translations and dynamics

8/9/2019 Translations and dynamics

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Translations and dynamics

Romualdo Tresguerres ∗Instituto de Matem´ aticas y F́ısica Fundamental Consejo Superior de Investigaciones Cient́ıcas

Serrano 113 bis, 28006 Madrid, SPAIN (Dated: October 7, 2008)

We analyze the role played by local translational symmetry in the context of gauge theories of fundamental interactions. Translational connections and elds are introduced, with special attentionbeing paid to their universal coupling to other variables, as well as to their contributions to eldequations and to conserved quantities.

PACS numbers: 04.20.Fy, 04.50.+h, 11.15.-qKeywords: Gauge theories, local translational symmetry, gauge translational connections and elds, space-time Noether currents, conservation of energy, translationally induced coupling of gravity to the remainingforces.

I. INTRODUCTION

Translational invariance is the main symmetry under-lying Classical Mechanics, being responsible for linearmomentum conservation, and thus for the law of actionand reaction and for inertial motion. Therefore, it isamazing to realize the nearly irrelevant role, if any, as-signed to such symmetry in other dynamical contexts,in particular in those concerned with basic interactions,such as General Relativity and gauge theories, where gen-eralized forms of momentum occur.

Global spacetime translations, as a constitutive part of the Poincaré group, are certainly recognized as essentialfor the spacetime conception of Special Relativity. But asfar as General Relativity makes appearance, general co-variance disguises the meaning of local translations. Fur-

thermore, local translational symmetry is usually ignoredin the context of gauge theories, with few exceptions pro-vided by a particular approach to gravity based on localspacetime groups such as the Poincaré or the affine one[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]. We con-clude that the central role of translational invariance asa foundational principle remains far from being univer-sally recognized, when it is not even explicitly refused byclaiming it to be necessarily violated [13].

The aim of the present paper is to uncover the hiddenpresence of local translational symmetry in the contextof gauge theories. This will be achieved by consider-ing the gauging of a spacetime group together with aninternal group, exploiting the virtualities of certain suit-able translational variables introduced in previous pa-pers [10] [11] [12] [14]. For the sake of simplicity wechoose Poincaré ⊗ U (1) as the gauge group, with elec-trodynamics taken as a characteristic representative of general Yang-Mills theories. However, the interplay weare going to show, concerning the universal coupling of the translational variables to gauge potentials and elds

∗ [email protected]

of the remaining symmetries, is easily generalizable toany internal group, so that all our results are applicableto the whole Standard Model by considering Poincaré ⊗SU (3) ⊗SU (2) ⊗U (1) ; a simple task which is left to the

reader. With special care in explicitly displaying the roleplayed by translations, we will begin closely following thesteps of Hehl et al. [7] to develop a Lagrangian formal-ism giving rise to the eld equations and to the Noetheridentities connected to the gauge symmetry. Then, anapparent digression on the rudiments of a Hamiltonianapproach leads us to the identication of a well behaved–automatically conserved– energy current 3-form relatedto the translational variables.

The paper is organized as follows. In Section II werecall the signicance of translations for Newtonian Me-chanics, showing the main lines of the way to go on. InSection III we discuss an exterior calculus reformulationof the standard variational principles. In IV we derivethe eld equations, and in V the Noether identities. InVI a Hamiltonian-like 3-form is introduced, and a de-nition of a conserved energy current–different from the(vanishing) Hamiltonian one– is suggested. In order toillustrate the previous results with more familiar formu-las, in VII we derive spacetime relations [ 15] between ex-citations and eld strengths generalizing the electromag-netic case ( 111), using several common Lagrangian piecesfor matter and for fundamental interactions. In VIII weoutline a Hamiltonian formalism containing a general-ized translational Gauss law as the constraint acting asgenerator of translations. The paper ends with severalnal remarks in IX and with the Conclusions. However,we still leave for the appendices some related commentson the geometrical and kinematical interpretation of theformalism.

http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2http://es.arxiv.org/abs/0707.0296v2mailto:[email protected]:[email protected]://es.arxiv.org/abs/0707.0296v2


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II. GLOBAL TRANSLATIONS IN NEWTONIANDYNAMICS

A. Laws of motion

In Classical Mechanics, linear momentum conserva-tion, as derived from global space translations with thehelp of Noether’s theorem, constitutes the ground whereNewton’s motion equations rest on. Actually, the law of inertia expresses conservation of the momentum of an iso-lated particle, while the law of action and reaction is thenecessary and sufficient condition for momentum conser-vation of a system consisting of two particles. As for theforces introduced by the second law, they are suitably de-ned as quantities measuring the mutually compensatingchange induced on the momenta of the individual bod-ies, in such a way that conservation of the total linearmomentum is guaranteed.

The fundamental role played by translations in New-tonian dynamics is explicitly shown by considering a sys-tem constituted by two particles, characterized by a La-grangian depending on their positions and velocities, thatis L = L (xa1 , x

a2 ; ẋ

a1 , ẋ

a2 ) (where the dot denotes as usual

derivation with respect to the time parameter t ), beingthe linear momenta of the particles respectively denedas p(1)a := ∂ L/∂ ẋa1 , p

(2)a := ∂ L/∂ ẋa2 . A generic variation

of such Lagrangian yields

δ L = δxa1 ∂ L∂x a1

− dp(1)a

dt+ δxa2

∂ L∂x a2

− dp(2)a

dt

+ ddt

δxa1 p(1)a + δxa2 p

(2)a . (1)

Assuming the derived term in ( 1) to vanish at the inte-gration limits, the principle of least action requiring theaction S = L dt to be extremal gives rise to the motionequations

∂ L∂x a1

− dp(1)a

dt = 0 ,

∂ L∂x a2

− dp(2)a

dt = 0 , (2)

where the gradients in ( 2) are identiable as forces, illus-trating Newton’s second motion equation for conservativeforces deriving from a potential.

Now we return back to ( 1) presupposing the motionequations ( 2) to hold, and instead of a general variation,we perform a rigid displacement of the whole system.That is, we consider a translational group variation char-acterized by the constant parameters ǫa . Since we aredealing with global transformations, we assume the vari-ation to be simultaneously well dened at distant places,being the same for both separated position variables xa1and xa2 , moved simultaneously as δx

a1 = δx

a2 = − ǫ

a . Soone gets

δ L = ddt

δxa1 p(1)a + δxa2 p

(2)a = − ǫa

ddt

p(1)a + p(2)a . (3)

From ( 3) we read out that invariance under translations

requires the conservation of linear momentum

ddt

p(1)a + p(2)a = 0 , (4)

which is a condition not contained in ( 2). Actually, byreplacing ( 2) in (4), we get the law of action and reaction

∂ L∂x a1 +

∂ L∂x a2 = 0 , (5)

affecting the forces appearing in ( 2). Eq.( 5) is a directconsequence of translational invariance, implying the La-grangian dependence on the individual positions xa1 , x

a2

to appear as dependence on the relative position xa1 − xa2

of both particles, that is L(xa1 , xa2 ; ...) = L(x

a1 − x

a2 ; ...) .

Besides Newton’s second law ( 2) and third law ( 5), onealso obtains the rst one by considering a system con-sisting of a single particle. Being the latter isolated inthe universe, no forces are present and ( 4) reduces todp(1)a /dt = 0 , expressing the principle of inertia concern-ing a single particle.

Historically, Descartes was pioneer in postulating arough (scalar) version of momentum conservation ( say p(1) + p(2) = const. ) in the context of contact interac-tions as occurring in collisions. The improved continu-ous (vector) formulation ( 4) of this principle –derivable,as already shown, from local translational invariance–suggests the introduction of Newton’s forces as quantita-tively reecting the soft changes of the momenta, see ( 2),thus being interpretable as measures of non-contact inter-actions. Since, according to ( 4), mutually compensatingchanges of momenta occur simultaneously at separatedplaces as a result of global space-translational symmetry,instantaneous action at a distance as admitted in New-tonian mechanics manifests itself as a byproduct of suchsymmetry.

B. The guiding principles

The previous derivation of Newton’s laws is based ona variational principle together with a symmetry prin-ciple, being both generalizable as powerful form-givinginstruments underlying diverse dynamical formulations.However, they don’t contain the complete physical infor-mation. Indeed, a third non actually existing principlewould be necessary to entirely deduce empirically mean-ingful equations, both in the Newtonian as much as inthe gauge-theoretical framework. When applied to theformer classical example, the lacking principle should beresponsible for justifying a Lagrangian

L = 12

m 1 ẋ21 +

12

m 2 ẋ22 − V (x

a1 − x

a2 ) , (6)

(where the potential V could also be specied), allowingto go beyond the mere form of equations ( 2) and ( 4) byyielding the empirically relevant quantities pa

(1)= m 1 ẋa1

and pa(2)

= m 2 ẋa2 characteristic for Classical Mechanics.


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Regarding gauge theories, the lacking principle would beexpected to provide a criterium to establish the form of the Lagrangian giving rise for instance to suitable space-time relations of the Maxwell-Lorentz type ( 111), deter-mining the generalized excitations studied in Section VII.Since we don’t have such a third principle, we are lim-ited to adopt several Lagrangian pieces as established byexperience.

Of course, we had avoided effort by having directlytaken ( 6) as the starting point to derive the classicaldynamical equations, since this Lagrangian resumes allthe information discussed previously. However, by do-ing so we had lost the possibility of studying separatelythe contributions to the conformation of physical lawscoming from each of the different principles invoked. Ac-tually, from our treatment of the example of NewtonianMechanics, we read out a general scheme to be kept inmind for what follows, consisting of three steps.

1.– First we consider the least action (in fact, the ex-tremal action) variational principle giving rise to the eldequations in terms of quantities to be determined. Theapplication of the principle does not require to know theparticular form of the Lagrangian. One merely has tochoose the dynamical variables, taking the Lagrangianto be a functional of them and of their (rst) derivatives.If symmetry conditions are still not taken into account,the resulting eld equations are trivially non covariant,see (23)–(28) below.

