) taking into account that R = trT = Tµ and putting F := dA that δF = J (47) − A where [36] 1 J A = T (A)+ trT A+ dδA + ∂ ∂A (48) − 2 g ·

19 µ Note that since δ(G Aµ) = 0 it follows from Eq.(44) that indeed δL = 0. g g

13 Eq.(48) gives the explicit form for the Komar current20. Moreover, taking into account that δF = ⋆d⋆F it is g g g

1 1 d ⋆ F = ⋆− T (A)+ trT A+ dδA + ∂ ∂A g − 2 g ·  1 = ⋆ T (A)+ trT A+ dδA + ∂ ∂A (49) g − 2 g ·  and thus taking into account Stokes theorem

d⋆ F = ⋆ F = ⋆J A g ∂ g g RV R V RV Moreover, since d⋆J A = 0 we have that g

0= d⋆J A = ⋆J A N g Z g R ∂N and thus ⋆J A (∂N = Σ1 Σ2 + Ξ) is a conserved quantity. We arrive at g ZΣ1 − the conclusion that Taking Σ1 (∂ = R) as a ball of radius R and making R the quantity V ⊂ V S → ∞ 1 (A) := ⋆F (50) g E≡ E 8π Z R S 1 1 = ⋆ T (A) AtrT dδA ∂ ∂A (51) 8π V g  − 2 − g − ·  R is conserved.

Remark 7 It is very important to realize that quantity defined by Eq.(50) is an scalar invariant, i.e., its value does not depend on theE particular reference frame Z and to the (nacs Z) (the naturally adpated coordinate chart adapted to Z)21. But, of course, |for each particular vector field A sec TM (which generates a one parameter group of diffeomorphisms) we have∈ a different. (A) and the different (A) ´s are not related as components of a covector. E E Remark 8 As we already remarked an equation equivalent to Eq.(50) has al- ready been obtained in [18] who called (as said above) that quantity the conserved generalized energy. But according to our best knowledge Eq.(51) is new and ap- pears for the first time in [36]. However, considering that for each A sec TM that generates a one pa- rameter group of diffeomorphisms of M we∈ have a conserved quantity it is not

20Something that is not given in [18]. 21Recall that in Relativity theory (both special and general) a reference frame is modelled by a time like vector field Z pointing into the future. A naturally adapted coordinate chart to Z (with coordinate functions {xµ} (denoted (nacs| Z)) is one such that the spatial components of Z are null. More details may be found, e.g., in Chapter 6 of [34].

14 in our opinion appropriate to think about this quantity as a generalized energy. Indeed, why should the energy depends on terms like dδA and ∂ ∂A if A is not g · a dynamical field? We know that [35] when A = K is a Killing vector field it is δA = 0 and g ∂ ∂A = T (A)+ 1 trT A and thus Eq.(51) reads · − 2 1 1 = ⋆ (T (A) AtrT ) (52) E 4π N g − 2 R which is a well known conserved quantity22. For a Schwarzschild spacetime, as well known, A = ∂/∂t is a timelike Killing vector field and in his case since T µ 8π µ i µ the components of are Tν = √ det g ρ(r)v vν and v vj = 0 (since v = − 1 (1, 0, 0, 0)) we get = m. √g00 E Remark 9 Note that that the conserved quantity given by Eq.(52) differs in general from the conserved quantity obtained with the current defined in Eq.(8) when V = K which holds in any structure (M, g,D) with the conditions given there. However, in the particular case analyzed above Eq.(52) and Eq.(23) give the same result.

Remark 10 Originally Komar obtained the same result as in Eq.(52) directly from Eq.(50) supposing that the generator of the one parameter group of dif- feomorphism was A = ∂/∂t. So, he got = m by pure chance. If he had picked another vector field generator of aE one parameter group of diffeomor- phisms A = ∂/∂t, he of course, would not obtained that result. 6 Remark 11 The previous remark shows clearly that the construction of Komar currents does not to solve the energy-momentum conservation problem for a system consisting of the matter and non gravitational fields plus the gravitational field in GRT. Indeed, too claim that a solution for a meaningful definition for the energy- momentum of the total system23 exist it is necessary to find a way to define a total conserved energy-momentum covector for the total system as it is possible to do in field theories in Minkowski spacetime (recall Section C.3.4). This can only be done if the spacetime structure modeling a gravitational field (generated by the matter fields energy-momentum tensor T ) possess appropriate additional structure, or if we interpret the gravitational field as a field in the Faraday sense living in Minkowski spacetime. More details in [34, 37].

22An equivalent formula appears, e.g., as Eq.(11.2.10) in [48]. However, it is to be empha- sized here the simplicity and transparency of our approach concerning traditional ones based on classical tensor calculus. 23The total system is the system consisting of the gravitational plus matter and non gravi- tational fields.

15 2.4.2 The Maxwell Like Equation ∂F = J A Encodes Einstein Equa- tion From Eq.(47) where F = dA (A = g(A,)) with A sec TM an arbitrary gener- ator of a one parameter group of diffeomorphisms∈ of M ( part of the structure (M, g,D,τg, ↑)) taking into account that dF = 0, we get the Maxwell like equation (MLE) ∂F = J A. (53) with a well defined conserved current. Of course, as we already said, there is an infinity of such equations. Each one encodes Einstein equation, i.e., given the form of J A (Eq.(48)) we can get back Eq.(42), which gives immediately Einstein equation (EE). In this sense we can claim:

EE G = T and the MLE ∂F = J A are equivalent.

Remark 12 Finally it is worth to emphasize that the above results show that in GRT there are infinity of conservation laws, one for each vector field generator of a one parameter group of diffeomorphisms and so, Noether’s theorem in GRT which follows from the supposition that the Lagrangian density is invariant under the diffeomorphism group gives only identities, i.e., an infinite set of conserved currents, each one encoding as we saw above Einstein equation.

It is now time to analyze the possible generalized conservation laws and their implications for the motions of probe single-pole particles in Lorentzian and teleparallel de Sitter spacetime structures, where these structures are not supposed to represent models of gravitational fields in GRT and compare these results with the ones in GRT. This will be one in the next sections.

3 The Lorentzian de Sitter M dSℓ Structure and its Conformal Representation

Let SO(1, 4) and SO(1, 3) be respectively the special pseudo-orthogonal groups in R1,4 = M˚ = R5,˚g and in R1,3 = R4, η where ˚g is a metric of signature (1, 4) and {η a metric of} signature (1, 3).{ The} manifold M = SO(1, 4)/SO(1, 3) will be called the de Sitter manifold. Since

M = SO(1, 4)/SO(1, 3) R S3 (54) ≈ × this manifold can be viewed as a brane (a submanifold) in the structure R1,4. We now introduce a Lorentzian spacetime, i.e., the structure M dSL = (M = 3 R S , g, D, τg, ) which will be called Lorentzian de Sitter spacetime structure × ↑ 3 5 where if ι : R S R , g = ι∗˚g and D is the parallel projection on M of the pseudo Euclidian× → metric compatible connection in R1,4 (details in [38]). As well known, M dSL is a spacetime of constant Riemannian curvature. It has ten Killing vector fields. The Killing vector fields are the generators of infinitesimal actions of the group SO(1, 4) (called the de Sitter group) in M = R S3 × ≈

16 SO(1, 4)/SO(1, 3). The group SO(1, 4) acts transitively24 in SO(1, 4)/SO(1, 3), which is thus a homogeneous space (for SO(1, 4)). We now recall the description of the manifold R S3 as a pseudo-sphere (a submanifold) of radius ℓ of the pseudo Euclidean space× R1,4 = R5,˚g . If (X0,X1,X2,X3,X4) are the global coordinates of R1,4 then the equation{ } rep- resenting the pseudo sphere is

(X0)2 (X1)2 (X2)2 (X3)2 (X4)2 = ℓ2. (55) − − − − − Introducing conformal coordinates25 xµ by projecting the points of R S3 from the “north-pole” to a plane tangent{ to the} “south pole” we see immediately× that xµ covers all R S3 except the “north-pole”. We immediately find that { } × 2 µ ν g = i ˚g =Ω ηµν dx dx (56) ∗ ⊗ where σ2 Xµ =Ωxµ, X4 = ℓΩ 1+ (57) −  4ℓ2  1 σ2 − Ω= 1 (58)  − 4ℓ2  and 2 µ ν σ = ηµν x x . (59) Since the north pole of the pseudo sphere is not covered by the coordinate functions we see that (omitting two dimensions) the region of the spacetime as seem by an observer living the south pole is the region inside the so called absolute of Cayley-Klein of equation

t2 x2 =4ℓ2. (60) − In Figure 1 we can see that all timelike curves (1) and (2) and lightlike (3) starts in the “past horizon” and end on the “future horizon”.

4 On the Geodesics of the M dSL

In a classic book by Hawking and Ellis [15] we can read on page 126 the following statement:

de Sitter spacetime is geodesically complete; however there are points in the space which cannot be joined to each other by any geodesic.

24A group G of transformations in a manifold M (σ : G × M → M by (g,x) 7→ σ(g,x)) is said to act transitively on M if for arbitraries x,y ∈ M there exists g ∈ G such that σ(g,x)= y. 25Figure 1 appears also in author’s paper [39].

17 Figure 1: Conformal Representation of de Sitter Spacetime. Note that the ”observer” spacetime is interior to the Caley-Klein absolute t2 ~x2 =4ℓ2. −

18 Unfortunately many people do not realize that the points that cannot be joined by a geodesic are some points which can be joined by a spacelike curve (living in region 3 in Figure 1). So, these curves are never the path of any particle. A complete and thoughtful discussion of this issue is given in an old an excellent article by Schmidt [42]. Remark 13 Having said that, let us recall that among the Killing vector fields26 of (M = R S3, g) there are one timelike and three spacelike vector fields (which are called the× translation Killing vector fields in physical literature). So, many people though since a long time ago that this permits the formulation of an energy-momentum conservation law in de Sitter spacetime M dSL structure . However, the fact is that what can really be done is the obtainment of conserved dSL quantities like the Pα in Eq.(26). But we cannot obtain in M structure an energy-momentum covector like P for a given closed physical system using an equation similar to Eq.(25). This is because de Sitter spacetime is not asymp- totically flat and so there is no way to physically determine a point to fix the γα to use in an equation similar to Eq.(25).

