# Notes on Conservation Laws, Equations of Motion of Matter, and Particle Fields in Lorentzian and Teleparallel de Sitter Space-Time Structures

Authors: Waldyr A. Rodrigues; Samuel A. Wainer
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

We discuss the physics of interacting fields and particles living in a de Sitter Lorentzian manifold (dSLM), a submanifold of a 5-dimensional pseudo-Euclidean (5dPE) equipped with a metric tensor inherited from the metric of the 5dPE space. The dSLM is naturally oriented and time oriented and is the arena used to study the energy-momentum conservation law and equations of motion for physical systems living there. Two distinct de Sitter space-time structuresMdSL and MdSTP are introduced given dSLM, the first equipped with the Levi-Civita connection of its metric field and the second with a metric compatible parallel connection. Both connections are used only as mathematical devices. Thus, for example, MdSL is not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Misconceptions appearing in the literature concerning the motion of free particles in dSLM are clarified. Komar currents are introduced within Clifford bundle formalism permitting the presentation of Einstein equation as a Maxwell like equation and proving that in GRT there are infinitely many conserved currents. We prove that in GRT even when the appropriate Killing vector fields exist it is not possible to define a conserved energy-momentum covector as in special relativistic theories.


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## Body

## 1. Introduction

In this paper we study some aspects of Physics of fields living and interacting in a manifoldM=SO(1,4)/SO(1,3)≃R×S3. We introduce two different geometrical space-time structures that we can form starting from the manifold M which is supposed to be a vector manifold, that is, 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 Sitter space-time MdSL=(M,g,D,τg,↑) and teleparallel de Sitter space-time MdSTP=(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 momentum) for a closed system of fields and moreover there are no genuine energy-momentum and angular momentum conservation laws for the system consisting of nongravitational plus the gravitational field. We discuss in Section 2.1 a pure mathematical result, namely, when there exist some conserved currents JV∈sec⁡T∗M in a Lorentzian space-time associated with a tensor field W∈sec⁡T11M 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 Relativity theories and how in that theory it is possible to construct a conserved energy-momentumcovector1 for the matter fields. After that we recall that in GRT we have a covariantly “conserved” energy-momentum tensor T∈sec⁡T11M (i.e., D•T=0) and so, using the results of Section 2.1, we can immediately construct conserved currents when the Lorentzian space-time modelling the gravitational field generated by T possesses Killing vector fields. However, we show that it is not possible in general in GRT even when some special conserved currents exist (associated with one timelike and three spacelike Killing vector fields) to build a conservedcovector for the system of fields, as it is the case in special relativistic theories. Immediately after showing that we ask the following question:Is it necessary to have Killing vector fields in a Lorentzian space-time modelling a given gravitational field in order to be possible to construct conserved currents?Well, we show that the answer isno. In GRT there are an infinite number of conserved currents. This is shown in Section 2.42 where we introduce the so-called Komar currents in a Lorentzian space-time modelling a gravitational field generated by a given (symmetric) energy-momentum tensor T and we show how any diffeomorphism associated with a one-parameter group generated by a vector field A leads to a conserved current. We show moreover using the Clifford bundle formalism recalled in Appendix A that F=dA∈sec⋀T2∗M, where A=gA,∈sec⋀T1∗M satisfy (with ∂ denoting the Dirac operator acting on sections of the Clifford bundle of differential forms) a Maxwell-like equation ∂F=JA (equivalent to dF=0 and δgF=-JA). The explicit form of JA as a function of the energy-momentum tensor is derived (see (50)) together with its scalar invariant. We establish that3  ∂F=JA encode the contents of Einstein equation. We show moreover that even if we can get four conserved currents given one timelike and three spacelike vector fields and thus get four scalar invariants, these objectscannot be associated with the components of a momentumcovector4 for the system of fields producing the energy-momentum tensor T. We also give the form of JA when A is a Killing vector field and emphasize that even if the Lorentzian space-time under consideration has one timelike 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=JA in Section 2.26. Lie derivatives and variations of tensor fields are discussed in Appendix B. In Appendix C.1 we derive from the Lagrangian formalism conserved currents for fields living in a general Lorentzian space-time structure and the corresponding generalized covariant energy-momentum “conservation” law. We compare these results in Appendix C.2 with the analogus ones for field theories in special relativistic theories where the Lorentzian space-time structure is Minkowski space-time. We show that despite the fact that we can derive conserved quantities for fields living and interacting in MdSL we cannot define in this structure a genuine energy-momentum conserved covector for the system of fields as it is the case in Minkowski space-time. A legitimate energy-momentum covector for the system of fields living in (M=SO(1,4)/SO(1,3),g) exists only in the teleparallel structure MdSTP. This is discussed in Section 5.2 after recalling the Lie algebra and the Casimir invariants of the Lie algebra of de Sitter group in Section 5.1. In Appendix D we derive for completeness and to insert Remark D.3 the so-called covariant energy-momentum conservation law in GRT. Appendix E recalls the intrinsic definition of relative tensors and their covariant derivatives. Appendix F presents 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 space-time structuresMdSL and MdSTP. In particular we want also to clarify some misunderstandings concerning the roles of geodesics in MdSL. So, in Section 3, we briefly recall the conformal representation of the de Sitter space-time structure MdSL and prove that the one timelike and the three spacelike “translation” Killing vector fields of (M,g) define a basis for almost all M. With this result we show in Section 4 that the method used in [1] to obtain the curves which minimizes or maximizes the length function of timelike curves in de Sitter space-time with the result that these curves are not geodesics is equivocated, since those authors useconstrained variations instead ofarbitrary variations of the length function. Even more, the equation obtained from the constrained variation in [1] 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 [2] 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 [1] it is not true that the equation of motion of a single-pole obtained from a method similar to Papapetrou’s in [2] but using the generalized energy-momentum tensor of matter fields in MdSL gives an equation of motion different from the geodesic equation in MdSL and in agreement with the one they derived from his constrained variation method. Indeed, we prove that from the equation describing the motion of a single-pole the geodesic equation follows automatically.


## 2. Preliminaries

Let(M,g,D,τg,↑) be a general Lorentzian space-time. Let U⊆M be an open set covered by coordinates {xμ}. Let {eμ=∂μ} be a basis of TU and {ϑμ=dxμ} the basis of T∗U dual to the basis {∂μ}; that is, ϑμ(∂ν)=δνμ. We denote by g a metric of the cotangent bundle such that if g=gμνϑμ⊗ϑν, then g=gμν∂μ⊗∂ν with gμρgρν=δνμ. We introduce also {∂μ} and {ϑμ}, respectively, as the reciprocal bases of {∂μ} and {ϑμ}; that is, we have(1)g∂ν,∂μ=δνμ,gϑμ,ϑν=δνμ.Next we introduce inTU the tetrad basis {eα=hαμ∂μ} and in T∗U the cotetrad basis {γα=hμαγμ} which are dual basis. We introduce moreover the bases {eα} and {γα} as the reciprocal bases of {eα} and {γα} satisfying(2)geα,eβ=δαβ,gγβ,γα=δαβ.Moreover recall that it is(3)g=ηαβγα⊗γβ=ηαβγα⊗γβ,g=ηαβeα⊗eβ=ηαβeα⊗eβ.


### 2.1. The CurrentsJV and JK

LetW=Wαβeα⊗eβ∈sec⁡T20M with Wαβ=Wβα and Wˇ=Wαβγα⊗γβ∈sec⁡T02M, Wαβ=ηαςηβτWςτ, and W=Wβαγβ⊗eα∈sec⁡T11M. For the applications we have in mind we will say that W,Wˇ, and W are physically equivalent.Note thatW (an example of an extensor field7) is such that(4)W:sec⁡⋀1T∗M⟶sec⁡⋀1T∗M,WV=VαWβαγβ.Define the divergence of W as the 1-form field(5)D•W≔DαWβαγβ,where(6)DαWβα≔DeαWβα=eαWβα+ΓαιαWβι-ΓαβιWια.Moreover, introduce the1-form fields(7)Wβ≔Wαβγα∈sec⁡⋀1T∗M.Remark 1.
Take notice for the developments that follow that the Hodge coderivative of the1-form fields Wβ is as follows (see Appendix): (8)δgWβ=-γκ⌟DeκWαβγα=-eκWαβγκ⌟γα-WαβΓκαιγκ⌟γι=-eαWαβ-WαβΓκακ.
So,D∙W=0 does not imply that δgWβ=0.Now, given a vector fieldV=Vαeα and the physically equivalent covector field V=Vαγα, define the current(9)JV=VαWα∈sec⁡⋀1T∗M↪sec⁡ClM,g.Of course, writing(10)JV=Jβγβ we have(11)Jβ=VαWαβ.Recalling (see AppendixA) that ⋆1=τg, define(12)
T
=
W
α
β
⋆
g
1
=
W
α
β
τ
g
∈
s
e
c
⋀
4
T
∗
M
↪
C
l
M
,
g
.Then, we have, with∂ denoting the Dirac operator,(13)d⋆gJV=∂∧⋆gJV=γα∧Deα⋆gJV=Deαγα∧⋆gJV-Deαγα∧⋆gJV.Taking into account thatγα∧⋆gJV=⋆g(γα⌟JV) and Deαγα∧⋆gJV=⋆g(Deαγα⌟JV) we can write (13) as(14)d⋆gJV=eαVκWκα+VκeαWκα+Γ·αβα··VκWκβτg.Also, we can easily verify from (5) that(15)⋆gD•WV=D•WVτg=VκeαWκα+Γ·αια··WκιVκ-Γ·ακι··WιαVκτg.Now, let£ be the (standard) Lie derivative operator. Let us evaluate the product of £Vg(eα,eβ) by T; that is,(16)£Vgeα,eβT∈sec⋀4T∗M.From Cartan magical formula we get (17)£Vγα=dVα+V⌟∂∧γα=eιVαγι-VςΓ·ςια··γι+VςΓ·ιςα··γι.Then,(18)£Vgeα,eβT=ηικ£Vγι⊗γκ+ηικγι⊗£Vγκeα,eβT=2eκVιWκι+2VςΓ·ιςκ··WκιTand we get from (14), (15), and (18) the important identity [3](19)£Vgeα,eβT=2d⋆JV-2⋆gD•WV.From (19) we see that if V is a conformal Killing vector field, that is, £Vg(eα,eβ)=2ληαβ, we have(20)⋆gλtrW=d⋆gJV-⋆D•WV,where trW is the trace of the matrix with entries Wβα.


