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 structures and 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, 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.

1. Introduction

In this paper we study some aspects of Physics of fields living and interacting in a manifold . We introduce two different geometrical space-time structures that we can form starting from the manifold which is supposed to be a vector manifold, that is, a submanifold of ( with and a metric of signature . If is the inclusion map the structures that will be studied are the Lorentzian de Sitter space-time and teleparallel de Sitter space-time , where , is the Levi-Civita connection of , and is a metric compatible teleparallel connection (see Section 4.1). Our main objective is the following: taking 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 in a Lorentzian space-time associated with a tensor field and a vector field . 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-momentum covector1 for the matter fields. After that we recall that in GRT we have a covariantly “conserved” energy-momentum tensor (i.e., ) 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 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 conserved covector 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 is no. 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 and we show how any diffeomorphism associated with a one-parameter group generated by a vector field leads to a conserved current. We show moreover using the Clifford bundle formalism recalled in Appendix A that , where satisfy (with denoting the Dirac operator acting on sections of the Clifford bundle of differential forms) a Maxwell-like equation (equivalent to and ). The explicit form of as a function of the energy-momentum tensor is derived (see (50)) together with its scalar invariant. We establish that3   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 objects cannot be associated with the components of a momentum covector4 for the system of fields producing the energy-momentum tensor . We also give the form of when 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 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 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 exists only in the teleparallel structure . 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 structures and . In particular we want also to clarify some misunderstandings concerning the roles of geodesics in . So, in Section 3, we briefly recall the conformal representation of the de Sitter space-time structure and prove that the one timelike and the three spacelike “translation” Killing vector fields of define a basis for almost all . 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 use constrained variations instead of arbitrary 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 gives an equation of motion different from the geodesic equation in 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 be a general Lorentzian space-time. Let be an open set covered by coordinates . Let be a basis of and the basis of dual to the basis ; that is, . We denote by a metric of the cotangent bundle such that if , then with . We introduce also and , respectively, as the reciprocal bases of and ; that is, we have

Next we introduce in the tetrad basis and in the cotetrad basis which are dual basis. We introduce moreover the bases and as the reciprocal bases of and satisfying

Moreover recall that it is

2.1. The Currents and

Let with and , , and . For the applications we have in mind we will say that , and are physically equivalent.

Note that (an example of an extensor field7) is such thatDefine the divergence of as the -form fieldwhere

Moreover, introduce the -form fields

Remark 1. Take notice for the developments that follow that the Hodge coderivative of the -form fields is as follows (see Appendix): So, does not imply that .

Now, given a vector field and the physically equivalent covector field , define the currentOf course, writing we have

Recalling (see Appendix A) that , define

Then, we have, with denoting the Dirac operator,

Taking into account that and we can write (13) asAlso, we can easily verify from (5) that

Now, let be the (standard) Lie derivative operator. Let us evaluate the product of by ; that is,From Cartan magical formula we get Then,and we get from (14), (15), and (18) the important identity [3]From (19) we see that if is a conformal Killing vector field, that is, , we havewhere is the trace of the matrix with entries .

2.2. Conserved Currents Associated with a Covariantly Conserved

Definition 2. One says that is “covariantly conserved” if

In this case, if is a Killing vector field, then and we haveand the current -form field is closed; that is, , or equivalently (taking into account the definition of the Hodge coderivative operator )

In resume, when we have Killing vector fields8  ,, “covariant conservation” of the tensor field , that is, , implies in genuine conservation laws for the currents that from , we can using Stokes theorem build the scalar conserved quantitieswhere is the region where has support and , where are spacelike surfaces and is null at (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 Appendix C.3.4 that in Minkowski space-time9 we can introduce global coordinates (in Einstein-Lorentz-Poincaré gauge) such that and , where is simultaneously a global tetrad and a coordinate basis. Also is a global cotetrad and a coordinate cobasis.

Moreover, are also Killing vector fields in 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 currents10are the conserved energy-momentum -form fields of the physical system under consideration, for which we know that the quantity (recall (C.38))withis the components of the conserved energy-momentum covector (CEMC) 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 is modelled by a Lorentzian space-time11   where the relation between and is given by Einstein equation which using the orthonormal bases and introduced above readsMoreover, defining and recalling that , it is

Based only on the contents of Section 2.1, given that , it may seem at first sight12 that the only possibility to construct conserved energy-momentum currents 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 in general does not have such Killing vector fields.

Remark 3. Moreover, even if it is the case that if in a particular model of GRT there exist one timelike and three spacelike Killing vector fields, we can construct the scalar invariants quantities , , given by (27), we cannot define an energy-momentum covector analogous to the one given by (26). This is so because in this case to have a conserved covector like it is necessary to select a at a fixed point of the manifold. But in general there is no physically meaningful way to do that, except if is asymptotically flat13 in which case we can choose a chart such that at spatial infinity and .