2.– Covariance is a consequence of the symmetry prin-ciple requiring the eld equations to be compatible withinvariance of the action under transformations of a par-ticular symmetry group, see ( 55)–(57) below. Dependingon the group parameters being constant quantities or not,symmetries are global or local, both relating to conser-vation laws through Noether’s theorem. (The symmetryprinciple in its local form is the gauge principle.)

3.– Finally, from the lacking third principle we wouldexpect a guide for establishing the fundamental space-time relations analogous to the Maxwell-Lorentz electro-magnetic one (111). As a succedaneum of such prin-ciple, we take as guaranteed by a long experience thewell established form of the Lagrangians of Dirac mat-ter and electromagnetism, while for gravity we choosefrom the literature [16] a reasonable generalization ( 120)of the Hilbert-Einstein Lagrangian for gravity, includingquadratic terms in the irreducible pieces of torsion andcurvature, constituting a tentative form to be adjustedby xing certain parameters. See Section VII.

III. VARIATIONAL TREATMENTS OF THEACTION

We use a formalism based on exterior calculus [7],with differential forms playing the role of dynamical vari-ables. The fundamental kinds of objects involved ingauge theories consist of connections ( 1-forms) and elds( 0-forms ), both of them (denoted generically as Q with

all indices suppressed for simplicity) being ber bundleconstitutive elements. In terms of Q and of their ex-act differentials, we build the Lagrangian density 4-formas a functional L (Q,dQ ), whose integral on a compactfour-dimensional region D of the bundle base space M constitutes the action

S :=

D

L (Q,dQ ) . (7)

The bundle structure provides a geometrical backgroundfor different variational and symmetry considerations. Infact, in a bundle, two mutually orthogonal sectors exist,being the bers regarded as vertical while the base spaceis conventionally taken as horizontal. Accordingly, twodifferent kinds of variations are distinguished, dependingon whether one moves vertically –by keeping xed theintegration domain–, or one alternatively considers hor-izontal displacements to neighboring integration regionsof the base space [17]. Each of these main categories of variations can be approached in different manners. So,

besides generic vertical variations of the elds required toleave (7) stationary in virtue of the principle of least ac-tion giving rise to the Euler-Lagrange equations, one hasto consider the important particular case of vertical au-tomorphisms along bers, providing the bundle interpre-tation of gauge transformations. On the other hand, theaction is not required to be left invariant under horizontalmotions in order to derive dynamical laws. Nevertheless,such invariance can actually occur. For instance, dis-placements along base space paths generated by Killingvectors play the role of base space symmetry transforma-tions.

A. Vertical variations

We rst consider variations of ( 7) affecting the vari-ables Q (transforming Q into Q̂ , say) while leaving thebase space integration domain D untouched, so that

δ S := D δL , (8)where the integrated variation is to be understood asthe innitesimal limit of the difference L ( Q̂ , dQ̂ ) −L (Q,dQ ). In view of the functional dependence of ( 7),the chain rule yields

δL = δQ ∧ ∂L∂Q

+ δdQ ∧ ∂L∂dQ

, (9)

which, being [ δ , d ] = 0, is trivially brought to the form

δL = δQ∧ ∂L∂Q

− (− 1) p d ∂L∂dQ

+ d δQ ∧ ∂L∂dQ

,

(10)analogous to ( 1) with p corresponding to the degree of the p-form Q. Variations of the action as given by ( 10)reveal to be useful to formalize both principles 1 and 2 of


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Section II B. These complementary impositions of verti-cal invariance of the action mainly differ from each otherin the kind of eld transformations considered in eachcase –namely generic variations versus group variations–as much as in the dissimilar treatments applied to theexact term in ( 10).

On the one hand, the variational principle of extremalaction demands the vertical invariance of the action ( 7)by simultaneously imposing boundary conditions. Ac-cording to Stokes’ theorem 1, (8) with ( 10) yields

δ S = D δQ∧ ∂L∂Q − (− 1) p d ∂L∂dQ + ∂ D δQ∧ ∂L∂dQ .(11)

Stationarity of the action is imposed inside the integra-tion domain D for generic variations δQ, arbitrary every-where but at the integration boundary, where they arexed (like the borders of a vibrating membrane, say) soas to cancel out the hypersurface term . In this way wederive the Euler-Lagrang equations

∂L∂Q − (− 1) p d ∂L∂dQ = 0 , (12)

generalizing ( 2). On the other hand, one can attend tothe symmetry principle by considering gauge group trans-formations instead of arbitrary variations by taking δQas describing vertical automorphisms on the bundle [18].By requiring the eld equations (12) still to hold, thevanishing of (10) then reduces to that of the exact term,yielding the symmetry induced current conservation

d δQ ∧ ∂L∂dQ

= 0 , (13)

according to Noether’s theorem. (Compare with eq.( 4) of Newtonian Mechanics.) The new result ( 13) replaces theboundary condition by a symmetry requirement whilekeeping vertical invariance. We will show immediatelyhow the consistence between ( 12) and ( 13) causes thecovariantization of the eld equations by imposing suit-able conditions on the partial derivatives ∂L∂Q occurringin (12).

B. Horizontal variations

In addition, one can alternatively evaluate horizon-tal diffeomorphisms f : M → M acting on points p ∈ M of the base space manifold [ 17]. (A horizon-tal displacement on the base space of a bundle implies

1 The Stokes theorem establishes

D d ω = ∂ D ω ,being ω a p-form on the ( p + 1)-dimensional compact integrationdomain D of the manifold M , with boundary ∂ D .

a displacement moving from bers to bers.) Usingthe notation L | p := L [Q( p ) , dQ( p )] , and L |f ( p ) :=L [Q(f ( p )) , dQ(f ( p )) ] , we dene the difference

∆ hor S := f ( D ) L |f ( p ) − D L | p (14)between the values of (7) at domains displaced with re-

spect to each other, where the notation ∆ hor indicatesthat we are considering horizontal (base-space) diffeo-morphisms. The pullback f ∗ : T

∗

f ( p) M → T ∗

p M induced by the diffeomorsm f on differential forms ωsatises f ( D ) ω = D f

∗ω , thus allowing to rewrite therst term in the r.h.s. of ( 14) on the integration domainD , so that it becomes comparable with the second one.By doing so while taking the diffeomorphism to dependon a parameter s as f s and to be generated by a vec-tor eld X , we nd the horizontal variation ( 14) in theinnitesimal limit to reduce to

δ hor S :=

D

lims

→0

1

sf ∗s L |f s ( p ) − L | p . (15)

In view of the identity of the integrand with the standarddenition of the Lie derivative [ 19], we nally get

δ hor S = D l x L . (16)The Lie derivative in ( 16) measures the horizontal varia-tion of the Lagrange density form along the vector eldX on the base space. For arbitrary p-forms α, the Liederivative takes the explicit form

l x α = X ⌋d α + d (X ⌋α) . (17)

A chain rule analogous to ( 9) holds for the LagrangianLie derivative in (16) as

l x L = l x Q ∧ ∂L∂Q

+ l x dQ ∧ ∂L∂dQ

. (18)

Only for certain vector elds X generating base spacesymmetries (Killing vectors), the Lie derivative (18) van-ishes. In general, displacements on M do not leave theLagrangian form invariant, but they change it as l x L = 0.In view of (17), we nd the Lie derivative of the 4-formLagrangian density to be l x L := d (X ⌋L ) . Thus from(18), being [ l x , d ] = 0, we nd the identities

0 = d l x Q ∧ ∂L∂dQ

− (X ⌋L ) + l x Q ∧ δLδQ

, (19)

where we introduced the shorthand notation that we willuse from now on for the variational derivative as appear-ing in (10)–(12), namely

δLδQ

:= ∂L∂Q

− (− 1) p d ∂L∂dQ

, (20)

whose vanishing means fulllment of the eld equations.Since we aren’t going to consider base space symmetries,


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the non-vanishing r.h.s. of ( 16) represents the effect of an admissible horizontal shift of the integration domain,while the horizontal identities (19) are merely a refor-mulation of the chain rule ( 18). However, provided theeld equations hold –in view of vertical stationarity– sothat ( 20) vanishes, the horizontal identities (19) loosethe last term, transforming into equations expressing thecompatibility conditions between vertical invariance andhorizontal displacements. The Noether type identities weare going to derive in Section V are of this kind .

IV. GAUGING THE POINCAR É GROUPTIMES AN INTERNAL SYMMETRY

The usually hidden role played by translations in gaugetheories will be revealed by applying step by step theguiding principles presented in Section II B. We choosethe Poincaré ⊗ U (1) group, giving rise to a gauge the-ory of gravity and electromagnetism, because of its sim-

plicity in considering together an internal and a space-time symmetry including translations. But our resultsare applicable to other spacetime symmetry groups suchas the affine group underlying metric-affine gravity [7],and to arbitrary internal groups yielding more generalYang-Mills theories such as the Standard Model or anyother.

A. The dynamical variables

Regarding the particular treatment given in the

present paper to translations, it may be clarifying toknow that the author worked for a long time on non-linear realizations of symmetries. It is in the context of nonlinear gauge approaches to several spacetime groups[10] [11] [12] [14] [20] that certain coordinate-like trans-lational Goldstone elds ξ α occur, playing an importantrole in allowing the interpretation of tetrads as modiedtranslative connections transforming as Lorentz covec-tors, thus making it possible to build Geometry entirelyin gauge-theoretical (dynamical) terms.

In a previous paper [14], the author proposed a com-posite ber bundle structure suitable to deal with non-linear realizations of symmetries, and in particular with

the gauge treatment of translations. The existence insuch bundle of three mutually orthogonal sectors has asa consequence that translational bers, although verti-cal when referred to the base space, may be regarded asdening an intermediate base space where other bers arevertically attached to, as to a horizontal basis. Localitywith respect to a given point x of the genuine base spaceis compatible with displacements moving from a positionξ α (x) to a different one ξ̂ α (x) . So to say, the transla-tional sector, characterized by the coordinate-like eldsξ α , provides a dynamical spacetime background for theremaining bundle constituents.

Nevertheless, for what follows we don’t need to sup-port the coordinate-like elds theoretically on compositebundles. One can simply introduce such variables ξ α ,transforming as in ( 30) below, regarding them as usefultools whose geometrical meaning as position vectors isdiscussed in Appendix B. In the following we will makean extensive use of these elds.