Now, the “translation” Killing vector fields of de Sitter spacetime are ex- ∂ µ pressed in the coordinate basis ∂µ = ∂xµ (where x are the conformal coordinates introduced in the last{ section) by}27: { } µ Πα = ξα∂µ (61) with (putting ℓ = 1, for simplicity) are η xν xµ σ2 ξµ = δµ ( αν δµ). (62) α α − 2 − 4 α 2 ν 0 2 0 0 σ ηαν x x σ 0 ξα = δα ( 4 2 4 δα), − 2 ν 1 − 2 1 1 σ ηαν x x σ 1 ξα = δα ( 4 2 4 δα), − 2 ν 2 − 2 (63) 2 2 σ ηαν x x σ 2 ξα = δα ( 4 2 4 δα), − 2 ν 3 − 2 ξ3 = δ3 ( σ ηαν x x σ δ3 ). α α − 4 2 − 4 α So, we want now to investigate if there is some region of de Sitter spacetime where the “translational” Killing vector fields are linearly independent. µ In order to proceed, we take an arbitrary vector field V = V ∂µ sec T . 3 ∈ U If the Πα are linearly independent in a region ′ R S then we can write U ⊂U ⊂ × µ α α µ V = V ∂µ = V Πα = V ξα∂µ. (64) So, the condition for the existence of a nontrivial solution for the V α is that 0 1 2 3 ξ0 ξ0 ξ0 ξ0 0 1 2 3  ξ1 ξ1 ξ1 ξ1  det 0 1 2 3 = 0 (65) ξ ξ1 ξ ξ 6  2 1 1   ξ0 ξ1 ξ2 ξ3   3 3 3 3  26More details in Section 5.1. 27We are using here a notation similar to the ones in [32] for compariosn of some of our results with the ones obtained there.

19 Thus, putting σ2 (xµ)2 σ2 χ := ( ) µ 4 2 − 4 we need to evaluate the determinant of the matrix

σ2 tx σ2 ty σ2 tz 1 χ0 ( 4 2 ) ( 4 2 ) ( 4 2 )  −σ2 xt σ2 xy σ2 xz  ( 4 2 ) 1 χ1 ( 4 2 ) ( 4 2 ) − σ2 yt −σ2 yx − − yz . (66)  ( 4 2 ) ( 4 2 ) 1 χ2 ( m2 )   − − − − 4 2   σ2 zt σ2 zx σ2 zy   ( ) ( ) ( ) 1 χ3   − 4 2 − 4 2 − 4 2 −  Then

1 3 1 3 1 det[ξµ]= σ10 σ6 + σ6 + σ2 +1 σ8 σ4 α −512 − 32 16 − 128 − 8 1 1 + σ8(3t2x2y2z2 + t2y2z2 x2y2z2 + y2z2)+ σ8(t2x2y2 + t2x2z2) 1024 − 512 1 1 + σ6t2x2(y2 + z2)+ σ6(t2y2z2 x2y2z2 +2y2z2) 128 256 − 1 1 + σ4(t2x2y2z2 + y2z2 x2y2z2)+ σ2(t2y2z2 2 x2y2z2 2y2z2) 64 − 16 − − 3 1 1 + σ4 y2z2 + σ8. (67) 8 − 4 256 In order to analyze this expression we put (without loss of generality, see the reason in [42]) y = z = 0. In this case

1 3 det[ξµ]= σ2 +4 σ4 +2σ2 +8 (68) α 512 −

which is null on the Caley-Klein absolute, i.e., at the points t2 ~x2 =4. (69) − and also on the spacelike hyperbolas given by t2 ~x2 = 2 and t2 ~x2 = 4 − − − − So, we have proved that the translational Killing vector fields are linearly independent in all the sub region inside the Cayley-Klein absolute except in the points of the hyperbolas t2 ~x2 = 2 and t2 ~x2 = 4. − − − − This result is very much important for the following reason. Timelike geodesics in de Sitter spacetime structure M dSL are (as well known) the curves σ : s σ(s), where s is propertime along σ, that extremizes the length function [28]→ , i.e., calling σ = u the velocity of a “particle” of mass m following a time like geodesic we have∗ that the equation of the geodesic is obtained by finding an extreme of the action, writing here in sloop notation, as

µ ν 1 I[σ]= m ds = m (gµν dx dx ) 2 . (70) − Zσ − Zσ

20 As well known, the determination of an extreme for I[σ] is given by evaluating the first variation28 of I[σ], i.e.,

0 µ ν 1 . δ I[σ]= m £Y (gµν dx dx ) 2 (71) Z and putting δ0I[σ] = 0. The result, as well known is the geodesic equation

Duu =0. (72)

Now, taking into account that the Killing vector fields determines a basis inside the Cayley-Klein absolute we write

µ α α µ u = u ∂µ = U Πα = U ξα∂µ. (73)

29 β We now define the “hybrid” connection coefficients Γ µ··α by: · β α α β D∂µ Πα := Γ µ··αΠβ , D∂µ Π := Γ µβ·· Π (74) · − · and write the geodesic equation as

µ α Duu = u D∂µ(U Πα) µ α µ α = (u ∂µU )Πα + u U D∂µΠα µ β µ α β = (u ∂µU )Πβ + u U Γ µα·· Πβ · β dU µ α β = ( + u U Γ µα·· )Πβ =0. (75) ds · or β dU µ α β + u U Γ µα·· =0, (76) ds · β β β which on multiplying by the “mass” m and calling π = mU and πρ = gρβπ can be written equivalently as

dπρ β µ Γ µ··ρu πβ =0, (77) ds − · which looks like Eq.(37) in [32]. We want now to investigate the question: is Eq.(77) the same as Eq.(37) in [32]? To know the answer to the above question recall that in [32] authors investi- gate the variation of I[σ] under a variation of the curves σ(s) σ(s,ℓ) induced µ µ µ 7→ by a coordinate transformation x x′ + δΠx where they put 7→ µ µ ρ δΠx = ξρ (x)δx (78)

28In Eq.(71) Y is the deformation vector field determining the curves σ(s, l) necessary to calculate the first variation of I[σ]. 29 Γβ·· β·· β·· Take notice that ·µα 6= Γ·µα where D∂µ ∂α = Γ·µα∂β, for otherwsise confusion will arise.

21 µ with ξρ (x) being the components of the Killing vector fields Πα (recall Eq.(62)) and where it is said that δxρ is an ordinary variation. However in [32] we cannot find what authors mean by ordinary variation, and so δxρ is not defined. So, to continue our analysis we recall, that if the δxρ = εα are constants (but µ arbitrary) then δΠx corresponds to a diffeomorphism generated by a Killing α ρ ρ vector field Π = ε Πα. However, if the δx = λ (x) are infinitesimal ar- ρ µ bitrary functions (i.e., λ (x) << 1), then the notation δΠx is misleading µ ρ 0 | | since ξρ λ (x) is the δ I[σ] variation generated by a quite arbitrary vector field ρ µ ρ µ 0 Y = λ (x)ξρ (x)∂µ = λ (x)Πρ = Y ∂µ. In this case we get from δ I[σ]=0 the geodesic equation.

4.1 Curves Obtained From Constrained Variations. So, let us study the constrained variation when the δxρ = εα are constants (but arbitrary). We denote the constrained variation by δcI[σ]. In this case starting from Eq.(70) c γ β ρ δ I[σ]= m u (Dγuβ)ξρ δx ds (79) − Zσ{ } where (taking account of notations already introduced) it is

γ β γ β u Deγ (uβe )= u (Dγ uβ)e , τ Dγuβ = ∂γ uβ Γ γβ·· uτ , − · τ β β τ Deγ eβ =Γ γβ·· eτ , Deγ e = Γ γτ·· e , (80) · − · we can write

c γ β ρ δ I[σ]= m u (Dγ uβ)ξ δx ds − Z { ρ } γ β ρ m γ β γ ρ = m u (Dγ(u ξρβ)) δx ds + u u (Dγ ξβρ + Dβξ δx ds − Z { { } 2 Z { ρ } γ ρ γ ρ = m u (Dγ(u·Πρ)) δx ds = m [u Dγ( ρ) δx ds. (81) − Z { } − Z U } and δcI[σ] = 0 implies γ u Dγ (mUρ)=0. (82)

Eq.(82) with πρ = mUρ can be written as

d γ β·· πρ u πβΓ γρ = 0 (83) ds − · which is Eq.(37) in [32]. Note that this equations looks like the geodesic equa- tion written as Eq.(77) above, but it is in fact different since of course, recalling β·· β·· Eq.(74) it is Γ γρ = Γ γρ. In [32] πρ = mUρ is unfortunately wrongly inter- preted as the components· 6 · of a covector field over σ supposed to be the energy- momentum covector of the particle, because authors of [32] supposed to have proved that this equation could be derived from Papapetrou’s method, what is not the case as we show in Section 6.

22 5 Generalized Energy-Momentum Conservation Laws in de Sitter Spacetime Structures 5.1 Lie Algebra of the de Sitter Group

5 Given a structure (M˚ R ,˚g, ) introduced in Section 3 define JAB sec T M˚ by ≃ ∈ ∂ ∂ J := η XC η XC . (84) AB AC ∂XB − BC ∂XA These objects are generators of the Lie algebra so(1, 4) of the de Sitter group. µ Using the bases ∂µ , dx introduced above the ten Killing vector fields of de Sitter spacetime{ are} the{ fields} J sec TM and J sec TM and it is30 α4 ∈ µν ∈ ∂ ∂ 1 J = η XC η XC = ℓP K (85) α4 αC ∂X4 − 4C ∂Xα α − 4ℓ 4α 1 = ℓ∂ (2η xλxν σ2δν )∂ . α − 4ℓ αλ − α ν ∂ ∂ J = η Xκ η Xκ = η xλP η xλP . (86) µν µκ ∂Xν − νκ ∂Xµ µλ ν − νλ µ

The Jα4 sec TM and Jµν sec TM satisfy the Lie algebra so(1, 4) of the de Sitter group,∈ this time acting∈ as a transformation group acting transitively in de Sitter spacetime. We have

[Jα4, Jβ4]= Jαβ, [J , J ]= η J η J , αβ λ4 λβ α4 − λα β4 [J , J ]= η J + η J η J η J . (87) αβ λτ αλ βτ βτ αλ − βλ ατ − ατ βλ It is usual in physical applications to define

Πα = Jα4/ℓ (88) for then we have 1 [Π , Π ]= J , α β ℓ2 αβ [J , Π ]= η Π η Π , αβ λ λβ α − λα β [J , J ]= η J + η J η J η J . (89) αβ λτ αλ βτ βτ αλ − βλ ατ − ατ βλ

The Killing vector fields Jαβ satisfy the Lie algebra so(1, 3) (of the special Lorentz group).

Remark 14 From Eq.(89) we see that when ℓ the Lie algebra of so(1, 4) goes into the Lie algebra of the Poincar´egroup P7→which ∞ is the semi-direct sum of the group of translations in R4 plus component of the special Lorentz group,

30 Proofs of Eqs.(85) and (86) are in Appendix F. Of course, in those equations, it is P α = eα (as introduced is Section 2).

