Conservation Laws on Riemann-Cartan, Lorentzian and Teleparallel Spacetimes

0.44 MB·English

by  Waldyr A. Rodrigues Jr

Conservation Laws on Riemann-Cartan,
Lorentzian and Teleparallel Spacetimes
∗
8
0
0
W. A. Rodrigues Jr. and Q. A.G. deSouza
2
Institute ofMathematics,Statistics andScientificComputation
n
a IMECC-UNICAMPCP6065
J 13083-859Campinas,SP,Brazil

2 and
1
R.da Rocha
3
1 InstitutodeF´ısicaTeo´rica,UNESP,RuaPamplona145
v 01405-900, Sa˜oPaulo,SP,Brazil.
8
and
0
0 DRCC-InstituteofPhysicsGlebWataghin, UNICAMPCP6165
5
13083-970Campinas,SP,Brazil
0
e-mails: [email protected]; [email protected]; [email protected]
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0
February 4, 2008
/
h
p
-
h
t Abstract
a
m
Using a Clifford bundleformalism, we examine: (a) the strong condi-
: tionsforexistenceofconservationlawsinvolvingonlytheenergy-momentum
v
andangularmomentumofthematterfieldsonageneralRiemann-Cartan
i
X spacetime and the particular cases of Lorentzian and teleparallel space-
r times and (b) the conditions for the existence of conservation laws of
a energy-momentum and angular momentum for the matter and gravita-
tionalfieldswhenthislaterconceptcanberigorouslydefined. Weexamine
inmoredetailssomestatementsconcerningtheissuesoftheconservation
laws inGeneralRelativityandRiemann-Cartan (includingtheparticular
case of the teleparallel ones) theories.
Contents
1 Introduction 2
∗ThecontentsofthispaperappearedintwopartsinBull. Soc. Sci. Lodz. Series 57,Ser.
Res. on Deformations, 52,37-65,66-77(2007). Inthepresentversionsomemisprintsofthe
publishedversionhavebeencorrected.
1
2 Some Preliminaries 5
2.1 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Vertical Variation . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Horizontal Variation . . . . . . . . . . . . . . . . . . . . . 5
2.2 Functional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Euler-LagrangeEquations from LagrangianDensities . . . . . . . 7
2.4 Invariance of the Action Integral under the Action of a Diffeo-
morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Covariant ‘Conservation’ Laws 9
4 When Genuine Conservation Laws Do Exist? 13
5 Pseudo Potentials in General Relativity 16
5.1 Is There Any Energy-Momentum Conservation Law in GRT? . . 18
6 Is there any Angular MomentumConservationlaw inthe GRT 22
7 Conservation Laws in the Teleparallel Equivalent of General
Relativity 23
8 Conclusions 25
A Clifford and Spin-Clifford Bundles 26
A.1 Clifford Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.1.1 Hodge Star Operator . . . . . . . . . . . . . . . . . . . . . 29
A.1.2 Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . 29
A.2 Dirac Operator Associated to a Levi-Civita Connection . . . . . 30
B Maxwell Theory in the Clifford Bundle 30
B.1 Energy-Momentum Densities ⋆ for the Electromagnetic Field . 30
a
T
C ExamplesofKillingVectorFieldsThatDoNotSatisfyEq.(102) 32
C.1 Teleparallel Schwarzschildspacetime . . . . . . . . . . . . . . . . 32
C.2 Teleparallel de Sitter spacetime . . . . . . . . . . . . . . . . . . . 33
C.3 Teleparallel Friedmann Spacetime. . . . . . . . . . . . . . . . . . 35
1 Introduction
Using the Clifford bundle formalism of differential forms (see Appendix A1) we
reexamine the origin and meaning of conservation laws of energy-momentum
and angular momentum and the conditions for their existence on a general
Riemann-Cartan spacetime (RCST)2 (M,g, ,τ , ) and also in the particular
g
∇ ↑
1In Appendix A we give a very short introduction to the main tools of the the Clifford
bundleformalismneededforthispaper. AdetailedanduptodatepresentationtotheClifford
bundleformalismisgiven,e.g.,in[55].
2SeedetailsinAppendixA.
2
cases of Lorentzian spacetimes M = (M,g,D,τ , ) which as it is well known
g
↑
modelgravitationalfieldsintheGeneralRelativityTheory(GRT)[57]. ARCST
is supposedto model a generalized gravitationalfield in the so calledRiemann-
Cartan theories [25]. The case of the so called teleparallel3 equivalent of GRT
[30]is alsoinvestigatedandtherecentclaim[12]thatthereis agenuineenergy-
momentum conservation law in that theory is investigated in more details.