In order to deal with the Poincaré ⊗ U (1) symmetry,we take as the fundamental dynamical variables Q actingas arguments of ( 7) the set

{Q} = { ξ α , ψ , ψ , A ,(T )Γα , Γαβ } . (21)

The quantities comprised in ( 21) are either elds (0-forms) or connections (1-forms). Among them we rec-ognize the previously discussed coordinate–like Gold-stone elds ξ α and the matter elds chosen in partic-ular to be Dirac spinors ψ and ψ –all of them 0-forms–and in addition we nd the electromagnetic potential

A = dx i Ai , a translational connection(T )Γα = dx i

(T )Γα

i , and

the Lorentz connection Γ αβ = dx i Γαβi , where the indexi refers to the underlying four-dimensional base space,while α = 0 , 1, 2, 3 are anholonomic Lorentz indices, be-ing the Lorentz connection antisymmetric in α , β .

B. Field equations and symmetry conditions

The variation ( 9) of a Lagrangian density 4–form de-pending on variables (21) and on their differentials reads

δL = δξ α ∂L

∂ξ α + δdξ

α∧

∂L

∂dξ α + δψ

∂L

∂ψ+ δdψ ∧

∂L

∂dψ+

∂L∂ψ

δψ + ∂L∂dψ

∧δdψ + δA ∧ ∂L∂A

+ δdA ∧ ∂L∂dA

+ δ Γα( T ) ∧ ∂L∂ Γα( T )

+ δdΓα( T ) ∧ ∂L∂dΓα( T )

+ δ Γαβ ∧ ∂L∂ Γαβ

+ δdΓαβ ∧ ∂L∂dΓαβ

. (22)

According to the extremal action principle, the eldequations ( 12) are found to be

∂L

∂ξ α − d

∂L

∂dξ α = 0 , (23)

∂L∂ψ

− d ∂L∂dψ

= 0 , (24)

∂L∂ψ

+ d ∂L∂dψ

= 0 , (25)

∂L∂A

+ d ∂L∂dA

= 0 , (26)

∂L∂ Γα( T )

+ d ∂L∂dΓα( T )

= 0 , (27)

∂L∂ Γαβ

+ d ∂L∂dΓαβ

= 0 . (28)


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(Notice in particular the similitude between (23) and(2).) On the other hand, according to the symmetryprinciple, the Noether conservation equation (13) takesthe explicit form

0 = d δξ α ∂L∂dξ α

+ δψ ∂L∂dψ

− ∂L∂dψ

δψ + δA ∧ ∂L∂dA

+ δ (T )

Γα ∧ ∂L∂dΓα( T ) + δ Γ

αβ

∧

∂L∂dΓαβ . (29)

For the Poincaré ⊗ U (1) symmetry we are considering,the local group variations of the quantities ( 21) are thoseof U (1) together with the Poincaré ones as derived forinstance in [ 14], that is

δξ α = − ξ β β β α − ǫα , (30)δψ = iλ + iβ αβ σαβ ψ , (31)

δψ = − ψ iλ + iβ αβ σαβ , (32)

δA = − 1e

dλ , (33)

δ (T )Γα = −

(T )Γβ β β α + Dǫα , (34)

δ Γα β = Dβ α β , (35)

with group parameters λ(x) , ǫα (x) , β αβ (x) (the lat-ter being antisymmetric in α , β ) depending on the basespace coordinates although not explicitly displayed 2, andbeing σαβ the Lorentz generators in terms of Diracgamma matrices. Intrinsic translations are not consid-ered here, but the interested reader is referred to [ 21] fora discussion on them. Rising and lowering of indices isperformed by means of the constant Minkowski metricoαβ = diag (− + ++) constituting the natural invariantof the Poincaré group. We remark the coordinate-likebehavior of ξ α under transformations ( 30), and we point

out the transformation ( 34) of (T )Γα as a connection, dis-

qualifying it as a candidate to be identied as a tetrad.Replacing in ( 29) the group variations ( 30)–(35) we get

0 = d λe

J + d ∂L∂dA

− ǫα ∂L∂dξ α

+ D ∂L

∂dΓα( T )

− β αβ τ αβ + ξ α∂L

∂dξ β +

(T )Γα ∧

∂L∂dΓβ( T )

+ D ∂L∂dΓαβ

,

(36)

where we introduced the denitions of electric current

J := − ie ψ ∂L∂dψ

+ ∂L∂dψ

ψ , (37)

2 The covariant differentials in (34) and ( 35) are dened respec-tively as

Dǫα := d ǫα + Γ β α ǫβ ,

Dβ αβ := d β αβ + Γ γ α β γβ + Γ γ β β αγ .

and of spin current

τ αβ := i ψ σαβ∂L

∂dψ+

∂L∂dψ

σαβ ψ . (38)

In order to deal with ( 36), we take from [7] the propertythat a zero exact differential d (µα Aα ) = dµα ∧ Aα +µα dAα = 0, with µα as much as dµα being pointwisearbitrary, implies the vanishing of both Aα and its dif-ferential. So from (36) we can derive the equations

J + d ∂L∂dA

= 0 , (39)

∂L∂dξ α

+ D ∂L∂dΓα( T )

= 0 , (40)

τ αβ + ξ [α∂L

∂dξ β ] +

(T )Γ[α ∧

∂L∂dΓβ ]( T )

+ D ∂L∂dΓαβ

= 0 , (41)

where the capital D stands for the covariant differentials;see footnote 3. The compatibility between ( 39)–(41) andthe eld equations ( 26)–(28) requires the following con-sistence conditions to hold

∂L∂A

= J , (42)

∂L∂ Γα( T )

= ∂L∂dξ α

− Γα β ∧ ∂L∂dΓβ( T )

, (43)

∂L∂ Γαβ

= τ αβ + ξ [α∂L

∂dξ β ] +

(T )Γ[α ∧

∂L∂dΓβ ]( T )

+2 Γ [α γ ∧ ∂L∂dΓβ ]γ

. (44)

Eq. (41) is not explicitly covariant, so that for the mo-ment it is not evident that ( 42)–(44), as derived with thehelp of the symmetry principle, just imply the covari-antization of the eld equations ( 26)–(28) obtained previ-ously. However, we are going to show that precisely thatis the case. A further consistence condition follows fromcovariantly differentiating the covariant equation ( 40) toget

D ∂L∂dξ α

− Rα β ∧ ∂L∂dΓβ( T )

= 0 , (45)

where Rαβ

is the Lorentz curvature 2-form dened in(A3). By comparing ( 45) with ( 23), we nd

∂L∂ξ α

= Γ α β ∧ ∂L∂dξ β

+ Rα β ∧ ∂L∂dΓβ( T )

. (46)

Notice that in ( 46) as much as in ( 42)–(44), and in ( 5) aswell, it is the value of ∂L/∂Q the relevant one to enablecovariance under the postulated symmetry. The covari-antized form of ( 23) obtained by replacing the condition(46) is identical with ( 45) derived from ( 40). Thus ( 23)–its covariant version in fact– results to be redundant.


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C. Fixing the notation

The variation ( 22) of the Lagrangian, together with thesymmetry conditions (42)–(44) and ( 46), yields what onewould obtain by varying a Lagrangian already dependingon covariant quantities, that is

δL = δϑα

∧

∂L∂dξ α + δDψ ∧

∂L∂dψ +

∂L∂dψ ∧δDψ

+ δψ ∂L∂ψ

− ∂Dψ

∂ψ∧

∂L∂dψ

+ ∂L∂ψ

− ∂L∂dψ

∧ ∂ Dψ

∂ψδψ

+ δF ∧ ∂L∂dA

+ δT α ∧ ∂L∂dΓα( T )

+ δR αβ ∧ ∂L∂dΓαβ

− ξ [α∂L

∂dΓβ ]( T ) , (47)

where the original variables ( 21) appear automatically re-arranged into a number of Lorentz covariant objects de-ned in Appendix A, namely the tetrads ϑα , the Lorentz⊗ U (1) covariant derivatives Dψ and Dψ of the matterelds, the electromagnetic eld strength F , the torsionT α and the Lorentz curvature R αβ .

In all these quantities, any vestige of explicit trans-lational symmetry is absent, see ( A7)–(A12), explainingwhy translations, although genuinely present in the the-ory, become hidden. The ultimate reason for it is thatthe only original elds affected by translations according

to (30)–(35), namely ξ α and(T )Γα , appear always joined

together into the translation-invariant combination

ϑα := Dξ α +

(T )Γα , (48)

shown with more detail in ( A1). Contrary to the original

translative connection(T )Γα , the modied one ( 48) trans-

forms as a Lorentz covector, see ( A7), making it possibleto identify ( 48) as a tetrad, with a geometrical meaningcompatible with its gauge-theoretical origin.

We further simplify the notation of several quantitiesalso involved in ( 47). Firstly we dene the canonicalenergy-momentum 3-form

Πα := ∂L∂dξ α , (49)

resembling the classical denition pa := ∂ L/∂ ẋa of or-dinary linear momentum. The symmetry condition ( 43)reveals a double character of ( 49) by showing its equality–up to terms having to do with covariance– with a trans-lational current. It is in this second interpretation as acurrent that Π α will behave as a source for gravitationalelds, see (56) below.

The ambiguity concerning the meaning of Π α becomesincreased by realizing, as we will do in Section V, that

all elds of the theory contribute to this quantity. De-composition (60) shows in fact that it consists of mate-rial, radiative and gravitational contributions, the doublemeaning affecting each of them. Usually it is illuminatingto separate these different pieces from each other, mainlybecause matter currents Σ α are naturally regarded assources, while pure gravitational contributions E α are of a different nature. But for the moment let us keep Π αunied as a whole. By doing so the notation becomessimplied; and on the other hand, it is the complete Π αthat will play a role in the denition of the conservedenergy current 3-form ( 84) to be dened in Section VI.

