Continuous body dynamics and the Mathisson-Papapetrou-Dixon equations

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by  S. P. Loomis

Continuous body dynamics and the Mathisson-Papapetrou-Dixon equations
S. P. Loomis∗ and J. David Brown†
Department of Physics, North Carolina State University, Raleigh, NC 27695
(Dated: January 9, 2017)
We show that an effective particle Lagrangian yields the Mathisson-Papapetrou-Dixon (MPD)
equations. The spin of theeffective particle is defined without any reference to a fixed body frame
orangularvelocityvariable. Wethendemonstratethatacontinuousbody,definedbyacongruence
ofworldlinesanddescribedbyageneralaction,canberewrittenasaneffectiveparticle. Weanalyze
thegaugefreedomofthebodyandshowthatanaturalcenterofmassconditionisrelatedtoaspin
supplementary condition.
PACSnumbers: 04.25.-g,04.40.-b
7
1 CONTENTS thetimeevolutionofthefourcomponentsofmomentum,
0 Pα, and the six components of spin, Sαβ. The thirteen
2
I. Introduction 1 unknownsarethemomentum,spin,andthethreedegrees
n offreedomcontainedintheparticle’spropervelocityUα.
a II. The MPD Equations for an effective particle 2 MathissonandlaterPirani[3]addressedthemismatch
J
between the number of equations and number of un-
6
III. Continuous bodies written as effective particles 4 knowns by introducing the spin supplementary condi-

tion UαS = 0. Pirani justified this choice by anal-
] αβ
c IV. Gauge conditions 8 ogy to a similar identity for the center of mass in special
q relativity. Papapetrou instead employed the spin sup-
r- V. Example: Dust 10 plementary condition VαSαβ = 0 where Vα is an ar-
g bitrary time flow vector field. In 1959 Tulczyjew sim-
[ VI. Discussion 12
plifiedMathisson’smultipoleformulation,againderiving

1 the same equations of motion at pole-dipole order but
VII. Acknowledgments 12
v choosing the spin supplementary condition PαS = 0,
αβ
5 arguing that Mathisson and Pirani’s condition did not
Appendices 12
4 uniquely determine the world line. A survey of the var-
5
A. Covariant Variations and the Exponential Map 12 ious spin supplementary conditions and how they relate
1
tooneanothercanbefoundin[4]. Aconcreteanalysisof
0
. B. Bitensors and Synge’s World Function 14 the relationship between spin supplementary conditions
1
and center of mass can be found in [5].
0
References 15 In a series of papers from 1970 to 1974, Dixon pre-
7
sented yet another reformulation of the multipole mo-
1
: ments in terms of a Fourier transformationof the stress-
v I. INTRODUCTION energytensor[6–8]. Thecompleteargumentisalsogiven
i
X in[9]. DixonfoundthatMathisson’svariationalprinciple
r The motion of extended bodies in general relativity yields dynamical equations for Pα and Sαβ, but leaves
a was first addressed by Mathisson in 1937 [1]. Mathisson the dynamical evolution of the quadrupole and higher
defined multipole moments for the stress-energy tensor order multipole moments undefined. His analysis places
expanded about a central world line and formulated restrictionsonthesymmetriesofthesemultipoles. Based
the conservation of stress-energy as a variational prin- onthesesymmetries,Dixondefinesthereducedmultipole
ciple. He derived what we now know as the Mathisson- moments Jµνρσα1···αn for n 0.
≥
Papapetrou-Dixon (MPD) equations to “pole–dipole”” The final form of the MPD equations through
order, and identified the quadrupole terms that had a quadrupole order, as given by Dixon, is [9]:
nonrelativistic analogue. Papapetrou, using a different
DP 1 1
definitionformultipolemoments,derivedthesameequa- α = R X˙βSµν R Jµνρσ , (I.1a)
αβµν α µνρσ
tions in 1951 [2]. Ds −2 − 6∇
DS 4
The analyses of Mathisson and Papapetrou yield ten αβ =2P X˙ + R J λνµ . (I.1b)
equationsforthirteenunknowns. The tenequationsgive Ds [α β] 3 µνλ[α β]
Here, the worldline is expressed as xα = Xα(s), where
xα are the spacetime coordinates and Xα are functions
∗ [email protected]; CurrentAddress: Department of Physics, of a worldline parameter s. The dot above a sym-
UniversityofCalifornia,Davis,California95616 bol denotes the time derivative d/ds and D/Ds is the
† david [email protected] covariant derivative along the worldline; for example,
2
DP /Ds = P˙ Γγ X˙βP . This will also be denoted positionvector. Thecorrespondingintegralsforthemul-
α α − αβ γ
with a circle above the symbol, so that P˚ =DP /Ds. tipolemomentsarecomplicatedandbeyondthe scopeof
α α
this introduction.
Dixon defines the momentum and spin in terms of in-
The multipole method of Mathisson, Papapetrou,
tegrals over the leaves of a foliation Σ(s) of the world
TulzcyjewandDixonisnottheonlyapproachthatyields
tube. The integrals,which involve the stress-energyten-
sor Tµ′ν′, Synge’s worldfunction σ and the Jacobiprop- the MPD equations. Bailey and Israel [10], extend-
agators Hµ′ and Kµ′ (described in Appendix B) are ing the work of Hanson and Regge [11], showed that
α α a form of the MPD equations could be derived from
[6]
any reparametrization–invariant Lagrangian involving a
worldlineXα(s),asetofLorentz–orthonormalbasisvec-
Pα(s)≡ZΣ(s)dΣν′Kµ′αTµν′′ , (I.2a) teoxrtserenaaαl(ste)ntsroarnsfipeoldrtsedthaaltoningttehraecwtowrlidthlinthee,abnoddya.seTthoef
basis vectors e α(s) define the orientation of the body,
Sαβ(s)≡2 dΣν′Hµ′[ασβ]Tµν′′ . (I.2b) although the exaact relationship for a physical (i.e. non–
ZΣ(s) rigid)bodyisnotaddressed. Byanalogywithrigidbody
motion in classical mechanics, one says that the index a
BothHµ′αandKµ′αarebitensorswithoneprimedindex labels the legs of a “body–fixed frame”.
