Multipolar test body equations of motion in generalized gravity theories
aa r X i v : . [ g r- q c ] M a y Multipolar test body equations of motion ingeneralized gravity theories
Yuri N. Obukhov ∗ Theoretical Physics LaboratoryNuclear Safety InstituteRussian Academy of SciencesB.Tulskaya 52, 115191 Moscow, RussiaDirk Puetzfeld † ZARMUniversity of BremenAm Fallturm, 28359 Bremen, GermanyJuly 22, 2018
Abstract
We give an overview of the derivation of multipolar equations ofmotion of extended test bodies for a wide set of gravitational theo-ries beyond the standard general relativistic framework. The classesof theories covered range from simple generalizations of General Rel-ativity, e.g. encompassing additional scalar fields, to theories with ad-ditional geometrical structures which are needed for the description ofmicrostructured matter. Our unified framework even allows to han-dle theories with nonminimal coupling to matter, and thereby for asystematic test of a very broad range of gravitational theories.
In this work we present a general unified multipolar framework, which en-ables us to derive equations of motion of extended test bodies for a widerange of gravitational theories. The framework presented here can be ap-plied to theories which significantly go beyond General Relativity (GR), andrange from the most straightforward extensions of GR, like scalar-tensortheories, to theories with additional geometrical structures and nonminimalcoupling.The multipolar method which we employ here can be thought of as thedirect generalization of the ideas pioneered by Mathisson [1], Papapetrou ∗ Email: [email protected] † Email: [email protected], URL: http://puetzfeld.org L , whereasthe original energy-momentum tensor T ab is substituted by a set of multipolemoments m ab ··· along this world-line. Such a multipolar description simpli-fies the equations of motion. This is achieved by consideration of only a finiteset of moments. Different flavors of multipolar approximation schemes existin the literature, in this work we define the moments `a la Dixon in [3].[2], and Dixon [3, 4, 5, 6] to the case of generalized gravity theories. Assketched in figure 1, the main aim of such methods is to find a simplified local description of the motion of extended test bodies in terms of a suitableset of multipolar moments, which catches the essential properties of thebody at the chosen order of approximation.In this work, we use a covariant multipolar description, based on Synge’sworld-function formalism [7, 8]. The multipolar moments to be introduced,can be viewed as a direct generalization of the moments first introduced byDixon in [3].Central to the derivation of the equations of motion, by means of amultipolar method, is the knowledge of the corresponding conservation lawsof the underlying gravity theory. In General Relativity, the starting point ofall methods mentioned so-far is the conservation of the (symmetric) energy-momentum ∇ j T ij = 0 . (1)Many generalized gravity theories allow for a more detailed description ofmatter than standard GR, e.g. also taking into account internal matterdegrees of freedom as source of the gravitational field. Consequently theconservation laws of such theories are more complex, and (1) has to bereplaced by a suitable generalization. Generalized gravity
A metric and connection are the two fundamentalgeometrical objects on a spacetime manifold. They play an important role in2he description of gravitational phenomena in the framework of what can bequite generally called an Einsteinian approach to gravity. The principles ofequivalence and general coordinate covariance are the cornerstones of thisapproach. As Einstein himself formulated, the crucial achievement of histheory was the elimination of the notion of inertial systems as preferredones among all possible coordinate systems.In Einstein’s theory, gravitation is associated with the metric tensoralone. Nevertheless, it is worthwhile to stress that Einstein clearly un-derstood the different physical statuses of the metric and the connection(Appendix II in [9]):“[...] at first Riemannian metric was considered the funda-mental concept on which the general theory of relativity and thusthe avoidance of the inertial system were based. Later, however,Levi-Civita rightly pointed out that the element of the theorythat makes it possible to avoid the inertial system is rather theinfinitesimal displacement field Γ ikj . The metric or the symmet-ric tensor field g ik which defines it is only indirectly connectedwith the avoidance of the inertial system insofar as it determinesa displacement field.”