Otherwise, we follow Hehl’s standard notation [7].Taking as a model the electromagnetic excitation 2-form

H := − ∂L∂dA

, (50)

(to be determined by the Maxwell-Lorentz spacetime re-lation ( 111)), we introduce its translative and Lorentziangauge analogs, dened respectively as the 2-forms

H α := − ∂L

∂dΓα( T ) , (51)

and

H αβ := −∂L

∂dΓαβ − ξ [α

∂L∂dΓβ ](T )

. (52)

The second term in the r.h.s. of ( 52) is due to the factthat, in view of (A2) with ( A1), the torsion reads T α :=

Dϑα = D (D ξ α +(T )Γα ) , so that δ T α = δ (Rβ α ξ β + D

(T )Γα ),

having as a consequence the occurrence of a contributionto (52) through the implicit dependence of T α on Rβ α .Comparison of ( 49),(51), (52) with ( 47) reveals that

Πα := ∂L∂ϑα

, H α := − ∂L∂T α

, H αβ := − ∂L∂R αβ

. (53)

In terms of these objects we are going to rewrite ( 39)–(41). However, rst we have to reformulate the non ex-plicitly covariant equation ( 41), making use of ( 48) anddenitions ( 49), (51), (52), as

DH αβ + ϑ[α ∧H β ] − τ αβ + ξ [α DH β ] − Πβ ] = 0 , (54)

where the term in parentheses is merely ( 40), thus van-ishing independently. So, the eld equations ( 39)–(41)take the form

dH = J , (55)DH α = Π α , (56)

DH αβ + ϑ[α ∧H β ] = τ αβ . (57)

All of them are explicitly Lorentz covariant 3, while withrespect to translations as much as to U (1), they are in-variant. In ( 55) we recognize the Maxwell equations up to

3 The covariant differentials in (56) and (57) are respectively de-


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the explicit form of H to be established in ( 111). The factthat ( 56) generalizes the gravitational Einstein equationsis less evident, but see Section VII C. Both ( 56) and ( 57)reproduce the standard form established by Hehl et al.[7], with the main difference that in (56) we do not sep-arate the different pieces of Π α , as discussed above. Theambiguity derived from considering such a source , whichis not a pure matter current – as the electric current J as well as the spin current τ αβ are–, is compensated bythe higher formal simplicity.

The redundant equation ( 45) constituting the covari-antized version of ( 23) is immediately deducible from ( 56)as

D (Πα − DH α ) = DΠα + Rα β ∧H β = 0 . (58)

Thus, the simultaneous application of the variationalprinciple yielding ( 23)–(28), and of the symmetry prin-ciple, is summarized by the matter equations ( 24)–(25)together with the covariant eld equations ( 55)–(57), thelatter ones being associated respectively to U (1), trans-

lations, and Lorentz symmetry.

V. NOETHER IDENTITIES

In the present section we consider separately the differ-ent pieces [7] into which one can meaningfully decomposethe Poincaré ⊗ U (1)–invariant Lagrangian, that is

L = Lmatt + Lem + Lgr , (59)

comprising on the one hand the material contribu-tion Lmatt ( ϑα , ψ , ψ , D ψ , D ψ ), plus an electromagneticpart Lem ( ϑα , F ) and a pure gravitational constituentLgr ( ϑα , T α , Rα β ). Notice that the matter part of theLagrangian depends basically on matter elds and theircovariant derivatives, and the electromagnetic and gravi-tational pieces on the eld strengths of the U (1) and thePoincaré symmetry respectively. But not only. The uni-versal ϑα –dependence is also displayed everywhere. Ac-tually, in Lagrangian pieces where the Hodge star oper-ator ∗ occurs, as it is the case for the physically realisticexamples ( 94), (110) and (120) to be considered later,this dependence is explicitly brought to light by the vari-ational formula ( F8). Here we realize for the rst timethe (nonminimal) universal coupling of the translationalvariables comprised in the tetrad ( 48) to the remainingquantities of the theory, having as a consequence that allpieces in (59) contribute to the energy-momentum ( 49).

We are going to study the conditions for the verticalinvariance of every separate part of ( 59) under Poincaré

ned asDH α := dH α − Γα β ∧ H β ,

andDH αβ := dH αβ − Γα γ ∧ H γβ − Γβ γ ∧ H αγ .

⊗U (1) gauge transformations ( 30)–(35) –and the derivedones (A7)–(A12)–, as well as the compatibility conditionswith the eld equations of the horizontal displacements(18) of each independent Lagrangian piece along a genericvector eld X . We follow Hehl et al. [7] in derivingsimultaneously the Noether type conservation equationsfor matter currents, as much as the form of the differentpieces

Πα = Σ mattα + Σemα + E α (60)

into which ( 49) becomes decomposed consistently with(59), with the obvious notation Σ mattα := ∂Lmatt /∂dξ α ,Σ emα := ∂L em /∂dξ α and E α := ∂L gr /∂dξ α , as read outfrom (49) and ( 59).

Let us start with the matter Lagrangian part Lmatt .For what follows, with the help of ( 42)–(44) we identifythe matter currents associated to the different symme-tries as the derivatives of the matter Lagrangian with re-spect to the corresponding connection, as usual in gaugetheories, that is

J = ∂L matt

∂A , Σmattα =

∂Lmatt

∂ Γα( T ),

τ αβ + ξ [α ∧Σ mattβ ] = ∂Lmatt

∂ Γαβ . (61)

Provided the eld equations are fullled, the gauge trans-formations ( 30)–(35) of Lmatt yield

δLmatt = λe

dJ − β αβ D τ αβ + ϑ[α ∧Σ mattβ ] . (62)

From the vanishing of ( 62), as required by its postulatedPoincaŕe ⊗ U (1) invariance, we read out rst the conser-vation of the electric current ( 37), namely

dJ = 0 , (63)

a result which looks trivial in view of being also obtain-able by merely differentiating ( 55). Furthermore we getalso the less simple conservation equation for the spincurrent

D τ αβ + ϑ[α ∧Σ mattβ ] = 0 , (64)

a result which is not a priori expected.On the other hand, we consider a horizontal displace-

ment of the matter part of the action, assuming simulta-neously its vertical invariance by supposing the symmetryconditions ( 63) and ( 64) to hold. The requirement of ver-tical invariance of the total action is also kept in mind,reecting itself in the eld equations. In this way weget new identities of the Noether type. For convenience,in our deduction we use ( 18) rather than the equivalentequation ( 19) due to the fact that the latter presents nocalculational advantage in the present case. Indeed, thevariational derivative term in ( 19) doesn’t vanish for eachLagrangian piece separately, since eld equations derivefrom the whole Lagrangian. The Lie derivative ( 18) of


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the matter piece of the Lagrangian satisfying the men-tioned conditions expands as

l x Lmatt = − X α D Σ mattα − ( eα ⌋T β ) ∧Σ mattβ

− ( eα ⌋Rβγ ) ∧ τ βγ − ( eα ⌋F ) ∧ J

+ d X α

Σmattα + ( X ⌋Dψ )

∂Lmatt

∂dψ

−∂L matt

∂dψ ( X ⌋Dψ ) . (65)

Due to the fact that l x Lmatt = d (X ⌋Lmatt ), as read outfrom (17) being Lmatt a 4–form, (65) can be reduced tothe form 0 = X α Aα + d (X α Bα ) = X α ( Aα + dBα ) +d X α Bα , so that –as before– for pointwise arbitrary X αand dX α , the vanishing of both Aα and Bα follows [7],implying

D Σ mattα = ( eα ⌋T β )∧Σ mattβ +( eα ⌋R

βγ )∧τ βγ +( eα ⌋F )∧J ,

(66)and

Σ mattα = − ( eα ⌋Dψ ) ∂Lmatt

∂dψ+

∂L matt

∂dψ ( eα ⌋Dψ )+ eα ⌋Lmatt .

(67)Eq. (66) is a sort of force equation; see ( 150). Indeed,in the last of the similar terms entering the r.h.s. werecognize the ordinary Lorentz force involving the elec-tromagnetic eld strength and the electric current. Theremaining pieces have the same structure, being builtfrom the eld strength and the matter current associ-ated to translational and Lorentz symmetry respectively.

On the other hand, ( 67) outlines the form of the matterpart of ( 60). (Recalling the previously discussed ambigu-ity of energy-momentum, notice that Σ mattα in the r.h.s.of (66) behaves as one of the three kinds of matter cur-rents present in the theory, while in the l.h.s. the samequantity is more naturally understood as matter momen-tum.)

Having nished our detailed study of the matter partof the Lagrangian, let us now briey summarize the re-sults obtained by proceeding analogously with the tworemaining pieces in ( 59). Regarding the electromagneticLagrangian constituent, its gauge transformation yieldsδLem = − β αβ ϑ[α ∧ Σ emβ ] , so that its invariance implies

the symmetry conditionϑ[α ∧Σ emβ ] = 0 . (68)

The equation analogous to ( 65) yields

D Σ emα = ( eα ⌋T β ) ∧Σ emβ − ( eα ⌋F ) ∧dH , (69)

as much as the form of the electromagnetic energy-momentum

Σ emα = ( eα ⌋F ) ∧H + eα ⌋Lem . (70)

Finally we consider the gravitational Lagrangian part.Its invariance condition

D DH αβ + ϑ[α ∧H β ] + ϑ[α ∧ DH β ] − E β ] = 0 , (71)

turns out to be redundant with previous results sinceit can be immediately derived from the eld equations(56), (57), together with ( 60), (64) and ( 68). The ( 65)–analogous equation gives rise to two different results. Onthe one hand, it yields

D DH α − E α − ( eα ⌋T β ) ∧ DH β − E β

− ( eα ⌋Rβγ ) ∧ DH βγ + ϑ[β ∧H γ ] = 0 , (72)

which is also redundant, derivable from the eld equa-tions ( 55)–(57) with ( 60), (66) and ( 69). On the otherhand, it provides the form of the pure gravitational con-tribution to energy-momentum, namely

E α = ( eα ⌋T β ) ∧H β + ( eα ⌋R βγ ) ∧H βγ + eα ⌋Lgr . (73)

The total momentum ( 60) entering the eld equation ( 56)is found by performing the sum of ( 67), (70) and ( 73) as

Πα = − ( eα ⌋Dψ ) ∂L∂dψ

+ ∂L∂dψ

( eα ⌋Dψ ) + ( eα ⌋F ) ∧H

+( eα ⌋T β ) ∧H β + ( eα ⌋Rβγ ) ∧H βγ + eα ⌋L . (74)

Written in this form, it will play a relevant role in whatfollows.