23 i.e., P = (R4 ⊞ SO(1, 3)). This is eventually the justification for physicists to call the Πα the “translation” generators of the de Sitter group. However, it is necessary to have in mind that whereas the translation sub- 4 group R of P acts transitively in Minkowski spacetime manifold, the set Πα does not close in a subalgebra of the de Sitter algebra and thus it is impossi-{ } α 3 ble in general to find exp(λ Πα) such that given arbitrary x, y R S it is y = exp(λαΠ )x. Only the whole group SO(1, 4) acts transitively∈ on ×R S3. α ×

Casimir Invariants Now, if EA = dXA is an orthonormal basis for the structure R1,4 = (R5,˚g) define the{ angular momentum} operator as the Clifford algebra valued operator 1 J = EA EBJ . (90) 2 ∧ AB Taking into account the results of Appendix A its (Clifford) square is

J2 = JyJ + J J = J J + J J. (91) ∧ − · ∧ It is immediate to verify that J2 is invariant under the transformations of the de Sitter group. JyJ is (a constant appart) the first invariant Casimir operator of the de Sitter group. The second invariant Casimir operator of the de Sitter is related to J J. Indeed, defining ∧ 1 1 W := ⋆ (J J)= (J J)yτ˚g. (92) ˚g 8ℓ ∧ 8ℓ ∧ one can easily show (details in [39]) that 1 1 W W = WW = W2 = (J J)y(J J)= (J J) (J J) · −64ℓ2 ∧ ∧ −64ℓ2 ∧ · ∧ 1 = (J J)(J J). (93) −64ℓ2 ∧ ∧ is indeed an invariant operator. As well known, the representations of the de Sitter group are classified by their Casimir invariants I1 and I2 which here following [14] we take as

1 1 I = JyJ = J JAB = ηαβΠ Π + ηαληβτ J J = M 2, 1 − 2ℓ2 AB α β 2ℓ2 αβ λτ 1 I = W2 = W AW = ηαβV V + (W )2, (94) 2 A α β ℓ2 4 where M R and the fields W and V are defined by: ∈ A α 1 W := ε JAB JDE A 8ℓ ABCDE 1 V := ε ηλρΠ Jµν , (95) α −2 4αλµν ρ

24 from where it follows that 1 W = ε Jµν Jµν . (96) 4 8 4µνρτ In the limit when ℓ we get the Casimir operators of the special Lorentz group 7→ ∞

I P P α = m2, 1 7→ α I ηαβV V = m2s(s +1) (97) 2 7→ α β where m R and s =0, 1/2, 1, 3/2, ... ∈ We see that Πα looks like the components of an energy-momentum vector α P = Pαϑ o of a closed physical system (see Eq.(25)) in the Minkowski space- time of Special| Relativity, for which P 2 = m2, with m the mass of the system. However, take into account that whereas the Pα are simple real numbers, the Πα are vector fields. αβ Moreover, take into account that η ΠαΠβ is not an invariant, i.e., it does not commute with the generators of the Lie algebra of the de Sitter group.

5.2 Generalized Energy-Momentum Covector for a Closed System in the Teleparallel de Sitter Spacetime Struc- ture

3 In this section we suppose that (M = R S , g, τg, ) is the physical arena where Physics take place. × ↑ We know that (M = R S3, g) has ten Killing vector fields and four of them × (one timelike and three spacelike, Πα, α =0, 1, 2, 3 ) generated “translations”. 3 Thus if we suppose that (M = R S , g, τg, ) is populated by interacting matter × ↑ fields φ1,..., φn with dynamics described by a Lagrangian formalism we can construct{ as described} in the Appendix C.1 the conserved currents

β µ Π = γ = ϑ (98) J α Jα β Jα µ 0 were taking into account that δΠ = £Π and denoting by δΠ each particular α − α α local variation generated by Πα (recall Eq.(61)) we have

0 ν Π δ α φA = δΠα φA + δx ∂ν φA (99) and thus we write

µ µ µ 0 µ δΠ δ α = πA α φA +Υν Πα x J µ ν µ ν =Λν ξα +Υν ξα (100) where µ ν µ Π Λν ξα := πAδ α φA. (101)

25 In Eq.(98) γβ is the dual basis of the orthonormal basis e defined by31 { } { α} Πα eα := . (102) g(Πα, Πα)

From the conserved currents Π we can obtain four conserved quantities (the J α Pα in Eq.(26)).

Remark 15 It is crucial to observe that the above results have been deduced 3 without introduction of any connection in the structure (M = R S , g,τg, ). × ↑ However that structure is not enough for using the Pα to build a (generalized) covector analogous to the energy-momentum covector P (see Eq.(25)) of special relativistic theories.

3 If we add D, the Levi-Civita connection of g to (M = R S , g,τg, ) we get the Lorentzian de Sitter spacetime structure M dSP and defining× a generalized↑ energy-momentum tensor

α 1 Θ = Π e sec T M (103) J α ⊗ ∈ 1 we know that δ Π = 0 implies D Θ = 0, a covariant “conservation” law. gJ α · However, the introduction of M dSP in our game is of no help to construct a covector like P since in M dSP vectors at different spacetime points cannot be directly compared.. So, the question arises: is it possible to define a structure where x, y R S3 we can define objects P dS such that ∀ ∈ × α P = P dS γα = P dS γα (104) dS α |x α |y defines a legitimate covector for a closed physical system living in a de Sitter 3 structure (M = R S , g,τg, ) and for which P dS can be said to be a kind of generalization of the× momentum↑ of the closed system in Minkowski spacetime? We show now the answer is positive. We recall that D has been introduced in our developments only as a useful mathematical device and is quite irrel- evant in the construction of legitimate conservation laws since the conserved currents Πα have been obtained without the use of any connection. So, we now introduceJ for our goal a teleparallel de Sitter spacetime, i.e., the structure dSTP 3 M = (M = R S , ∇, τg, ) where ∇ is a metric compatible teleparallel connection defined by× ↑ κ ∇eα eβ = ω αβ·· eκ =0. (105) · Under this condition we know that we can identify all tangent and all cotangent spaces. So, we have for x, y R S3, ∀ ∈ × e e and γα γα = Eα, α|x ≃ α|y |x ≃ |y 31 3 Recall that the fields eα are only defined in subset {S −north pole}.

26 where Eα is a basis of a vector space R4. Thus,{ in} the structure M dSTP Eq.(104)V ≃ defines indeed a legitimate covector in R4 and thus permits a legitimate generalization of the concepts of energy- momentumV ≃ covector obtained for physical theories in Minkowski spacetime. The term generalization is a good one here because in the limit where ℓ , Π ∂/∂xα, Λκ = 0 and thus Θκ =Υκ . → ∞ α 7→ α α α

5.3 The Conserved Currents J Πα = g(Πα, ) To proceed we show that the translational Killing vector fields of the de Sitter structure (M, g) determines trivially conserved currents

J Πα = g(Πα, ) Indeed using the M dSTP structure as a convenient device we recall the result proved in [35] that for each vector Killing K the one form field K = g(K, ) is such that δK = 0. So, it is g δJ Π =0. (106) g α and of course, also δJ Π =0 (107) g where α J Π := g(Π, ), Π := ε Πα (108) with the εα real constants such that εα << 1. Recalling from Eq.(86) that in projective conformal coordinate bases| | ( e = ∂ and ϑµ = dxµ ) the { µ µ} { } components of Πα are 1 ξµ = δµ (2η xρxµ σ2δµ) (109) α α − 4ℓ2 αρ − α we get the conserved current J Π is

α α µ J Π = ε J Πα = ε ξαϑµ. (110)

Remark 16 Take notice that of course, this current is not the conserved current that we found in the previous section.

Remark 17 Take notice also that in [32] authors trying to generalize the re- sults that follows from the canonical formalism for the case of field theories µ in Minkowski spacetime suppose that they can eliminate the term πAδφA from Eq.(100) by decree postulating a “new kind of “local variation”, call it δ′ for which δ′φA = 0. The fact is that such a “new kind of local variation” never appears in the canonical Lagrangian formalism, only δφA appears and in general it is not zero.

27 Now, the generalized canonical de Sitter energy-momentum tensor is 1 α κ α Θ = Πα e =Θαϑκ e sec T ∗M TM (111) J ⊗ ⊗ ∈ ^ ⊗ κ and making analogy with the case of Minkowski spacetime where the Υα have been defined as the components of the canonical energy-momentum tensor32 Υ, κ we write Θα, as 1 Θκ =Υκ Kκ +Λκ , (112) α α − 4ℓ2 α α Thus each one of the genuine conservation laws d⋆ Π = 0, α =0, 1, 2, 3 read g J α in coordinate basis components

κ λ κ 1 µ λ κ κ λ κ (∂µΥα Γ λν·· Υα) (∂µKα Γ λν·· Kα) (∂µΛα Γ λν·· Λα) − · − 4ℓ2 − · − − · 1 µ 1 1 µ = ∂µ( det gΥ ) ∂µ( det gK ) (113) √ det g − α − 4ℓ2 √ det g − α − p − p 1 µ + ∂µ( det gΛ )=0 √ det g − α − p or 1 ∂ det g Υµ Kµ +Λµ =0 D Θµ =0. (114) µ   α 4ℓ2 α α µ α p− − ⇔ Remark 18 Now, recalling the relation of the connection coefficients of the κ bases eα of the Levi-Civita connection D of g (denoted Γ αβ·· ) and the coeffi- { } · ¯κ cients of the basis eα of the teleparallel connection of g (denoted Γ αβ·· ) are [34] { } ∇ · ¯κ κ κ Γ αβ·· = Γ αβ·· + αβ·· (115) · · △· where κ 1 κ κ κ αβ·· := Tα· ·β +Tβ· ·α T αβ·· (116) △· −2 · · − · κ
are the components of the contorsion tensor and T αβ·· are the components of the torsion tensor of the connection , we can write D· Θ = 0 (taking into account ¯κ ∇ • that Γ αβ·· = 0) in components relative to orthonormal basis as · α α α α ι ι α D Θ := (De Θ) = e (Θ )+ Θ Θ =0. α β α β α β △αι β −△αβ ι α α α On the other hand since Θ := ( e Θ) = e (Θ ) we have ∇α β ∇ α β α β Θα = α Θι + ι Θα, (117) ∇α β −△αι β △αβ ι which means that although ( Θ)α := Θα =0 we can generate the conserved ∇• β ∇α β 6 currents J Πα in the teleparallel de Sitter spacetime structure if Eq.(117) is satisfied. Remark 19 To end this section and for completeness of the article it is nec- essary to mention that in a remarkable paper ([9]) the gravitational energy- momentum tensor in teleparallel gravity is discused in details. Also related papers are ([10, 24]). A complete list of references can be found in ([1])

32 κ The explict form of the Kα can be determined withouth difficulty if needed.