In what follows, we suppose that a set of dynamic fields live and interact
in (M,g, ,τ , ) (or M). Of course, we want that the RCST admits spinor
g
∇ ↑
fields,whichimplies accordingtoGeroch’stheoremthattheorthonormalframe
bundle must be trivial [21, 38, 55]. This permits a great simplification in our
calculations, in particular if use is made of the calculation procedures of the
Clifford bundle formalism. Moreover, we will suppose, for simplicity that the
dynamic fields of the theory φA, A = 1,2,...,n, are r-forms4, i.e., each φA
sec rT M ֒ sec ℓ(M,g), for some r =0,1,...,4. ∈
∗
→ C
VWe recall the very important fact that there are in such theories a set of
‘covariantconservationlaws’whichareidentitieswhichresultfromthefactthat
Lagrangian densities of relativistic field theories are supposed to be invariant
under diffeomorphisms and active local Lorentz rotations5. These covariant
conservation laws do not express in general any genuine conservation law of
energy-momentumorangularmomentum. Weprovemoreover,asfirstshownby
[6]thatgenuineconservationlawsofenergy-momentumandangularmomentum
for only the matter fields exist for a field theory in a RCST only if there exists
a set of6 m appropriate vector fields ξ(a), a = 1,2,...,m such that £ g = 0
ξ(a)
and £ Θ=0, where Θ is the torsion tensor.
ξ(a)
Thus, we show in Section 6 that in the teleparallel version of GRT, the ex-
istence of Killing vector fields does not warrant (contrary to the case of GRT)
the existenceofconservationlawsinvolvingonly the energy-momentumtensors
of the matter fields. We show moreover, still in Section 6, that in the telepar-
allel version of GRT (with null or non null cosmological constant) there is a
genuine conservationlaw involvingthe energy-momentumtensor of matter and
the energy-momentum tensor of the gravitational field, which in that theory is
a well defined object.
Althoughthisisawellknownresult,wethinkthatourformalismputsitina
newperspective. Indeed, inourapproach,theteleparallelequivalentofGeneral
Relativity as formulated, e.g., by [30] or [12], is easily seem as consisting in
the introduction of: (a) a bilinear form (a deformed metric tensor [52, 55])
3A teleparallel spacetime is a particular Riemann-Cartan spacetime with null curvature
andnonnulltorsiontensor[1,2,3].
4Thisisnotaseriousrestrictionintheformalismsinceasitisshownindetailsin[38,55],
one can represent spinor fields by sums of even multiform fields once a spinorial frame is
given. Thefunctional derivativeofnon-homogeneous multiformfields isdeveloped indetails
in,e.g.,[55].
5Satisfyingsuchaconditionimpliesingeneralintheuseofgeneralizedgaugeconnections,
implyingasortofequivalencebetweenspacetimesequippedwithconnectionshavingdifferent
curvatureand/or torsiontensors[51,14].
6The maximum possible value of the integer number m in a 4-dimensional spacetime is
ten.
3
g = η θa θb, (b) a teleparallel connection (necessary to make the theory
ab
invariantun⊗der active local Lorentz transformations7) in the manifold M R4
≃
of Minkowski spacetime structure, and (c) a Lagrangian density differing from
the Einstein-Hilbert Lagrangiandensity by an exact differential.
The paper is organizedas follows:
Section2 andAppendix Aareaimedto giveto the readersomebackground
information needed to better understand our developments. In Section 2 we
recall some mathematical preliminaries as the definition of vertical and hori-
zontal variations, the concept of functional derivatives of functionals on a 1-jet
bundle, the Euler-Lagrange equations (ELE) and the fact that the action of
any theory formulated in terms of differential forms is invariant under diffeo-
morphisms, whereas in Appendix A we briefly describe the Clifford bundle for-
malism used throughout the paper. Appendix A also provides a derivation of
the energy-momentum 3-forms for the electromagnetic field which in the Clif-
ford bundle formalism (and our conventions) are expressed very elegantly by
⋆Ta =⋆ a = 1 ⋆(FθaF˜).
− T −2
In Section 3 we recall the proof of a set of identities called ‘covariant con-
servation laws’ valid in a RCST [6], which as already mentioned above do not
encode, in general, any genuine energy-momentum and/or angular momentum
conservation laws.