### 2.2. Conserved Currents Associated with a Covariantly ConservedW

Definition 2.
One says thatW is “covariantly conserved” if(21)D•W=0.In this case, ifV=K is a Killing vector field, then £Kg=0 and we have(22)d⋆gJK=⋆gD•WKand the current 3-form field ⋆gJK is closed; that is, d⋆gJK=0, or equivalently (taking into account the definition of the Hodge coderivative operator δg)(23)δgJK=0.In resume, when we have Killing vector fields8  Ki,i=1,2,…,n, “covariant conservation” of the tensor field W, that is, D•W=0, implies ingenuine conservation laws for the currents JKi that from δgJKi=0, we can using Stokes theorem build thescalar conserved quantities(24)EKi≔18π∫Σ′⋆gJKi,where N is the region where JK has support and ∂N=Σ+Σ′+Ξ, where Σ,Σ′ are spacelike surfaces and JKi is null at Ξ (spatial infinity).


### 2.3. Conserved Currents in GRT Associated with Killing Vector Fields

Before studying the conditions for the existence or not of genuine energy-momentum conservation laws in GRT, let us recall from AppendixC.3.4 that in Minkowski space-time9(M≃R4,η,D,τη,↑) we can introduce global coordinates {xμ} (in Einstein-Lorentz-Poincaré gauge) such that η(eμ,eν)=ημν and Deμeν=0, where {eμ=∂/∂xμ} is simultaneously a global tetrad and a coordinate basis. Also {ϑμ=dxμ} is a global cotetrad and a coordinate cobasis.Moreover,eμ=∂/∂xμ are also Killing vector fields in (M≃R4,η) and thus we have for a closed physical system (consisting of particles and fields in interaction living in Minkowski space-time and whose equations of motion are derived from a variational principle with a Lagrangian density invariant under space-time translations) that the currents10(25)Tα≔J∂/∂xα=Tανϑν∈sec⋀1T∗M↪ClM,η,M≃R4are the conserved energy-momentum 1-form fields of the physical system under consideration, for which we know that the quantity (recall (C.38))(26)P=Pαϑαo=PαEα,with(27)Pα=∫⋆gTαis the components of theconserved energy-momentumcovector (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-momentumT is modelled by a Lorentzian space-time11  (M,g,D,τg,↑) where the relation between g and T is given by Einstein equation which using the orthonormal bases {eα} and {γα} introduced above reads(28)Gβα=Rβα-12δβαR=-Tβα.Moreover, defining G=Gβαγβ⊗eα and recalling that T=Tβαγβ⊗eα, it is(29)D•G=0,D•T=0.Basedonly on the contents of Section 2.1, given that D•T=0, it may seem at first sight12 that the only possibility to construct conserved energy-momentumcurrentsTα in GRT is for models of the theory where appropriate Killing vector fields (such that one is timelike and the other three are 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 in 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 (27), we cannot define an energy-momentum covector P analogous to the one given by (26). 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 asymptotically flat13 in which case we can choose a chart such that at spatial infinity limx→→∞γα(x0,x→)=ϑα=dxα and limx→→γgμν=ημν.Thus, parodying Sachs and Wu [4] we must say that nonexistence 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 Section2.1 we next show that there exists trivially an infinity of conserved currents (the Komar currents) in any Lorentzian space-time 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 Cl(M,g).Remark 5.
We end this subsection recalling 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 space-time structure of GRT. Details of such a theory may be found in [5].


### 2.4. Komar Currents: Their Mathematical and Physical Meaning

LetA∈sec⁡TM be the generator of a one-parameter group of diffeomorphisms of M in the space-time structure 〈M,g,D,τg,↑〉 which is a model of a gravitational field generated by T∈sec⁡T11M (the matter fields energy-momentum tensor) in GRT. It is quite obvious that if we define F=dA, where A=g(A,)∈sec⋀1T∗M↪Cl(M,g), then the current(30)JA=-δgFis conserved; that is,(31)δgJA=0.Surprisingly such a trivialmathematical result seems to be very important for people working in GRT who call JA the Komar current14 [6]. Komar called15(32)E=∫V⋆gJA=∫∂V⋆gFthegeneralized energy.To understand whyJA is considered important in GRT write the action for the gravitational plus matter and nongravitational fields as(33)A=∫Lg+∫Lm=-12∫Rτg+∫Lm.Now, the equations of motion forg can be obtained considering its variation under an (infinitesimal) diffeomorphism h:M→M generated by A. We have that g↦g′=h∗g=g+δ0g where16 the variation δ0g=-£Ag. Taking into account Cartan’s magical formula (£AM=A⌟gdM+d(A⌟gM), for any M∈sec⋀T∗M), we have(34)δ0A=∫δ0Lg+∫δ0Lm=-∫£ALg-∫£ALm=-∫dA⌟Lg-∫dA⌟Lm≔∫d⋆gC,where(35)⋆gC=-A⌟Lg-A⌟Lm+Kwith dK=0.To proceed introduce a coordinate chart with coordinates{xμ} for the region of interest U⊂M. Recall that Gμ=Rμ-1/2Rϑμ are the Einstein 1-form fields17, with Rμ=Rνμϑμ being the Ricci 1-forms and R the curvature scalar. Einstein equation obtained from the variation principle δ0A=0 is Eμ:=Gμ+Tμ=0, with Tμ=Tνμϑν being the energy-momentum 1-form fields and moreover DμGμν=0=DμTμν.Next write explicitly the action as [7](36)A=-12∫R-det⁡gdx0dx1dx2dx3+∫Lm-det⁡gdx0dx1dx2dx3.We have immediately(37)δ0A=-12∫Eμν£Agμν-det⁡gdx0dx1dx2dx3=-∫EμνDμAν-det⁡gdx0dx1dx2dx3=-∫DμEμνAν-det⁡gdx0dx1dx2dx3=-∫∂⌟gEνAντg=∫⋆gδgEνAν=-∫d⋆gEνAν.From (34) and (37) we have(38)∫d⋆gEνAν+d⋆gC=0,and thus(39)δgEνAν+δgC=0.Thus, the current C∈sec⋀1T∗M is conserved if the field equations Eν=0 are satisfied. An equation (in component form) equivalent to (39) already appears in [6] (and also previously in [8]) that took C=EνKν+N with δgN=0.Here, to continue, we prefer to write an identity involving onlyδ0Ag=∫‍δ0Lg. Proceeding exactly as before we get putting G(A)=GμAμ that there exists N∈sec⋀1T∗M such that(40)δgGA+δgN=0and we see that we can identify(41)N≔-GμAμ+L,where δgL=0. Now, we claim the following.Proposition 6.
There existsL∈sec⋀1T∗M such that(42)N=-GμAμ+L=δgdA=-JA,where JA was defined in (30).Proof.
To prove our claim we suppose from now on that⋀T∗M↪Cl(M,g)18.
Then it is possible to write(43)GμAμ=RμAμ-12RA=∂∧∂A-12RA=∂∧∂A+∂·∂A-12RA-∂·∂A=∂2A-12RA-∂·∂A=-δgdA-dδgA-12RA-∂·∂A,where ∂∧∂ is the Ricci operator and ∂·∂=□ is the D’Alembertian operator. Then we take(44)-GμAμ-dδgA-12RA-∂·∂A=δgdAand of course it is19(45)L=-dδgA-12RA-∂·∂Aproving the proposition.Now that we found aL satisfying (42) we investigate if we can give some nontrivial physical meaning to such N=-JA∈sec⋀1T∗M.