“located” at the point of integration xµ′ Σ(s) and one These results were independently replicated without
unprimed index located at the world line∈Xα(s). These theexternalfieldsbyPorto[12]. Morerecently,Steinhoff
bitensors act to transfer vectors from one tangent space [13] has reformulated Bailey and Israel’s full result in
into the other. The derivative of Synge’s world function newer notation. We give a brief overview of Steinhoff’s
σβ = βσ points opposite to the vector tangent to the presentation below.
unique∇geodesic connecting Xα(s) and xµ′ and acts as a Steinhoff begins by considering an action of the form
S[X,e,Φ ]= dsL g (X),Φ (X),X˙α(s),e α(s),Ωαβ(s),Φ (s) , (I.3)
I αβ A a I
Z (cid:16) (cid:17)
wheretheintegrationisalongtheworldlineXα(s). Here alongtheworldline,andasetofexternalfieldsφ (x). We
A
g (x)isthemetricofthespacetimemanifoldM,Φ (x) call this the “effective particle” model, and show that it
αβ A
arefieldsonthismanifoldandΦ (s)arescalardynamical yields the MPD equations.1 Ouraction does not depend
I
variables defined along the world line. Note that the Φ on an orthonormal body frame e α or angular velocity
I a
arefunctionsofsonly,andtheindexI canincludeabody Ωαβ. We show in III that continuous bodies (defined
§
frameindexa. Theangularvelocityisdefinedintermsof as a congruence of world lines minimizing a particular
the basis vectors e α(s) by Ωαβ(s) ηabe αD(e β)/Ds, action) can be expressed in terms of the effective parti-
a a b
where ηab is the Minkowski metric.≡ cle action. The definitions for momentum and spin that
Steinhoff defines the momentum and spin as emerge from this analysis coincide with the definitions
given by Dixon [6]. In IV we discuss the gauge con-
∂L §
P , (I.4a) straints that can be placed on the effective particle, and
α ≡ ∂X˙α explore the relation between spin supplementary condi-
∂L tion and center of mass. Finally, in V we apply our
Sαβ ≡2∂Ωαβ . (I.4b) results to analyze a continuous body §of noninteracting
particles—a “dust cloud.” As an effective particle, the
With these definitions, variation of the action (I.3) with
dust cloud satisfies the geodesic deviation equations.
respect to Xα(s) and e α(s) yields the equations of mo-
a
tion
DP 1 ∂L II. THE MPD EQUATIONS FOR AN
α = RαβµνX˙βSµν +( αΦA) , (I.5a) EFFECTIVE PARTICLE
Ds −2 ∇ ∂Φ
A
DS ∂L
Dsαβ =2P[αX˙β]−2(G[αβ]ΦA)∂Φ . (I.5b) In this section we generalize the method for deriving
A the MPD equations used by Bailey and Israel [10].
Here,G isalinearoperatordefinedsuchthat Φ =
αβ α A
∂ Φ +Γγ Gβ Φ . The action of Gα depend∇s on the
α A αβ γ A β
tensor type of Φ .
A
In II we consider a general action that depends only 1 After obtaining this result, we became aware of a similarresult
onaw§orldlinexα =Xα(s),asetoftensorsψ (s)defined byH.Fuchs [14].
I
3
The system consists of an effective particle in a man- Because φ are external sources, their variations are
A
ifold M with position xα = Xα(s) and a collection of given by δφ = (∂φ /∂xα)δXα. Then the covariant
A A
tensors ψ (s). The tensors ψ take the place of the or- variationsofthesefieldsare∆φ = φ δXα. Alsoob-
I I A α A
thonormalbasise αandbody–framevariablesφ usedin servethatthecovariantvariationof∇Xα isdefinedbythe
a I
Eq. (I.3). The index I can denote tensor indices as well vector ∆Xα δxα. That is, the worldline coordinates
≡
as functional dependence. The evolution of the effective behave as spacetime scalars under covariant variation.
particle system is described by the action TobringδS intoaformthatwillprovidetheequations
of motion, we must swap the order of the variations and
s1
S[X,ψ ]= dsL(φ (X),X˙α(s),ψ (s),ψ˚(s)) . time derivatives in ∆X˙α and ∆ψ˚α. The covariantvaria-
I A I I
Zs0 tionandcovariantderivativedonotcommute;ingeneral,
(II.1)
we have the following relation from Eq. (A.21):
The φ (x) are any collection of spacetime fields. For
A
example,φ canincludetheelectromagneticfieldandits D
A ∆, =Rσ δXµX˙νGρ . (II.7)
derivatives, the metric tensor gαβ, the curvature tensor Ds ρµν σ
(cid:20) (cid:21)
R anditssymmetrizedderivatives R .
αβγδ (α γ) µνρσ
In this paper we treat these fields as e∇xter·n·a·l∇sources— This yields the results
they are not varied in the variational principle.
D
Recall that the notation ψ˚ is an abbreviation for the ∆X˙α = (δXα) , (II.8a)
I Ds
covariant derivative along the worldline,
D
∆ψ˚ = (∆ψ )+Rσ δXµX˙νGρ ψ , (II.8b)
ψ˚ DψI = dψI +Γν X˙αGµ ψ . (II.2) I Ds I ρµν σ I
I ≡ Ds ds αµ ν I sincetheworldlinecoordinatesXαbehaveasscalarfields
The operator Gµ , discussed more fully in Appendix A, under covariant differentiation and covariantvariation.