InSection4weassumethattheLagrangiandensityisinvariantunderactive
local Lorentz transformations and diffeomorphisms and then recall the condi-
tions for the existence of genuine conservation laws in a RCST which involve
only the energy-momentum and angular momentum tensor of the matter fields
[6].
Next,inSection5,werecall(forcompleteness)withourformalismthetheory
of pseudo-potentials and pseudo energy-momentum tensors in GRT, and show
thatthereareingeneralnoconservationlawsofenergy-momentumandangular
momentuminthistheory[54]. Wealsodiscusssomemisleadingandevenwrong
statements concerning this issue that appear in the literature.
Finally, in Section 6 we discuss the conservation laws in the teleparallel
equivalent of General Relativity, as already mentioned above.
Our conclusions canbe found in Section 7. To better illustrate the meaning
of our results, we also present, in Appendix B, various examples showing that
notallKilling vector fields ofa teleparallelspacetime (Schwarzschild,de Sitter,
Friedmann) satisfy Eq.(39) meaning that in a model of the teleparallel ‘equiv-
alent’ of GRT there are, in general, fewer conservation laws involving only the
matter fields than in the corresponding model of GRT.
7OntheissueonactivelocalLorentzinvariance,seealso[14,51].
4
2 Some Preliminaries
2.1 Variations
2.1.1 Vertical Variation
Let X sec ℓ(M,g), be a Clifford (multiform) field8. An active local Lorentz
∈ C
transformation sends X X sec ℓ(M,g), with
′
7→ ∈ C
X =UXU˜. (1)
′
Each U secSpine (M) can be written (see, e.g., [55]) as the expo-
∈ 1,3 ±
nential of a 2-form field F sec 2T M ֒ sec ℓ(M,g). For infinitesimal
∗
∈ → C
transformations we must choose thVe + sign and write F =αf, α 1, F2 =0.
≪ 6
Definition 1 Let X be a Clifford field. The vertical variation of X is the field
δ X (of the same nature of X) such that
v
δvX =X′ X. (2)
−
Remark 2 The case where F is independent of x M is said to be a gauge
∈
transformation of the first kind, and the general case is said to be a gauge
transformation of the second kind.
2.1.2 Horizontal Variation
Let σ be a one-parameter group of diffeomorphisms of M and let ξ secTM
t
∈
be the vector field that generates σ , i.e.,
t
dσµ(x)
ξµ(x)= t . (3)
dt (cid:12)
(cid:12)t=0
(cid:12)
Definition 3 Wecallthehorizontal variation(cid:12) ofX inducedbyaone-parameter
group of diffeomorphisms of M to be the quantity
σ X X
δ X = lim t∗ − = £ X. (4)
h ξ
t 0 t −
→
Definition 4 We call total variation of a multiform field X to the quantity
δX =δ X+δ X =δ X £ X. (5)
v h v ξ
−
It is crucial to distinguish between the two variations defined above.
8If X = ψ ∈ secCℓ(0)(M,g) (where Cℓ(0)(M,g) is the even subbundle of Cℓ(M,g)) is a
representative of a Dirac-Hestenes spinor field in a given spin frame, then an active local
transformationsendsψ7→ψ′,withψ′=Lψ [55].
5
2.2 Functional Derivatives
Let J1( T M) be the 1-jet bundle over T M ֒ ℓ(M,g), i.e., the vector
∗ ∗
→ C
bundle dVefined by V
J1( T∗M)= (x,φ(x),dφ(x)); x M, φ sec T∗M ֒ sec ℓ(M,g) .
{ ∈ ∈ → C }
^ ^ (6)
Then, with each local section φ sec T M ֒ sec ℓ(M,g) , we may
∗
∈ → C }
associate a local section j1(φ) secJ1( T∗VM).
∈
Let θa , θa sec 1T∗M ֒ sec ℓV(M,g), a=0,1,2,3,be an orthonormal
{ } ∈ → C
basis of T∗M dual toVthe basis {ea} of TM and let ωba ∈ sec 1T∗M ֒→
sec ℓ(M,g) be the connection 1-forms of the connection in a gViven gauge.
C ∇
Weintroducealsothe1-jetbundleJ1[( T M)n+2]overtheconfigurationspace
∗
( T∗M)n+2 ֒ ( ℓ(M,g))n+2 of a fiVeld theory describing n different fields
→ C
φVA sec TpM ֒ sec ℓ(M,g) on a RCST, where for each different value of
∈ → C
A we havVe in general a different value of p.