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 Section 2.1 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 .

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

Let be the generator of a one-parameter group of diffeomorphisms of in the space-time structure which is a model of a gravitational field generated by (the matter fields energy-momentum tensor) in GRT. It is quite obvious that if we define , where , then the currentis conserved; that is,Surprisingly such a trivial mathematical result seems to be very important for people working in GRT who call the Komar current14 [6]. Komar called15the generalized energy.

To understand why is considered important in GRT write the action for the gravitational plus matter and nongravitational fields as

Now, the equations of motion for can be obtained considering its variation under an (infinitesimal) diffeomorphism generated by . We have that where16 the variation . Taking into account Cartan’s magical formula (, for any ), we havewherewith

To proceed introduce a coordinate chart with coordinates for the region of interest . Recall that are the Einstein -form fields17, with being the Ricci -forms and the curvature scalar. Einstein equation obtained from the variation principle is , with being the energy-momentum -form fields and moreover .

Next write explicitly the action as [7]

We have immediately

From (34) and (37) we haveand thusThus, the current is conserved if the field equations are satisfied. An equation (in component form) equivalent to (39) already appears in [6] (and also previously in [8]) that took with .

Here, to continue, we prefer to write an identity involving only . Proceeding exactly as before we get putting that there exists such thatand we see that we can identifywhere . Now, we claim the following.

Proposition 6. There exists such thatwhere was defined in (30).

Proof. To prove our claim we suppose from now on that 18.
Then it is possible to writewhere is the Ricci operator and □ is the D’Alembertian operator. Then we takeand of course it is19proving the proposition.

Now that we found a satisfying (42) we investigate if we can give some nontrivial physical meaning to such .

2.4.1. Determination of the Explicit Form of

We recall that the extensor field acts on as Thus, since we have from (44)We can write (48) taking into account that and putting and thatwhere [9]Equation (50) gives the explicit form for the Komar current20. Moreover, taking into account that , it isand thus taking into account Stokes theorem Moreover, since we have that and thus () is a conserved quantity. We arrive at the conclusion that taking () as a ball of radius and making the quantityis conserved.

Remark 7. It is very important to realize that quantity defined by (54) is a scalar invariant, that is, its value does not depend on the particular reference frame and (the naturally adapted coordinate chart adapted to )21. But, of course, for each particular vector field (which generates a one-parameter group of diffeomorphisms) we have a different and the different ’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 each that generates a one-parameter group of diffeomorphisms of 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 and if is not a dynamical field?
We know that [10] when is a Killing vector field it is and and thus (55) readswhich is a well known conserved quantity22. For a Schwarzschild space-time, as is well known, is a timelike Killing vector field and in this case since the components of are and (since ) we get .

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 which holds in any structure 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 . So, he got by pure chance. If he had picked another vector field generator of a one-parameter group of diffeomorphisms , 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 ) 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 = Encodes Einstein Equation

From (49) where with being an arbitrary generator of a one-parameter group of diffeomorphisms of (part of the structure ) taking into account that , we get the Maxwell-like equation (MLE)with a well defined conserved current. Of course, as we already said, there is an infinity of such equations. Each one encodes Einstein equation; that is, given the form of (see (50)), we can get back (43), which gives immediately Einstein equation (EE). In this sense we can claim that

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 in Lorentzian and teleparallel de Sitter space-time 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 of the next sections.

3. The Lorentzian de Sitter Structure and Its Conformal Representation

Let and be, respectively, the special pseudoorthogonal groups in and in , where is a metric of signature and is a metric of signature . The manifold will be called the de Sitter manifold. Sincethis manifold can be viewed as a brane (a submanifold) in the structure . We now introduce a Lorentzian space-time, that is, the structure which will be called Lorentzian de Sitter space-time structure where if , and is the parallel projection on of the pseudo-Euclidian metric compatible connection in (details in [12]). As is well known, 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 (called the de Sitter group) in . The group acts transitively24 in , which is thus a homogeneous space (for ).

We now recall the description of the manifold as a pseudosphere (a submanifold) of radius of the pseudo-Euclidean space . If are the global coordinates of , then the equation representing the pseudosphere is

Introducing conformal coordinates25   by projecting the points of from the “north pole” to a plane tangent to the “south pole” we see immediately that covers all except the “north pole.” We immediately find thatwhere

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 of Cayley-Klein of equation

In Figure 1 we can see that all timelike curves () and () and lightlike () 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 absolute .