ν
acts on tensor indices [15]. For example, we have Now integrate by parts. The endpoint terms vanish if
we assume that Xα and ψ are fixed at s and s . The
Gµ ψρ =δρψµ , (II.3a) I 0 1
ν ν variation of the action becomes
Gµ ψ = δµψ , (II.3b)
ν ρ − ρ ν s1
δS = ds P˚ +Rσ X˙βπIGρ ψ
forcontravariantandcovariantvectors. Theextensionof − α ραβ σ I
Gµ to higher rank tensors is straightforward. Also note Zs0 (cid:20)(cid:18)
ν ∂L ∂L
thatthe covariantderivativeofXα isdefinedbythevec- + αφA δXα+ ˚πI ∆ψI .
tor DXα/Ds X˙α. Thus, the worldline coordinates ∇ ∂φA(cid:19) (cid:18)∂ψI − (cid:19) (cid:21)
≡ (II.9)
behave as spacetime scalars under covariant differentia-
tion. The coefficients of δXα and ∆ψ give the equations of
I
The Lagrangian(II.1) is a function over a tensor bun-
motion
dle. That is, L depends on the position X(s) as well as
∂L
tensors in the tangent space of x=X(s). The variation P˚ = R ρσX˙βπIG ψ + φ , (II.10a)
δψI isnotcovariantwheneverthebasepointxisalsovar- α − αβ ρσ I ∇α A∂φA
ied. As a result, the functional derivatives δS/δXα and ∂L
˚πI = . (II.10b)
δS/δψI yield the equations of motion in non–covariant ∂ψI
combinations. Here we use the results of Appendix A to
vary the action in a covariant manner. Equation(II.6)showsthatthe variation∆ψI isasumof
To begin, let us define the momentum variables non–covarianttermsproportionaltoδψI andδXα. From
thisweseethattheequationofmotionδS/δψ =0(with
I
∂L Xα fixed) coincides with (II.10b), while the equation of
P , (II.4a)
α ≡ ∂X˙α motion δS/δXα = 0 (with ψ fixed) is a non–covariant
I
∂L combination of the covariant Eqs. (II.10)
πI . (II.4b)
≡ ∂ψ˚ Spin is defined as
I
Usingtheresult(A.15a)fromAppendix A,thecovariant Sρσ 2πIG[ρσ]ψI . (II.11)
≡
variation of the action is
This puts the equation of motion (II.10a) into the form
δS = s1ds P ∆X˙α+ ∂L φ δXα of the MPD equation (I.5a). The equation of motion
α α A
∂φ ∇ for the spin variable itself is obtained from the covariant
Zs0 (cid:18) A (II.5) derivative of the definition (II.11), which yields S˚ =
∂L ρσ
+πI∆ψ˚I + ∂ψ ∆ψI 2˚πIG[ρσ]ψI +2πIG[ρσ]ψ˚I. With the equation of motion
I (cid:19) (II.10b) this becomes
Here, ∆ is the covariantvariation defined by [13]
∂L
∆≡δ+ΓναµδXαGµν . (II.6) S˚ρσ =2∂ψIG[ρσ]ψI +2πIG[ρσ]ψ˚I . (II.12)
4
We now make use of (A.15b), which follows from the The derivatives of Xµ and (Z0,Zi) satisfy [18]
requirement of general covariance. This equation tells
us that the operator Gαβ acting on the LagrangianL of X˙µZ,0ν +X,µiZ,iν =δνµ , (III.1a)
(II.1) follows a “chainrule”. Since L is a scalar,we have XµZj =δj , (III.1b)
,i ,µ i
X˙µZ0 =1 , (III.1c)
∂L ∂L ,µ
0=∂φAGαβφA+ ∂X˙µGαβX˙µ X,µiZ,0µ =0 , (III.1d)
(II.13)
+ ∂∂ψLIGαβψI + ∂∂ψ˚LIGαβψ˚I X˙µZ,iµ =0 , (III.1e)
where X˙µ =∂Xµ/∂s.
We can lower the index α on Gα and antisymmetrize. We introduce the notations
β
Using the notation (II.4) for the momenta, we have ∂
D X˙µ = +Γσ X˙µGρ , (III.2a)
s ≡ ∇µ ∂s µρ σ
0=∂∂φLAG[αβ]φA+P[βX˙α] (II.14) DDDζi ≡X,µi∇µ = ∂∂ζi +ΓσµρX,µiGρσ , (III.2b)
∂L
+ ∂ψ G[αβ]ψI +πIG[αβ]ψ˚I usingthegeneratorsGρσfromAppendixA,andfollowing
I Vines’ use of as a covariantpartial derivative [19]. As
D
before,wemayalsoabbreviate / swiththecircle(e.g.
This result can be used to rewrite the time derivative of A˚) and / ζi with semicolonsD(e.Dg. A ).
;i
the spin from (II.12) as the MPD equation (I.5b). D D
We consider a fairly general action assuming that the
To summarize, we have shown that the equations of body is subject to no external forces, though it may be
motion (II.10) that follow from the action (II.1), along subject to internal forces mediated by the first spatial
withthe definition(II.11)forspin,yieldthe MPDequa- derivative Xµ:
,i
tions
s1
S[X]= ds d3ζ ζ,g (X),X˙µ,Xµ . (III.3)
P˚α =−21RαβµνX˙βSµν +(∇αφA)∂∂φL , (II.15a) Zs0 ZS L(cid:16) µν ,i(cid:17)
A
Wewillgenerallyrestrictthisactiontobereparametriza-
∂L
S˚ =2P X˙ 2(G φ ) (II.15b) tioninvariant,althoughthe resultsofthis sectiondo not
αβ [α β]− [αβ] A ∂φ
A dependonthatassumption. Whenreparametrizationin-
varianceis enforced,the above actioncoincides with De-
in the form (I.5). Witt’s elastic body [20, 21]
The key difference between this result and previous
s1
analyses is that we do not require a body frame or basis S[X]= ds d3ζαρ(ζ,f ) , (III.4)
ij
vectors eaα to define the orientation of the body, and −Zs0 ZS
we have no need for an angular velocity variable Ω .
αβ
The spin, as defined in (II.11), does not rely on these where α = X˙µX˙ and f is the fleet metric f =
µ ij ij
−
constructions. (g +U Uq)XµXν, with Uµ = X˙µ/α being the four-
µν µ ν ,i ,j
velocity field.
In this section we study some basic properties of the
action (III.3) and show that it can be expressed in the
III. CONTINUOUS BODIES WRITTEN AS
formoftheeffectiveparticleaction(II.1). Doingsogives
EFFECTIVE PARTICLES
definitions ofP andS identicaltoDixon’s definitions
α αβ
(I.2) [6].