J1[( T∗M)n+2]:=J1( T∗M T∗M ... T∗M)
× × ×
^ ^ ^ ^
= (x,θa(x),dθa(x),ωa(x),dωa(x),φA(x),dφA(x), A=1,...,n
{ b b }
(7)
Sections of J1[( T M)n+2] will be denoted by j (θa,ωa,φ) or simply by j (φ)
∗ 1 b 1
when no confusVion arises.
A functional for a field φ sec T M ֒ sec ℓ(M,g) in J1( T M) is a
∗ ∗
∈ → C
mapping :secJ1( T∗M) sec VT∗M, j1(φ) (j1(φ)). V
F → 7→F
A Lagrangian deVnsity mappingVfor a field theory described by fields φA
∈
sec T M, A=1,2,...,n over a Riemann-Cartan spacetime is a mapping
∗
V
4
:secJ1[( T M)n+2] sec T M, (8)
m ∗ ∗
L →
^ ^
j (θa,ωa,φ) (j (θa,ωa,φ)). (9)
1 b 7→Lm 1 b
Remark 5 When convenient the image of , i.e., (j (θa,ωa,φ)) (called
Lm Lm 1 b
Lagrangian density) will be represented by the sloppy notation (x,θa,ωa,φ)
Lm b
or, when the Lagrangian density does not depend explicitly on x, (θa,ωa,φ)
Lm b
or simply (φ) and even just . The same observation holds for any other
m m
L L
functional.
To simplify the notation even further consider in the next few definitions of
afieldtheorywithonlyonefieldφ sec rT M ֒ sec ℓ(M,g),inwhichcase
∗
∈ → C
m is a functional on J1[( T∗M)3]. V
L
terGfieilvdenφaLsaegcranrgTianMde֒nVsisteycLℓm(M(j1,(gθ)a,oωvebar,φa)g)efnoerraalgRivieenmhaonmn-oCgaernteaonusspmaacet--
∗
∈ → C
time, we shallneVed (in orderto apply the variationalactionprinciple)to calcu-
late some algebraic derivatives of Lm. These are terms such as ∂L∂mφ(φ),∂L∂mdφ(φ)
6
which appears in the variation of , i.e.,
m
L
∂ (φ) ∂ (φ)
δ (φ)=δφ Lm +δ(dφ) Lm
m
L ∧ ∂φ ∧ ∂dφ
∂ (φ) ∂ (φ)
=δφ Lm +d(δφ) Lm
∧ ∂φ ∧ ∂dφ
∂ (φ) ∂ (φ) ∂ (φ)
=δφ Lm ( 1)rd Lm +d δφ Lm
∧(cid:18) ∂φ − − (cid:18) ∂dφ (cid:19)(cid:19) (cid:18) ∧ ∂dφ (cid:19)
∂ (φ)
=δφ ⋆Σ(φ)+d δφ Lm . (10a)
∧ (cid:18) ∧ ∂dφ (cid:19)
Definition 6 The terms ∂∂Lφm and ∂∂Ldmφ are called in what follows algebraic
derivatives of 9 and ⋆ (φ) sec 3T M ֒ sec ℓ(M,g),
m ∗
L ± ∈ → C
V
∂ (φ) ∂ (φ)
⋆Σ(φ)= Lm ( 1)rd Lm (11)
∂φ − − (cid:18) ∂dφ (cid:19)
is called the Euler-Lagrange functional of the field φ. Some authors call it the
functional derivative of and in this case write
m
L
δ (φ)
m
⋆Σ(φ)= L (12)
δφ
In working with these objects it is necessary to keep in mind that for
φ sec rT M, (φ) (j (φ)) sec pT M and (φ) (j (φ))
∗ 1 ∗ 1
sec∈ qT∗VM, F ≡ F ∈ V K ≡ K ∈
V
∂ ∂ ∂
[ (φ) (φ)]= (φ) (φ)+( 1)pr (φ) (φ). (13)
∂φ F ∧K ∂φF ∧K − F ∧ ∂φK
We recall also that if (j (φ)) sec pT M is an arbitrary functional and
1 ∗
G ∈
σ : M M a diffeomorphism, then (jV1(φ)) is said to be invariant under σ if
→ G
and only if σ (j (φ))= (j (φ)). Also, it is a well knownresult that (j (φ))
∗ 1 1 1
G G G
is invariantunder the action of a one parameter group of diffeomorphisms σ if
t
and only if
£ (j (φ))=0, (14)
ξ 1
G
where ξ secTM is the infinitesimal generatorof the groupσ and£ denotes
t ξ
∈
the Lie derivative.