#### 2.4.1. Determination of the Explicit Form ofJA

We recall that the extensor fieldT acts on A as(46)TA=TμAμ. Thus, since(47)GμAμ=-TA we have from (44)(48)δgdA=TA-dδgA-12RA-∂·∂A.We can write (48) taking into account that R=trT=Tμμ and putting F:=dA and that(49)δF=-JA,where [9](50)JA=-TA+12tr⁡TA+dδgA+∂·∂A.Equation (50) gives the explicit form for the Komar current20. Moreover, taking into account that δgF=⋆gd⋆gF, it is(51)d⋆F=⋆g-1-TA+12tr⁡TA+dδgA+∂·∂A=⋆g-TA+12tr⁡TA+dδgA+∂·∂Aand thus taking into account Stokes theorem(52)∫Vd⋆gF=∫∂V⋆gF=∫V⋆gJA. Moreover, since d⋆gJA=0 we have that(53)0=∫Nd⋆gJA=∫∂N⋆gJA and thus ∫Σ1‍⋆gJA(∂N=Σ1-Σ2+Ξ) is a conserved quantity. We arrive at the conclusion that taking V⊂Σ1 (∂V=SR) as a ball of radius R and making R→∞ the quantity(54)E≡EA≔18π∫SR⋆gF(55)=18π∫V⋆gTA-12AtrT-dδgA-∂·∂Ais conserved.Remark 7.
It is very important to realize that quantityE defined by (54) is a scalar invariant, that is, its value does not depend on the particular reference frame Z and (nacs∣Z) (the naturally adapted 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 E(A) and the different EA’s are not related as components of a covector.Remark 8.
As we already remarked an equation equivalent to (54) has already been obtained in [6] that called (as said above) that quantity the conserved generalized energy. But to the best of our knowledge (55) is new and appears for the first time in [9].
However, considering that for eachA∈sec⁡TM that generates a one-parameter group of diffeomorphisms of M we have a conserved quantity it is not in our opinion appropriate to think about this quantity as a generalized energy. Indeed, why should the energy depend on terms like dδgA and ∂·∂A if A is not a dynamical field?
We know that [10] when A=K is a Killing vector field it is δgA=0 and ∂·∂A=-T(A)+1/2trTA and thus (55) reads(56)E=14π∫N⋆gTA-12Atr⁡Twhich is a well known conserved quantity22. For a Schwarzschild space-time, as is well known, A=∂/∂t is a timelike Killing vector field and in this case since the components of T are Tνμ=8π/-det⁡gρ(r)vμvν and vivj=0 (since vμ=1/g00(1,0,0,0)) we get E=m.Remark 9.
Note that the conserved quantity given by (56) differs in general from the conserved quantity obtained with the current defined in (9) when V=K which holds in any structure (M,g,D) with the conditions given there. However, in the particular case analyzed above, (56) and (24) give the same result.Remark 10.
Originally Komar obtained the same result as in (56) directly from (54) supposing that the generator of the one-parameter group of diffeomorphisms was A=∂/∂t. So, he got E=m by pure chance. If he had picked another vector field generator of a one-parameter group of diffeomorphisms A≠∂/∂t, he, of course, would have not obtained that result.Remark 11.
The previous remark shows clearly that the construction of Komar currents does not solve the energy-momentum conservation problem for a system consisting of the matter and nongravitational fields plus the gravitational field in GRT.
Indeed, to 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 space-time (recall Section C.3.4). This can only be done if the space-time structure modelling a gravitational field (generated by the matter fields energy-momentum tensor T) possesses appropriate additional structure, or if we interpret the gravitational field as a field in the Faraday sense living in Minkowski space-time. More details are in [5, 11].


#### 2.4.2. The Maxwell-Like Equation∂F=JA Encodes Einstein Equation

From (49) where F=dA(A=g(A,)) with A∈sec⁡TM being an arbitrary generator 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)(57)∂F=JAwith a well defined conserved current. Of course, as we already said, there is an infinity of such equations. Each oneencodes Einstein equation; that is, given the form of JA (see (50)), we can get back (43), which gives immediately Einstein equation (EE). In this sense we can claim that(58)EEG=Tand theMLE∂F=JAare equivalent.Remark 12.
Finally it is worthy 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, that is, 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 inLorentzian andteleparallel de Sitter space-time structures, where these structures arenot supposed to represent models of gravitational fields in GRT, and compare these results with the ones in GRT. This will be one of the next sections.


## 2.1. The CurrentsJV and JK

LetW=Wαβeα⊗eβ∈sec⁡T20M with Wαβ=Wβα and Wˇ=Wαβγα⊗γβ∈sec⁡T02M, Wαβ=ηαςηβτWςτ, and W=Wβαγβ⊗eα∈sec⁡T11M. For the applications we have in mind we will say that W,Wˇ, and W are physically equivalent.Note thatW (an example of an extensor field7) is such that(4)W:sec⁡⋀1T∗M⟶sec⁡⋀1T∗M,WV=VαWβαγβ.Define the divergence of W as the 1-form field(5)D•W≔DαWβαγβ,where(6)DαWβα≔DeαWβα=eαWβα+ΓαιαWβι-ΓαβιWια.Moreover, introduce the1-form fields(7)Wβ≔Wαβγα∈sec⁡⋀1T∗M.Remark 1.
Take notice for the developments that follow that the Hodge coderivative of the1-form fields Wβ is as follows (see Appendix): (8)δgWβ=-γκ⌟DeκWαβγα=-eκWαβγκ⌟γα-WαβΓκαιγκ⌟γι=-eαWαβ-WαβΓκακ.
So,D∙W=0 does not imply that δgWβ=0.Now, given a vector fieldV=Vαeα and the physically equivalent covector field V=Vαγα, define the current(9)JV=VαWα∈sec⁡⋀1T∗M↪sec⁡ClM,g.Of course, writing(10)JV=Jβγβ we have(11)Jβ=VαWαβ.Recalling (see AppendixA) that ⋆1=τg, define(12)
T
=
W
α
β
⋆
g
1
=
W
α
β
τ
g
∈
s
e
c
⋀
4
T
∗
M
↪
C
l
M
,
g
.Then, we have, with∂ denoting the Dirac operator,(13)d⋆gJV=∂∧⋆gJV=γα∧Deα⋆gJV=Deαγα∧⋆gJV-Deαγα∧⋆gJV.Taking into account thatγα∧⋆gJV=⋆g(γα⌟JV) and Deαγα∧⋆gJV=⋆g(Deαγα⌟JV) we can write (13) as(14)d⋆gJV=eαVκWκα+VκeαWκα+Γ·αβα··VκWκβτg.Also, we can easily verify from (5) that(15)⋆gD•WV=D•WVτg=VκeαWκα+Γ·αια··WκιVκ-Γ·ακι··WιαVκτg.Now, let£ be the (standard) Lie derivative operator. Let us evaluate the product of £Vg(eα,eβ) by T; that is,(16)£Vgeα,eβT∈sec⋀4T∗M.From Cartan magical formula we get (17)£Vγα=dVα+V⌟∂∧γα=eιVαγι-VςΓ·ςια··γι+VςΓ·ιςα··γι.Then,(18)£Vgeα,eβT=ηικ£Vγι⊗γκ+ηικγι⊗£Vγκeα,eβT=2eκVιWκι+2VςΓ·ιςκ··WκιTand we get from (14), (15), and (18) the important identity [3](19)£Vgeα,eβT=2d⋆JV-2⋆gD•WV.From (19) we see that if V is a conformal Killing vector field, that is, £Vg(eα,eβ)=2ληαβ, we have(20)⋆gλtrW=d⋆gJV-⋆D•WV,where trW is the trace of the matrix with entries Wβα.


## 2.2. Conserved Currents Associated with a Covariantly ConservedW

Definition 2.
One says thatW is “covariantly conserved” if(21)D•W=0.In this case, ifV=K is a Killing vector field, then £Kg=0 and we have(22)d⋆gJK=⋆gD•WKand the current 3-form field ⋆gJK is closed; that is, d⋆gJK=0, or equivalently (taking into account the definition of the Hodge coderivative operator δg)(23)δgJK=0.In resume, when we have Killing vector fields8  Ki,i=1,2,…,n, “covariant conservation” of the tensor field W, that is, D•W=0, implies ingenuine conservation laws for the currents JKi that from δgJKi=0, we can using Stokes theorem build thescalar conserved quantities(24)EKi≔18π∫Σ′⋆gJKi,where N is the region where JK has support and ∂N=Σ+Σ′+Ξ, where Σ,Σ′ are spacelike surfaces and JKi is null at Ξ (spatial infinity).


## 2.3. Conserved Currents in GRT Associated with Killing Vector Fields

Before studying the conditions for the existence or not of genuine energy-momentum conservation laws in GRT, let us recall from AppendixC.3.4 that in Minkowski space-time9(M≃R4,η,D,τη,↑) we can introduce global coordinates {xμ} (in Einstein-Lorentz-Poincaré gauge) such that η(eμ,eν)=ημν and Deμeν=0, where {eμ=∂/∂xμ} is simultaneously a global tetrad and a coordinate basis. Also {ϑμ=dxμ} is a global cotetrad and a coordinate cobasis.Moreover,eμ=∂/∂xμ are also Killing vector fields in (M≃R4,η) and thus we have for a closed physical system (consisting of particles and fields in interaction living in Minkowski space-time and whose equations of motion are derived from a variational principle with a Lagrangian density invariant under space-time translations) that the currents10(25)Tα≔J∂/∂xα=Tανϑν∈sec⋀1T∗M↪ClM,η,M≃R4are the conserved energy-momentum 1-form fields of the physical system under consideration, for which we know that the quantity (recall (C.38))(26)P=Pαϑαo=PαEα,with(27)Pα=∫⋆gTαis the components of theconserved energy-momentumcovector (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-momentumT is modelled by a Lorentzian space-time11  (M,g,D,τg,↑) where the relation between g and T is given by Einstein equation which using the orthonormal bases {eα} and {γα} introduced above reads(28)Gβα=Rβα-12δβαR=-Tβα.Moreover, defining G=Gβαγβ⊗eα and recalling that T=Tβαγβ⊗eα, it is(29)D•G=0,D•T=0.Basedonly on the contents of Section 2.1, given that D•T=0, it may seem at first sight12 that the only possibility to construct conserved energy-momentumcurrentsTα in GRT is for models of the theory where appropriate Killing vector fields (such that one is timelike and the other three are 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 in 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 (27), we cannot define an energy-momentum covector P analogous to the one given by (26). 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 asymptotically flat13 in which case we can choose a chart such that at spatial infinity limx→→∞γα(x0,x→)=ϑα=dxα and limx→→γgμν=ημν.Thus, parodying Sachs and Wu [4] we must say that nonexistence 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 Section2.1 we next show that there exists trivially an infinity of conserved currents (the Komar currents) in any Lorentzian space-time 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 Cl(M,g).Remark 5.
We end this subsection recalling 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 space-time structure of GRT. Details of such a theory may be found in [5].