LetM beaRiemannianmanifoldwithcoordinatesxµ. We start by varyingthe action to determine the equa-
We consider a continuous body in the sense defined by tionsofmotion. Withtheexceptionofitsdependenceon
Carter and Quintana [16], as a congruence of worldlines ζ (which does not affect the variation), the Lagrangian
represented by the smooth mapping X : R S M, density is a function of X through the metric field
where S is a differentiable manifold whose p×oints→repre- g andLis also a function of vectors X˙µ and Xµ. This
sent the worldlines, and the real numbers R label points mµeνans we can use the methods of Appendix A,,ispecif-
along each world line. S is called the “matter space” ically Eq. (A.15a), to vary the action. We define the
and is given coordinates ζi [17]. The coordinate on R is canonical momentum density and stress density as
s. Thus,functionsoverthecongruencemaybewrittenin
terms of coordinates (s,ζi) on R S, or in terms of the ∂L , (III.5a)
manifold coordinates xµ. We intr×oduce the inverse map- Pµ ≡ ∂X˙µ
ping (Z0(x),Zi(x)) that satisfies s = Z0(X(s,ζ)) and i ∂L . (III.5b)
ζi =Zi(X(s,ζ)). Sµ ≡ ∂Xµ
,i
5
Then the variation with respect to Xµ is We now turn to the primary goal of this section: to
showthatthecontinuousbodyaction(III.3)canbewrit-
s1
δS = ds d3ζ ∆X˙µ+ i∆Xµ . (III.6) tenintheformoftheeffectiveparticleaction(II.1). This
Zs0 ZS hPµ Sµ ,ii roefqgueioredsesmicasptpointhgethtaenignefnotrmspaaticoenoafbaopuotinthteoncoangfirduuecnicael
Notethat,sincethemetricistreatedasanexternalfield, world line. To prepare for this analysis, we first make a
its covariantvariation ∆gµν(X)= αgµνδXα vanishes. changeofnotationby placing primeson coordinatesand
∇
For the ∆X˙µ term in Eq. (III.6), integration by parts tensors associatedwith the continuous body. In particu-
gives lar,wewilluseX′ :R S M ratherthanX todenote
× →
thecongruenceofworldlines. Fortensorsassociatedwith
s1 s1
ds ∆X˙µ = dsDPµδXµ the body, we place the prime on the indices: for exam-
µ
Zs0 P +−Z[Ps0µδXµ]Dss10s. (III.7) pstlreTe,shtshedefie(ndusunictniyaolramrweaolXri˙zldµe′d,l)iPnveµe′lwoacinliltdyb,Semµi′od,merneeosnptteeudcmtiXvdeµel(nys.s)i.tyTanhde
Thelasttermintheaboveequationvanisheswhenwefix fiducualworldlinedoesnotneedtoliewithinthematerial
the initial and final configurations Xµ(s ) and Xµ(s ); body. If it does, the spacetime point Xµ(s) (for a given
0 1
thatis,wesetδXµ(s )=δXµ(s )=0. Wecansimilarly value of s) does not need to lie on the s=const surface.
0 1
integrate by parts for the Xµ term, with the result The introduction of the fiducual worldline adds a
,i
gauge freedom to the system, in additional to the
i reparametrization invariance that is already present. In
d3ζ i∆Xµ = d3ζDSµδXµ
ZS Sµ ,i −ZS Dζi (III.8) lbaetefirxseedc.tions we will discuss how the gauge freedom can
+ d2ζη iδXµ . We makeuse ofthe bitensor and exponentialmapfor-
Z∂S iSµ malism of Appendix B. We begin by defining the expo-
nentialmapfromapointXµ(s)onthe fiducialworldline
where η is the outward-pointing vector field normal to
the bounidary surface ∂S. The variation of the action and a vector ξµ in the tangent space at Xµ(s), to the
with Xµ fixed at s and s is then point Xµ′(s,ζ) in the continuous body:
0 1
s1 X′(s,ζ)=exp(X(s),ξ(s,ζ)) . (III.13)
δS = ds d3ζ ˚ i δXµ
Zs0 ZS h−Pµ−Sµ;ii (III.9) Here, ξµ(s,ζ) = σµ(X(s),X′(s,ζ)) is the vector at
s1 −
+ ds d2ζ η i δXµ . X(s) tangent to the geodesic γ(u) connecting X(s) and
Zs0 Z∂S iSµ X′(s,ζ), affinely parametrized such that γ(0) = X and
(cid:2) (cid:3) γ(1) = X′. Note that the point X′(s,ζ) defined by the
SinceδXµ isarbitrary,thisgivestheequationsofmotion
exponentialmapdependsonthegeometryinaneighbor-
and boundary conditions
hood of X(s). To ensure that there is a unique geodesic
connecting X(s) and X′(s,ζ), the size of the body must
˚ i =0 (ζ S) , (III.10a)
−Pµ−Sµ;i ∈ berestrictedbythecurvatureanditsderivativesatX(s).
η i =0 (ζ ∂S) . (III.10b) Theregiontowhichitisrestrictedistermedthe“normal
iSµ ∈
convex neighborhood” (NCN).
Byintegrating(III.10a)overthematterspaceS,andus- With the placement of primes on the coordinates and
ing the boundary (III.10b), we obtain a continuity equa- tensors associated with points in the continuous body,
tion for the total momentum within the body. the action (III.3) becomes
The stress–energy tensor is defined by
Tµν = √2 g∂∂gLµν . (III.11) S[X′]=Zs0s1dsZS d3ζL(cid:16)ζ,gµ′ν′,X˙µ′,X,µi′(cid:17) . (III.14)
−
Equation(III.13) defines a change of variables in the ac-
This can be rewritten in terms of the canonical momen- tion from Xµ′(s,ζ) to Xµ(s) and ξµ(s,ζ). We can use
tum and stress densities as
Eq. (A.7) to write δXµ′ in terms of δXµ and the covari-
Tµν = 1 µX˙ν + µiXν . (III.12) ant variation ∆ξµ. The result,
√ g P S ,i
− (cid:16) (cid:17) δXµ′ =Kµ′ δXα+Hµ′ ∆ξα , (III.15)
α α
byusingthechainrule(A.15b)forGα . FromEq.(III.1)
β
weseethat µ = √hTµνn wherehisthedeterminant makes use of the Jacobi propagators, Kµ′ and Hµ′ .