4. On the Geodesics of

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 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 and excellent article by Schmidt [14].

Remark 13. Having said that, let us recall that among the Killing vector fields26 of 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 structure. However, the fact is that what can really be done is the obtainment of conserved quantities like in (27). But we cannot obtain in structure an energy-momentum covector like 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 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 (where are the conformal coordinates introduced in the last section) by27:with (putting , for simplicity) 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 field . If are linearly independent in a region , then we can writeSo, the condition for the existence of a nontrivial solution for the is that

Thus, putting we need to evaluate the determinant of the matrixThen

In order to analyze this expression we put (without loss of generality; see the reason in [14]) . In this casewhich is null on the Caley-Klein absolute, that is, at the pointsand also on the spacelike hyperbolas given by

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 hyperbolas and .

This result is very important for the following reason. Timelike geodesics in de Sitter space-time structure are (as is well known) the curves , where is proper time along , which minimizes or maximizes the length function [15], that is, calling the velocity of a “particle” of mass 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, asAs is well known, the determination of an extreme for is given by evaluating the first variation28 of , that is,and putting . The result, as is well known, is the geodesic equation

Now, taking into account that the Killing vector fields determine a basis inside the Cayley-Klein absolute we writeWe now define the “hybrid” connection coefficients29   byand write the geodesic equation asorwhich on multiplying by the “mass” and calling and can be written equivalently aswhich looks like Equation () in [1].

We want now to investigate the following question: is (82) the same as Equation () in [1]?

To know the answer to the above question recalls that in [1] authors investigate the variation of under a variation of the curves induced by a coordinate transformation , where they putwith being the components of the Killing vector fields (recall (65)) and where it is said that is an ordinary variation. However in [1] we cannot find what authors mean by ordinary variation, and so is not defined.

So, to continue our analysis we recall that if are constants (but arbitrary), then corresponds to a diffeomorphism generated by a Killing vector field . However, if are infinitesimal arbitrary functions (i.e., ), then the notation is misleading since is the variation generated by a quite arbitrary vector field . In this case we get from the geodesic equation.

4.1. Curves Obtained from Constrained Variations

So, let us study the constrained variation when are constants (but arbitrary). We denote the constrained variation by . In this case starting from (75)where (taking account of notations already introduced) We can writeand implies

Equation (87) with can be written aswhich is Equation () in [1]. Note that this equations looks like the geodesic equation written as (82) above, but it is in fact different since of course, recalling (79), it is . In [1] is unfortunately wrongly interpreted as the components of a covector field over 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 introduced in Section 3 define byThese objects are generators of the Lie algebra of the de Sitter group.

Using the bases introduced above the ten Killing vector fields of de Sitter space-time are the fields and and it is30 and satisfy the Lie algebra of the de Sitter group, this time acting as a transformation group acting transitively in de Sitter space-time. We haveIt is usual in physical applications to definefor then we have

The Killing vector fields satisfy the Lie algebra (of the special Lorentz group).

Remark 14. From (94) we see that when the Lie algebra of goes into the Lie algebra of the Poincaré group which is the semidirect sum of the group of translations in plus component of the special Lorentz group; that is, . This is eventually the justification for physicists to call the “translation” generators of the de Sitter group.
However, it is necessary to have in mind that whereas the translation subgroup of acts transitively in Minkowski space-time manifold, the does not close in a subalgebra of the de Sitter algebra and thus it is impossible in general to find such that given arbitrary it is . Only the whole group acts transitively on .

Casimir Invariants. Now, if is an orthonormal basis for the structure define the angular momentum operator as the Clifford algebra valued operator

Taking into account the results of Appendix A its (Clifford) square is

It is immediate to verify that is invariant under the transformations of the de Sitter group. 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 . Indeed, definingone can easily show (details in [16]) thatis indeed an invariant operator.

As is well known, the representations of the de Sitter group are classified by their Casimir invariants and which here following [17] we take aswhere and the fields and are defined byfrom where it follows thatIn the limit when we get the Casimir operators of the special Lorentz groupwhere and

We see that looks like the components of an energy-momentum vector of a closed physical system (see (26)) in the Minkowski space-time of Special Relativity, for which , with being the mass of the system. However, take into account that whereas are simple real numbers, are vector fields.

Moreover, take into account that 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 is the physical arena where Physics take place.

We know that ) has ten Killing vector fields and four of them (one timelike and three spacelike, , ) generated “translations.” Thus if we suppose that is populated by interacting matter fields with dynamics described by a Lagrangian formalism we can construct as described in Appendix C.1 the conserved currentswhere taking into account that and denoting by each particular local variation generated by (recall (64)) we haveand thus we writewhereIn (103) is the dual basis of the orthonormal basis defined by31

From the conserved currents we can obtain four conserved quantities ( in (27)).