## 2.3.1. Limited Possibility to Construct a CEMC in GRT

Now, recall that in GRT a gravitational field generated by an energy-momentumT is modelled by a Lorentzian space-time11  (M,g,D,τg,↑) where the relation between g and T is given by Einstein equation which using the orthonormal bases {eα} and {γα} introduced above reads(28)Gβα=Rβα-12δβαR=-Tβα.Moreover, defining G=Gβαγβ⊗eα and recalling that T=Tβαγβ⊗eα, it is(29)D•G=0,D•T=0.Basedonly on the contents of Section 2.1, given that D•T=0, it may seem at first sight12 that the only possibility to construct conserved energy-momentumcurrentsTα in GRT is for models of the theory where appropriate Killing vector fields (such that one is timelike and the other three are 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 in 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 (27), we cannot define an energy-momentum covector P analogous to the one given by (26). 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 asymptotically flat13 in which case we can choose a chart such that at spatial infinity limx→→∞γα(x0,x→)=ϑα=dxα and limx→→γgμν=ημν.Thus, parodying Sachs and Wu [4] we must say that nonexistence 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 Section2.1 we next show that there exists trivially an infinity of conserved currents (the Komar currents) in any Lorentzian space-time 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 Cl(M,g).Remark 5.
We end this subsection recalling 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 space-time structure of GRT. Details of such a theory may be found in [5].


## 2.4. Komar Currents: Their Mathematical and Physical Meaning

LetA∈sec⁡TM be the generator of a one-parameter group of diffeomorphisms of M in the space-time structure 〈M,g,D,τg,↑〉 which is a model of a gravitational field generated by T∈sec⁡T11M (the matter fields energy-momentum tensor) in GRT. It is quite obvious that if we define F=dA, where A=g(A,)∈sec⋀1T∗M↪Cl(M,g), then the current(30)JA=-δgFis conserved; that is,(31)δgJA=0.Surprisingly such a trivialmathematical result seems to be very important for people working in GRT who call JA the Komar current14 [6]. Komar called15(32)E=∫V⋆gJA=∫∂V⋆gFthegeneralized energy.To understand whyJA is considered important in GRT write the action for the gravitational plus matter and nongravitational fields as(33)A=∫Lg+∫Lm=-12∫Rτg+∫Lm.Now, the equations of motion forg can be obtained considering its variation under an (infinitesimal) diffeomorphism h:M→M generated by A. We have that g↦g′=h∗g=g+δ0g where16 the variation δ0g=-£Ag. Taking into account Cartan’s magical formula (£AM=A⌟gdM+d(A⌟gM), for any M∈sec⋀T∗M), we have(34)δ0A=∫δ0Lg+∫δ0Lm=-∫£ALg-∫£ALm=-∫dA⌟Lg-∫dA⌟Lm≔∫d⋆gC,where(35)⋆gC=-A⌟Lg-A⌟Lm+Kwith dK=0.To proceed introduce a coordinate chart with coordinates{xμ} for the region of interest U⊂M. Recall that Gμ=Rμ-1/2Rϑμ are the Einstein 1-form fields17, with Rμ=Rνμϑμ being the Ricci 1-forms and R the curvature scalar. Einstein equation obtained from the variation principle δ0A=0 is Eμ:=Gμ+Tμ=0, with Tμ=Tνμϑν being the energy-momentum 1-form fields and moreover DμGμν=0=DμTμν.Next write explicitly the action as [7](36)A=-12∫R-det⁡gdx0dx1dx2dx3+∫Lm-det⁡gdx0dx1dx2dx3.We have immediately(37)δ0A=-12∫Eμν£Agμν-det⁡gdx0dx1dx2dx3=-∫EμνDμAν-det⁡gdx0dx1dx2dx3=-∫DμEμνAν-det⁡gdx0dx1dx2dx3=-∫∂⌟gEνAντg=∫⋆gδgEνAν=-∫d⋆gEνAν.From (34) and (37) we have(38)∫d⋆gEνAν+d⋆gC=0,and thus(39)δgEνAν+δgC=0.Thus, the current C∈sec⋀1T∗M is conserved if the field equations Eν=0 are satisfied. An equation (in component form) equivalent to (39) already appears in [6] (and also previously in [8]) that took C=EνKν+N with δgN=0.Here, to continue, we prefer to write an identity involving onlyδ0Ag=∫‍δ0Lg. Proceeding exactly as before we get putting G(A)=GμAμ that there exists N∈sec⋀1T∗M such that(40)δgGA+δgN=0and we see that we can identify(41)N≔-GμAμ+L,where δgL=0. Now, we claim the following.Proposition 6.
There existsL∈sec⋀1T∗M such that(42)N=-GμAμ+L=δgdA=-JA,where JA was defined in (30).Proof.
To prove our claim we suppose from now on that⋀T∗M↪Cl(M,g)18.
Then it is possible to write(43)GμAμ=RμAμ-12RA=∂∧∂A-12RA=∂∧∂A+∂·∂A-12RA-∂·∂A=∂2A-12RA-∂·∂A=-δgdA-dδgA-12RA-∂·∂A,where ∂∧∂ is the Ricci operator and ∂·∂=□ is the D’Alembertian operator. Then we take(44)-GμAμ-dδgA-12RA-∂·∂A=δgdAand of course it is19(45)L=-dδgA-12RA-∂·∂Aproving the proposition.Now that we found aL satisfying (42) we investigate if we can give some nontrivial physical meaning to such N=-JA∈sec⋀1T∗M.


### 2.4.1. Determination of the Explicit Form ofJA

We recall that the extensor fieldT acts on A as(46)TA=TμAμ. Thus, since(47)GμAμ=-TA we have from (44)(48)δgdA=TA-dδgA-12RA-∂·∂A.We can write (48) taking into account that R=trT=Tμμ and putting F:=dA and that(49)δF=-JA,where [9](50)JA=-TA+12tr⁡TA+dδgA+∂·∂A.Equation (50) gives the explicit form for the Komar current20. Moreover, taking into account that δgF=⋆gd⋆gF, it is(51)d⋆F=⋆g-1-TA+12tr⁡TA+dδgA+∂·∂A=⋆g-TA+12tr⁡TA+dδgA+∂·∂Aand thus taking into account Stokes theorem(52)∫Vd⋆gF=∫∂V⋆gF=∫V⋆gJA. Moreover, since d⋆gJA=0 we have that(53)0=∫Nd⋆gJA=∫∂N⋆gJA and thus ∫Σ1‍⋆gJA(∂N=Σ1-Σ2+Ξ) is a conserved quantity. We arrive at the conclusion that taking V⊂Σ1 (∂V=SR) as a ball of radius R and making R→∞ the quantity(54)E≡EA≔18π∫SR⋆gF(55)=18π∫V⋆gTA-12AtrT-dδgA-∂·∂Ais conserved.Remark 7.
It is very important to realize that quantityE defined by (54) is a scalar invariant, that is, its value does not depend on the particular reference frame Z and (nacs∣Z) (the naturally adapted 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 E(A) and the different EA’s are not related as components of a covector.Remark 8.
As we already remarked an equation equivalent to (54) has already been obtained in [6] that called (as said above) that quantity the conserved generalized energy. But to the best of our knowledge (55) is new and appears for the first time in [9].
However, considering that for eachA∈sec⁡TM that generates a one-parameter group of diffeomorphisms of M we have a conserved quantity it is not in our opinion appropriate to think about this quantity as a generalized energy. Indeed, why should the energy depend on terms like dδgA and ∂·∂A if A is not a dynamical field?
We know that [10] when A=K is a Killing vector field it is δgA=0 and ∂·∂A=-T(A)+1/2trTA and thus (55) reads(56)E=14π∫N⋆gTA-12Atr⁡Twhich is a well known conserved quantity22. For a Schwarzschild space-time, as is well known, A=∂/∂t is a timelike Killing vector field and in this case since the components of T are Tνμ=8π/-det⁡gρ(r)vμvν and vivj=0 (since vμ=1/g00(1,0,0,0)) we get E=m.Remark 9.
Note that the conserved quantity given by (56) differs in general from the conserved quantity obtained with the current defined in (9) when V=K which holds in any structure (M,g,D) with the conditions given there. However, in the particular case analyzed above, (56) and (24) give the same result.Remark 10.
Originally Komar obtained the same result as in (56) directly from (54) supposing that the generator of the one-parameter group of diffeomorphisms was A=∂/∂t. So, he got E=m by pure chance. If he had picked another vector field generator of a one-parameter group of diffeomorphisms A≠∂/∂t, he, of course, would have not obtained that result.Remark 11.
The previous remark shows clearly that the construction of Komar currents does not solve the energy-momentum conservation problem for a system consisting of the matter and nongravitational fields plus the gravitational field in GRT.
Indeed, to 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 space-time (recall Section C.3.4). This can only be done if the space-time structure modelling a gravitational field (generated by the matter fields energy-momentum tensor T) possesses appropriate additional structure, or if we interpret the gravitational field as a field in the Faraday sense living in Minkowski space-time. More details are in [5, 11].


### 2.4.2. The Maxwell-Like Equation∂F=JA Encodes Einstein Equation

From (49) where F=dA(A=g(A,)) with A∈sec⁡TM being an arbitrary generator 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)(57)∂F=JAwith a well defined conserved current. Of course, as we already said, there is an infinity of such equations. Each oneencodes Einstein equation; that is, given the form of JA (see (50)), we can get back (43), which gives immediately Einstein equation (EE). In this sense we can claim that(58)EEG=Tand theMLE∂F=JAare equivalent.Remark 12.
Finally it is worthy 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, that is, 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 inLorentzian andteleparallel de Sitter space-time structures, where these structures arenot supposed to represent models of gravitational fields in GRT, and compare these results with the ones in GRT. This will be one of the next sections.