28 6 Equation of Motion for a Single-Pole Mass in a GRT Lorentzian Spacetime

In a classical paper Papapetrou derived the equations of motion of single pole and spinning particles in GRT. Here we recall his derivation for the case of a single pole-mass. We start recalling that in GRT the matter fields is described by a energy-mometum tensor that satisfies the covariant conservation law D T = 0. If we introduce the relative tensor •

1 4 T =T ⊗ τ sec T M T ∗M (118) g ∈ 1 ⊗ V we have recalling Appendix E and that D T = 0 that • µν µ αν Dν T +Γ να·· T =0. (119) · From Eq.(119) we have

α µν µα α µ αν ∂ν (x T )= T x Γ να·· T . (120) − · To continue we suppose that a single-pole mass (considered as a probe particle) is modelled by the restriction of the energy-mometum tensor T inside a “narrow” tube in the Lorentzian spacetime representing a given gravitational field. Let us call T 0 that restriction and notice that D T 0 = 0. Inside the tube a timelike line γ is chosen to represent the particle motion.• We restrict our analysis to hyperbolic Lorentzian spacetimes for which a foliation R ( a 3-dimensional manifold) exists. We choose a parametrization for γ such that×S theS its coordinates are xµ(γ(t)) = Xµ(t) where t = x0. The probe particle is characterized by taking the coordinates of any point in the world tube to satisfy

δxµ = xµ Xµ << 1. (121) − According to Papapetrou a single-pole particle is one for which the integral33

Tµν dv =0, (122) Z o 6 and all other integrals

δxαTµν dv, δxαδxβTµν dv, ... (123) Z o Z o are null. We now evaluate

d µ0 µ αν To = Γ να·· To dv (124) dt Z − Z · and d α µ0 µα α µ αν x To = To dv x Γ να·· To dv. (125) dt Z Z − Z · 33The integration is to be evaluated at a t = const spaceline hypersurface S.

29 Inside the world tube modelling the particle we can expand the connection coefficients as µ µ µ κ Γ να·· = oΓ να·· + oΓ να,κ·· δx (126) · · · µ with oΓ να·· the components of the connection in the worldline γ. Then according to the definition· of a single-pole particle we get from Eq.(124) and Eq.(125) along γ: d µ0 µ αν To + oΓ να·· To =0, (127) dt Z · Z dXµ Tµα = Tν0dv. (128) Z o dt Z o Now, put dXµ γ = u := eµ (129) ∗ ds where ds is proper time along γ and define

M µα = u0 Tµαdv. (130) Z o Eq.(127) and Eq.(128) become

µ0 d M µ αν + oΓ να·· M =0 (131) ds  u0  · and M α0 M µα = uµ . (132) u0 µ0 µ M 00 So, M = u u0 from where it follows that putting M 00 m := (133) (u0)2 it is M α0 M µα = uµ = muµuα (134) u0 and we get d µ µ ν α (mu )+ oΓ να·· mu u =0. (135) ds ·

Now, the acceleration of the probe particle is a := Duu and thus g(a,u)= 0, i.e., d µ µ ν α uµ u + oΓ να·· mu u uµ =0. (136) ds ·

Multiplying Eq.(135) by uµ and using Eq.(136) gives

d m =0 (137) ds

30 and then Eq.(135) says that γ is a geodesic of the Lorentzian spacetime struc- ture, i.e.,

Dγ∗ γ =0 (138) ∗ or µ du µ ν α + oΓ να·· mu u =0. ds · 7 Equation of Motion for a Single-Pole Mass in a de Sitter Lorentzian Spacetime

In this section we suppose that the arena where physical events take place is the de Sitter spacetime structure M dSℓ where fields live and interact, without never changing the metric g, which we emphasize do not represent any gravitational field here, i.e., we do not suppose here that M dSℓ is a model of a gravitational field in GRT. As we learned in Section 4 since de Sitter spacetime has one timelike and three spacelike Killing vector fields Πα we can construct the con- µ served currents Πα =Θαϑµ (see Eq.(98)) from where we get D Θ = 0 with α J κ α • Θ = Πα e =Θαϑκ e . Now,J if⊗ we suppose that⊗ a probe free single-pole particle (i.e., one for which its interaction with the remaining fields can be despised) is described by a covariant conserved tensor Θ in a narrow tube like the one introduced in the previous section, we can derive (using analog notations for γ, etc...) an equation like Eq.(127), i.e.,

d µ0 µ αν Θo det godv + oΓ να·· Θo det godv =0. (139) dt − · − Z p Z p Now, we obtain an equation analogous to Eq.(131) with M αν substituted by N αν i.e., µ0 d N µ αν + oΓ να·· N =0 (140) ds  u0  · with N αν := u0 Θµα det g dv. (141) o − o Z p Putting this time N 00 m := (142) (u0)2

µα µ N α0 µ α we get N = u u0 = mu u and

d µ µ ν α (mu )+ oΓ να·· mu u =0, (143) ds · from where we get exactly as in the previous section that m = const and

Dγ∗ γ = 0. ∗

31 Conclusion 20 Papapetrou method applied to the M dSℓ structure gives for the motion of a free single pole particle the geodesic equation Dγ∗ γ =0. Moreover, Eq.(143) is of course, different from Eq.(83) contrary to conclusions∗ of authors of [32]. It is also to be noted here that in [32] authors inferred correctly that Papapetrou method leads to an equation that looks like to Eq.(139) for a single- pole particle moving in the M dSℓ structure. The equation that looks like Eq.(139) αν αν in [32] is the Eq.(37) there, but where in the place of Θo they used Υo 1 αν − 4ℓ2 Υo , because they believe to be possible to use a local variation of the fields µ that results in Λν =0.

A Clifford Bundle Formalism

Let (M, g,D,τg, ) be an arbitrary Lorentzian or Riemann-Cartan spacetime ↑ structure. The quadruple (M, g, τg, ) denotes a four-dimensional time-oriented ↑ 0 and space-oriented Lorentzian manifold [34, 43]. This means that g sec T2 M 4 ∈ is a Lorentzian metric of signature (1, 3), τg sec T ∗M and is a time- ∈ ↑ orientation (see details, e.g., in [43]). Here, T ∗M [TMV ] is the cotangent [tan- 1,3 gent] bundle. T ∗M = x M Tx∗M, TM = x M TxM, and TxM Tx∗M R , where R1,3 is the Minkowski∪ ∈ vector space∪34.∈D is a metric compatible≃ connec-≃ tion, i.e., Dg = 0. When D = D is the Levi-Civita connection of g , RD = 0, and ΘD = 0, RD and ΘD being respectively the curvature and torsion6 ten- sors of the connection. D = ∇ is is a Riemann-Cartan connection , R∇ = 0, ∇ 35 2 6 and Θ = 0 Let g sec T0 M be the metric of the cotangent bundle. The Clifford bundle6 of differential∈ forms ℓ(M, g) is the bundle of algebras, i.e., C ℓ(M, g)= x M ℓ(Tx∗M, g), where x M, ℓ(Tx∗M, g)= R1,3, the so called Cspacetime algebra∪ ∈ C[34]. Recall also that∀ ∈ℓ(M, gC) is a vector bundle associated to C 36 the orthonormal frame bundle, i.e., we have ℓ(M, g)= PSOe (M) Ad′ R1,3 C (1,3) × [21, 25]. Also, when (M, g) is a spin manifold we can show that37 ℓ(M, g) C = PSpine (M) Ad R1,3 For any x M, ℓ(T ∗M, g ) as a linear space over the (1,3) × ∈ C x |x real field R is isomorphic to the Cartan algebra Tx∗M of the cotangent space. 4 k k 4 We have that ∗M = T ∗M, where V T ∗M is the -dimensional x ⊕k=0 x x k space of k-forms.V Then, sectionsV of ℓ(M, g) canV be represented as
a sum of non homogeneous differential forms, thatC will be called Clifford (multiform) fields. In the Clifford bundle formalism, of course, arbitrary basis can be used, but in this short review of the main ideas of the Clifford calculus we use mainly orthonormal basis. Let then e be an orthonormal basis for TU TM, i.e., { α} ⊂ 34Not to be confused with Minkowski spacetime [34, 43]. 35Minkowski spacetime is the particular case of a Lorentzian spacetime structure for which M ≃ R4. and the curvature and torsion tensors of the Levi-Civita connection of Minkowski metric are null. a teleparallel spacetime is a particualr Riemann-Cartan spacetime such that RD = 0, and ΘD 6= 0. 36 e R −1 Ad: Spin1,3 → Aut( 1,3) by Adux = uxu . 37 e R R −1 Take notice that Ad : Spin1,3 → Aut( 1,3) such that for any C ∈ 1,3 it is AduC =uCu . e Since Ad−1 = Ad1 = identity, Ad descends to a representation of SO1,3 that we denoted by Ad′.

32 α 1 (g(eα, eβ) = ηαβ = diag(1, 1, 1, 1). Let γ sec T ∗M ֒ sec ℓ(M, g − − − α ∈ → C (α =0, 1, 2, 3) be such that the set γ is the dual basisV of eα . Also, γα α { }α α α { } { } is the reciprocal basis of γ , i.e., g(γ , γβ) = δβ and e is the reciprocal basis of e , i.e., g(eα, e{ )=} δα. { } { α} β β A.1 Clifford Product The fundamental Clifford product (in what follows to be denoted by juxtaposi- tion of symbols) is generated by

γαγβ + γβγα =2ηαβ (144) and if sec ℓ(M, g) we have C ∈ C 1 1 = s + v γα + f γαγβ + t γαγβγγ + pγ5 , (145) C α 2! αβ 3! αβγ 5 0 1 2 3 where τg = γ = γ γ γ γ is the volume element and s, vα, fβ, tαβγ, p 0 ∈ .(sec T ∗M ֒ sec ℓ(M, g → rC s VFor Ar sec T ∗M ֒ sec ℓ(M, g),Bs sec T ∗M ֒ sec ℓ(M, g) we ∈ → C ∈ → C define the exteriorV product in ℓ(M, g) ( r, s =0, 1,V2, 3) by C ∀ A B = A B , (146) r ∧ s h r sir+s k where k is the component in T ∗M of the Clifford field. Of course, h i rs Ar Bs = ( 1) Bs Ar, and the exteriorV product is extended by linearity to all sections∧ of− ℓ(M, g∧). C r s .(Let A sec T ∗M ֒ sec ℓ(M, g),B sec T ∗M ֒ sec ℓ(M, g r ∈ → C s ∈ → C We define a scalarV product in ℓ(M, g) (denoted by )V as follows: 1 C · ,(i) For a,b sec T ∗M ֒ sec ℓ(M, g) ∈ → C V 1 a b = (ab + ba)= g(a,b). (147) · 2