Remark 15. It is crucial to observe that the above results have been deduced without introduction of any connection in the structure . However that structure is not enough for using the to build a (generalized) covector analogous to the energy-momentum covector (see (26)) of special relativistic theories.

If we add , the Levi-Civita connection of , to we get the Lorentzian de Sitter space-time structure and defining a generalized energy-momentum tensorwe know that implies , a covariant “conservation” law.

However, the introduction of in our game is of no help to construct a covector like since in vectors at different space-time points cannot be directly compared.

So, the question arises: is it possible to define a structure where we can define objects such thatdefines a legitimate covector for a closed physical system living in a de Sitter structure and for which 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 that 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 have been obtained without the use of any connection. So, we now introduce for our goal a teleparallel de Sitter space-time, that is, the structure , where is a metric compatible teleparallel connection defined byUnder this condition we know that we can identify all tangent and all cotangent spaces. So, we have, , where is a basis of a vector space .

Thus, in the structure (109) defines indeed a legitimate covector in and thus permits a legitimate generalization 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 , and thus .

5.3. The Conserved Currents

To proceed we show that the translational Killing vector fields of the de Sitter structure determine trivially conserved currents

Indeed using the structure as a convenient device we recall the result proved in [10] that for each vector Killing the one form field is such that . So, it isand, of course, alsowherewith being real constants such that . Recalling from (91) that in projective conformal coordinate bases ( and ) the components of arewe get that the conserved current is

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 from (105) by decree postulating a new kind of “local variation,” call it , for which . The fact is that such a “new kind of local variation” never appears in the canonical Lagrangian formalism; only appears and in general it is not zero.

Now, the generalized canonical de Sitter energy-momentum tensor isand making analogy with the case of Minkowski space-time where have been defined as the components of the canonical energy-momentum tensor32  , we write , asThus each one of the genuine conservation laws , reads in coordinate basis components as or

Remark 18. Now, recalling that the relation of the connection coefficients of the bases of the Levi-Civita connection of (denoted ) and the coefficients of the basis of the teleparallel connection of (denoted ) are [11]whereare the components of the contorsion tensor and are the components of the torsion tensor of the connection , we can write (taking into account that ) in components relative to orthonormal basis as On the other hand since we havewhich means that although we can generate the conserved currents 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 law . If we introduce the relative tensorwe have, recalling Appendix E and that , thatFrom (127) we haveTo continue we suppose that a single-pole mass (considered as a probe particle) is modelled by the restriction of the energy-momentum tensor inside a “narrow” tube in the Lorentzian space-time representing a given gravitational field. Let us call the restriction and notice that . Inside the tube a timelike line is chosen to represent the particle motion. We restrict our analysis to hyperbolic Lorentzian space-times for which a foliation ( is a -dimensional manifold) exists. We choose a parametrization for such that its coordinates are , where . The probe particle is characterized by taking the coordinates of any point in the world tube to satisfyAccording to Papapetrou a single-pole particle is one for which the integral33and all other integralsare null. We now evaluate

Inside the world tube modelling the particle we can expand the connection coefficients aswith being the components of the connection in worldline . Then according to the definition of a single-pole particle we get from (132) along

Now, putwhere is proper time along , and defineEquations (134) and (135) becomeSo, from where it follows that puttingit isand we get

Now, the acceleration of the probe particle is and thus ; that is,

Multiplying (142) by and using (143) giveand then (142) says that is a geodesic of the Lorentzian space-time structure; that is,or

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 structure where fields live and interact, without never changing the metric , which as we emphasize does not represent any gravitational field here; that is, we do not suppose here that 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 we can construct the conserved currents (see (103)) from where we get with .

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 tensor in a narrow tube like the one introduced in the previous section, we can derive (using analog notations for , etc.) an equation like (134); that is,Now, we obtain an equation analogous to (138) with substituted by ; that is,with

Putting this timewe get andfrom where we get exactly as in the previous section that and .

Conclusion 1. Papapetrou method applied to the structure gives for the motion of a free single-pole particle the geodesic equation . 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 structure. The equation that looks like (147) in [1] is Equation () there, but where in place of they used , because they believe it is possible to use a local variation of the fields that results in .