ν α α
of the inducPed me−tric h = g XµXν on surfaces Σ(s) The Jacobi propagators are defined in terms of deriva-
ij µν ,i ,j
of constants. Also, n is the future-pointing unit vector tives of the exponential map. They depend on Xα(s),
µ
field normal to Σ(s). ξα(s,ζ)andthegeometryinaneighborhoodofx=X(s).
6
They are introduced in Appendix A and further devel- ξα and Xµ′.
oped in Appendix B. Note that Hµ′ is invertible in the As a functional of Xµ(s) and ξµ(s,ζ), the variation of
α
NCN,sincetheremustbeaone-to-onemappingbetween the action is
δS =Z+s0Zs1s0sd1sdZsSZ∂dS3ζd(cid:2)2−ζP(cid:2)ηµi′S−µi′S(cid:3)µi(cid:16)′K;i(cid:3)µ(cid:16)′αKδµX′ααδ+XαH+µ′αH∆µξ′αα∆(cid:17)ξ,α(cid:17) (III.16)
wherewehaveusedtheresultsfromEq.(III.9). Extrem- (III.18).
ization of S with respect to ξα yields the equations of Theanalysisaboveshowsthattheequationsofmotion
motion obtainedbyvaryingXµ andξµ arenotindependent;this
is a consequence of the fact that Xµ and ξµ constitute a
−P˚µ′ −Sµi′;i Hµ′α =0 (ζ ∈S) , (III.17a) lcaorngtearinseatonfevwargiaaubgleesftrheaednoXmµ,′n.oTthperevsaerniatbilnesthXeµoaringdinξaµl
h ηiSµi′iHµ′α =0 (ζ ∈∂S) . (III.17b) variablesXµ′,whichisthefreedomtochoosethefiducial
world line.
(cid:2) (cid:3)
The propagator Hµ′ is invertible in the NCN, so these Theaction(III.14)canbewrittenexplicitlyintermsof
α
the new variables Xµ and ξµ. To show this, we first use
equations are clearly equivalent to the equations of mo-
tion obtained by extremizing the action with respect to thegeneralvariation(III.15)toexpandthederivativesof
Xµ′ (Eqs.(III.10)withprimesonthespacetimeindices). Xµ′ as
The variationofthe actionwithrespecttothe fiducial X˙µ′ =Kµ′ X˙α+Hµ′ ˚ξα , (III.19a)
worldline Xα must be handled with care, since δXα on α α
the boundary∂S is notindependent ofδXα in the bulk X,µi′ =Hµ′αξ,αi . (III.19b)
S. To isolate δXα in Eq. (III.16), we must convert the As a functional of Xα(s) and ξα(s,ζ), the Lagrangian
surface term to a volume integral. This yields density from Eq. (III.14) becomes
ZS d3ζh−P˚µ′Kµ′α+Sµi′Kµ′α;ii=0 (ζ ∈S(II)I.18) whLer=e Lth(cid:16)eζa,rggµu′νm′,eKntµ′oαfX˙gαµ′+ν′(HXµ′)′αi˚ξsαw,Hritµt′eαnξ,αii(cid:17)n te(rImII.s20o)f
for the equation of motion that comes from extremiza- X(s) and ξ(s,ζ) using the exponential map (III.13).
tion of S with respect to Xµ. This equation is a com- Next, we make use of the parallel propagator gαµ′ de-
bination of the equations (III.17). To see this, we first fined in Appendix B. Since the Lagrangian density is
multiply Eq. (III.17a) by −H1αν′Kν′β and integrate over a scalar, it depends only on scalar combinations ofLits
the matter space S. We then integrate by parts on the arguments. Furthermore,itisapropertyofthepropaga-
term i Kµ′ . The boundary term vanishes by virtue tor that contractions between contravariant and covari-
of Eq.S(µI′I;iI.17bβ) (multiplied by −H1αν′Kν′β). The remain- Tanhteirnedfoicree,sawreepsereesetrhvaetd;thi.ee.LAagµr′gaαngµ′iagnν′αdBenνs′i=tyAcµa′nBµb′e.
ing volume integral is precisely the equation of motion written as
L=L ζ,gαβ,gαµ′Kµ′βX˙β +gαµ′Hµ′β˚ξβ,gαµ′Hµ′βξ,βi . (III.21)
(cid:16) (cid:17)
The action for the continuous body is
S[X,ξ]=Zs0s1dsL=Zs0s1ds ZS d3ζL(cid:16)ζ,gαβ,gαµ′Kµ′βX˙β +gαµ′Hµ′β˚ξβ,gαµ′Hµ′βξ,βi(cid:17) , (III.22)
with the Lagrangiandefined by L= d3ζ . fromtensor fields defined in the tangentspaces of points
S L Xµ(s) along the fiducial world line. Hence, the ac-
R
TheLagrangiandensity(III.21)isconstructedentirely
7
tion(III.22)describesaneffectiveparticlewithworldline TheseresultsaremosteasilyderivedwiththeLagrangian
Xµ(s), carryinginternaldegreesoffreedomdescribedby density written in the form of Eq. (III.20). We also note
the vectors ξµ(s). The effective particle interacts nonlo- the result
cally with the geometry in a neighborhood of the world-
line;thisdependenceiscontainedinthetensorsgαµ′Hµ′β
sainodnsgfαoµr′gKαµµ′′βH.µI′nβ Eanqds.g(αBµ.′7K)µw′βe.sEhoxwprtehsseedseirnietsheisxpwaany-, δδξL,αi =Hµ′αSµi′ (III.25)
gαµ′Hµ′β and gαµ′Kµ′β depend on the Riemann tensor
anditsderivativesevaluatedalongthefiducialworldline.