## 2.4.1. Determination of the Explicit Form ofJA

We recall that the extensor fieldT acts on A as(46)TA=TμAμ. Thus, since(47)GμAμ=-TA we have from (44)(48)δgdA=TA-dδgA-12RA-∂·∂A.We can write (48) taking into account that R=trT=Tμμ and putting F:=dA and that(49)δF=-JA,where [9](50)JA=-TA+12tr⁡TA+dδgA+∂·∂A.Equation (50) gives the explicit form for the Komar current20. Moreover, taking into account that δgF=⋆gd⋆gF, it is(51)d⋆F=⋆g-1-TA+12tr⁡TA+dδgA+∂·∂A=⋆g-TA+12tr⁡TA+dδgA+∂·∂Aand thus taking into account Stokes theorem(52)∫Vd⋆gF=∫∂V⋆gF=∫V⋆gJA. Moreover, since d⋆gJA=0 we have that(53)0=∫Nd⋆gJA=∫∂N⋆gJA and thus ∫Σ1‍⋆gJA(∂N=Σ1-Σ2+Ξ) is a conserved quantity. We arrive at the conclusion that taking V⊂Σ1 (∂V=SR) as a ball of radius R and making R→∞ the quantity(54)E≡EA≔18π∫SR⋆gF(55)=18π∫V⋆gTA-12AtrT-dδgA-∂·∂Ais conserved.Remark 7.
It is very important to realize that quantityE defined by (54) is a scalar invariant, that is, its value does not depend on the particular reference frame Z and (nacs∣Z) (the naturally adapted 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 E(A) and the different EA’s are not related as components of a covector.Remark 8.
As we already remarked an equation equivalent to (54) has already been obtained in [6] that called (as said above) that quantity the conserved generalized energy. But to the best of our knowledge (55) is new and appears for the first time in [9].
However, considering that for eachA∈sec⁡TM that generates a one-parameter group of diffeomorphisms of M we have a conserved quantity it is not in our opinion appropriate to think about this quantity as a generalized energy. Indeed, why should the energy depend on terms like dδgA and ∂·∂A if A is not a dynamical field?
We know that [10] when A=K is a Killing vector field it is δgA=0 and ∂·∂A=-T(A)+1/2trTA and thus (55) reads(56)E=14π∫N⋆gTA-12Atr⁡Twhich is a well known conserved quantity22. For a Schwarzschild space-time, as is well known, A=∂/∂t is a timelike Killing vector field and in this case since the components of T are Tνμ=8π/-det⁡gρ(r)vμvν and vivj=0 (since vμ=1/g00(1,0,0,0)) we get E=m.Remark 9.
Note that the conserved quantity given by (56) differs in general from the conserved quantity obtained with the current defined in (9) when V=K which holds in any structure (M,g,D) with the conditions given there. However, in the particular case analyzed above, (56) and (24) give the same result.Remark 10.
Originally Komar obtained the same result as in (56) directly from (54) supposing that the generator of the one-parameter group of diffeomorphisms was A=∂/∂t. So, he got E=m by pure chance. If he had picked another vector field generator of a one-parameter group of diffeomorphisms A≠∂/∂t, he, of course, would have not obtained that result.Remark 11.
The previous remark shows clearly that the construction of Komar currents does not solve the energy-momentum conservation problem for a system consisting of the matter and nongravitational fields plus the gravitational field in GRT.
Indeed, to 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 space-time (recall Section C.3.4). This can only be done if the space-time structure modelling a gravitational field (generated by the matter fields energy-momentum tensor T) possesses appropriate additional structure, or if we interpret the gravitational field as a field in the Faraday sense living in Minkowski space-time. More details are in [5, 11].


## 2.4.2. The Maxwell-Like Equation∂F=JA Encodes Einstein Equation

From (49) where F=dA(A=g(A,)) with A∈sec⁡TM being an arbitrary generator 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)(57)∂F=JAwith a well defined conserved current. Of course, as we already said, there is an infinity of such equations. Each oneencodes Einstein equation; that is, given the form of JA (see (50)), we can get back (43), which gives immediately Einstein equation (EE). In this sense we can claim that(58)EEG=Tand theMLE∂F=JAare equivalent.Remark 12.
Finally it is worthy 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, that is, 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 inLorentzian andteleparallel de Sitter space-time structures, where these structures arenot supposed to represent models of gravitational fields in GRT, and compare these results with the ones in GRT. This will be one of the next sections.


## 3. The Lorentzian de SitterMdSL Structure and Its Conformal Representation

LetSO(1,4) and SO(1,3) be, respectively, the special pseudoorthogonal groups in R1,4={M∘=R5,g∘} and in R1,3={R4,η}, where g∘ is a metric of signature (1,4) and η is a metric of signature (1,3). The manifold M=SO(1,4)/SO(1,3) will be called thede Sitter manifold. Since(59)M=SO1,4/SO1,3≈R×S3this manifold can be viewed as a brane (a submanifold) in the structure R1,4. We now introduce a Lorentzian space-time, that is, the structure MdSL=(M=R×S3,g,D,τg,↑) which will be calledLorentzian de Sitter space-time structure where if ι:R×S3→R5, g=ι∗g∘ and D is the parallel projection on M of the pseudo-Euclidian metric compatible connection in R1,4 (details in [12]). As is well known, MdSL is a space-time 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≈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 manifoldR×S3 as a pseudosphere (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 representing the pseudosphere is(60)X02-X12-X22-X32-X42=-l2.Introducingconformal 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(61)g=i∗g∘=Ω2ημνdxμ⊗dxν,where(62)Xμ=Ωxμ,X4=-lΩ1+σ24l2,Ω=1-σ24l2-1,σ2=ημνxμxν.Since the north pole of the pseudosphere is not covered by the coordinate functions, we see that (omitting two dimensions) the region of the space-time as seen by an observer living in the south pole is the region inside the so-called absolute ofCayley-Klein of equation(63)t2-x2=4l2.In Figure1 we can see that all timelike curves (1) and (2) and lightlike (3) curve start in the “past horizon” and end on the “future horizon.”Figure 1
Conformal representation of de Sitter space-time. Note that the “observer” space-time is the interior of the Cayley-Klein absolutet2-x→2=4l2.


## 4. On the Geodesics ofMdSL

In a classic book by Hawking and Ellis [13] 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.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 Figure1). So, these curves are never the path of any particle. A complete and thoughtful discussion of this issue is given in an old and excellent article by Schmidt [14].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 have thought for a long time that this permits the formulation of an energy-momentum conservation law in de Sitter space-time MdSL structure. However, the fact is that what can really be done is the obtainment of conserved quantities like Pα in (27). But we cannot obtain in MdSL structure an energy-momentum covector like P for a given closed physical system using an equation similar to (26). This is because de Sitter space-time is not asymptotically flat and so there is no way to physically determine a point to fix the γα to use in an equation similar to (26).Now, the “translation” Killing vector fields of de Sitter space-time are expressed in the coordinate basis{∂μ=∂/∂xμ} (where {xμ} are the conformal coordinates introduced in the last section) by27:(64)Πα=ξαμ∂μwith (puttingl=1, for simplicity) (65)ξαμ=δαμ-ηανxνxμ2-σ24δαμ,(66)ξα0=δα0-σ24ηανxνx02-σ24δα0,ξα1=δα1-σ24ηανxνx12-σ24δα1,ξα2=δα2-σ24ηανxνx22-σ24δα2,ξα3=δα3-σ24ηανxνx32-σ24δα3.So, we want now to investigate whether there is some region of de Sitter space-time where the “translational” Killing vector fields are linearly independent.In order to proceed, we take an arbitrary vector fieldV=Vμ∂μ∈sec⁡TU. If Πα are linearly independent in a region U′⊂U⊂R×S3, then we can write(67)V=Vμ∂μ=VαΠα=Vαξαμ∂μ.So, the condition for the existence of a nontrivial solution for the Vα is that(68)det⁡ξ00ξ01ξ02ξ03ξ10ξ11ξ12ξ13ξ20ξ21ξ22ξ23ξ30ξ31ξ32ξ33≠0.Thus, putting(69)χμ≔σ24xμ22-σ24 we need to evaluate the determinant of the matrix(70)
1
-
χ
0
σ
2
4
t
x
2
σ
2
4
t
y
2
σ
2
4
t
z
2
-
σ
2
4
x
t
2
1
-
χ
1
-
σ
2
4
x
y
2
-
σ
2
4
x
z
2
-
σ
2
4
y
t
2
-
σ
2
4
y
x
2
1
-
χ
2
-
y
z
m
2
/
4
2
-
σ
2
4
z
t
2
-
σ
2
4
z
x
2
-
σ
2
4
z
y
2
1
-
χ
3
.Then(71)det⁡ξαμ=-1512σ10-332σ6+116σ6+σ2+1-3128σ8-18σ4+11024σ83t2x2y2z2+t2y2z2-x2y2z2+y2z2+1512σ8t2x2y2+t2x2z2+1128σ6t2x2y2+z2+1256σ6t2y2z2-x2y2z2+2y2z2+164σ4t2x2y2z2+y2z2-x2y2z2+116σ2t2y2z2-x2y2z2-2y2z2+38σ4-14y2z2+1256σ8.In order to analyze this expression we put (without loss of generality; see the reason in [14]) y=z=0. In this case(72)det⁡ξαμ=1512σ2+43-σ4+2σ2+8which is null on the Caley-Klein absolute, that is, at the points(73)t2-x→2=4,and also on the spacelike hyperbolas given by(74)t2-x→2=-2,t2-x→2=-4.So, we have proved that the translational Killing vector fields are linearly independent in all the subregions inside the Cayley-Klein absolute except in the points of the hyperbolast2-x→2=-2 and t2-x→2=-4.This result is very important for the following reason. Timelike geodesics in de Sitter space-time structureMdSL are (as is well known) the curves σ:s→σ(s), where s is proper time along σ, which minimizes or maximizes the length function [15], that is, calling σ∗=u the velocity of a “particle” of mass m following a timelike geodesic; we have that the equation of the geodesic is obtained by finding an extreme of the action, written here in sloop notation, as(75)Iσ=-m∫σds=-m∫σgμνdxμdxν1/2.As is well known, the determination of an extreme for I[σ] is given by evaluating the first variation28 of I[σ], that is,(76)δ0Iσ=m∫£Ygμνdxμdxν1/2,and putting δ0I[σ]=0. The result, as is well known, is the geodesic equation(77)Duu=0.Now, taking into account that the Killing vector fields determine a basis inside the Cayley-Klein absolute we write(78)u=uμ∂μ=UαΠα=Uαξαμ∂μ.We now define the “hybrid” connection coefficients29  Γ·μαβ·· by(79)D∂μΠα≔Γ·μαβ··Πβ,D∂μΠα≔-Γ·μβα··Πβand write the geodesic equation as(80)Duu=uμD∂μUαΠα=uμ∂μUαΠα+uμUαD∂μΠα=uμ∂μUβΠβ+uμUαΓ·μαβ··Πβ=dUβds+uμUαΓ·μαβ··Πβ=0or(81)dUβds+uμUαΓ·μαβ··=0,which on multiplying by the “mass” m and calling πβ=mUβ and πρ=gρβπβ can be written equivalently as(82)dπρds-Γ·μρβ··uμπβ=0,which looks like Equation (37) in [1].We want now to investigate the following question: is (82) the same as Equation (37) in [1]?To know the answer to the above question recalls that in [1] authors investigate the variation of I[σ] under a variation of the curves σ(s)↦σ(s,l) induced by a coordinate transformation xμ↦x′μ+δΠxμ, where they put(83)δΠxμ=ξρμxδxρwith ξρμ(x) being the components of the Killing vector fields Πα (recall (65)) and where it is said that δxρ is anordinary variation. However in [1] we cannot find what authors mean by ordinary variation, and so δxρ is not defined.So, to continue our analysis we recall that ifδxρ=εα are constants (but arbitrary), then δΠxμ corresponds to a diffeomorphism generated by a Killing vector field Π=εαΠα. However, if δxρ=λρ(x) are infinitesimalarbitrary functions (i.e., λρ(x)≪1), then the notation δΠxμ is misleading since ξρμλρ(x) is the δ0I[σ] variation generated by a quitearbitrary vector field Y=λρ(x)ξρμ(x)∂μ=λρ(x)Πρ=Yμ∂μ. In this case we get from δ0I[σ]=0 the geodesic equation.