VI. ENERGY CONSERVATION

In Section III B we introduced equation ( 19) governinghorizontal displacements along arbitrary vector elds Xon the base space, and we discussed the compatibility of such displacements with vertical invariance of the action(that is, with fulllment of the eld equations). Nowwe are going to particularize to the case of the promi-nent vector eld n characterized as follows. On the basespace we introduce a 1-form ω satisfying the Frobenius’foliation condition ω ∧ dω = 0, whose general solutionreads ω = N dτ . With the help of τ obtained in this way,taken to be –at least locally– a monotone increasing vari-

able, it becomes possible to parametrize nonintersecting3-dimensional base space hypersurfaces. This justies toregard τ as parametric time , while N is the so calledlapse function xing a time scale . The vector n acquiresits temporal meaning through the condition n⌋(Ndτ ) = 1relating it to the parametric time variable. The conceptof temporality thus emerges from the foliation of the basespace. (The same holds for spatiality , the latter howeveras a secondary result.) Indeed, in principle no a time coordinate is identiable as such in the base space. Itis through the foliation that parametric time τ appears,conforming its associated parametric time vector eld n .


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Horizontal displacements along n given by the Liederivative of any variable are to be understood as para-metric time evolution . Being normal to the spatial hyper-surfaces, the vector eld n is tangential to a congruenceof worldlines. The direction dened by the parametric time vector on the base space allows to perform a de-composition [ 15] of any p-form α into two constituents,respectively longitudinal and transversal to n as

α = N dτ ∧ α⊥ + α , (75)

being the longitudinal component

α⊥ := n⌋α (76)

the projection of α along n , and the transversal part

α := n⌋(Ndτ ∧ α ) , (77)

an orthogonal projection on the spatial sheets.The longitudinal part of the tetrad ( 48) will play a

singular role due to the following formal reason. As dis-

cussed in Appendix B, one can introduce a vector basis eαdual to the coframes ( 48) in the sense that eα ⌋ϑβ = δ βα .Thus, being ϑα⊥ := n⌋ϑα according to ( 76), one can ex-press the vector eld n = n i ∂ i alternatively as n = ϑα⊥ eα .The fact that n itself must be trivially time-like has itsformal plasmation in the property oαβ ϑα⊥ ⊗ ϑβ⊥ = − 1read out from ( E9).

A. Vanishing Hamiltonian-like 3-form

Starting with the identity ( 19) valid for arbitrary vec-tor elds, we apply it in particular to the time vector n. We do not perform here a complete foliation of theequations as we will do in Section VIII, where we willtotally separate longitudinal and transversal parts fromeach other; but we make use of the notation ( 76) as aconvenient shorthand for quantities such as n⌋Q := Q⊥or n⌋L =: L⊥ . Analogously, applying ( 17) particularizedfor the parametric time vector n , we denote

l n Q := ( n⌋d Q ) + d (n⌋Q ) =: ( d Q )⊥ + d Q⊥ , (78)

compare with ( D2). Using (78) we rewrite ( 19) as

0 = d Q⊥ ∧ ∂L

∂Q + ( d Q)⊥ ∧

∂L

∂dQ − L⊥ − Q⊥ ∧

δL

δQ+ l n Q ∧

δLδQ

. (79)

By dening the Hamiltonian-like 3-form

H := Q⊥ ∧ ∂L∂Q

+ ( d Q)⊥ ∧ ∂L∂dQ

− L⊥ − Q⊥ ∧ δLδQ

, (80)

eq. (79) becomes

dH + l n Q ∧ δLδQ

= 0 . (81)

Thus, provided the eld equations ( 12) hold, ( 81) seemsto yield a continuity equation dH = 0 affecting the quan-tity H, the latter being a sort of energy current 3-form.Unfortunately, we are going to prove that such equationtrivializes since H itself vanishes. To arrive at such con-clusion, we evaluate ( 80) explicitly for the variables ( 21).Although not immediately evident, the rst terms in ther.h.s. of ( 80) can be rearranged into covariant expressionsby replacing the symmetry conditions ( 42)–(44), so thatfor fullled eld equations, ( 80) takes the gauge invariantform

H = ϑα⊥∂L

∂dξ α + Ln ψ

∂L∂dψ

− ∂L∂dψ

Ln ψ

+ F ⊥ ∧ ∂L∂dA

+ T α⊥ ∧ ∂L∂dΓα(T )

+ Rαβ⊥ ∧∂L

∂dΓαβ − ξ [α

∂L∂dΓβ ](T )

− L⊥ , (82)

where we used denitions Ln ψ := n⌋Dψ = ( Dψ )⊥

, com-pare with (C1), and F ⊥ := n⌋F , etc.; see (76). By re-turning back now to the previous result ( 74), contractingit with ϑα⊥ and recalling that n = ϑα⊥ eα , we nd

0 = ϑα⊥ Πα + Ln ψ ∂L∂dψ

− ∂L∂dψ

Ln ψ − F ⊥ ∧H

− T α⊥ ∧H α − Rαβ⊥ ∧H αβ − L⊥ , (83)

revealing that ( 82) reduces to zero. So, instead of a con-tinuity equation dH = 0 , we merely have a relation be-tween the different terms in ( 82), namely H = 0 or ( 83).This result holds independently of the particular form

of the Lagrangian, and it is in close relationship withthe well known vanishing of any possible Hamiltonian of General Relativity.

B. A well behaved energy current

Since dH = 0 cannot play the role of a law of conser-vation of energy because of its triviality, we look for analternative formulation of such law, if possible. At thisrespect, let us recall the singular role played by trans-lational variables as compared with the remaining con-stituents of the theory, in the sense that ξ α and Γ α(T ) ,

conned together in the translation-invariant combina-tion constituting the tetrad ( 48), couple to any otherphysical quantity (usually through the ϑα –terms in (F8 ),provided the Hodge dual operator occurs, as alreadymentioned). The universal coupling of translations com-pels information relative to any other quantity to becomestored in the (translational) energy-momentum ( 74). Ac-cordingly, in ( 83) each contribution appears twice, so tosay: once explicitly and once through Π α , with the re-sult that the total sum cancels out. Having this fact inmind, we propose to identify in ( 83) a meaningful expres-sion to be dened as (translational) energy, balancing the


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joint amount of the remaining energy contributions. Thepossible energy candidate is expected to be conserved.

We nd such a quantity effectively to exist, consistingin the energy current 3-form

ǫ := − ( ϑα⊥ Πα + Dϑα⊥ ∧H α ) , (84)

which in view of (56) satises the nontrivial continuity

equationdǫ = 0 , (85)

with the meaning of local conservation of energy. Byrewriting ( 83) in terms of ( 84) while taking into account(C2), we get

ǫ = Ln ψ ∂L∂dψ

− ∂L∂dψ

Ln ψ − F ⊥ ∧H

− Ln ϑα ∧H α − Rαβ⊥ ∧H αβ − L⊥ , (86)

where the total –nonvanishing– energy ǫ in the l.h.s. of (86) resumes the whole information concerning the re-maining elds displayed in the r.h.s., as already com-mented.

We conclude that the singularity of the Hamiltonian(82) is a consequence of the presence of translations, evenif hidden, in the scheme. This result is unavoidable asfar as gravitation is taken into account, since modiedtranslational connections ( 48) –that is tetrads, or theRiemannian metric built from them– are to be treated asdynamical variables, thus giving rise to the occurrence of a contribution ( 84) leading to the vanishing of H . No-tice that a nonvanishing Hamiltonian-like 3-form H withthe ordinarily expected meaning of a nonvanishing en-ergy current only would make sense in contexts wheregravitational contributions (and thus translations) weredisregarded.

According to ( 84) and taking the decomposition ( 60)into account, we introduce three different contributionsto energy, namely

ǫ = ǫmatt + ǫem + ǫgr , (87)

respectively dened as

ǫmatt := −ϑα⊥ Σ mattα , (88)ǫ

em := −ϑα⊥ Σ emα , (89)ǫ

gr := − ( ϑα⊥ E α + Dϑα⊥ ∧H α ) . (90)

None of them is a conserved quantity. Actually, from(88) with ( 66) we get for instance

dǫmatt := − Ln ϑα ∧Σ mattα − Rαβ⊥ ∧τ αβ − F ⊥ ∧ J . (91)

The non zero r.h.s. of (91) may be partially illuminatedwith the help of the remaining contributions to the totalenergy conservation ( 85). Indeed, ( 89) with ( 69) yields

dǫem := − Ln ϑα ∧Σ emα + F ⊥ ∧dH = − Ln ϑα ∧Σ emα + F ⊥ ∧ J , (92)

which we are going to compare in Section VII B with thewell known electromagnetic energy conservation equationinvolving the Poynting vector and Joule’s heat. (In thestandard electromagnetic formulation, the rst term inthe r.h.s. of ( 92) is absent.) If desired, one can consider(92) as a modied form of the rst law of Thermodynam-ics, an idea which is generalizable to the previous and thenext case. For the gravitational energy ( 90) with ( 72) wenally nd

dǫgr := − Ln ϑα ∧ ( E α − DH α )+ Rαβ⊥ ∧ DH αβ + ϑ[α ∧H β ]

= Ln ϑα ∧ Σ mattα + Σemα + R

αβ⊥ ∧τ αβ . (93)

So, the energy exchange is performed in such a waythat not the different types of energy separately, butonly the sum ( 84) of all of them, (88), (89) and ( 90),is conserved. The reason for it is that, in virtue of (56), the total energy ( 84), although composed of threehighly nontrivial pieces, reduces to an exact form as

ǫ = − d (ϑα⊥ H α )+ ϑ

α⊥ (DH α − Πα ) , which in general, con-trarily to H , is different from zero.