The effective particle action (III.22) takes the form of satisfied by the momentum density. From the definition
Eq. (II.1) with the correspondences:
(II.11), the spin is
ψ ξα(ζ) , (III.23a)
I
↔
φ g ,R , R ,... , (III.23b)
A αβ µνρσ α µνρσ
↔{ ∇ }
S =2 d3ζ π G ξµ =2 d3ζ ξ π . (III.26)
wherethedotsdenotehigherorderderivativesoftheRie- αβ µ [αβ] [α β]
S S
Z Z
mann tensor. Note that the index I now represents the
continuouslabels ζi,aswellasthediscretespacetimein-
dex α. This requires us to replace certain partialderiva-
btievceosmweisthδLfu/nδcξtαi(oζn)a.l dFeurritvhaetrivmeos;ref,oraerxeapmeaptleed, ∂inLd/e∂xψII U−s√inhg′ndνΣ′Tνµ′ν′′=, a−nndν′t√heh′dde3fiζn,itoiuorneξaαrli=er−reσsαu,ltwtehasteePµth′a=t
must include an integral over the matter space S. the above definitions for momentum and spin coincide
The momenta as defined in Eqs. (II.4) are with Dixon’s definitions in Eqs. (I.2).
Pα ≡ ∂∂X˙Lα =ZS d3ζKµ′αPµ′ , (III.24a) dthearTitvhetehdeeiqniun§adItIei,oxmnIsusontfobwmeoignteiconlunerdafeolsirztethdheetoceoaffncetcciontiuuvnoetupsfoalrrattbihceellesf,aζacist.
πα(ζ)≡ δ˚ξδαL(ζ) =Hµ′αPµ′ . (III.24b) The variation of the action, from Eq. (II.5), becomes
s1 ∂L δL δL
δS = ds P ∆X˙α+ R δXα+ + d3ζ π ∆˚ξα+ ∆ξα+ ∆ξα . (III.27)
Zs0 α ∂Rµνρσ∇α µνρσ ··· ZS " α δξα δξ,αi ,i#!
After integration by parts, we obtain the generalization of Eq. (II.9):
s1 ∂L
δS = ds P˚ + R + +Rµ X˙σ d3ζπ ξν δXα
α α µνρσ νασ µ
Zs0 (cid:18)− ∂Rµνρσ∇ ··· ZS (cid:19)
s1 δL δL s1 δL
+ ds d3ζ ˚π + D ∆ξα+ ds d2ζ η ∆ξα . (III.28)
Zs0 ZS − α δξα − Dζi δξ,αi!! Zs0 Z∂S iδξ,αi!
TheequationsofmotionobtainedbyextremizingS with of S with respect to the fiducial particle worldline Xα is
respect to ξα(ζ) are
∂L
P˚ = R ρσX˙β d3ζξ π + R + .
α αβ ρ σ α µνρσ
− S ∂Rµνρσ∇ ···
Z
(III.30)
δL δL
˚πα = δξα − DDζi δξ,αi! (ζ ∈S) , (III.29a) TEqh.is(IIe.1q0uaa)t.ionItgiesneerqauliizveaslentthteo Eeffqe.ct(iIvIeI.18p).articAles
δL shown before, Eq. (III.30) is redundant; it can be
η =0 (ζ ∂S) . (III.29b)
iδξα ∈ derived from Eqs. (III.29).
i,
We have shown that the continuous body, described
by the action (III.14), can be interpreted as an effective
singleparticlewithaction(III.22). Thisresultstillholds
These equations generalize the effective particle
when the Lagrangian density depends on higher-order
Eq. (II.10b). They are equivalent to Eqs. (III.17).
spatial derivatives of Xµ′. In that case, the formulae
Theequationofmotionthatcomesfromextremization for converting derivatives of Xµ′ into derivatives of ξα
8
and bitensors are more complicated. Nevertheless, it is action,
still true that the final Lagrangian density can be writ-
tenintermsofaXµ(s), ξα(s,ζ)andtheirderivatives,in δS = ds d3ζ δS δXµ′ , (IV.2)
combinations that depend on the geometry in a neigh- S δXµ′
borhood of the fiducal worldline. In terms of a series Z Z (cid:18) (cid:19)
expansion, the dependence on geometry appears as the where δXµ′ depends on δXα and δξα through the expo-
Riemann tensor and its derivatives evaluated at Xµ(s).
nentialmap(III.13). UsingtheresultsofAppendixB,we
We can also generalize the action (III.14) by allow- find δXµ′ =Kµ′ δXα+Hµ′ ∆ξα. This shows that the
ing for further dependence on X′ through some exterior α α
action is invariant, δS =0, for any variation satisfying
fields. This would not change the main result—we sim-
ply use the parallel propagator to express these fields in
terms of tensors defined along the fiducial worldline. ∆ξβ =σβµ′Kµ′αδXα . (IV.3)
This invariance holds independent of the equations of
IV. GAUGE CONDITIONS motion.
Fortheeffectiveparticle,thereparametrizationinvari-
In this section we assume the action (III.3) for the ance (IV.1) and fiducial worldline invariance (IV.3) are
continuous body is reparametrization invariant (RP- independentsymmetries,eachrequiringtheirowncondi-
invariant). A reparametrization s˜ = F(s,ζ) consists tionsforgaugefixing. RP–invariance ischaracterizedby
in replacing the coordinates Xµ′ with X˜µ′ such that 1realfunctionof4realparameters. Worldlineinvariance
X˜µ′(s˜,ζ) = Xµ′(s,ζ). In this case, the action can be is characterized by 4 real functions of 1 real parameter.
written in the form (III.4) for an elastic body [21]. There are many ways to fix RP-invariance for the ef-
A natural way to remove the gauge freedom for the fective particle. We will consider two different condi-
continuous body is to choose the “proper time gauge” tions,namely,the“propertime”gaugeandthe“normal”
tXw˙µe′eXn˙µn′e=igh−b1o.riWngitchotnhsitsacnotnsdsituirofna,ctehsecoseinpcairdaetsiownitdhstbhee- gaalluygceo.nFvoernitehnetfitoduicmiaplowseorXl˙dµliXn˙eµi=nvari1a.ncTe,hiist ipsagrteinaellry-
−
proper time interval measured along each of the world- fixes the gauge by setting the fiducial worldline parame-
lines. ter equal to proper time. The remaining freedom in the
The effective particle action (III.22) inherits RP– fiducialworldlineisremovedbyimposingacenterofmass
invariance from the continuous body (III.3). Recall the condition. We show that for a specific definition of the
change of variables from Xµ′ to Xµ and ξµ, defined by center of mass,the center of mass condition is relatedto
a spin supplementary condition.