2.3 Euler-Lagrange Equations from Lagrangian Densities
Recallnowthattheprincipleofstationaryactionisthestatementthatthevaria-
tionoftheactionintegralwrittenintermsofaLagrangiandensity (j (θa,ωa,φ))
Lm 1 b
9This terminology was originally introduced in [64]. The exterior product δφ∧ ∂ is a
∂φ
particularinstanceoftheA∧ ∂ directionalderivativesintroducedinthemultiformcalculus
∂φ
developedin[55]withδφ=A.
7
isnullforarbitraryvariationsofφwhichvanishintheboundary∂U oftheopen
set U M (i.e., δφ =0)
⊂ |∂U
δ (φ)=δ (j (θa,ωa,φ))= δ (j (θa,ωa,φ))=0. (15)
A Z Lm 1 b Z Lm 1 b
U U
A trivial calculation gives
δ (φ)= δφ ⋆Σ(φ). (16)
A Z ∧
U
Since δφ is arbitrary,the stationary action principle implies that
∂ (φ) ∂ (φ)
⋆Σ(φ)= Lm ( 1)rd Lm =0. (17)
∂φ − − (cid:18) ∂dφ (cid:19)
Theequation⋆Σ(φ)=0isthecorrespondingELE forthefieldφ sec rT M ֒
∗
∈ →
sec ℓ(M,g). V
C
2.4 Invariance of the Action Integral under the Action of
a Diffeomorphism
Proposition 7 The action (φ) for any field theory formulated in terms of
A
fields that are differential forms is invariant under the action of one parameters
groups of diffeomorphisms if (j (θa,ωa,φ)) =0 on the boundary ∂U of a
Lm 1 b |∂U
domain U M.
⊂
Proof. Let (j (θa,ωa,φ)) be the Lagrangian density of the theory. The
Lm 1 b
variation of the action which we are interested in is the horizontal variation,
i.e.:
δ (φ)= £ (j (θa,ωa,φ)) (18)
hA Z ξLm 1 b
U
Let
1
ξ =g(ξ, ) sec T M ֒ sec ℓ(M,g). (19)
∗ ∗
· ∈ → C
^
ThenwehavefromawellknownpropertyoftheLiederivative(Cartan’smagical
formula) that
£ =d(ξ y )+ξ y(d ). (20)
ξ m ∗ m ∗ m
L L L
But, since (j (θa,ωa,φ)) sec 4T M ֒ sec ℓ(M,g) we have d = 0
and then £Lξmm 1=d(ξ∗by m).∈It follVows,∗usin→g StokeCs theorem that Lm
L L
£ (j (θa,ωa,φ))= d[ξ y (j (θa,ωa,φ))]
Z ξLm 1 b Z ∗ Lm 1 b
U U
= ξ y (j (θa,ωa,φ))=0, (21)
Z ∗ Lm 1 b
∂U
since (j (θa,ωa,φ)) =0.
Lm 1 b |∂U
Remark 8 It is important to emphasize that the action integral is always in-
variant under the action of a one parameter group of diffeomorphisms even if
the corresponding Lagrangian density is not invariant (in the sense of Eq.(14))
under the action of that same group.
8
3 Covariant ‘Conservation’ Laws
Let (M,g, ,τ , ) denote a general Riemann-Cartan spacetime. As stated
g
∇ ↑
above we suppose that the dynamic fields φA, A = 1,2,...,n, are r-forms, i.e.,
each φA sec rT M ֒ sec ℓ(M,g), for some r=0,1,...,4.
∗
∈ → C
Let ea bVe an arbitrary global orthonormal basis for TM, and let θa
{ } { }
be its dual basis. We suppose that θa sec 1T M ֒ sec ℓ(M,g). Let
∗
∈ → C
moreover θa be the reciprocal basis of θa . VAs it is well known (see, e.g.,
{ } { }
[64, 52, 53, 55]) it is possible to representthe gravitationalfield using θa and
{ }
itisalsopossibletowritedifferentialequationsequivalenttoEinsteinequations
for such objects.10
Here, we make the hypothesis that a Riemann-Cartan spacetime models a
generalized gravitational field which must be described by θa,ωa , where ωa
{ b} b
aretheconnection1-forms(inagivengauge). Thus,wesupposethatadynamic
theory for the fields φA sec rT M (called in what follows matter fields) is
∗
∈
obtainedthroughtheintroductVionofaLagrangiandensity,whichisafunctional
on J1[( T M)2+n] as previously discussed.