Appendix

A. Clifford Bundle Formalism

Let be an arbitrary Lorentzian or Riemann-Cartan space-time structure. The quadruple denotes a four-dimensional time-oriented and space-oriented Lorentzian manifold [4, 11]. This means that is a Lorentzian metric of signature , , and is a time-orientation (see details, e.g., in [4]). Here, [] is the cotangent [tangent] bundle. , , and , where is the Minkowski vector space34. is a metric compatible connection; that is, . When is the Levi-Civita connection of , , and , with and being, respectively, the curvature and torsion tensors of the connection. is a Riemann-Cartan connection, , and 35. Let be the metric of the cotangent bundle. The Clifford bundle of differential forms is the bundle of algebras, that is, , where, , , the so-called space-time algebra [11]. Recall also that is a vector bundle associated with the orthonormal frame bundle; that is, we have36   [22, 23]. Also, when is a spin manifold we can show that37. For any , as a linear space over the real field is isomorphic to the Cartan algebra of the cotangent space. We have that , where is the -dimensional space of -forms. Then, sections of can be represented as a sum of nonhomogeneous differential forms, which 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 be an orthonormal basis for ; that is, . Let () be such that the set is the dual basis of . Also, is the reciprocal basis of , that is, , and is the reciprocal basis of ; that is, .

A.1. Clifford Product

The fundamental Clifford product (in what follows to be denoted by juxtaposition of symbols) is generated byand if we havewhere is the volume element and , , , , .

For and we define the exterior product in ( bywhere is the component in of the Clifford field. Of course, , and the exterior product is extended by linearity to all sections of .

Let and . We define a scalar product in (denoted by ) as follows.

(i) For ,

(ii) For and , ,

We agree that if , the scalar product is simply the ordinary product in the real field.

Also, if , then . Finally, the scalar product is extended by linearity for all sections of .

For , , , we define the left contraction bywhere is the reverse mapping (reversion) defined by . For any , ,We agree that for the contraction is the ordinary (pointwise) product in the real field and that if , , then . Left contraction is extended by linearity to all pairs of sections of ; that is, for ,

It is also necessary to introduce the operator of right contraction denoted by . The definition is obtained from the one presenting the left contraction with the imposition that and taking into account that now if , , then . See also the fourth formula in (A.9).

The main formulas used in this paper can be obtained from the following ones:Two other important identities used in the main text arefor any and .

A.1.1. Hodge Star Operator

Let be the Hodge star operator, that is, the mapping , . For we havewhere is a standard volume element. We have,where as noted before, in this paper, denotes the reverse of . Equation (A.12) permits calculation of Hodge duals very easily in an orthonormal basis for which . Let be the dual basis of (i.e., it is a basis for ) which is either an orthonormal or a coordinate basis. Then writing , with , and , we have from (A.12)where denotes the determinant of the matrix with entries , that is, We also define the inverse of the Hodge dual operator, such that . It is given bywhere denotes the sign of the determinant of .

Some useful identities (used in the text) involving the Hodge star operator, the exterior product, and contractions are

A.1.2. Dirac Operator Associated with a Levi-Civita Connection

Let and be, respectively, the differential and Hodge codifferential operators acting on sections of . If , then .

The Dirac operator acting on sections of associated with the metric compatible connection is the invariant first-order differential operatorwhere is an arbitrary (coordinate or orthonormal) basis for and is a basis for dual to the basis ; that is, , . The reciprocal basis of is denoted by and we have . Also, when and we have and when , are orthonormal basis we haveWe define the connection38-forms in the gauge defined by asMoreover, we write for an arbitrary tensor field in a coordinate basisand also when we write we also write So, please pay attention when reading a particular formula to certificate the meaning of , that is, whether we are using in that formula coordinate or orthonormal frames.

We have also the important results (see, e.g., [11]) for the Dirac operator associated with the Levi-Civita connection acting on the sections of the Clifford bundle39

We shall need the following identity valid for any

A.2. Covariant D’Alembertian, Hodge D’Alembertian, and Ricci Operators

The square of the Dirac operator is called Hodge D’Alembertian and we have the following noticeable formulas:where is called the covariant D’Alembertian and is called the Ricci operator40. If , we haveAlso for in an arbitrary basis (coordinate or orthonormal)

In particular we have [11]where are the Ricci -form fields, such that if are the components of the Riemann tensor we use the convention that are the components of the Ricci tensor.

Applying this operator to the 1-forms of the basis , we get

is an extensor operator; that is, for , it is

Remark A.1. We remark that covariant Dirac spinor fields used in almost all Physics texts books and research papers can be represented as certain equivalence classes of even sections of the Clifford bundle . These objects are now called Dirac-Hestenes spinor fields (DHSF) and a thoughtful theory describing them can be found in [11, 23, 24]. Moreover, in [25], using the concept of DHSF a new approach is given to the concept of Lie derivative for spinor fields, which does not seem 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 publication.