the exponential map, Eq. (III.13). RP–invariance con-
sists in replacing the vector ξα(s,ζ) with ξ˜α(s˜,ζ) such The proper time gauge for an effective particle is an
that Xapµp′roisxdimefiantieodnintotetrhmescoofnXditαioannsdXξ˙αµ′bX˙yµt′h=e e−xp1o,nwenhteirael
X˜′(s˜,ζ)=exp X(s˜),ξ˜(s˜,ζ) . (IV.1) map. In our examples in the next sections, we will take
thisapproximationtothirdorderinǫ,wherewe suppose
(cid:16) (cid:17)
ξ /ℓ < ǫ for any relevant length scale ℓ. For example, ℓ
Notethatthereparametrizations˜=F(s,ζ)onlychanges
| |
theparametrizationoftheparticleworldlines. Itdoesnot would include the radius of curvature 1/ R. This
∼ | |
affect the parametrization of the fiducial world line. See condition allows for convergence of series expansions in
Figure 1. ξ. We also impose ˚ξ /α < ǫ. This seconpd condition
| |
The fiducial worldline can be chosen arbitrarily. This requires the relative motions of the particles to be much
freedom appears as a gauge symmetry for the effective less than the speedof light(so that vibrationalenergyis
particleaction(III.22),inadditiontotheRP–invariance. not comparable to the total mass–energy).
This gauge symmetry can be identified by varying the Using Eqs. (III.19) and (B.7), we have
gαµ′X˙µ′ = δβα− 12Rαξβξ− 61Rαξβξ;ξ X˙β+ δβα− 61Rαξβξ ˚ξα+O(ǫ4)
(cid:18) (cid:19) (cid:18) (cid:19) (IV.4)
1 1 1
=X˙α+˚ξα Rα Rα Rα + (ǫ4) ,
− 2 ξX˙ξ− 6 ξ˚ξξ− 6 ξX˙ξ;ξ O
which gives
X˙µ′X˙µ′ =X˙αX˙α+2X˙α˚ξα+˚ξ2−RX˙ξX˙ξ− 34RX˙ξ˚ξξ− 13RX˙ξX˙ξ;ξ+O(ǫ4) . (IV.5)
9
X(·) X′(·,ζ) X(·) X′(·,ζ)
3 s˜=2 3
s=2
2 X′(1,ζ) 2 X˜′(2,ζ)
s˜=1
s=1 ξ˜(2,ζ)
1 ξ(1,ζ) 1
s˜=0
s=0
0 0
FIG. 1. Each figure shows the fiducial worldline X(s) and the worldline for a generic particle X′(s,ζ) in the body. The dots
on the fiducialworldline indicate theparameter valuesfor X(s). In theleft figure, theparticle worldlines are parametrized by
s; in the right figure, the particle worldlines are parametrized by s˜. The spacetime point X′(1,ζ) = exp(X(1),ξ(1,ζ)) in the
left figurecoincides with thespacetime pointX˜′(2,ζ)=exp(X(2),ξ˜(2,ζ))in theright figure. Thedashedlinein each figureis
thegeodesic that definestheexponential map.
TheoccurrenceofX˙ andξasindicesontheRiemanncur- The most useful form of the center of mass condition
vature tensor indicates contraction of those indices with is found by setting ∂ f = 0 at the minimum. With the
α
the specified vectors. We now partially fix the fiducial relation (B.3) this yields Ξ =0, where
α
worldline by setting X˙ X˙α = 1. Then the proper time
α
gauge condition X˙µ′X˙µ′ =−1−yields Ξα ≡ S d3ζwξα (IV.9)
Z
4 1
2X˙ ˚ξα+˚ξ2 R R R + (ǫ4)=0, defines the center of mass
α − X˙ξX˙ξ−3 X˙ξ˚ξξ−3 X˙ξX˙ξ;ξ O
One natural choice for the density weight is w =
(IV.6)
to third order in ǫ. √h′nµ′nν′Tµ′ν′,wherenµ′ is the unitvectorfieldnormal
to the surface Σ(s) of constant s and h′ is the determi-
Analternativetothepropertimegaugeisthe“normal
gauge”, in which we simply suppose the vectors ξα(s,ζ) nantoftheinducedmetriconΣ(s). Thisw istheenergy
density relative to a fleet of observers at rest in Σ(s).
inhabitthesubspacenormaltosometimelikevectorfield
Vα(s) defined along the world line, that is, If we choose the normal gauge, then Σ(s) is (a subset
of) the surface of geodesics passing through X(s) which
Vα(s)ξα(s,ζ)=0 . (IV.7) are normal to Vα at X(s). This gauge choice has been
considered by Costa and Natario [5], but using different
The vector field can be a defined in various ways; for definitions of P and S than those used in this pa-
α αβ
example, Vα might equal four–velocity Uα or the mo- per. Theyshowedthatwiththenormalgauge(withtheir
mentum Pα. definitions of spin and momentum), the center of mass
toWbeitfihxXe˙dαfXo˙rαt=he−fid1u,ctiharleweodreldglrienees. oTfwfroeeodpotimonrseamraeina ceoqnudiviatiloenntΞtαoa=s0piwnistuhpwple=m−en√tahr′yZc0,oµn′ndνit′iToµn′νV′ iSsαexβa=ctl0y.