### 4.1. Curves Obtained from Constrained Variations

So, let us study theconstrained variation when δxρ=εα are constants (but arbitrary). We denote the constrained variation by δcI[σ]. In this case starting from (75)(84)δcIσ=-m∫σuγDγuβξρβδxρds,where (taking account of notations already introduced) (85)uγDeγuβeβ=uγDγuβeβ,Dγuβ=∂γuβ-Γ·γβτ··uτ,Deγeβ=Γ·γβτ··eτ,Deγeβ=-Γ·γτβ··eτ.We can write(86)δcIσ=-m∫uγDγuβξρβδxρds=-m∫uγDγuβξρβδxρds+m2∫uγuβDγξβρ+Dβξργδxρds=-m∫uγDγu·Πρδxρds=-m∫uγDγUρδxρdsand δcI[σ]=0 implies(87)uγDγmUρ=0.Equation (87) with πρ=mUρ can be written as(88)ddsπρ-uγπβΓ·γρβ··=0which is Equation (37) in [1]. Note that this equations looks like the geodesic equation written as (82) above, but it is in factdifferent since of course, recalling (79), it is Γ·γρβ··≠Γ·γρβ··. In [1] πρ=mUρ is unfortunately wrongly interpreted as the components of a covector field over σ which is supposed to be the energy-momentum covector of the particle, because authors of [1] were supposed to prove that this equation could be derived from Papapetrou’s method, which is not the case as we show in Section 6.


## 4.1. Curves Obtained from Constrained Variations

So, let us study theconstrained variation when δxρ=εα are constants (but arbitrary). We denote the constrained variation by δcI[σ]. In this case starting from (75)(84)δcIσ=-m∫σuγDγuβξρβδxρds,where (taking account of notations already introduced) (85)uγDeγuβeβ=uγDγuβeβ,Dγuβ=∂γuβ-Γ·γβτ··uτ,Deγeβ=Γ·γβτ··eτ,Deγeβ=-Γ·γτβ··eτ.We can write(86)δcIσ=-m∫uγDγuβξρβδxρds=-m∫uγDγuβξρβδxρds+m2∫uγuβDγξβρ+Dβξργδxρds=-m∫uγDγu·Πρδxρds=-m∫uγDγUρδxρdsand δcI[σ]=0 implies(87)uγDγmUρ=0.Equation (87) with πρ=mUρ can be written as(88)ddsπρ-uγπβΓ·γρβ··=0which is Equation (37) in [1]. Note that this equations looks like the geodesic equation written as (82) above, but it is in factdifferent since of course, recalling (79), it is Γ·γρβ··≠Γ·γρβ··. In [1] πρ=mUρ is unfortunately wrongly interpreted as the components of a covector field over σ which is supposed to be the energy-momentum covector of the particle, because authors of [1] were supposed to prove that this equation could be derived from Papapetrou’s method, which is not the case as we show in Section 6.


## 5. Generalized Energy-Momentum Conservation Laws in de Sitter Space-Time Structures

### 5.1. Lie Algebra of the de Sitter Group

Given a structure(M∘≃R5,g∘,) introduced in Section 3 define JAB∈sec⁡TM∘ by(89)JAB≔ηACXC∂∂XB-ηBCXC∂∂XA.These objects are generators of the Lie algebra so(1,4) of the de Sitter group.Using the bases{∂μ},{dxμ} introduced above the ten Killing vector fields of de Sitter space-time are the fields Jα4∈sec⁡TM and Jμν∈sec⁡TM and it is30(90)Jα4=ηαCXC∂∂X4-η4CXC∂∂Xα=lPα-14lK4α=l∂α-14l2ηαλxλxν-σ2δαν∂ν,(91)Jμν=ημκXκ∂∂Xν-ηνκXκ∂∂Xμ=ημλxλPν-ηνλxλPμ.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 space-time. We have(92)Jα4,Jβ4=Jαβ,Jαβ,Jλ4=ηλβJα4-ηλαJβ4,Jαβ,Jλτ=ηαλJβτ+ηβτJαλ-ηβλJατ-ηατJβλ.It is usual in physical applications to define(93)Πα=Jα4lfor then we have(94)Πα,Πβ=1l2Jαβ,Jαβ,Πλ=ηλβΠα-ηλαΠβ,Jαβ,Jλτ=ηαλJβτ+ηβτJαλ-ηβλJατ-ηατJβλ.The Killing vector fieldsJαβ satisfy the Lie algebra so(1,3) (of the special Lorentz group).Remark 14.
From (94) we see that when l↦∞ the Lie algebra of so(1,4) goes into the Lie algebra of the Poincaré group P which is the semidirect sum of the group of translations in R4 plus component of the special Lorentz group; that is, P=(R4⊞SO(1,3)). This is eventually the justification for physicists to call Πα the “translation” generators of the de Sitter group.
However, it is necessary to have in mind that whereas the translation subgroupR4 of P acts transitively in Minkowski space-time manifold, the set{Πα} does not close in a subalgebra of the de Sitter algebra and thus it is impossible in general to find exp(λαΠα) such that given arbitrary x,y∈R×S3 it is y=exp(λαΠα)x. Only the whole group SO(1,4) acts transitively on R×S3.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(95)J=12EA∧EBJAB.Taking into account the results of AppendixA its (Clifford) square is(96)J2=J⌟J+J∧J=-J·J+J∧J.It is immediate to verify thatJ2 is invariant under the transformations of the de Sitter group. J⌟J is (a constant apart) 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(97)W≔⋆g∘18lJ∧J=18lJ∧J⌟τg∘,one can easily show (details in [16]) that(98)W·W=WW=W2=-164l2J∧J⌟J∧J=-164l2J∧J·J∧J=-164l2J∧JJ∧Jis indeed an invariant operator.As is well known, the representations of the de Sitter group are classified by their Casimir invariantsI1 and I2 which here following [17] we take as(99)I1=-J⌟J=12l2JABJAB=ηαβΠαΠβ+12l2ηαληβτJαβJλτ=M2,I2=W2=WAWA=ηαβVαVβ+1l2W42,where M∈R and the fields WA and Vα are defined by(100)WA≔18lεABCDEJABJDE,Vα≔-12ε4αλμνηλρΠρJμν,from where it follows that(101)W4=18ε4μνρτJμνJμν.In the limit when l↦∞ we get the Casimir operators of the special Lorentz group(102)I1⟼PαPα=m2,I2⟼ηαβVαVβ=m2ss+1,where m∈R and s=0,1/2,1,3/2,….We see thatΠα looks like the components of anenergy-momentum vectorP=Pαϑαoof a closed physical system (see (26)) in the Minkowski space-time of Special Relativity, for which P2=m2, with m being the mass of the system. However, take into account that whereas Pα are simple real numbers, Πα are vector fields.Moreover, take into account thatηαβΠαΠβ is not an invariant; that is, 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 Space-Time Structure