B. Lie Derivatives and Variations

In modern field theory the physical fields are tensor and spinor fields living on a structure 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 [11]) 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), a Lorentzian space-time , where is the Levi-Civita connection of , and a teleparallel space-time , where is a metric compatible teleparallel connection. Minkowski space-time structure will be denoted by . 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 is well known, diffeomorphism invariance is a crucial ingredient of any physical theory. This means that if a physical phenomenon is described by fields, say, (defined in ), satisfying equations of motion of the theory (with appropriated initial and boundary conditions), then if is a diffeomorphism, then the fields (where is the pullback mapping) describe the same physical phenomenon41 in .

Suppose that fields (in what follows called simply matter fields42) are arbitrary differential forms. Their Lagrangian density will here be defined as the functional mapping43where is here supposed to be constructed using the Hodge star operator . The action of the system is

Choose a chart of covering and with coordinate functions . Then under an infinitesimal mapping , generated by a one-parameter group of diffeomorphisms associated with the vector field we have (with being the coordinate representative of the mapping )In Physics textbooks given an infinitesimal diffeomorphism several different kinds of variations (for each one of the fields ) are defined.

Let be one of the fields and recall that the Lie derivative of in the direction of the vector field is given byAs an example, take as -form. Then, in the chart introduced above using the definition of the pullbackit isThen, (B.4) can be written in components as

Now, to first order in , we haveSo,

Then,

Now, we define following Physics textbooks the horizontal variation44 by

This definition (with the negative sign) is used by physicists because they usually work only with the components of the fields and diffeomorphism invariance is interpreted as invariance under choice of coordinates. Then, they interpret (B.3) as a coordinate transformation between two charts whose intersection of domains cover the regions and of interest with coordinate functions and such thatand thenThe field at has the representations and in first order in it is and on the other hand sincewe have in first order in thatfrom where we get

Remark B.1. The above calculation can be done in a while recalling Cartan’s “magical” formula, which with readsIn components we havefrom which the substituting of these results in (B.19) and (B.10) follows immediately.

Remark B.2. If we have chosen the coordinate functions and related by45 we would get that .

Remark B.3. Take into account for applications that for any

Now, physicists introduce another two variations and defined by called, for example, in [26], local variation46. We have

Remark B.4. 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 still a subject of recent research with many conflicting views. In [25] we present a novel geometrical approach to this subject using the theory of Clifford and spin-Clifford bundles which seems to lead to consistent results.

Remark B.5. Definingwe have returning to (B.2) that Stokes theorem permit us to write

C. The Generalized Energy-Momentum Current in (, g, τg, )

C.1. The Case of a General Lorentzian Space-Time Structure

In this subsection is an arbitrary oriented and time-oriented Lorentzian manifold which will be supposed to be the arena where physical phenomena take place. We choose coordinate charts and with coordinate functions and covering . We call , . We take such that , that is, is bounded from above and below by spacelike surfaces and such that and , and moreover we suppose that set of the matter fields in interaction denoted byliving in satisfy in (a timelike boundary)In what follows the action functional for the fields is writtenUnder a coordinate transformation corresponding to a diffeomorphism generated by a one-parameter group of diffeomorphisms,we already know that the fields suffer the variation

We have in first order in , recalling that and commute, thatPuttingthe second term in (C.6) can be written using Gauss theorem as

Recalling the concept of local variation introduced above we havePuttingwe callthe canonical momentum canonically conjugated to the field . Moreover, puttingwe callthe canonical energy-momentum tensor of the closed physical system described by the fields.

Now, we can write (C.7) asMoreover, definingwe can rewrite (C.6):Now, the action principle establishes that and then we must havewhich are the Euler-Lagrange equations satisfied by each one of the fields and alsoNow, if is the volume element, taking into account that we took where is a timelike surface such that and introducing the currentwe can rewrite (C.18) using Stokes theorem as

C.2. Introducing and the Covariant “Conservation” Law for

If we add , the Levi-Civita connection of to , we get a Lorentzian space-time structure . Then recalling from Appendix A the definitions of the Hodge coderivative and of the Dirac operator we can writeand we arrive at the conclusion that implies that

Remark C.1. Recalling the definition of the canonical energy-momentum tensor (see (C.13)) gives a covariant “conservation” law for , that is,only if the term in the current is null. This, of course, happens if the local variation of the fields , something that cannot happen in an arbitrary structure . So, we need to investigate when this occurs.

Remark C.2. We observe here that comparison of (C.16) with (B.26) permits us to write

C.3. The Case of Minkowski Space-Time
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 space-time . In this case we can introduce global coordinates in Einstein-Lorentz-Poincaré gauge (see Section 2.3).