∗
ActiVveLocal Lorentztransformationsarerepresentedbyeven sectionsofthe
CliffordbundleU secSpine (M)֒ sec ℓ(0)(M,g),suchthatUU˜ =U˜U =1,
i.e., U(x) Spine∈ Sl(2,C1,)3. Und→er a loCcal Lorentz transformation the fields
∈ 1,3 ≃
transform as
θa θa =UθaU 1 =Λaθb,
7→ ′ − b
ωba 7→ωb′a =Λacωdc(Λ−1)db+Λac(dΛ−1)cb, (22)
φA φA =UφAU 1,
′ −
7→
whereΛa(x) SOe . Inourformalismitisatrivialitytoseethat (θa,ωa,φ)
b ∈ 1,3 Lm b ∈
sec 4T M ֒ sec ℓ(M,g) is invariant under local Lorentz transformations.
∗
→ C
IndVeed, since τg = θ5 = θ0θ1θ2θ3 sec 4T∗M ֒ sec ℓ(M,g) commutes
∈ → C
with even multiform fields, we have that Va local Lorentz transformation pro-
duces no changes in , i.e.,
m
L
(θa,ωa,φ) U (θa,ωa,φ)U 1 = (θa,ωa,φ). (23)
Lm b 7→ Lm b − Lm b
However,this does not implies necessarily that the variation of the Lagrangian
density (θa,ωa,φ)obtainedby variationofthe fields (θa,ωa,φ) isnull,since
Lm b b
δ = (θa+δ θa,ωa+δ ωa,φ+δ φ) (θa,ωa,φ)=0, (24)
vLm Lm v b v b v −Lm b 6
unless it happens that for an infinitesimal Lorentz transformation,
(θa+δ θa,ωa+δ ωa,φ+δ φ)
Lm v b v b v
= (UθaU 1,UωaU 1,UφU 1)=U U 1 = . (25)
Lm − b − − Lm − Lm
10The Lagrangian density for the {θa} for the case of General Relativity is recalled in
Section5.
9
Inwhatfollowswesupposethatthe Lagrangianofthe matterfieldisinvari-
antunderlocalLorentztransformations11,i.e.,δ =0. Also dependson
v m m
L L δ
theθaandωba,butnotondθaanddωba (minimalcoupling).12 Then, δLθma = ∂∂Lθma
δ
and δLωmba = ∂∂Lωmba and we can write
∂ ∂
δ = δθa Lm +δωa Lm +δφA ⋆Σ , (26)
Z Lm Z (cid:20) ∧ ∂θa b∧ ∂ωa ∧ A(cid:21)
b
where Σ are the Euler-Lagrangefunctionals of the fields φA.
A
As we just showed above the action of any Lagrangian density is invariant
under diffeomorphisms. Let us now calculate the total variation of the La-
grangian density , arising from a one-parameter group of diffeomorphisms
m
L
generated by a vector field ξ secTM and by a local Lorentz transformation,
∈
when we vary θa,ωa,φA,dφA independently. We have
b
δ =δ £ . (27)
m v m ξ m
L L − L
Under the (nontrivial) hypothesis [51, 14] that δ =0,
v m
L
δ = £ = ⋆ £ θa ⋆Jb £ ωa ⋆Σ £ φA, (28)
Lm − ξLm − Ta∧ ξ − a ∧ ξ b− A∧ ξ
where we have:
Definition 9 The coefficients of δθa = £ θa, i.e.
ξ
−
∂ m 3
⋆Ta = ∂Lθa ∈sec T∗M (29)
^
are called the energy-momentum densities of the matter fields, and the
a
sec 1T M are called the energy-momentum density 1-forms of the mTatte∈r
∗
fieldVs. The coefficients of δωa, i.e.,
b
⋆Jba = ∂∂Lωma ∈sec 3T∗M, (30)
b ^
are called the angular momentum densities of the matter fields.
Taking into account that each one of the fields φA obey a Euler-Lagrange
equation, ⋆Σ =0, we can write
A
£ = ⋆ £ θa+⋆Jb £ ωa (31)
Z ξLm Z Ta∧ ξ a ∧ ξ b
Now, since all geometrical objects in the above formulas are sections of the
Clifford bundle, we can write
£ξθa =ξ∗ydθa+d(ξ∗yθa). (32)
11We discuss further the issue of local Lorentz invariance and its hidden consequence in
[51,14].
12SeeexampleoftheelectromagneticfieldinAppendiceB.
10