1 ֒ ii) For A = a a ,B = b . b , a ,b sec T ∗M) r 1 ∧···∧ r r 1 ∧ · · · ∧ r i j ∈ → sec ℓ(M, g), i, j =1, ..., r, V C A B = (a a ) (b b ) r · r 1 ∧···∧ r · 1 ∧···∧ r a b .... a b 1 · 1 1 · r = ...... (148)

ar b1 .... ar br · ·

We agree that if r = s = 0, the scalar product is simply the ordinary product in the real field. Also, if r = s, then Ar Bs = 0. Finally, the scalar product is extended by linearity for all6 sections of · ℓ(M, g). C

33 For r s, A = a a , B = b b , we define the left contraction ≤ r 1 ∧···∧ r s 1 ∧···∧ s y : (Ar,Bs) AryBs by 7→ g

i1...is AryBs = ǫ (a1 ar) (b ... bi )∼bi +1 bi (149) g ∧···∧ · i1 ∧ ∧ r r ∧···∧ s i1 <...X

0 We agree that for α, β sec T ∗M the contraction is the ordinary (pointwise) ∈ 0 r product in the real field andV that if α sec T ∗M, Xr sec T ∗M, Ys s ∈ ∈ ∈ sec T ∗M ֒ sec ℓ(M, g) then (αXr)yBsV= Xry(αYs). LeftV contraction → C is extendedV by linearity to all pairs of sections of ℓ(M, g), i.e., for X, Y sec ℓ(M, g) C ∈ C XyY = X ry Y s, r s. (151) g h i gh i ≤ Xr,s It is also necessary to introduce the operator of right contraction denoted by x . The definition is obtained from the one presenting the left contraction with g r the imposition that r s and taking into account that now if Ar sec T ∗M, s ≥ ∈ Bs sec T ∗M then Arx(αBs) = (αAr)xBs. See also the third formulaV in ∈ g g Eq.(152).V The main formulas used in this paper can be obtained from the following ones

a s = ay s + a s, sa = sxa + s a, B B ∧B B B g B ∧ 1 s ay s = (a s ( 1) sa), gB 2 B − − B r(s r) ry s = ( 1) − sx r, A gB − B gA 1 a = (a + ( 1)s a), ∧Bs 2 Bs − Bs r s = r s r s + r s r s +2 + ... + r s r+s A B hA B i| − | hA B i| − | hA B i| | m

= r s r s +2k hA B i| − | Xk=0

r r = r r = ry r = rx r = r r 0 = r r 0. (152) A ·B B · A A gB A gB hA B i hA B i e e e e Two other important identities used in the main text are:

34 ay( ) = (ay ) + ˆ (ay ), (153) g X ∧ Y gX ∧ Y X ∧ gY y( y ) = ( )y , (154) X g Y gZ X ∧ Y gZ

1 .(for any a sec T ∗M ֒ ℓ(M, g) and , , sec T ∗M ֒ ℓ(M, g ∈ ^ →C X Y Z ∈ ^ →C A.1.1 Hodge Star Operator k 4 k − Let ⋆ be the Hodge star operator, i.e., the mapping ⋆ : T ∗M T ∗M, g g → k ^ ^ Ak ⋆Ak. For Ak sec T ∗M ֒ sec ℓ(M, g) we have 7→ g ∈ → C V k (Bk Ak]τg = Bk ⋆Ak, Bk sec T ∗M ֒ sec ℓ(M, g). (155] g · ∧ ∀ ∈ ^ → C 5 4 where τg = θ sec T ∗M ֒ sec ℓ(M, g) is a standard volume element. We ∈ → C have, V ⋆ Ak = Akτg = Akyτg. (156) g g e e where as noted before, in this paper k denotes the reverse of k. Eq.(156) permits calculation of Hodge duals veryA easily in an orthonormal basisA for which 5 α e 1 τg = γ . Let ϑ be the dual basis of e (i.e., it is a basis for T ∗U T ∗U) { } { α} ≡ which is either an orthonormal or a coordinate basis. Then writing g(ϑVα, ϑβ)= αβ αβ β µ1...µp µ1 µp νp+1...νn νp+1 νn g , with g gαρ = δρ , and ϑ = ϑ ϑ , ϑ = ϑ ϑ we have from Eq.(156) ∧···∧ ∧···∧

µ1...µp 1 µ1ν1 µpνp νp+1...νn ⋆ ϑ = det g g g ǫν ...ν ϑ . (157) g (n p)! | | ··· 1 n − p where det g denotes the determinant of the matrix with entries gαβ = g(eα,eβ), 1 i.e., det g = det[gαβ]. We also define the inverse ⋆− of the Hodge dual operator, g 1 1 such that ⋆− ⋆ = ⋆⋆− = 1. It is given by: g g gg

r n r 1 − ⋆− : sec T ∗M sec T ∗M, g ^ → ^ 1 r(n r) ⋆− Ar = ( 1) − sgn det g ⋆ Ar, (158) g − g where sgn det g = det g/ det g denotes the sign of the determinant of g. Some useful identities| (used| in the text) involving the Hodge star operator, the exterior product and contractions are:

35 Ar ⋆Bs = Bs ⋆Ar; r = s ∧ g ∧ g Ar ⋆Bs = Bs ⋆Ar; r + s = n · g · g r(s 1) ˜ Ar ⋆Bs = ( 1) − ⋆ (AryBs); r s (159) ∧ g − g g ≤ rs Ary ⋆ Bs = ( 1) ⋆ (A˜r Bs); r + s n g g − g ∧ ≤ ⋆τg = sign g; ⋆ 1= τg. g g

A.1.2 Dirac Operator Associated to a Levi-Civita Connection D Let d and δ be respectively the differential and Hodge codifferential operators p = acting on sections of ℓ(M, g). If Ap sec T ∗M ֒ sec ℓ(M, g), then δAp C ∈ → C g p 1 V ( 1) ⋆− d⋆ Ap. − g g The Dirac operator acting on sections of ℓ(M, g) associated with the metric compatible connection D is the invariant firstC order differential operator

α ∂ = ϑ Deα , (160) where eα is an arbitrary (coordinate or orthonormal) basis for TU TM {α } β ⊂ α and ϑ is a basis for T ∗U T ∗M dual to the basis eα , i.e., ϑ (eα) = δβ , { } ⊂ α { } α, β = 0, 1, 2, 3. The reciprocal basis of ϑ is denoted ϑα and we have ϑ ϑ = g . Also, when e = ∂ and {ϑα }= dxα we have{ } α · β αβ { α α} { } ε β β ε D∂α ∂β = Γ αβ·· ∂β, D∂α dx = Γ αε·· dx (161) · − · and when e = e , ϑα = γα are orthonormal basis we have { α α} { } λ β β λ Deα eβ = ω αβ·· eλ , Deα γ = ω αλ·· γ (162) · − · We define the connection38 1-forms in the gauge defined by γα as { } α α λ ω β· := ω λβ·· γ . (163) · ·

µ1...µr ν1 νs Moreover, we write for an arbitrary tensor field Y = Yν1...νs ϑ ... ϑ ∂ ... ∂ in a coordinate basis ⊗ ⊗ ⊗ µ1 ⊗ ⊗ µr

µ1...µr ν1 νs De Y := (DαY )ϑ ... ϑ ∂µ ... ∂µ (164) α ν1...νs ⊗ ⊗ ⊗ 1 ⊗ ⊗ r

µ1...µr ν1 νs and also when we write Y = Y γ ... γ eµ ... eµ we also write ν1...νs ⊗ ⊗ ⊗ 1 ⊗ ⊗ r µ ...µ ν ν De Y := (D Y 1 r )γ 1 ... γ s e ... e (165) α α ν1...νs ⊗ ⊗ ⊗ µ1 ⊗ ⊗ µr so please pay attention when reading a particular formula to certificate the D µ1...µr meaning of αYν1...νs , i.e., if we are using in that formula coordinate or or- thonormal frames. 38Also called “spin connection 1-forms”.

36 We have also the important results (see, e.g., [34]) for the Dirac operator associated with the Levi-Civita connection D acting on the sections of the Clifford bundle39

∂Ap = ∂ Ap + ∂yAp = dAp δAp, (166a) ∧ − g

∂ Ap = dAp, ∂yAp = δAp. (166b) ∧ g −g

1 ,(We shall need the following identity valid for any a,b sec T ∗M ֒ ℓ(M, g ∈ →C V ∂(a b) = (a ∂)b + (b ∂)a + ay(∂ b)+ by(∂ a). (167) · · · ∧ ∧ A.2 Covariant D’ Alembertian, Hodge D’Alembertian and Ricci Operators The square of the Dirac operator ♦ = ∂2 is called Hodge D’Alembertian and we have the following noticeable formulas:

∂2 = dδ δd, (168) − g − g and ∂2A = ∂ ∂ A + ∂ ∂ A (169) p · p ∧ p where ∂ ∂ is called the covariant D’Alembertian and ∂ ∂ is called the Ricci ·40 1 µ µ ∧ operator If A = A ϑ 1 ϑ p , we have p p! µ1...µp ∧···∧

αβ ρ 1 αβ α1 αp ∂ ∂ Ap = g (D∂α D∂β Γ αβ·· D∂ρ )Ap = g DαDβAα1...αp ϑ ϑ , · − · p! ∧···∧ (170) Also for ∂ ∂ in an arbitrary basis (coordinate or orthonormal) ∧

1 α β ρ ρ ∂ ∂ Ap = ϑ ϑ ([Deα , Deβ ] (Γ αβ·· Γ βα·· )Deρ )Ap. (171) ∧ 2 ∧ − · − · In particular we have [34]

∂ ∂ ϑµ = µ, (172) ∧ R µ µ ν 1 ,where = Rν ϑ sec T ∗M ֒ sec ℓ(M, g) are the Ricci 1-form fields R µ ∈ → C such that if R νσµ··· are theV components of the Riemann tensor we use the con- · µ vention that Rνσ = R νσµ··· are the components of the Ricci tensor. · 39For a general metric compatible Riemann-Cartan connection the formula in Eq.(166b) is not valid, we have a more general relation involving the torsion tensor that will not be used in this paper. The interested reader may consult [34]. 40For more details concerning the square of Dirac (and spin-Dirac operators) on a general Riemann-Cartan spacetime, see [26].