VII. EXPLICIT LAGRANGIAN PIECES

All the previous results were derived by invoking onlythe least action variational principle together with a sym-metry principle. That is, until now we took into ac-count two of the principles of Section II B, but we didn’tmiss the lacking principle expected to provide the formof the fundamental Lagrangian. However, in order tophysically complete the formal scheme deduced previ-

ously, we nally have to introduce explicit gauge invari-ant Lagrangian pieces ( 59), built from the covariant objects (A1)–(A6), in order to derive the form of energy-momentum ( 49) and of the generalized excitations ( 50)–(52), as much as of concrete matter equations. In partic-ular, for Dirac matter and for Maxwell electromagnetismwe will use the corresponding standard Lagrangians, andfor gravity a generalization of the Hilbert-Einstein one.

A. Dirac matter

Let us introduce the Dirac Lagrangian

Lmatt = i2

( ψ ∗γ ∧Dψ + Dψ ∧ ∗γψ ) + ∗m ψ ψ , (94)

see [22], built with the Poincaré ⊗ U (1) covariant deriva-tives ( A5) and ( A6), using the notation γ := ϑα γ α , withγ α as the Dirac gamma matrices, so that ∗γ := ηα γ α ;see (F4). For a discussion on the absence of intrinsictranslational contributions in such derivatives, see [21].From ( 94) we nd

∂L∂dψ

= i2∗γ ψ ,

∂L∂dψ

= i2

ψ ∗γ , (95)


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see [15], actually yields for the electromagnetic excitation(50) the explicit form

H = − ∂L∂dA

= ∗F , (111)

constituting the Maxwell-Lorentz electromagnetic space-time relations [15]. It is by replacing ( 111) in (55) that

we get the fundamental Maxwell equations.From ( 111) with ( D3) follows H ⊥ = # F , and H =− # F ⊥ , so that ( 105) and ( 106) become actually uniedas

Σ emα = ϑ⊥ α ǫem − ϑα ∧F ⊥ ∧ # F . (112)

On the other hand, the electromagnetic part ( 70) of themomentum derived from the explicit Lagrangian ( 110)reads

Σ emα = 12

[ (eα ⌋F ) ∧H − F ∧(eα ⌋H ) ] , (113)

so that ( 107) becomes nally

ǫem = − ϑα⊥ Σ emα = −

12

[F ⊥ ∧H − F ∧H ⊥ ] . (114)

In order to compare this result with more familiar no-tations [15], we nd the components of the electromag-netic energy current 3-form ( 114) analogous to the onesin (102) to be the energy ux or Poynting 2-form

ǫem⊥ = F ⊥ ∧ # F , (115)

being identiable as the exterior calculus version of thestandard Poynting vector

→

E ×→

B , and the energy density

3-form

ǫem =

12

F ⊥ ∧ # F ⊥ + F ∧# F , (116)

equal to the electromagnetic eld energy which in stan-dard vector notation reads 12 ( E

2 + B 2 ) dV . The con-servation equation (92) can then be brought to a moreexplicit form by decomposing on the one hand

dǫem = N dτ ∧ l n ǫem − 1N

d (Nǫem⊥ ) , (117)

using (D2), and on the other hand the remaining terms

according to ( 75), to get

l n ǫem − 1N

d (Nǫem⊥ ) = − Ln ϑα⊥ Σemα − F ⊥ ∧J ⊥ . (118)

Replacing ( 105), and invoking the formal relation N dτ =− ϑα ϑα⊥ found at the end of Appendix E to deducedN/N = ϑα⊥ ( T ⊥ α − Ln ϑα ) , we transform ( 118) into

l n ǫem = d ǫem⊥ + ϑα⊥ ( T ⊥ α − 2 Ln ϑα ) ∧ǫem⊥ − F ⊥ ∧ J ⊥ .(119)

The time derivative of the energy density equals the di-vergence of the Poynting 2-form, plus additional terms

having to do with the underlying geometry, plus a termwhich in standard notation reads

→

E ·→

j dV , being inter-pretable as Joule’s heat produced by the electric current[24].

In parallel to the matter case, from ( 113) we nd ϑα ∧Σ emα = 0, while Lem = − 12 F ∧ H , being in principlel n Lem = 0 , so that the electromagnetic action, according

to (16), evolves in time. (Actually, l

n Lem

= dǫem

−12 F ∧J ⊥ + F ⊥ ∧ J .)

C. Gravitation

Since no universally accepted action exists for gravity,we take from Ref. [16] a quite general Lagrangian densityincluding, besides a term of the Hilbert-Einstein type anda cosmological term, additional contributions quadraticin the Lorentz–irreducible pieces of torsion and curva-ture as established by McCrea [ 7] [25]. The gravitationalLagrangian reads

Lgr = 1κ

a02

Rαβ ∧ηαβ − Λ η

−12

T α ∧ 3

I =1

a I κ

∗( I ) T α

−12

Rαβ ∧ 6

I =1

bI ∗( I ) Rαβ , (120)

with κ as the gravitational constant, and a0 , aI , bI as dimensionless constants. Denitions ( F2)–(F5) areused, and the quadratic expressions are written takinginto account that McCrea’s irreducible torsion pieces( I ) T α are mutually orthogonal, so that (I ) T α ∧ ∗( I ) T α =T α ∧∗( I ) T α . The same holds for the irreducible curvaturepieces (I ) Rαβ . From ( 120) we calculate the translationaland Lorentz excitations ( 51) and ( 52) respectively to be

H α =3

I =1

a I κ

∗( I ) T α , (121)

H αβ = −a02κ

ηαβ +6

I =1

bI ∗( I ) Rαβ , (122)

and we nd the pure gravitational contribution ( 73) to

the energy momentum

E α = a04κ

eα ⌋ Rβγ ∧ηβγ − Λκ

ηα

+12

eα ⌋T β ∧H β − T β ∧ (eα ⌋H β )

+12

eα ⌋Rβγ ∧H βγ − Rβγ ∧ (eα ⌋H βγ ) .

(123)

(Notice the resemblance between ( 123) and ( 113).) Forcompleteness, let us also calculate the formulas analogous


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to (106) and ( 107) respectively. From ( 73) we get

E α = ϑ⊥ α ǫgr + Dϑ β⊥ ∧H β

+( eα ⌋T β ) ∧H β + ( eα ⌋Rβγ ) ∧H βγ , (124)

while (90) with ( 73) takes the form

ǫ

gr= − Ln ϑ

α

∧H α − Rαβ⊥ ∧H αβ − L

gr⊥ . (125)

On the other hand, the gravitational Lagrangian reducesto

Lgr = 14

ϑα ∧E α − 12

T α ∧H α − 12

Rαβ ∧H αβ , (126)

with ϑα ∧E α = 1κ a0 Rαβ∧ηαβ − 4 Λ η .

For readers which are not familiar with exterior cal-culus notation [ 7], it may be useful to show how ordi-nary general relativistic Einstein equations are comprisedas a particular case of the gauge-theoretical equations(56) and ( 57) with ( 121), (122) and ( 123). The Hilbert-

Einstein theory in vacuum with cosmological constant de-rives from the pure gravitational Lagrangian ( 120) withthe constants xed as a0 = 1 , aI = 0 , bI = 0 . Accord-ingly, (121) vanishes, ( 122) reduces to H αβ = − 12κ ηαβ ,and ( 123), coinciding with the whole energy-momentum(60) due to the absence of matter and radiation, becomes

E α = 1κ

12

Rβγ ∧ηβγα − Λ ηα . (127)

The eld equations ( 57) then read

0 = DH αβ = − 1

2κ Dηαβ = −

1

2κ ηαβγ ∧T γ , (128)

implying vanishing torsion, so that equations ( 56) reduceto

0 = Π α = E α = 1κ

12

Rβγ ∧ηβγα − Λ ηα

= −1κ

ei α R ij − 12

g ij R + Λ g ij ηj , (129)

constituting a well known reformulation of the ordinaryEinstein equations in vacuum, which for clarity we alsogive in their standard form. For more details see forinstance [ 12].

VIII. HAMILTONIAN APPROACH TODYNAMICS

In Section IV we discussed covariant eld equationsas conditions derived from two complementary ways of imposing vertical invariance of the action, namely theprinciple of extremal action and the symmetry principle.In our exterior calculus notation, the coordinate inde-pendent eld equations ( 55)–(57) do not display any ex-plicit reference to the base space. But in Section VI we

introduced a base space foliation becoming actually re-ected in the notation (even in the language of differentialforms) by distinguishing from each other the projectionsrespectively longitudinal and transversal with respect toa certain parametric time direction (dened in the basespace).

Associated with such foliation, we presented paramet-

ric time evolution as a form of horizontal displacementon the base space, compatible with the eld equationsguaranteeing vertical invariance. So to say, vertical in-variance guides horizontal motions. Bundle connections(that is, gauge potentials) are known to dene horizon-tality in ber bundles. Thus, provided the eld equa-tions hold, connections become responsible for maintain-ing several vertical features along horizontal paths in thebase space. Vertical invariance conditions act as forces orinteractions inuencing the quantities subjected to hor-izontal evolution displacements. Here we briey outlinea Hamiltonian formalism suitable to deal with evolutionunderstood in this manner.

A. The Hamiltonian evolution equations

In the present approach, a central role is played by thevanishing Hamiltonian-like 3-form ( 80), whose transver-sal part reads

H = Q⊥ ∧ ∂L⊥∂Q ⊥

+ ( d Q)⊥ ∧ ∂L⊥∂ l n Q

− L⊥ − Q⊥ ∧ δL⊥δQ⊥

,

(130)being covariant as a consequence of the symmetry condi-tions ( 42)–(44) and ( 46); compare with the non-foliatedexpression ( 82). The relevance of the quantity ( 130) de-rives from the fact that it results to occur in the vari-ational formula ( 9) when foliated as ( D12), so that bytaking into account ( D2) as much as the foliated eldequations ( D17) and (D18), (D12) yields

0 = Ndτ ∧ δ H + δQ ∧ l n ∂L⊥∂ l n Q

− l n Q ∧δ ∂L⊥∂ l n Q

+ δQ⊥ ∧ δL⊥δQ⊥ + δQ ∧

δL⊥δQ

− d Ndτ ∧ Q⊥ ∧δ ∂L⊥∂ l n Q

+ δQ ∧ ∂L⊥∂dQ

.