In this section we suppose that(M=R×S3,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.” Thus if we suppose that (M=R×S3,g,τg,↑) is populated by interacting matter fields {ϕ1,…,ϕn} with dynamics described by a Lagrangian formalism we can construct as described in Appendix C.1 the conserved currents(103)JΠα=Jαβγβ=Jαμϑμ,where taking into account that δΠα0=-£Πα and denoting by δΠα each particularlocal variation generated by Πα (recall (64)) we have(104)δΠαϕA=δΠα0ϕA+δxν∂νϕAand thus we write(105)Jαμ=πAμδΠαϕA+ΥνμδΠα0xμ=Λνμξαν+Υνμξαν,where(106)Λνμξαν≔πAμδΠαϕA.In (103) {γβ} is the dual basis of the orthonormal basis {eα} defined by31(107)eα≔ΠαgΠα,Πα.From the conserved currentsJΠα we can obtain four conserved quantities (Pα in (27)).Remark 15.
It is crucial to observe that the above results have been deduced without introduction of any connection in the structure(M=R×S3,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 (26)) of special relativistic theories.If we addD, the Levi-Civita connection of g, to (M=R×S3,g,τg,↑) we get the Lorentzian de Sitter space-time structure MdSP and defining a generalized energy-momentum tensor(108)Θ=JΠα⊗eα∈sec⁡T11Mwe know that δgJΠα=0 implies D·Θ=0, a covariant “conservation” law.However, the introduction ofMdSP in our game is of no help to construct a covector like P since in MdSP vectors at different space-time 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 PαdS such that(109)PdS=PαdSγαx=PαdSγαydefines a legitimate covector for a closed physical system living in a de Sitter structure (M=R×S3,g,τg,↑) and for which PdS can be said to be a kind of generalization of the momentum of the closed system in Minkowski space-time?We show now that the answer is positive. We recall thatD has been introduced in our developments only as a useful mathematical device and is quite irrelevant in the construction of legitimate conservation laws since the conserved currents JΠα have been obtained without the use of any connection. So, we now introduce for our goal ateleparallel de Sitter space-time, that is, the structure MdSTP=(M=R×S3,∇,τg,↑), where ∇ is a metric compatible teleparallel connection defined by(110)∇eαeβ=ω·αβκ··eκ=0.Under this condition we know that we can identify all tangent and all cotangent spaces. So, we have, ∀x,y∈R×S3,(111)eαx≃eαy,γαx≃γαy=Eα, where {Eα} is a basis of a vector space V≃R4.Thus, in the structureMdSTP (109) defines indeed a legitimate covector in V≃R4 and thus permits a legitimategeneralization of the concepts of energy-momentum covector obtained for physical theories in Minkowski space-time. The term generalization is a good one here because in the limit, where l→∞, Πα↦∂/∂xα,Λακ=0 and thus Θακ=Υακ.


### 5.3. The Conserved CurrentsJΠα=g(Πα,)

To proceed we show that the translational Killing vector fields of the de Sitter structure(M,g) determine trivially conserved currents(112)JΠα=gΠα,.Indeed using theMdSTP structure as a convenient device we recall the result proved in [10] that for each vector Killing K the one form field K=g(K,) is such that δgK=0. So, it is(113)δgJΠα=0and, of course, also(114)δgJΠ=0,where(115)JΠ≔gΠ,,Π≔εαΠαwith εα being real constants such that εα≪1. Recalling from (91) that in projective conformal coordinate bases ({eμ=∂μ} and {ϑμ=dxμ}) the components of Πα are(116)ξαμ=δαμ-14l22ηαρxρxμ-σ2δαμwe get that the conserved current JΠ is(117)JΠ=εαJΠα=εαξαμϑμ.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 [1] authors trying to generalize the results that follow from the canonical formalism for the case of field theories in Minkowski space-time suppose that they can eliminate the term πAμδΠαϕA from (105) 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.Now, thegeneralized canonical de Sitter energy-momentum tensor is(118)Θ=JΠα⊗eα=Θακϑκ⊗eα∈sec⋀1T∗M⊗TMand making analogy with the case of Minkowski space-time where Υακ have been defined as the components of the canonical energy-momentum tensor32  Υ, we write Θακ, as(119)Θακ=Υακ-14l2Kακ+Λακ.Thus each one of the genuine conservation laws d⋆gJΠα=0, α=0,1,2,3 reads in coordinate basis components as(120)∂μΥακ-Γ·λνλ··Υακ-14l2∂μKαμ-Γ·λνλ··Kακ-∂μΛακ-Γ·λνλ··Λακ=1-det⁡g∂μ-det⁡gΥαμ-14l21-det⁡g∂μ-det⁡gKαμ+1-det⁡g∂μ-det⁡gΛαμ=0 or(121)∂μ-det⁡gΥαμ-14l2Kαμ+Λαμ=0⟺DμΘαμ=0.Remark 18.
Now, recalling that the relation of the connection coefficients of the bases{eα} of the Levi-Civita connection D of g (denoted Γ·αβκ··) and the coefficients of the basis {eα} of the teleparallel connection ∇ of g (denoted Γ¯·αβκ··) are [11](122)Γ¯·αβκ··=Γ·αβκ··+Δ·αβκ··,where(123)Δ·αβκ··≔-12Tα·β·κ·+Tβ·α·κ·-T·αβκ··are the components of the contorsion tensor and T·αβκ·· are the components of the torsion tensor of the connection ∇, we can write D•Θ=0 (taking into account that Γ¯·αβκ··=0) in components relative to orthonormal basis as(124)DαΘβα≔DeαΘβα=eαΘβα+ΔαιαΘβι-ΔαβιΘια=0. On the other hand since ∇αΘβα:=(∇eαΘ)βα=eα(Θβα) we have(125)∇αΘβα=-ΔαιαΘβι+ΔαβιΘια,which means that although (∇•Θ)βα:=∇αΘβα≠0 we can generate the conserved currents JΠα in the teleparallel de Sitter space-time structure if (125) is satisfied.Remark 19.
To end this section and for completeness of the article it is necessary to mention that in a remarkable paper [18] the gravitational energy-momentum tensor in teleparallel gravity is discussed in detail. Also related papers are [19, 20]. A complete list of references can be found in [21].


## 5.1. Lie Algebra of the de Sitter Group

Given a structure(M∘≃R5,g∘,) introduced in Section 3 define JAB∈sec⁡TM∘ by(89)JAB≔ηACXC∂∂XB-ηBCXC∂∂XA.These objects are generators of the Lie algebra so(1,4) of the de Sitter group.Using the bases{∂μ},{dxμ} introduced above the ten Killing vector fields of de Sitter space-time are the fields Jα4∈sec⁡TM and Jμν∈sec⁡TM and it is30(90)Jα4=ηαCXC∂∂X4-η4CXC∂∂Xα=lPα-14lK4α=l∂α-14l2ηαλxλxν-σ2δαν∂ν,(91)Jμν=ημκXκ∂∂Xν-ηνκXκ∂∂Xμ=ημλxλPν-ηνλxλPμ.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 space-time. We have(92)Jα4,Jβ4=Jαβ,Jαβ,Jλ4=ηλβJα4-ηλαJβ4,Jαβ,Jλτ=ηαλJβτ+ηβτJαλ-ηβλJατ-ηατJβλ.It is usual in physical applications to define(93)Πα=Jα4lfor then we have(94)Πα,Πβ=1l2Jαβ,Jαβ,Πλ=ηλβΠα-ηλαΠβ,Jαβ,Jλτ=ηαλJβτ+ηβτJαλ-ηβλJατ-ηατJβλ.The Killing vector fieldsJαβ satisfy the Lie algebra so(1,3) (of the special Lorentz group).Remark 14.
From (94) we see that when l↦∞ the Lie algebra of so(1,4) goes into the Lie algebra of the Poincaré group P which is the semidirect sum of the group of translations in R4 plus component of the special Lorentz group; that is, P=(R4⊞SO(1,3)). This is eventually the justification for physicists to call Πα the “translation” generators of the de Sitter group.
However, it is necessary to have in mind that whereas the translation subgroupR4 of P acts transitively in Minkowski space-time manifold, the set{Πα} does not close in a subalgebra of the de Sitter algebra and thus it is impossible in general to find exp(λαΠα) such that given arbitrary x,y∈R×S3 it is y=exp(λαΠα)x. Only the whole group SO(1,4) acts transitively on R×S3.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(95)J=12EA∧EBJAB.Taking into account the results of AppendixA its (Clifford) square is(96)J2=J⌟J+J∧J=-J·J+J∧J.It is immediate to verify thatJ2 is invariant under the transformations of the de Sitter group. J⌟J is (a constant apart) 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(97)W≔⋆g∘18lJ∧J=18lJ∧J⌟τg∘,one can easily show (details in [16]) that(98)W·W=WW=W2=-164l2J∧J⌟J∧J=-164l2J∧J·J∧J=-164l2J∧JJ∧Jis indeed an invariant operator.As is well known, the representations of the de Sitter group are classified by their Casimir invariantsI1 and I2 which here following [17] we take as(99)I1=-J⌟J=12l2JABJAB=ηαβΠαΠβ+12l2ηαληβτJαβJλτ=M2,I2=W2=WAWA=ηαβVαVβ+1l2W42,where M∈R and the fields WA and Vα are defined by(100)WA≔18lεABCDEJABJDE,Vα≔-12ε4αλμνηλρΠρJμν,from where it follows that(101)W4=18ε4μνρτJμνJμν.In the limit when l↦∞ we get the Casimir operators of the special Lorentz group(102)I1⟼PαPα=m2,I2⟼ηαβVαVβ=m2ss+1,where m∈R and s=0,1/2,1,3/2,….We see thatΠα looks like the components of anenergy-momentum vectorP=Pαϑαoof a closed physical system (see (26)) in the Minkowski space-time of Special Relativity, for which P2=m2, with m being the mass of the system. However, take into account that whereas Pα are simple real numbers, Πα are vector fields.Moreover, take into account thatηαβΠαΠβ is not an invariant; that is, 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 Space-Time Structure