37 Applying this operator to the 1-forms of the a 1-form of the basis ϑµ , we get: { } µ 1 µ α β ρ µ ∂ ∂ ϑ = R ραβ··· (ϑ ϑ )ϑ = ρ ϑρ. (173) ∧ −2 · ∧ R 1 is an extensor operator, i.e., for A sec T ∗M ֒ sec ℓ(M, g) it is ∂ ∂ ∧ ∈ → C V ∂ ∂ A = A ∂ ∂ϑµ. (174) ∧ µ ∧ Remark 21 We remark that covariant Dirac spinor fields used in almost all Physics texts books and research papers can be represented as certain equiva- lence classes of even sections of the Clifford bundle ℓ(M, g). These objects are now called Dirac-Hestenes spinor fields (DHSF ) andC a thoughtful theory de- scribing them can be found in [33, 25, 34]. Moreover, in [20] using the concept of DHSF a new approach is given to the concept of Lie derivative for spinor fields, which does not seems to have the objections of previous approaches to the subject. Of course, a meaningful definition of Lie derivative for spinor fields is a necessary condition for a formulation of conservation laws involving bosons and fermion fields in interaction in arbitrary manifolds. We will present the complete Lagrangian density involving the gravitation field (interpreted as fields in the Faraday sense and described by cotetrad fields), the electromagnetic and the DHSF living in a parallelizable manifold and its variation in another pub- lication.

B Lie Derivatives and Variations

In modern field theory the physical fields are tensor and spinor fields living on a structure (M, g, τg, ) and interacting among themselves. Note that at this point we did not introduce↑ any connection in our game, since according to our view (see, e.g., Chapter 11 of [34]) the introduction of a particular connection to describe Physics is only a question of convenience. For the objective of this paper we shall consider two structures (already introduced in the main text), dSℓ a Lorentzian spacetime M = (M, g, D, τg, ) where D is the Levi-Civita dSTP↑ connection of g and a teleparallel spacetime M = (M, g, , τg, ) where is a metric compatible teleparallel connection. Minkowski spacetime∇ ↑ structure∇ will be denoted by (M, η, D, τη, ). The equations of motion are derived from a variational principle once a given↑ Lagrangian density is postulated for the interacting fields of the theory. As well known, diffeomorphism invariance is a crucial ingredient of any phys- ical theory. This means that if a physical phenomenon is described by fields, say, φ1, ...., φn (defined in M) satisfying equations of motion of the theory (with appropriated initial andU ⊂ boundary conditions) then if h : M M is a dif- 7→ feomorphism then the fields h∗φ1, ...., h∗φN (where, h∗ is the pullback mapping) describe the same physical phenomenon41 in h . U 41 ∗ ∗ Of course, the fields h φ1,...., h φN must satisfy deformed initial conditions and deformed boundary conditions.

38 42 Suppose that fields φ1, ....,(in what follows called simply matter fields ) are arbitrary differential forms. Their Lagrangian density will here be defined as the functional mapping43

: (φ , ..., φ , dφ , ...., dφ ) (φ, dφ) sec 4TM (175) Lm 1 N 1 N 7→ Lm ∈ V where m(φ, dφ) is here supposed to be constructed using the Hodge star oper- ator ⋆.L The action of the system is g

= m(φ, dφ). (176) A Z L U Choose a chart of M covering and h with coordinate functions xµ . U U { } Then under an infinitesimal mapping hε : M M , x x′ = hε(x) generated by a one parameter group of diffeomorphisms7→ associated7→ to the vector field ξ sec TM we have (with hµ the coordinate representative of the mapping h ) ∈ ε ε µ µ µ µ µ α µ µ x = x (x) x′ = x (h (x)) = h (x )= x + εξ , ε << 1 (177) 7→ ε ε | |

In Physics textbooks given an infinitesimal diffeormorphism hε several different kinds of variations (for each one of the fields φi) are defined. Let φ be one of the fields φ1, ..., φN and recall that the Lie derivative of φ in the direction of the vector field ξ is given by

hε∗ φ hε φ £ξφ = lim ◦ ◦ − (178) ε 0 ε → As an example, take φ as 1-form. Then, in the chart introduced above using the definition of the pullback

κ hε∗φµ (x ) := [hε∗(φ(hε(x)))]µ (179) it is κ κ µ κ κ ∂x′ µ h∗φ (x) = h∗φ (x )dx := φ (x′ (x )) dx . (180) ε ε µ κ ∂xµ Then, Eq.(178) can be written in components as

κ κ κ hε∗φµ (x ) φµ(x ) (£ξφ(x )) = lim − (181) µ ε 0 → ε Now, to first order in ε we have ∂x κ ′ = δκ + ε∂ ξκ (182) ∂xµ µ µ and κ k κ κ κ α κ φκ(x′ (x )) = φκ(x + εξ )= φκ(x )+ εξ ∂αφκ(x ) (183)

42In truth, by matter fields we understand fields of two kinds, fermion fields (electrons, neutrinos, quarks) and boson fields (electromagnetic, gravitational, weak and strong fields). 43A rigorous formulation needs the introduction of jet bundles (see, e.g., [12]). We will not need such sophistication for the goals of this paper.

39 So,

κ κ κ κ ∂x′ h∗φ (x )= φ (x′ (x )) ε µ κ ∂xµ κ α κ κ κ = (φκ(x )+ εξ ∂αφκ(x )) (δµ + ε∂µξ ) κ κ κ α κ = φµ(x )+ ε∂µξ φκ(x )+ εξ ∂αφµ(x ) (184) Then, κ κ κ α κ (£ξφ(x ))µ = ∂µξ φκ(x )+ ξ ∂αφµ(x ) (185) Now, we define following Physics textbooks the horizontal variation44 by

0 δ φ := £ξφ. (186) − This definition (with the negative sign) is used by physicists because they usually work only with the components of the fields and diffeomorphism in- variance is interpreted as invariance under choice of coordinates. Then, they interpret Eq.(177) as a coordinate transformation between two charts whose intersection of domains cover the regions and h of interest with coordinate µ µ U U functions x and x′ such that { } { } µ µ x′ := x h (187) ◦ ε and then µ µ µ x′ (x)= x + εξ (188) The field φ at x M has the representations ∈U⊂ ι µ ι κ φ(x)= φµ′ (x′ )dx′ = φκ(x )dx and in first order in ε it is κ ι ∂x ι φµ′ (x′ )= µ φκ(x ) (189) ∂x′ = φ (xι) ε∂ ξκφ (xι) µ − µ κ and on the other hand since

ι ι ι ι κ ι φµ′ (x′ )= φµ′ (x + εξ )= φµ′ (x )+ εξ ∂κφµ′ (x ). (190) we have in first order in ε that

ι ι κ ι κ ι φ′ (x )= φ (x ) ε∂ ξ φ (x ) εξ ∂ φ (x ). (191) µ µ − µ κ − κ µ from where we get

κ κ 0 φµ′ (x ) φµ(x ) κ δ φµ(x) = lim − = (£ξφ(x )) . (192) ε 0 µ → ε − 44 In [34] the horizontal variation is denoted by δh, where a vertical variation denoted by 0 δv (associated with gauge transformations) is also introduced. Moreover, let us recall that δ has been usely extensively after a famous paper by L. Rosenfeld [41], but appears also for the best of our knowledge in Section 23 of Pauli’s book [30] on Relativity theory.

40 Remark 22 The above calculating can be done in a while recalling Cartan’s ‘magical’ formula, which with ξ := g(ξ, ) reads:

£ξφ = ξydφ + d(ξyφ) (193)

In components we have

µ µ (ξydφ)α = ξ ∂µφα ξ ∂αφµ, µ − µ (d(ξyφ))α = ∂αξ φµ + ξ ∂αφµ (194) from where the substituting these results in Eq.(193), Eq.(185) follows immedi- ately.

µ µ Remark 23 If we have chosen the coordinate functions x and x′ related by45 { } { } µ µ 1 x′ := x h− (195) ◦ ε 0 κ we would get that δ φµ(x) = (£ξφ(x ))µ.

Remark 24 Take into account for applications that for any sec T ∗M C ∈ V d£ξ = £ξd (196) C C Now, physicists introduce another two variations δa and δ defined by

κ κ a φµ(x′ ) φµ(x ) ι κ δ φµ(x) := lim − = ξ ∂ιφµ(x ) (197) ε 0 ε → and κ κ φµ′ (x′ ) φµ(x ) δφµ(x) := lim − ε 0 → ε called, e.g., in [40] local variation46. We have

κ ι κ κ φµ′ (x )+ εξ ∂ιφµ(x ) φµ(x ) 0 κ ι κ (δφ(x)) = lim − = δ φ(x ) + ξ ∂ιφµ(x ) µ ε 0 ε µ →
= ∂ ξιφ (xκ) ξα∂ φ (xκ)+ ξι∂ φ (xκ)= ∂ ξιφ (xκ) (198) − µ ι − α µ ι µ − µ ι Remark 25 In what follows we shall use the above terminology for the various variations introduced above for an arbitrary tensor field. The definition of the Lie derivative of spinor fields is is still a subject of recent researh with many conflicting views. In [20] we present a novel geoemtrical aprroach to this subject using the theory of Clifford and spin-Clifford bundles which seems to lead to a consistent results. 45As, e.g., in [46]. 46 Some authors call δφµ(x) the total variation, but we think that this is not an appropriate name.

41 Remark 26 Defining

0 δ := £ξ m(φ, dφ), (199) A − Z L U we have returning to Eq.(176) that Stokes theorem permit us to write

0 δ = £ξ m(φ, dφ) A − Z L U

= d(ξy m(φ, dφ)) ξyd m(φ, dφ) − Z L − Z L U U

= ξy m(φ, dφ). (200) − Z∂ L U C The Generalized Energy-Momentum Current in (M, g,τg, ) ↑ C.1 The Case of a General Lorentzian Spacetime Struc- ture

In this subsection (M, g, τg, ) is an arbitrary oriented and time oriented Lorentzian manifold (M, g) which will↑ be supposed to be the arena where physical phe- nomena takes place. We choose coordinate charts ( 1,χ1) and ( 2,χ2) with µ µ U U coordinate functions x and x′ covering = . We call χ ( )= U, { } { } U1 ∩U2 U 1 U χ ( ) = U ′. We take such that ∂ = Σ Σ + Ξ , i.e., is bounded 2 U U U 2 − 1 U from above and below by spacelike surfaces Σ1 and Σ2 such that σ1 = χ1(Σ1) and σ2 = χ1(Σ2) and moreover we suppose that set of the N matter fields in interaction denoted φ = φ , A =1, 2, ..., N (201) { A} living in satisfy in Ξ (a timelike boundary) U φ Ξ =0. (202) A| In what follows the action functional for the fields is written

4 4 = Lm(φ , ∂µφ)d x = Lm(φ , ∂µφ) det gd x (203) A Z µ Z µ − U U p Under a coordinate transformation corresponding to a diffeomorphism generated by a one parameter group of diffeomorphisms,

µ µ µ µ µ µ x x′ = x + εξ = x + εξ [x ] 7→ µ µ µ µ = x £ ξx = x + δx (204) − ε we already know that the fields suffers the variation