We now construct the conserved current associated with the diffeomorphisms generated by the vector fields which are Killing vector fields on . Consider then the Killing vector fieldwhere are constants such that and the coordinate transformation

Recalling the definitions of , the momentum canonically conjugated to the field (see (C.11)) and of (see (C.13)) and recalling that in the present case it is we have that the conserved Noether current isand the canonical energy-momentum tensor of the physical system described by the fields is conserved; that is,

Remark C.3. Of course, if we introduce an arbitrary coordinate system covering an open set of the Minkowski space-time manifold , (C.28) reads

C.3.2. The Energy-Momentum 1-Form Fields in Minkowski Space-Time

Recall that the objectsare conserved currents. They may be called the generalized energy-momentum -form fields of the physical system described by the fields . We have

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, that is,But, of course, if is conserved, so it iswherewith each one of the . So, it is always possible for any field theory to find47 a condition on the such that the components of satisfy the symmetry condition.When this is the case will be called the Belinfante energy-momentum tensor of the system.

C.3.4. The Energy-Momentum Covector in Minkowski Space-Time

Since Minkowski space-time is parallelizable we can identify all tangent and cotangent spaces and thus define a covector in a vector space . Fixing (global) coordinates in Einstein-Lorentz-Poincaré gauge a vector can be identified by a pair [27] , where and . If two vectors , are such thatthat is, they have the same vector part, we will say that they can be identified as a a vector of some vector space With these considerations we write where is a basis of and is a basis of a . Then we can write (with an arbitrary point, taken in general, for convenience as origin of the coordinate system)as the energy-momentum covector of the closed physical system described by the fields .

Remark C.4. Note that under a global (constant) Lorentz transformation we have that and it results in ; that is, 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 space-time structure the canonical Lagrangian formalism does not give a covariant “conserved” energy-momentum tensor unless the local variations of the matter fields are null. So, in GRT the matter energy-momentum tensor that enters Einstein equation is symmetric (i.e., ) and it is obtained in the following way. We start with the matter action

Remark D.1. Consider, as above, a diffeomorphism generated by a one-parameter group associated with a vector field (such that components and at ) and a corresponding coordinate transformationwith and study the variation of that induced the variation of the (gravitational) field (without changing the fields ) induced by the coordinate transformation of (D.2). We have immediately thatTo first order in it isThen, under the above conditions, using Gauss theorem and supposing that vanishes at , it iswithSince we can write

Remark D.2. Contrary to what is stated in many textbooks in GRT we cannot conclude with the ingredients introduced in this section that .
However, if we take into account that in GRT the total action describing the mater fields and the gravitational field is we get from the variation induced by the variation of the (gravitational) field (without changing the fields ) and induced by the coordinate transformation given by (D.2) the Einstein field equation which reads in components asSince it is it follows that in GRT we have

Remark D.3. It is opportune to recall that as observed, for example, by Weinberg [28] for the case of Minkowski space-time the symmetric energy-momentum tensor 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 procedure eliminates a legitimate conserved current introducing a covariant “conserved” energy-momentum tensor that does not give any legitimate energy-momentum conserved current for the matter fields, except for the particular Lorentzian space-times containing appropriate Killing vector fields. And even in this case no energy-momentum covector as it exists in special relativistic theories can be defined. Moreover, at this point we cannot forget the existence of the quantum structure of matter fields which experimentally says that the Minkowskian concept of energy and momentum is carried out by field excitations that one calls particles. This strongly suggests that, parodying (again) Sachs and Wu [4], it is really a shame to lose the special relativistic conservations laws in .

E. Relative Tensors and Their Covariant Derivatives

Now, recall that given arbitrary coordinates covering and covering () a relative tensor of type and weight48   is a section of the bundle49  .

We have with . The set of functions is said to be the components of the relative tensor field and under a coordinate transformation with Jacobian these functions transform as [29, 30]On a manifold equipped with a metric tensor field we can write , where are the components of a tensor field .

The covariant derivative of a relative tensor field relative to a given arbitrary connection defined on such that is given (as the reader may easily find) bywhere

In particular for the Levi-Civita connection of we have for the relative tensor that

F. Explicit Formulas for and in Terms of Projective Conformal Coordinates

We have taking into account (62) the following identities: where and . We want to prove that

Proof. (a) One has(b) One has where we used the fact that

Competing Interests

The authors declare that they have no competing interests regarding the publication of this paper.