α
“centerofmass”conditionandora“spinsupplementary” WeproduceasimilarresultforourdefinitionsofP ,S
α αβ
condition VαSαβ =0. We will consider a natural formu- and w.
lationofthecenterofmassconditionwhichis equivalent Thequickestwaytoderivetheresultisbyusingthese-
tdoera, tshpoinugshupfuprletmheernmtaurylticpoonledictoiornrecutpiotnosqeuxaisdtr.upole or- riesexpansionof−H1αµ′nµ′. Ifwedifferentiatethenormal
Thecenterofmassatparametertime scanbedefined gauge condition (IV.7) with respect to ζi, we see that it
asthe Frechet-Karchermeanofthe surfaceΣ(s). Specif- satisfiesthepropertyVαξ,αi =0. Wealsohavenµ′X,µi′ =0
itchaellfyu,nXct(iso)ni[s2t2h]e center of mass of Σ(s) if it minimizes everywhereonΣ(s). Usingthe identity −H1αµ′Hν′α =δνµ′′
and the result (III.19b), we can write nν′δµν′′X,µi′ =0 as
f(y) d3ζw(ζ)σ(y,X′(s,ζ)) , (IV.8)
≡ZS Hν′αnν′ −H1αµ′X,µi′ =Hν′αnν′ξ,αi =0 . (IV.10)
wherew(ζ)canbeanydensitydefinedonS. Essentially, (cid:16) (cid:17)(cid:18) (cid:19)
this definition tells us that the center of mass minimizes This means that Hµ′αnµ′ is parallel to Vα. Next, set
aweightedaverageofthesquareddistancebetweenitself Hµ′αnµ′ =AVα and write
and every other point on the surface. It may be the case
othnatthtehesuprofainctexΣ(=s)X. (s)thatminimizesf(y)doesnotlie gµ′ν′nµ′nν′ =A2 −H1αµ′−H1βν′gµ′ν′ VαVβ . (IV.11)
(cid:18) (cid:19)
10
Using the series expansion (B.6) and nµ′nµ′ = 1 we have
−
−1
1 1 2 1 1
A= 1 R R + (ǫ4) =1+ R + R + (ǫ4) . (IV.12)
VξVξ ξ VξVξ VξVξ ξ VξVξ
− 3 − 6∇ O 6 12∇ O
(cid:18) (cid:19)
Now we can calculate
−H1αµ′nµ′ =−H1αµ′gµ′ν′nν′ =−H1αµ′gµ′ν′ −H1βν′Hλ′β nλ′ . (IV.13)
(cid:18) (cid:19)
Using the results above for Hλ′βnλ′ and the expansion (B.6b) from Appendix B, we have
−H1αµ′nµ′ = gαβ + 1Rαξβξ+ 1 ξRαξβξ+ (ǫ4) 1+ 1RVξVξ+ 1 ξRVξVξ+ (ǫ4) Vβ
3 6∇ O 6 12∇ O
(cid:18) (cid:19)(cid:18) (cid:19) (IV.14)
1 1
=Vα+ ψαβ R + R + (ǫ4) .
βξVξ ξ βξVξ
3 2∇ O
(cid:18) (cid:19)
where ψαβ gαβ + 1VαVβ.
Now set w≡=√h′n2µ′nν′Tµ′ν′ = nµ′ µ′, where the second equality follows from the results of III, to obtain
− P §
d3ζwξα = d3ζnµ′−H1βµ′Hν′β ν′ξα = 2 d3ζ−H1βµ′nµ′ξ[αHν′β] ν′ . (IV.15)
S − S P − S P
Z Z Z
Hereweutilize theresult−H1βµ′nµ′ξβ =nµ′σµβ′σβ =nµ′σµ′ =0,derivedfromtherelationsinAppendix Bandthefact
thatσµ′ istangenttoΣ(s)andthereforeorthogonaltonµ′. Nowreplacing−H1βµ′nµ′ withtheexpansioninEq.(IV.14)
and using the definition of spin from Eqs. (III.24b) and (III.26), we find that the center of mass satisfies
2
Ξ d3ζwξ =VβS ψβγVδR Oρσ + (ǫ4) . (IV.16)
α α βα γρδσ αβ
≡ S − 3 O
Z
Here, Thus, for the dust cloud, there is no interaction energy
between particles.
Oρσαβ ≡ZS d3ζξρξσξ[αHµ′β]Pµ′ (IV.17) actWiointhisthe definition α = q−X˙µ′X˙µ′, the dust cloud
is an octupole moment in the sense that it is an integral
s1
over a density field with three position vectors. It is not S[X′]= ds d3ζρ X˙µ′X˙µ′ . (V.1)
clear to us how this term relates to the gravitational or −Zs0 ZS q−
reduced multipoles. In any case, Eq. (IV.16) shows that
the center of mass condition Ξ = 0 with appropriately Theequationsofmotionaregivenin(III.10),withprimes
α
chosen weight w is equivalent to a spin supplementary placed on the spacetime indices. In this case the stress
condition up to second order in ǫ, with an octupole cor- density i ∂ /∂Xµ′ vanishes,sotheequationsofmo-
Sµ′ ≡ L ,i
rection at third order. tionreduceto ˚µ′ =0. The momentumdensity is µ′
P P ≡
∂ /∂X˙µ′ = ρUµ′, where Uµ′(s,ζ) X˙µ′/ X˙µ′X˙µ′ is
L ≡ −
V. EXAMPLE: DUST the four–velocity of the particle with labelqζi. Therefore
theequationsofmotionforthedustcloudimplyU˚µ′ =0;
An extremely simple example of a continuous body is as expected, each dust particle follows a geodesic.
a cloud of dust. This is a special case of an elastic body We now use the exponential map to write the action
Eq. (III.4) with the Lagrangiandensity = αρ, where interms ofafiducialworldline Xµ andvectorsξµ. From
L −
ρ(ζ,f ) = ρ(ζ) is independent of the fleet metric f . Eq. (IV.5) we have
ij ij
∗ 1 ∗ ∗ ∗ 2 1
X˙µ′X˙µ′ = X˙2 1 U ξ 1+U ξ fαβξαξβ RUξUξ + R ∗ + RUξUξ;ξ + (ǫ4) (V.2)
− − − · − 2 · − 3 Uξξξ 6 O
q p (cid:26) (cid:18) (cid:19)(cid:18) (cid:19) (cid:27)