(131)

Now we introduce the momentum notation

# π Q := ∂L⊥∂ l n Q

, (132)


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The details of the dynamical approach based on ( 141) willbe developed elsewhere, constituting a modied versionof the Hamiltonian formalism already published in Refs.[12] [20], with the difference that the present formalismis adapted to a different explicit covariance.

The last terms in ( 141) constitute the explicit expan-sion of the term Q⊥ ∧ δL⊥ /δQ ⊥ in (130), proportionalto the transversal parts of the eld equations. The latterones vanish separately as much as the remaining Hamil-tonian 3-form (130) does. The main reason for keepingthem in ( 141) is that they play the role of rst classconstraints, and thus of generators of the symmetries in-volved in the theory [ 28]. Since the Q’s given in (21) areeither 0-forms or 1-forms, the transversal parts Q⊥ onlyexist as 0-forms. In particular, they are the longitudinal

parts of the gauge potentials, that is A⊥ ,(T )Γα⊥ and Γ αβ⊥ ,

playing the role of Lagrange multipliers. The constraintspresent in ( 141) as the generators of Poincaré ⊗ U (1)read

Ĉ := d # πA − ie ψ # πψ

+ # πψ

ψ , (142)

P̂ α := D # π( T )Γ

α + # πξα , (143)

L̂αβ := D # πΓαβ +

(T )Γ [α ∧

# π( T )Γ

β ] + ξ [α# πξβ ]

+ i ψ σαβ # πψ + # πψ σαβ ψ , (144)

being identical with the transversal part of the eld equa-tions in their form ( 39)–(41). The covariantized form of (144) is obtained by combining ( 143) and (144) into

L̂αβ − ξ [α P̂ β ] = D # πΓαβ − ξ [α

# π( T )Γ

β ] + ϑ[α ∧# π

( T )Γ

β ]

+ i ψ σαβ # πψ + # πψ σαβ ψ .

(145)

(Compare with the transversal part of (57).) By buildinga symmetry generator with the form of the last terms in(141) with the Q⊥ ’s replaced by the usual group param-eters, that is

Ĝ := − λe

Ĉ + ǫµ P̂ µ + β µν L̂µν , (146)

variations of any dynamical variable can be obtained withthe help of Poisson brackets ( 138) as

δα = { α , Ĝ} . (147)

In particular mainly due to the fact that –up to Diracdeltas–

{ ξ α , P̂ β } = { ξ α , # πξβ } = δ αβ , (148)

we are able to reproduce ( 30) as

δξ α = { ξ α , Ĝ} = − ξ β β β α − ǫα , (149)

and analogously we can calculate the remaining varia-tions ( 31)–(35).

IX. FINAL REMARKS

At the end of Section VIII A, we mentioned an ex-ample of loss of explicit symmetry –without symmetrybreaking– associated with the choice of ξ 0 as clock time .At this point, let us mention further cases of explicit sym-metry loss which also result to be useful. For instance,one can nd certain similitudes between the gauge equa-tions introduced above and related equations of ClassicalMechanics. We begin by reformulating ( 66) as a forcelaw

D Σ mattα = f α , (150)

(obtained by applying the symmetry principle separatelyto the matter Lagrangian) with f α being understood asan external force 4-form generalizing the Lorentz force.Using (150) and ( 48), it is also possible to rewrite ( 64)as an equation for generalized angular momentum

D τ αβ + ξ [ α ∧Σ mattβ ] +(T )Γ[ α ∧Σ mattβ ] = ξ [ α f β ] , (151)

where the term in the r.h.s. behaves as a generalizedtorque.

Renouncing to explicit covariance also helps in nd-ing strictly conserved currents from the covariant quasi-conservation equations ( 55)–(57). Actually, true conser-vation as expressed by the continuity equations ( 63) and(85) involve ordinary differentials rather than covariantones, so that exact conservation of tensor quantities can-not be formulated covariantly. Thus let us reformulatethe covariant equations ( 55)–(57) in terms of suitable cur-rents as follows. With the help of denitions ( 49)–(52),we leave (42) as it is but from ( 43) and ( 44) we denerespectively the noncovariant linear momentum current

J α := ∂L∂ Γα( T )

= Π α + Γ α β ∧H β , (152)

and the noncovariant angular momentum current

J αβ := ∂L∂ Γαβ

= τ αβ + ξ [ α Πβ ] −(T )Γ[ α ∧H β ]

+Γ α γ ∧ H γβ + ξ [γ H β ]− Γβ γ ∧ H γα + ξ [γ H α ] , (153)

so that the covariant eld eqs. ( 55)–(57) become express-

ible asdH = J , (154)

dH α = J α , (155)d H αβ + ξ [α H β ] = J αβ + ξ [ α dH β ] − J β ] .(156)

Obviously, from ( 154)–(156) follow the true conservationequations

dJ = 0 , (157)dJ α = 0 , (158)

dJ αβ = 0 . (159)


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On the other hand, let us end this section mentioningthe possible relevance of translations for interpreting theposition-momentum commutation relations of QuantumMechanics. Indeed, the analogy between ( 148) and thecommutation relations

[ Ξα , P β ] = i δ αβ , (160)

might allow to regard ( 160) as the reformulation of atranslational property concerning ξ α and P̂ β –or maybe# π ξβ – into the language of operators, with Ξ

α as the op-erator version of our position vector ξ α . Notice in factthat, by introducing G trans := i ǫµ P µ as the generator of translations, similar to the corresponding piece in ( 146),we get

δ Ξα = [ Ξ α , G trans ] = − ǫα , (161)

as a translational-like variation, while a Poincaré gen-erator, say GPoinc := i ( ǫµ P µ + β µν Lµν ), with Lµν :=Ξ[ µ P ν ], yields

δ Ξα = [ Ξ α , GPoinc ] = − β β α Ξβ − ǫα , (162)

analogous to the eld variation ( 149).

X. CONCLUSIONS

Translations are an usually forgotten symmetry in thecontext of gauge-theoretical dynamics of fundamental in-teractions. We have shown that, although hidden, theyare present in a variety of physical contexts. So, in New-

tonian Mechanics, global space translations are respon-sible for linear momentum conservation. Due to the factthat the same rigid displacement ǫa in (3) makes sensesimultaneously at distant positions, a momentum inter-change is predicted to occur between far separated bod-ies, thus providing a basis for action at a distance. Ingauge theories instead, group parameters depend on basespace coordinates. Local spacetime translations ǫα (x)are different at different points, so that only local inter-changes –say ”collisions”– are admissible. Consequently,action at a distance abandons the scene in favor of aninterchange of linear momentum affecting elds locally:interactions mediated by gauge potentials replace remote

inuence in Newton’s manner.Fiber bundles are known to be the geometrical struc-tures underlying Yang-Mills theories of internal localgroups [2] [18] [30] [31] [32] [33]. A slight modicationof them also constitutes the implicit geometrical back-ground of the present paper. Bundles merely have to bemade enough exible to accommodate local translationsconveniently. Indeed, by embracing the translationalgroup as a gauge symmetry, we accept it to be fully dis-tinguished from horizontal (base space) diffeomorphisms,since gauge transformations are vertical. Notwithstand-ing, there is possible for translations to actively move

from a spacetime position to another provided the af-fected points aren’t presupposed to be identical with basespace ones. (See Appendices B and C.) In Ref. [ 14] weproposed a certain composite bundle as the geometricalframework suitable to deal with the local realization of translations. The bers of composite bundles are to bevisualized as broken lines, with the translational sector(attached itself to the base space) acting as an intermedi-ate base space where other ber sectors orthogonal to itare attached to. Translations become unied with any in-ternal symmetry, so that all interactions including gravi-tation can be treated in a homogeneous gauge-theoreticalway within a unique structure. As shown in the presentpaper, geometry and light –gravity and radiation– appearas different aspects of the same unied bundle approachto spacetime and internal forces, all of them obeying sim-ilar eld equations; see ( 55)–(57).

The bundle is equipped with an action required to bevertically invariant. Horizontal displacements are subjected to interactions in the sense that they must respectthe vertical invariance conditions, that is the (covariant)eld equations. However, in our proposal the base spaceis not dynamical , but it plays the role of a sort of in-ert screen. Not a base space metric, but tetrads denedon the bres, and in general quantities built from theber variables ( 21), are the physical objects affected bydynamical laws. This makes a major difference with re-spect to ordinary General Relativity, where spacetime ismodelized by a manifold equipped with a dynamical met-ric, being such dynamical spacetime expected to act asthe base space of Yang-Mills theories of internal groupswhen gravity is present. Certainly, in our case as wellas in General Relativity, dynamics manifests itself on thebase space, where evolution occurs as a horizontal conse-quence of vertical invariance; also a Riemannian metric(dynamically determined), and thus a full Riemanniangeometry, can be dened on our base space. But, re-markably, only as the result of the pullback of the verti-cal structures considered in the present paper. See [ 14]for more details.

The coordinate-like translational Goldstone elds ξ αtaken from the nonlinear Poincaré Gauge Theory [12][14] were shown to play a central role due to the nonmin-

imal universal coupling of these translational variables–and thus of gravity– to any other quantity. This factmainly manifests itself in the contribution of all dynam-ical elds to the energy-momentum Π α and accordinglyto the energy current 3-form ( 84). (See also AppendixC for a discussion on the relevance of the elds ξ α forthe description of motion.) Finally, we recall that we ex-plained why translations, despite their fundamental con-tributions as made manifest in the present paper, remaina hidden and commonly ignored symmetry, as a conse-quence of the translation-invariant structure ( 48) of thetetrads.


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Acknowledgments

The author wants to thank Alfredo Tiemblo for pastcollaboration.