0 φ φ′ = φ + δ φ. (205) 7→

42 0 We have in first order in ε and recalling that δ and ∂µ commutes that

0 4 4 δ = Lm(φ′ , ∂µφ′)d x′ Lm(φ , ∂µφ)d x A Z µ − Z µ U U ∂L ∂L ∂δxµ m δ0 m δ0 L 4 = φA + ∂µ φA + m µ d x Z  ∂φA ∂∂µφA ∂x  U ∂Lm ∂Lm 0 ∂Lm 0 µ 4 = ∂µ δ φA + ∂µ δ φA + Lmδx d x (206) Z  ∂φA − ∂∂µφA  ∂∂µφA  U Putting µ ∂Lm 0 µ := δ φA + Lmδx (207) J ∂∂µφA the second term in Eq.(206) can be written using Gauss theorem as

∂Lm 0 µ 4 µ µ ∂µ δ φA + Lmδx d x = J dσµ J dσµ (208) Z ∂∂µφA  Zσ − Zσ U 2 1 Recalling the concept of local variation introduced above we have

0 ν δφA = δ φA + δx ∂ν φA (209)

Putting µ ∂Lm πA = (210) ∂∂µφA we call µ πA := πA∂µ (211) the canonical momentum canonically conjugated to the field φA. Moreover, putting Υµ := πµ ∂ φ δµL (212) ν A ν A − ν m we call Υ := Υµdxν ∂ (213) ν ⊗ µ the canonical energy-momentum tensor of the closed physical system described by the fields. Now, we can write Eq.(207) as

µ := πµ δφ Υµδxν . (214) J A A − ν Moreover, defining

µ µ ν F (σ) := (πAδφA Υν δx ) dσµ (215) Zσ − we can rewrite Eq.(206)

0 ∂Lm ∂Lm 0 4 δ = ∂µ δ φAd x + F (σ2) F (σ1) (216) A ZU  ∂φA − ∂∂µφA  −

43 Now, the action principle establishes that δ0 = 0 and then we must have A ∂Lm ∂Lm ∂µ =0, (217) ∂φA − ∂∂µφA which are the Euler-Lagrange equations satisfied by each one of the fields φA and also F (σ ) F (σ )=0 (218) 2 − 1 Now, if τg is the volume element, taking into account that we took Σ Σ +Ξ = 2 − 1 ∂ where Ξ is a timelike surface such that Ξ = 0 and introducing the current U J | 1 µ ν µ (µdx = gµν dx sec T ∗M ֒ sec ℓ(M, g) (219 =: J J J ∈ ^ → C we can rewrite Eq.(218) using Stokes theorem as

µ µ dσµ dσµ = ⋆ = d⋆ . (220) Zσ J − Zσ J Z∂ gJ ZU g J 2 1 U C.2 Introducing D and the Covariant “Conservation” Law for Υ

If we add D, the Levi-Civita connection of g to (M, g, τg, ) we get a Lorentzian ↑ spacetime structure (M, g, D, τg, ). Then recalling from Appendix A the defi- nitions of the Hodge coderivative↑ and of the Dirac operator we can write:

1 d⋆ = ⋆ ⋆− d⋆ Z J Z g g g J U U

= δ τg − Z gJ  U

= ∂y τg Z J U µ = Dµ τg (221) ZU J and we arrive at the conclusion that δ0 = 0 implies that A µ 1 µ d⋆ =0 ⇔ Dµ =0 ∂µ( det g )=0. (222) g J J ⇔ √ det g − J − p

Remark 27 Recalling the definition of the canonical energy-momentum tensor Υ (Eq.(213)) gives a covariant “conservation” law for Υ, i.e.,

D Υ =0 (223) • µ µ only if the term πAδφA in the current J is null. This, of course, happens if the local variation of the fields δφA = 0, something that cannot happens in an arbitrary structure (M, g, τg, ). So, we need to investigate when this occurs. ↑

44 Remark 28 We observe here that comparison of Eq.(216) with Eq.(200) per- mit us to write

⋆ = d⋆ = ξy (φ, dφ)= d(ξy (φ, dφ)). (224) Z∂ gJ Z g J − Z∂ L − Z L U U U U C.3 The Case of Minkowski Spacetime C.3.1 The Canonical Energy-Momentum Tensor in Minkowski Space- time We now, apply the results of the last section to the case where the fields live in Minkowski spacetime (M, η, D, τη, ). In this case we can introduce global coordinates xµ in Einstein-Lorentz-Poincar´egauge↑ (see Section 2.3). We now{ construct} the conserved current associated to the diffeomorphisms µ generated by the vector fields eµ = ∂/∂x which are Killing vector fields on (M, η). Consider then the Killing vector field

µ ξ := ε eµ (225) where εµ R are constants such that εµ << 1 and the coordinate transfor- mation ∈ | | µ µ µ µ µ µ x x′ = x + £ξx = x + ε . (226) 7→ Recalling the definitions of πA, the momentum canonically conjugated to the field φA (Eq.(211)) and of the Υ (Eq.(213)) and recalling that in the present case it is δφA = 0 we have that the conserved Noether current is: µ = ενΥµ. (227) J − ν and the canonical energy-momentum tensor of the physical system described by the fields φA is conserved, i.e., µ ∂µΥν =0. (228) Remark 29 Of course, if we introduce an arbitrary coordinate system xµ covering an open set of the Minkowski spacetime manifold M R4 Eq.(228){ } reads U ≃ D Υ =0 D Υµ =0. (229) • ⇔ µ ν C.3.2 The Energy-Momentum 1-Form Fields in Minkowski Space- time Recall that the objects 1 µ µ µ ν µ ν (Υ := (πA∂ν φA δν ℓ) dx =Υν dx sec T ∗M ֒ sec ℓ(M, η) (230 − ∈ ^ → C are conserved currents. They may be called the generalized energy momentum 1-form fields of the physical system described by the fields φA. We have δΥµ =0 d⋆ Υµ =0. (231) η ⇔ η

45 C.3.3 The Belinfante Energy-Momentum Tensor in Minkowski Space- time It happens that given an arbitrary field theory the canonical energy-momentum tensor is in general not symmetric, i.e.,

Υµν =Υνµ. (232) 6 But, of course, if Υµ is conserved, so it is

µ := Υµ + ̥µ (233) T where ̥µ := δΨµ (234) η

µ 2 with each one of the Ψ sec T ∗M ֒ sec ℓ(M, g). So, it is always possible ∈ → C for any field theory to find47 aV condition on the Ψµ such that the components T µ of µ = T µdxν satisfy the symmetry condition. ν T ν T µν = T νµ. (235)

µ ν When this is the case T = Tν dx ∂µ will be called the Belinfante energy- momentum tensor of the system. ⊗

C.3.4 The Energy-Momentum Covector in Minkowski Spacetime Since Minkowski spacetime is parallelizable we can identify all tangent and cotangent spaces and thus define a covector in a vector space R4. Fixing µ V ≃ (global) coordinates in Einstein-Lorentz-Poincar´egauge x a vector vx { }4 4 ∈ TxM can be identified by a pair [27] (x, v) where (x, v) R R and x = (x0, x1, x2, x3). If two vectors v T M, v T M are such∈ that× x ∈ x y ∈ y

vx = (x, v), vy = (y, v), (236) i.e., they have the same vector part we will say that they can be identified as a a vector of some vector space V R4. With these considerations we write x, y M R4 ≃ ∀ ∈ ≃ ∂ ∂ = E and dxµ dxµ = Eµ (237) ∂xµ ≈ ∂xµ µ |x ≈ |y x y

47For example, in [19] the condition is fixed in such a way that the orbital angu- 1 µ ν µν lar momentum tensor of the system defined as M = 2 Mµν dx |o ∧ dx |o (with M = µ νκ ν µκ R (x T − x T )dσκ) be automatically conserved. However take into account that since the fields possess in general intrinsic spin an angular momentum conservation law can only be formulated by taking into account the orbital and spin angular momenta. It can be shown (see Chapter 8 of [34] that the antisymmetric part of the canonical energy-momentum tensor is the source of the spin tensor of the field. In [3] it is shown how to obtain a conserved symmetrical energy-momentum tensor by studying the conservation laws that come from a general Poincar´evariation, which involves translations and general Lorentz transformations.

46 µ 4 where Eµ is a basis of V and E is a basis of a R . Then we can write (with o{ M} an arbitrary point,{ taken} in general, forV ≃ convenience as origin of the coordinate∈ system)

µ µ µ P = PµE = Pµ dx := ⋆ µ dx (238) |0 Z ηT  |0 as the energy-momentum covector of the closed physical system described by the fields φA.

Remark 30 Note that under a global (constant) Lorentz transformation ∂/∂xµ µ ν ν ν 7→ ∂/∂x′ =Λ ∂/∂x we have that ⋆ µ ⋆ ′ = ⋆ ν Λ and it results Pµ P ′ = µ ηT 7→ ηTµ ηT µ 7→ µ ν Pν Λµ, i.e., the Pµ are indeed the components of a covector under any global (constant) Lorentz transformation.

D The Energy-Momentum Tensor of Matter in GRT

The result of the previous section shows that in a general Lorentzian spacetime structure the canonical Lagrangian formalism does not give a covariant “con- served” energy-mometum tensor unless the local variations of the matter fields µ ν ∂ δφA are null. So, in GRT the matter energy momentum tensor T = Tν dx ∂xµ that enters Einstein equation is symmetric (i.e., T µν = T νµ) and it is obtained⊗ in the following way. We start with the matter action

4 m = Lm(φ , ∂µφ) det gd x (239) A Z µ − U p Remark 31 Consider as above, a diffeomorphism x x′ = hε(x) generated 7→ µ by a one parameter group associated to a vector field ξ = ξ ∂µ (such that com- ponents ξµ <<1 and the ξµ 0 at Ξ) and a corresponding coordinate trans- formation| | → µ µ µ µ x x′ = x + εξ (240) 7→ with ε << 1 and study the variation of induced the variation of the (grav- | | Am itational) field g (without changing the fields φA) induced by the coordinate transformation of Eq.(240). We have immediately that

µν κ µν κ µν κ µι κ ν νι κ µ g (x ) g′ (x′ )= g (x )+ εg (x )∂ ξ + εg (x )∂ ξ . (241) 7→ ι ι To first order in ε it is

µν κ µν κ µι κ ν νι κ µ ι µν g′ (x )= g (x )+ εg (x )∂ ξ + εg (x )∂ ξ εξ ∂ g (242) ι ι − ι and

0 µν µν ν µ µ ν 0 δ g = £ξg = D ξ + D ξ and δ gµν = £ξgµν = Dν ξµ Dµξν . − − − − (243)