Endnotes

  1. The energy-momentum covector is an element of a vector space and is not a covector field.
  2. This section is an improvement of results first presented in [9].
  3. The symbol denotes the Dirac operator acting on sections of the Clifford bundle . See Appendix A.
  4. Not a covector field.
  5. Some of the material of the appendices is well known, but we think that despite this fact their presentation here will be useful for most of our readers.
  6. The 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 and give meaningful definition for such an object, something necessary to study conservation laws in Lorentzian space-time structures when spinor fields are present. Our approach to the subject is described in [25] and a thoughtful derivation of Dirac equation in de Sitter structure using DHSFs is given in [16].
  7. See Chapter of [11].
  8. The maximum number is 10 when and that maximum number occurs only for space-times of constant curvature.
  9. See [11] for the rest of the notation.
  10. Keep in mind that in (25) are the -component of the current and moreover are here taken as symmetric. See Appendix C.3.3.
  11. In fact, by an equivalence classes of quintuples are modulo diffeomorphisms.
  12. See however Section 2.4 to learn that this naive expectation is incorrect.
  13. The concept of asymptotically flat Lorentzian manifold can be rigorously formulated without the use of coordinates, as, for example, in [31]. However we will not need to go into detail here.
  14. Komar called a related quantity the generalized flux.
  15. denotes a spacelike hypersurface and its boundary. Usually the integral is calculated at a constant time hypersurface and the limit is taken for being the boundary at infinity.
  16. Please do not confuse with .
  17. , where are the components of the Einstein tensor. Moreover, we write .
  18. is the Clifford bundle of differential forms; see Appendix and if more details are necessary, consult, for example, [11].
  19. Note that since it follows from (45) that indeed .
  20. Something that is not given in [6].
  21. Recall that in relativity theory (both special and general) a reference frame is modelled by a timelike vector field pointing into the future. A naturally adapted coordinate chart to (with coordinate functions (denoted )) is one such that the spatial components of are null. More details may be found, for example, in Chapter 6 of [11].
  22. An equivalent formula appears, for example, as Equation (11.2.10) in [31]. However, the simplicity and transparency of our approach concerning traditional ones based on classical tensor calculus is to be emphasized here.
  23. The total system is the system consisting of the gravitational plus matter and nongravitational fields.
  24. A group of transformations in a manifold ( by ) is said to act transitively on if for arbitraries there exists such that .
  25. Figure 1 appears also in author’s paper [16].
  26. More details are in Section 5.1.
  27. We are using here a notation similar to the ones in [1] for comparison of some of our results with the ones obtained there.
  28. In (76) is the deformation vector field determining the curves necessary to calculate the first variation of .
  29. Take notice that , where , for otherwise confusion will arise.
  30. Proofs of (90) and (91) are in Appendix F. Of course, in those equations, it is (as introduced is Section 2).
  31. Recall that the fields are only defined in subset .
  32. The explict form of the can be determined without difficulty if needed.
  33. The integration is to be evaluated at a space line hypersurface .
  34. Not to be confused with Minkowski space-time [4, 11].
  35. Minkowski space-time is the particular case of a Lorentzian space-time structure for which and the curvature and torsion tensors of the Levi-Civita connection of Minkowski metric are null. A teleparallel space-time is a particular Riemann-Cartan space-time such that and .
  36. by .
  37. Take notice that such that for any it is . Since identity, descends to a representation of that we denoted by .
  38. Also called “spin connection 1-forms.”
  39. For a general metric compatible Riemann-Cartan connection the formula in (A.22b) 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 [11].
  40. For more details concerning the square of Dirac (and spin-Dirac operators) on a general Riemann-Cartan space-time, see [32].
  41. Of course, the fields must satisfy deformed initial conditions and deformed boundary conditions.
  42. In truth, by matter fields, we understand fields of two kinds, fermion fields (electrons, neutrinos, and quarks) and boson fields (electromagnetic, gravitational, weak, and strong fields).
  43. A rigorous formulation needs the introduction of jet bundles (see, e.g., [33]). We will not need such sophistication for the goals of this paper.
  44. In [11] the horizontal variation is denoted by , where a vertical variation denoted by (associated with gauge transformations) is also introduced. Moreover, let us recall that has been used extensively after a famous paper by Rosenfeld [34] but appears also to the best of our knowledge in Section of Pauli’s book [35] on relativity theory.
  45. As, for example, in [36].
  46. Some authors call the total variation, but we think that this is not an appropriate name.
  47. For example, in [7], the condition is fixed in such a way that the orbital angular momentum tensor of the system defined as (with ) is 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 [11]) that the antisymmetric part of the canonical energy-momentum tensor is the source of the spin tensor of the field. In [37] how to obtain a conserved symmetrical energy-momentum tensor by studying the conservation laws that come from a general Poincaré variation is shown, which involves translations and general Lorentz transformations.
  48. The number is an integer. Of course, if we are back to tensor fields.
  49. The notation means the -fold tensor product of with itself.