APPENDIX A: DEFINITIONS OF DERIVEDDYNAMICAL QUANTITIES

In the main text we made use of the following deni-tions. On the one hand the combination

ϑα := D ξ α +

(T )Γα = d ξ α + Γ β α ξ β +

(T )Γα (A1)

provides us with a modied translational gauge poten-tial which turns out to be invariant under translations,transforming as a Lorentz covector, see ( A7) below. In[10] [11] [12] [14] we discussed (A1) as the components of translational nonlinear connections, which we identiedas tetrads. So we do here. Notice that in the absence of connections and thus of gravity, that is, in the Minkowskispace, ( A1) reduces to the trivial tetrad d ξ α . Torsion isdened [4] [5] [6] [7] as the covariant differential of tetrads(A1), namely

T α := D ϑα = d ϑα + Γ β α ∧ϑβ , (A2)

while the denition of the Lorentzian curvature reads

Rα β := d Γα β + Γ γ β ∧Γα γ , (A3)

being antisymmetric in the indices α , β . Such anholo-nomic Lorentz indices are rised and lowered with the helpof the anholonomic constant Minkowski metric oαβ =diag (− + ++) which is assumed to exist as the naturalinvariant of the local Poincaré group ( δoαβ = 0 ).

Besides these quantities, we also dene the ordinaryelectromagnetic eld strength

F := dA , (A4)

and the covariant derivatives of matter elds

Dψ := dψ + i eA − Γαβ σαβ ψ , (A5)

Dψ := dψ − i ψ eA − Γαβ σαβ . (A6)

In analogy to the electromagnetic eld strength ( A4), tor-sion (A2) and curvature ( A3) are to be regarded as theeld strengths of translations and of the Lorentz grouprespectively. The variations of all these ob jects are sum-marized as

δϑα = − ϑ β β β α , (A7)δDψ = iλ + iβ αβ σαβ Dψ , (A8)

δDψ = − Dψ iλ + iβ αβ σαβ , (A9)δF = 0 , (A10)

δT α = − T β β β α , (A11)δRα β = β α γ Rγ β − β γ β Rα γ , (A12)

calculated from ( 30)–(35) applied to denitions ( A1)–(A6).

APPENDIX B: GEOMETRICAL MEANING OFSOME DYNAMICAL VARIABLES

Tetrads (48) (the pullback of ( 48) to the base space,in fact) can be chosen as a 1-form basis of the cotan-gent space. The corresponding affine space dual basis istaken to be the local reference frame ( ox , eα ) , attachedto each point x of the base space [34] [35], consisting of an origin ox , together with a vector basis eα dened bythe condition eα ⌋ϑβ = δ βα . Locally, the coordinate-likeelds ξ α allow to dene the position relative to a frame( ox , eα ) as

px := ox + ξ α eα , (B1)

so that the ξ α ’s, although gauge-theoretical in origin, re-veal to be interpretable as the components of a relativespacetime position vector. By taking

δox = ǫα eα , δeα = β α β eβ , (B2)

together with ( 30), the position ( B1) results to be gaugeinvariant, that is

δpx = 0 . (B3)

On the other hand, by introducing the translational andLorentz connections, related respectively to the originand to the vector basis as

∇ox =(T )Γα ⊗ eα , ∇eα = Γ α β ⊗ eβ , (B4)

we nd

∇ px = ϑα

⊗ eα , (B5)with ϑα given by (48), showing the tetrad to origi-nate from position transport. The line element ds2 :=oαβ ϑα ⊗ ϑβ , built with the Minkowski metric and thetetrads (and equal to the standard Riemannian line ele-ment ds2 = gij dx i dx j ), can be regarded as a sort of ( B5)squared. By acting again on ( B5) one generates torsion

∇∇ px = T α ⊗ eα , (B6)

while a double action on the basis vectors produces cur-vature

∇∇ eα = Rα β ⊗ eβ . (B7)

Formulas ( B1)–(B7) provide a simple geometrical mean-

ing to the dynamical objects ξ α ,(T )Γα , and Γ αβ in (21),

as much as to ( A1)–(A3).

APPENDIX C: ON MOTION

Let us notice that the structure ( 48) of tetrads, withthe help of denition ( B1) and ( B5), allows to outline


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a mathematical description of motion in terms of thecoordinate-like elds ξ α . As a useful notational tool, weintroduce besides ordinary Lie derivatives ( 17) the co-variant Lie derivatives [ 7], which for the particular caseof the time vector n are dened as

Ln α A := n⌋Dα A + D (n⌋α A ) . (C1)

For instance

Ln ϑα := n⌋Dϑ α + D (n⌋ϑα ) = T α⊥ + Dϑα⊥ . (C2)

Vertical gauge variations along bers don’t affect the po-sition points (sections) px due to their invariance ( B3).However, when a horizontal displacement occurs betweenneighboring bers from position px to px + dx along aworldline (that is, along a path parametrized by τ havingn as its tangent vector), then according to ( B5)

∇ n px = ∇ n ( ox + ξ α eα ) = ϑα⊥ ⊗ eα , (C3)

with the quantity

ϑ

α⊥ := Ln ξ

α+

(T )

Γα⊥ , (C4)

acting as the covariant four-velocity. As read out from(C4), horizontal displacements cause the measurable rel-ative position vector ξ α in (B1) to evolve with respect toparametric time , while a contribution due to the changeof origin ensures covariance. From ( C4) we get the co-variant acceleration

Ln ϑα⊥ = l n ϑα⊥ + Γ ⊥ β α ϑβ⊥

= Ln Ln ξ α + Ln(T )Γα⊥ , (C5)

including a sort of force contribution associated to the

origin. (In principle, it should be possible to reexpressparametric time evolution as evolution with respect toclock time, say to ξ 0 .)

Einstein’s general relativistic geodesic equations forclassical test particles establish the vanishing of ( C5).To get such a simple equation from ( 66), we have toconsider phenomenological matter, for instance that de-scribed by a dust model with matter currents J = 0 ,τ αβ = 0 and Σ mattα = ρϑ⊥ α ϑ

β⊥ ηβ , being ρ a ow den-

sity 0-form. Then, taking into account, as derived from(F3) and ( F4) respectively, that η⊥ α = − # ϑα and ηα =− # ϑ⊥ α , and on the other hand η⊥ αβ = # ( ϑα ∧ ϑβ )and η

α = − # ( ϑ⊥ α ϑβ − ϑ⊥ β ϑα ) , eq. (66) yields

Ln ϑ⊥ α # ρ = − ρ # ϑα ∧ϑ⊥ β T β⊥ , (C6)

which for vanishing torsion and l n # ρ = 0 reproduces thedesired result Ln ϑα⊥ = 0 . Fundamental matter gives riseto more complicated equations involving Ln ϑα⊥ by usingeither ( 66) with ( 100), or (69) with ( 105), or (56) withthe transversal part of the total energy-momentum

Πα = n⌋ ϑα ∧ ǫ+ Dϑβ⊥ ∧H β

− 2ϑβ⊥ D τ αβ + a02κ

Dηαβ , (C7)

(identical with # π ξα in (139) ), resulting from putting to-gether ( 100), (105) and ( 124), the latter one evaluatedfor (120), with (60) and (87). Let us conclude claim-ing that there are the dynamical relative positions ξ αinvolved in the eld equations, rather than the underly-ing quite metaphysical base space points (or their coor-dinates), that describe observable spacetime.

APPENDIX D: CONSEQUENCES OF THE BASESPACE FOLIATION

The foliation of the base space considered by us restson the introduction of a time-like vector eld n = n i ∂ i ,tangent to a congruence of worldlines, whose direction isxed with respect to the 1-form Ndτ by requiring both tosatisfy the condition n⌋(Ndτ ) = 1 . In terms of the lapseN and the shift N a functions, it is possible to rewritethe parametric time vector eld as n = 1N (∂ τ − N

a ∂ a ) ,with ∂ a as space derivatives. In the present appendix weextend the decomposition (75) of any p-form into longi-tudinal and transversal parts ( 76) and (77) respectively,to both, exterior derivatives of forms and Hodge dualforms (F7), and then we present a foliated version of thevariations presented in Section III A.

In analogy to ( 75), exterior derivatives decompose as

d α = N dτ ∧(d α )⊥ + d α , (D1)

with

(d α)⊥ = l n α − d α⊥ = l n α − 1N

d Nα ⊥ . (D2)

On the other hand, the Hodge dual of an arbitrary p-formα, as dened by ( F7), decomposes as

∗α = ( − 1) p Ndτ ∧ # α − # α⊥ , (D3)

being # the Hodge dual operator in the 3-dimensionalspatial sheets. Taking ( D3) into account, we derive thefollowing results, which are useful to reproduce the calcu-lations of the main text. In the four–dimensional space-time with Lorentzian signature, the double applicationof the Hodge dual operator reproduces α itself up to thesign as ∗∗α = − (− 1) p α. From this relation we deduce

## α⊥ = α⊥ , (D4)

## α = α . (D5)

On the other hand, from ϑα ∧ eα ⌋α = p α we nd

ϑα∧ eα ⌋α⊥ = ( p − 1 ) α⊥ , (D6)ϑ

α∧ eα ⌋α = p α , (D7)

and from the further relation ∗(α ∧ϑα ) = eα ⌋ ∗α involv-ing Hodge duality we get

# (α⊥ ∧ϑα ) = eα ⌋ # α⊥ , (D8)# (α ∧ϑα ) = eα ⌋

# α . (D9)


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where (F5) is the four–dimensional volume element. Be-ing tetrads ϑα chosen as a basis of the cotangent space,an arbitrary p-form α reads

α = 1 p !

ϑα 1 ∧ ... ∧ϑα p (eα p ⌋...e α 1 ⌋α ) , (F6)

while its Hodge dual becomes expressed in terms of the

eta basis ( F1)–(F5) as∗α =

1 p !

ηα 1 ...α p (eα p ⌋...eα 1 ⌋α ) . (F7)

Notice that comparison of the variations of ( F6) and ( F7)to each other yields the relation

δ ∗α = ∗δα − ∗ (δϑα ∧ eα ⌋α ) + δϑα ∧ (eα ⌋∗α ) , (F8)

which has a decisive relevance in showing that the varia-tion of forms affected by the Hodge star operator involve

variation of the tetrads ( 48) and thus of ξ α and(T )Γα . It

is through the coupling of the translational variables toany other quantity that the universal inuence of grav-ity on other elds takes place. This fact reveals itself indynamics in the existence of contributions to the total

energy-momentum ( 60) arising from any matter or forceparts of the Lagrangian. The foliated version of ( F2)–(F5), as much as of ( F8), are relevant for the completeHamiltonian approach to be published elsewhere.

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