In this section we suppose that(M=R×S3,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.” Thus if we suppose that (M=R×S3,g,τg,↑) is populated by interacting matter fields {ϕ1,…,ϕn} with dynamics described by a Lagrangian formalism we can construct as described in Appendix C.1 the conserved currents(103)JΠα=Jαβγβ=Jαμϑμ,where taking into account that δΠα0=-£Πα and denoting by δΠα each particularlocal variation generated by Πα (recall (64)) we have(104)δΠαϕA=δΠα0ϕA+δxν∂νϕAand thus we write(105)Jαμ=πAμδΠαϕA+ΥνμδΠα0xμ=Λνμξαν+Υνμξαν,where(106)Λνμξαν≔πAμδΠαϕA.In (103) {γβ} is the dual basis of the orthonormal basis {eα} defined by31(107)eα≔ΠαgΠα,Πα.From the conserved currentsJΠα we can obtain four conserved quantities (Pα in (27)).Remark 15.
It is crucial to observe that the above results have been deduced without introduction of any connection in the structure(M=R×S3,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 (26)) of special relativistic theories.If we addD, the Levi-Civita connection of g, to (M=R×S3,g,τg,↑) we get the Lorentzian de Sitter space-time structure MdSP and defining a generalized energy-momentum tensor(108)Θ=JΠα⊗eα∈sec⁡T11Mwe know that δgJΠα=0 implies D·Θ=0, a covariant “conservation” law.However, the introduction ofMdSP in our game is of no help to construct a covector like P since in MdSP vectors at different space-time 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 PαdS such that(109)PdS=PαdSγαx=PαdSγαydefines a legitimate covector for a closed physical system living in a de Sitter structure (M=R×S3,g,τg,↑) and for which PdS can be said to be a kind of generalization of the momentum of the closed system in Minkowski space-time?We show now that the answer is positive. We recall thatD has been introduced in our developments only as a useful mathematical device and is quite irrelevant in the construction of legitimate conservation laws since the conserved currents JΠα have been obtained without the use of any connection. So, we now introduce for our goal ateleparallel de Sitter space-time, that is, the structure MdSTP=(M=R×S3,∇,τg,↑), where ∇ is a metric compatible teleparallel connection defined by(110)∇eαeβ=ω·αβκ··eκ=0.Under this condition we know that we can identify all tangent and all cotangent spaces. So, we have, ∀x,y∈R×S3,(111)eαx≃eαy,γαx≃γαy=Eα, where {Eα} is a basis of a vector space V≃R4.Thus, in the structureMdSTP (109) defines indeed a legitimate covector in V≃R4 and thus permits a legitimategeneralization of the concepts of energy-momentum covector obtained for physical theories in Minkowski space-time. The term generalization is a good one here because in the limit, where l→∞, Πα↦∂/∂xα,Λακ=0 and thus Θακ=Υακ.


## 5.3. The Conserved CurrentsJΠα=g(Πα,)

To proceed we show that the translational Killing vector fields of the de Sitter structure(M,g) determine trivially conserved currents(112)JΠα=gΠα,.Indeed using theMdSTP structure as a convenient device we recall the result proved in [10] that for each vector Killing K the one form field K=g(K,) is such that δgK=0. So, it is(113)δgJΠα=0and, of course, also(114)δgJΠ=0,where(115)JΠ≔gΠ,,Π≔εαΠαwith εα being real constants such that εα≪1. Recalling from (91) that in projective conformal coordinate bases ({eμ=∂μ} and {ϑμ=dxμ}) the components of Πα are(116)ξαμ=δαμ-14l22ηαρxρxμ-σ2δαμwe get that the conserved current JΠ is(117)JΠ=εαJΠα=εαξαμϑμ.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 [1] authors trying to generalize the results that follow from the canonical formalism for the case of field theories in Minkowski space-time suppose that they can eliminate the term πAμδΠαϕA from (105) 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.Now, thegeneralized canonical de Sitter energy-momentum tensor is(118)Θ=JΠα⊗eα=Θακϑκ⊗eα∈sec⋀1T∗M⊗TMand making analogy with the case of Minkowski space-time where Υακ have been defined as the components of the canonical energy-momentum tensor32  Υ, we write Θακ, as(119)Θακ=Υακ-14l2Kακ+Λακ.Thus each one of the genuine conservation laws d⋆gJΠα=0, α=0,1,2,3 reads in coordinate basis components as(120)∂μΥακ-Γ·λνλ··Υακ-14l2∂μKαμ-Γ·λνλ··Kακ-∂μΛακ-Γ·λνλ··Λακ=1-det⁡g∂μ-det⁡gΥαμ-14l21-det⁡g∂μ-det⁡gKαμ+1-det⁡g∂μ-det⁡gΛαμ=0 or(121)∂μ-det⁡gΥαμ-14l2Kαμ+Λαμ=0⟺DμΘαμ=0.Remark 18.
Now, recalling that the relation of the connection coefficients of the bases{eα} of the Levi-Civita connection D of g (denoted Γ·αβκ··) and the coefficients of the basis {eα} of the teleparallel connection ∇ of g (denoted Γ¯·αβκ··) are [11](122)Γ¯·αβκ··=Γ·αβκ··+Δ·αβκ··,where(123)Δ·αβκ··≔-12Tα·β·κ·+Tβ·α·κ·-T·αβκ··are the components of the contorsion tensor and T·αβκ·· are the components of the torsion tensor of the connection ∇, we can write D•Θ=0 (taking into account that Γ¯·αβκ··=0) in components relative to orthonormal basis as(124)DαΘβα≔DeαΘβα=eαΘβα+ΔαιαΘβι-ΔαβιΘια=0. On the other hand since ∇αΘβα:=(∇eαΘ)βα=eα(Θβα) we have(125)∇αΘβα=-ΔαιαΘβι+ΔαβιΘια,which means that although (∇•Θ)βα:=∇αΘβα≠0 we can generate the conserved currents JΠα in the teleparallel de Sitter space-time structure if (125) is satisfied.Remark 19.
To end this section and for completeness of the article it is necessary to mention that in a remarkable paper [18] the gravitational energy-momentum tensor in teleparallel gravity is discussed in detail. Also related papers are [19, 20]. A complete list of references can be found in [21].


## 6. Equation of Motion for a Single-Pole Mass in a GRT Lorentzian Space-Time

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 are described by an energy-momentum tensor that satisfies the covariant conservation lawD•T=0. If we introduce the relative tensor(126)T=T⊗τg∈sec⁡T11M⊗⋀4T∗Mwe have, recalling Appendix E and that D•T=0, that(127)DνTμν+Γ·ναμ··Tαν=0.From (127) we have(128)∂νxαTμν=Tμα-xαΓ·ναμ··Tαν.To continue we suppose that a single-pole mass (considered as a probe particle) is modelled by the restriction of the energy-momentum tensor T inside a “narrow” tube in the Lorentzian space-time representing a given gravitational field. Let us call T0 the restriction and notice that D•T0=0. Inside the tube a timelike line γ is chosen to represent the particle motion. We restrict our analysis to hyperbolic Lorentzian space-times for which a foliation R×S (S is a 3-dimensional manifold) exists. We choose a parametrization for γ such that 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(129)δxμ=xμ-Xμ≪1.According to Papapetrou a single-pole particle is one for which the integral33(130)∫Toμνdv≠0and all other integrals(131)∫δxαToμνdv,∫δxαδxβToμνdv⋮are null. We now evaluate(132)ddt∫Toμ0=-∫Γ·ναμ··Toανdv,ddt∫xαToμ0=∫Toμαdv-∫xαΓ·ναμ··Toανdv.Inside the world tube modelling the particle we can expand the connection coefficients as(133)Γ·ναμ··=Γo·ναμ··+Γo·να,κμ··δxκwith Γo·ναμ·· being the components of the connection in worldline γ. Then according to the definition of a single-pole particle we get from (132) along γ(134)ddt∫Toμ0+Γo·ναμ··∫Toαν=0,(135)∫Toμα=dXμdt∫Toν0dv.Now, put(136)γ∗=u≔dXμdseμ,where ds is proper time along γ, and define(137)Mμα=u0∫Toμαdv.Equations (134) and (135) become(138)ddsMμ0u0+Γo·ναμ··Mαν=0,(139)Mμα=uμMα0u0.So, Mμ0=uμM00/u0 from where it follows that putting(140)m≔M00u02it is(141)Mμα=uμMα0u0=muμuαand we get(142)ddsmuμ+Γo·ναμ··muνuα=0.Now, the acceleration of the probe particle isa:=Duu and thus g(a,u)=0; that is,(143)uμddsuμ+Γo·ναμ··muνuαuμ=0.Multiplying (142) by uμ and using (143) give(144)ddsm=0and then (142) says that γ is a geodesic of the Lorentzian space-time structure; that is,(145)Dγ∗γ∗=0or(146)duμds+Γo·ναμ··muνuα=0.


## 7. Equation of Motion for a Single-Pole Mass in a de Sitter Lorentzian Space-Time

In this section we suppose that the arena where physical events take place is the de Sitter space-time structureMdSL where fields live and interact, without never changing the metric g, which as we emphasize does not represent any gravitational field here; that is, we do not suppose here that MdSL is a model of a gravitational field in GRT. As we learned in Section 4 since de Sitter space-time has one timelike and three spacelike Killing vector fields Πα we can construct the conserved currents JΠα=Θαμϑμ (see (103)) from where we get D•Θ=0 with Θ=JΠα⊗eα=Θακϑκ⊗eα.Now, 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 tensorΘ in a narrow tube like the one introduced in the previous section, we can derive (using analog notations for γ, etc.) an equation like (134); that is,(147)ddt∫Θoμ0-det⁡godv+Γo·ναμ··∫Θoαν-det⁡godv=0.Now, we obtain an equation analogous to (138) with Mαν substituted by Nαν; that is,(148)ddsNμ0u0+Γo·ναμ··Nαν=0with(149)Nαν≔u0∫Θoμα-det⁡godv.Putting this time(150)m≔N00u02we get Nμα=uμNα0/u0=muμuα and(151)ddsmuμ+Γo·ναμ··muνuα=0,from where we get exactly as in the previous section that m=const and Dγ∗γ∗=0.Conclusion 1.
Papapetrou method applied to theMdSL structure gives for the motion of a free single-pole particle the geodesic equation Dγ∗γ∗=0. Moreover, (151) is, of course, different from (88) contrary to conclusions of authors of [1]. It is also to be noted here that in [1] authors inferred correctly that Papapetrou method leads to an equation that looks like (147) for a single-pole particle moving in the MdSL structure. The equation that looks like (147) in [1] is Equation (37) there, but where in place of Θoαν they used Υoαν-1/4l2Υoαν, because they believe it is possible to use a local variation of the fields that results in Λνμ=0.



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