47 Then, under the above conditions, using Gauss theorem and supposing that δ0gµν vanishes at Ξ it is √ g √ g 0 ∂Lm det 0 µν ∂Lm det 0 µν 4 δ m := − δ g + − δ (∂ιg ) d x A Z  ∂gµν ∂∂ιgµν  U √ g √ g ∂Lm det 0 µν ∂Lm det 0 µν 4 = − δ g + − ∂ιδ g d x Z  ∂gµν ∂∂ιgµν  U ∂L √ det g ∂ ∂L √ det g m m δ0 µν 4 = − ι − g d x Z  ∂gµν − ∂x  ∂∂ιgµν  U 1 0 µν 4 1 µν 0 4 =: Tµν δ g det gd x = T δ gµν det gd x. (244) 2 Z − −2 Z − U p U p with 1 ∂Lm√ det g ∂ ∂Lm√ det g det gTµν := − − (245) 2 ∂gµν ∂xι  ∂∂ gµν  p− − ι Since Tµν = Tνµ we can write

0 1 0 µν 4 δ m = Tµν δ g det gd x A 2 Z − U p 1 µν 4 = T (Dν ξµ + Dµξν ) det gd x 2 Z − U p µν 4 = T Dν ξµ det gd x Z − U p ν µ 4 ν µ 4 = Dν (T ξ µ) det gd x (Dν T )ξ det gd x Z µ − − Z µ − U p U p ν µ 4 ν µ 4 = ∂ν ( det gT ξ µ)d x (Dν T )ξ det gd x Z − µ − Z µ − U p U p ν µ 4 = (Dν T )ξ det gd x (246) − Z µ − U p Remark 32 Contrary to what is stated in many textbooks in GRT we cannot 0 conclude with the ingredients introduced in this section that δ m =0. However, if we take into account that in GRT the total actionA describing the mater fields and the gravitational field is 1 = + = Rτ g + A Lg Lm −2 Lm R R R R we get from the variation δ0 induced by the variation of the (gravitational) A field g (without changing the fields φA) and induced by the coordinate transfor- mation given by Eq.(240) the Einstein field equation G = −T which reads in components as 1 Gµ = Rµ Rδµ = T µ. (247) ν ν − 2 ν − ν Since it is D • G =0 it follows that in GRT we have D • T =0. (248)

48 Remark 33 It is opportune to recall that as observed, e.g., by Weinberg [47] that for the case of Minkowski spacetime the symmetric energy-momentum ten- sor obtained by the above method is always equal to a convenient symmetrization of the canonical energy-momentum tensor. But it is necessary to have in mind that the GRT procedure eliminates a legitimate conserved current introduc- ing a covariant “conserved” energy-momentum tensor that does notJ give any legitimate energy-momentum conserved current for the matter fields, except for the particular Lorentzian spacetimes containing appropriate Killing vector fields. And even in this case no energy-momentum covector as it exists in special rel- ativistic theories can be defined. Moreover, at this point we cannot forget the existence of the quantum structure of matter fields which experimentally says the Minkowskian concept of energy and momentum (in general quantized) being carried by field excitations that one calls particles. This strongly suggests that parodying (again) Sachs and Wu [43] it is really a shame to loose the special relativistic conservations laws in GRT.

E Relative Tensors and their Covariant Deriva- tives

α α Now, recall that given arbitrary coordinates x covering U M and x′ covering covering V M (U V = ∅) a relative{ } tensor A of⊂ type (r, s){ and} 48 ⊂ ∩ 6 49 p 4 w weight w is a section of the bundle T M ( T ∗M)⊗ . q ⊗ We have V

µ1...µr α ν1 νs w A = A (x )∂µ ∂µ dx dx (τ)⊗ , ν1...νs 1 ⊗···⊗ r ⊗ ⊗···⊗ ⊗ with τ := dx0 dx3. The set of functions ∧···∧ w µ1...µr α g µ1...µr α Aν1...νs (x )= det Aν1...νs (x ) p−  r 4 w is said to be the components of the relative tensor field A sec(Ts M ( T ∗M) ) ∈ ⊗ ′α α β V ∂x and under a coordinate transformation x x′ with Jacobian J = det ∂xβ 7→   these functions transform as [23, 45]

λ1 λ1 ν1 νs λ1...λr β w ∂x′ ∂x′ ∂x ∂x µ1...µr α Aκ′ ...κ (x′ )= J ... Aν ...ν (x ). (249) 1 s ∂xµ1 ··· ∂xµ1 ∂xκ1 ∂xκs 1 s g Aµ1...µr α On a manifold M equipped with a metric tensor field we can write ν1...νs (x )= w √ g µ1 ...µr α µ1...µr α det Aν1...νs (x ) where the Aν1...νs (x ) are the components of a tensor − r field A sec
Ts M. The∈covariant derivative of a relative tensor field relative to a given arbitrary µ µ α connection defined on M such that ∂ dx = ℓ να·· dx is given (as the ∇ ∇ ∂xν − · reader may easily find) by

µ ...µr ν νs w A := ( A 1 )∂ ∂ dx 1 dx (τ)⊗ , (250) ∇∂κ ∇κ ν1...νs µ1 ⊗···⊗ µr ⊗ ⊗···⊗ ⊗ 48The number w is an integer. Of course, if w = 0 we are back to tensor fields. 49 4 4 The notation (V T ∗M)⊗w means the w-fold tensor product of V T ∗M with itself.

49 where

·· µ1...µr ∂ µ1...µr µp µ1...µp−1ιµp+1...µr κAν ...ν = Aν ...ν + ℓ ικ Aν ...... ν ∇ 1 s ∂xκ 1 s · 1 s ι µ1...... µr σ µ1...µr ℓ ν··qκAν ...νq− ινq ...νs wℓ κσ·· Aν ...νs . (251) − · 1 1 +1 − · 1 In particular for the Levi-Civita connection D of g we have for the relative tensor 0 3 τg = det g dx dx (252) p− ⊗ ∧···∧ that:

ρ Dα det g = ∂γ det g Γ γρ·· det g =0, p−  p  − · p 1 1 ρ 1 Dα = ∂γ + Γ γρ·· =0. (253) √ det g  √ det g  · √ det g − − −

F Explicit Formulas for Jµν and Jµ4 in Terms of Projective Conformal Coordinates

We have taking into account Eqs.(57), (58) and (59) the following identities

∂Xκ Ω2 = x xκ +Ωδκ, ∂xα 2ℓ2 α α ∂X4 Ω2 = x , Xµ =Ωxµ. ∂xα − ℓ α ν ν where xµ := ηµν x and Xµ := ˚ηµν X . We want to prove that:

∂ ∂ ∂ ∂ (a) J = η xβ η xβ = ˚η Xβ ˚η Xβ , (254) µν µβ ∂xν − νβ ∂xµ µβ ∂Xν − νβ ∂Xµ ∂ 1 ∂ ∂ ∂ (b) J = ℓ 2η xν xλ σ2δλ = X4 + X . µ4 ∂xµ − 4ℓ µν − µ ∂xλ − ∂Xµ µ ∂X4
(255)

Proof of (a):

50 ∂Xκ ∂ ∂X4 ∂ ∂Xκ ∂ ∂X4 ∂ J = η xβ + η xβ η xβ η xβ µν µβ ∂xν ∂Xκ µβ ∂xν ∂X4 − νβ ∂xµ ∂Xκ − νβ ∂xµ ∂X4 ∂Xκ ∂ ∂Xκ ∂ ∂X4 ∂ ∂X4 ∂ = x x + x x µ ∂xν ∂Xκ − ν ∂xµ ∂Xκ µ ∂xν ∂X4 − ν ∂xµ ∂X4 Ω2 ∂ Ω2 ∂ = x x xκ +Ωδκ x x xκ +Ωδκ µ −2ℓ2 ν ν  ∂Xκ − ν −2ℓ2 µ µ ∂Xκ Ω2 Ω2 ∂ + x x x x  ℓ ν µ − ℓ µ ν  ∂X4 ∂ ∂ Ω2 ∂ Ω2 ∂ = X X x x xκ + x x xκ µ ∂Xν − ν ∂Xµ − 2ℓ2 ν µ ∂Xκ 2ℓ2 ν µ ∂Xκ ∂ ∂ ∂ ∂ = X X = ˚η Xβ ˚η Xβ . µ ∂Xν − ν ∂Xµ µβ ∂Xν − νβ ∂Xµ Proof of (b):

∂ 1 ∂ J = ℓ 2η xν xλ σ2δλ µ4 ∂xµ − 4ℓ µν − µ ∂xλ ∂ 1 ∂ 1
∂ 1 ∂ 1 ∂ = ℓ + σ2 2η xν xλ = X4 η xν xλ ∂xµ 4ℓ ∂xµ − 4ℓ µν ∂xλ −Ω ∂xµ − 2ℓ µν ∂xλ 1 Ω2 ∂ 1 ∂X4 ∂ = X4 x xκ +Ωδκ X4 −Ω 2ℓ2 µ µ ∂Xκ − Ω ∂xµ ∂X4 1 Ω2 ∂ 1 ∂X4 ∂ x xλ x xκ +Ωδκ η xν xλ − 2ℓ µ 2ℓ2 λ λ ∂Xκ − 2ℓ µν ∂xλ ∂X4 ∂ Ω ∂ 1 ∂ 1 ∂ = X4 X4x xκ x xλΩ Ω2σ2x xκ − ∂Xµ − 2ℓ2 µ ∂Xκ − 2ℓ µ ∂Xλ − 4ℓ3 µ ∂Xλ ∂ 1 ∂X4 ∂ + X4Ωx + x Ω2xλ µ ∂X4 2ℓ2 µ ∂xλ ∂X4 ∂ 1 ∂ 1 1 1 ∂ = X4 + X4Ω+ Ω2σ2 x X4 +1+ Ωσ2 Ωx xλ − ∂Xµ  2ℓ2  µ ∂X4 −  ℓ 2ℓ2  2ℓ µ ∂Xλ ∂ ∂ σ2 1 1 1 ∂ = X4 + X 1+ + + σ2 Ω2x xλ − ∂Xµ µ ∂X4 − −  4ℓ2  Ω 2ℓ2  2ℓ µ ∂Xλ ∂ ∂ 1 1 1 ∂ = X4 + X 1+ σ2 +1 σ2 Ω2x xλ − ∂Xµ µ ∂X4 − − 4ℓ2 − 4ℓ2  2ℓ µ ∂Xλ ∂ ∂ = X + X . − 4 ∂Xµ µ ∂X4 where we used that:

51 1 1 ∂ X4Ω + Ω2σ2 x  ℓ 2ℓ2  µ ∂X4 1 2 ∂ = 1+ σ2 + σ2 Ω2x −  4ℓ2  4ℓ2  µ ∂X4 1 2 ∂ 1+ σ2 + σ2 Ω2x −  4ℓ2  4ℓ2  µ ∂X4 1 ∂ ∂ = Ω2x = X . −Ω µ ∂X4 − µ ∂X4 References

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