.There exists a variety of gravitational theories that generalize or extendthe physical and mathematical structure of GR. Among these theories thereare large classes of so-called f ( R ) models, and of theories with nonmini-mal coupling to matter; they are developed in particular in the context ofrelativistic cosmology (but not only there), see [10, 11, 12, 13, 14]. The so-called Palatini approach represents another class of widely discussed theoriesin which the metric and the connection are treated as independent variablesin the action principle [15, 16, 17]. Another early example of a generalizedgravity theory with balance laws different from Einstein’s gravity can befound in [18, 19]. Last but not least, we should mention the vast family ofthe gauge gravity theories constructed using a Yang-Mills type of approach[20, 21].Since our main aim is to present a unified multipolar framework for avery wide range of gravity theories, the theory underlying our analysis has tobe sufficiently general. In this work we choose metric-affine gravity (MAG)as the foundation for the derivation of the equations of motion of extendeddeformable test bodies.In MAG, matter is characterized by three fundamental Noether currents– the canonical energy-momentum current, the canonical hypermomentumcurrent, and the metrical energy-momentum current. These objects satisfy aset of conservation laws (or, more exactly, balance equations). In view of themulti-current characterization of matter in metric-affine gravity, we developa general approach which is applicable to an arbitrary set of conservationlaws for any number of currents. The latter can include gravitational, elec-tromagnetic, and other physical currents if they are relevant to the model3igure 2: In contrast to GR, MAG also allows to take into account themicrostructural properties of matter (spin, dilation current, proper hyper-charge). These additional currents couple to the post-Riemannian geometricdegrees of freedom of the underlying spacetime. Our unified description ofextended test bodies allows to cover a wide range of gravitational theoriesand matter models. 4nder consideration. In particular, our approach is flexible enough to beapplied to the case in which there is a general nonminimal coupling betweengravity to matter. Our presentation here is mainly based on the originalworks [22, 23, 24]. Structure of the paper
The structure of the paper is as follows: Insection 2 we give an overview of the metric-affine theory of gravity. In par-ticular we cover its geometrical and dynamical aspects. This is followed bya general Lagrange-Noether analysis in section 3. The results of this gen-eral analysis are then used to derive the conservation laws of a metric-affinegravity with nonminimal coupling in section 4. In section 5 a unified multi-polar framework is presented. This framework is then used 6 to derive thegeneral equations of motion of extended test bodies in nonminimal metric-affine gravity. Several specializations to other gravity theories are discussed.Furthermore, in section 7 the equations of motion in scalar-tensor theoriesare worked out. Section 8 contains a summary of our results.Our conventions are those of [25]. In particular, the basic geometricalquantities such as the curvature, torsion, and nonmetricity are defined asin [25], and we use the Latin alphabet to label the spacetime coordinateindices. Furthermore, the metric has the signature (+ , − , − , − ). It shouldbe noted that our definition of the metrical energy-momentum tensor differsby a sign from the definition used in [12, 14, 26]. A summary of our notationand conventions, as well as additional material for comparison to previousworks can be found in the appendix. Metric-affine gravity [25] is a natural extension of Einstein’s general rela-tivity theory. It is based on gauge-theoretic principles [20, 21], and it takesinto account microstructural properties of matter (spin, dilation current,proper hypercharge) as possible physical sources of the gravitational fieldon an equal footing with macroscopic properties (energy and momentum)of matter. The formalism of MAG makes it possible to study all the above-mentioned alternative theories in a unified framework. The correspondingspacetime landscape [27] includes as special cases the geometries of Riemann,Riemann-Cartan, Weyl, Weitzenb¨ock, etc. (c.f. figure 3).The standard understanding of metric-affine gravity is based on thegauge-theoretic approach for the general affine symmetry group. In thisframework, we can naturally distinguish the kinematics of the gravity the-ory and its dynamics . The kinematics embraces all aspects that are relatedto the description of the fundamental variables, their mathematical prop-erties and physical interpretation. The dynamics studies the choice of theLagrangian and the field equations. We collect in this section all the relevantmaterial. Although the Lagrange-Noether machinery is deeply interrelatedwith both kinematics and dynamics of the theory, we will discuss it in a sep-5rate section which will ultimately underlie the derivation of the multipoleequations of motion.
We model spacetime as a four-dimensional smooth manifold. In this studywe are not interested in the global (topological) aspects, and we will confineour attention only to local issues. The local coordinates x i , i = 0 , , , x i → x i + δx i , where δx i = ξ i ( x ) . (2)The four arbitrary functions ξ i ( x ) parametrize an arbitrary local diffeomor-phism. In the metric-affine theory of gravity, the gravitational physics is encoded intwo fields: the metric tensor g ij and an independent linear connection Γ kij .The latter is not necessarily symmetric and compatible with the metric.From the geometrical point of view, the metric introduces lengths and anglesof vectors, and thereby determines the distances (intervals) between pointson the spacetime manifold. The connection introduces the notion of paralleltransport and defines the covariant differentiation ∇ k of tensor fields.Under the spacetime diffeomorphisms (2), these geometrical variablestransform as δg ij = − ( ∂ i ξ k ) g kj − ( ∂ j ξ k ) g ik , (3) δ Γ kij = − ( ∂ k ξ l ) Γ lij − ( ∂ i ξ l ) Γ klj + ( ∂ l ξ j ) Γ kil − ∂ ki ξ j . (4)In general, the geometry of a metric-affine manifold is exhaustively char-acterized by three tensors: the curvature, the torsion and the nonmetricity.They are defined [27] as follows: R klij := ∂ k Γ lij − ∂ l Γ kij + Γ knj Γ lin − Γ lnj Γ kin , (5) T kli := Γ kli − Γ lki , (6) Q kij := − ∇ k g ij = − ∂ k g ij + Γ kil g lj + Γ kjl g il . (7)The curvature and the torsion tensors determine the commutator of thecovariant derivatives. For a tensor A c ...c k d ...d l of arbitrary rank and indexstructure: ( ∇ a ∇ b − ∇ b ∇ a ) A c ...c k d ...d l = − T abe ∇ e A c ...c k d ...d l + k X i =1 R abec i A c ...e...c k d ...d l − l X j =1 R abd j e A c ...c k d ...e...d l . (8)6igure 3: Different spacetime types as special cases of a general metric-affinespacetime ( R klij = 0, T kli = 0, Q kij = 0). The abbreviations R (curvature), T (torsion), and Q (nonmetricity) over the arrows denote the vanishing ofthe corresponding geometrical object.The Ricci tensor is introduced by R ij := R kijk , and the curvature scalar is R := g ij R ij .A general metric-affine spacetime ( R klij = 0, T kli = 0, Q kij = 0) incor-porates several other spacetimes as special cases, see figure 3 for an overview.The Riemannian connection e Γ kji is uniquely determined by the conditionsof vanishing torsion and nonmetricity which yield explicitly e Γ kji = 12 g il ( ∂ j g kl + ∂ k g lj − ∂ l g kj ) . (9)Here and in the following, a tilde over a symbol denotes a Riemannianobject (such as the curvature tensor) or a Riemannian operator (such asthe covariant derivative) constructed from the Christoffel symbols (9). Thedeviation of the geometry from the Riemannian one is then convenientlydescribed by the distortion tensor N kji := e Γ kji − Γ kji . (10)The system (6) and (7) allows to find the distortion tensor in terms of thetorsion and nonmetricity. Explicitly: N kji = −
12 ( T kji + T ikj + T ijk ) + 12 ( Q ikj − Q kji − Q jki ) . (11)Conversely, one can use this to express the torsion and nonmetricity tensorsin terms of the distortion, T kji = − N [ kj ] i , (12) Q kij = − N k ( ij ) . (13)7ubstituting (10) into (5), we find the relation between the non-Riemannianand the Riemannian curvature tensors R adcb = e R adcb − e ∇ a N dcb + e ∇ d N acb + N anb N dcn − N dnb N acn . (14)Applying the covariant derivative to (5)-(7) and antisymmetrizing, wederive the Bianchi identities [27]: ∇ [ n R kl ] ij = T [ klm R n ] mij , (15) ∇ [ n T kl ] i = R [ kln ] i + T [ klm T n ] mi , (16) ∇ [ n Q k ] ij = R nk ( ij ) . (17) Along with tensors, an important role in physics is played by densities. Afundamental density √− g is constructed from the determinant of the metric, g =det g ij . Under diffeomorphisms (2) it transforms as δ √− g = − ( ∂ i ξ i ) √− g. (18)This is a direct consequence of (3). From any tensor B i...j... one can constructa density B i...j... = √− gB i...j... . Although in this paper we will encounteronly such objects, it is worthwhile to notice that not all densities are of thistype – see the exhaustive presentation in the book of Synge and Schild [28].There are two kinds of covariant differential operators on the spacetimemanifold, depending on whether the connection is involved or not. TheLie derivative L ζ is defined along any vector field ζ i and it maps tensors(densities) into tensors (densities) of the same rank. Let us recall the explicitform of the Lie derivative of the metric and the distortion: L ζ g ij = ζ k ∂ k g ij + ( ∂ i ζ k ) g kj + ( ∂ j ζ k ) g ik , (19) L ζ N kji = ζ n ∂ n N kji + ( ∂ k ζ n ) N nj i + ( ∂ j ζ n ) N kni − ( ∂ n ζ i ) N kjn . (20)In contrast, a covariant derivative ∇ k raises the rank of tensors (densi-ties) and it is determined by the linear connection Γ kji . Moreover, there aredifferent covariant derivatives which arise for different connections that maycoexist on the same manifold.A mathematical fact is helpful in this respect: Every third rank tensor X kj i defines a map of one connection into a different new connectionΓ kji −→ Γ kji + X kji . (21)There are important special cases of such a map. One example is obtainedfor X kji = N kji : then the connection Γ kji is mapped into the RiemannianChristoffel symbols, e Γ kji = Γ kji + N kji , in accordance with (10).Another interesting case arises for X kji = T jki . The result of the map-ping Γ kji = Γ kji + T jki = Γ kji + Γ jki − Γ kji = Γ jki . (22)8s then called a transposed connection , or associated connection, see [29, 30].The importance of the transposed connection is manifest in the followingobservation. Although the Lie derivative is a covariant operator – this is notapparent since it is based on partial derivatives – one can make everythingexplicitly covariant by noticing that it is possible to recast (19) and (20)into equivalent form L ζ g ij = ζ k ∇ k g ij + ( ∇ i ζ k ) g kj + ( ∇ j ζ k ) g ik , (23) L ζ N kji = ζ n ∇ n N kji + ( ∇ k ζ n ) N nj i + ( ∇ j ζ n ) N kni − ( ∇ n ζ i ) N kjn . (24)By the same token we can “covariantize” the Lie derivatives for all othertensors of any structure and of arbitrary rank.A more nontrivial (and less known) fact is that we can define the Liederivatives also for objects which are not tensors. In particular, the Liederivative of the connection then reads [29]: L ζ Γ kji = ∇ k ∇ j ζ i − R klji ζ l . (25)This quantity measures the noncommutativity of the Lie derivative with thecovariant derivative( L ζ ∇ k − ∇ k L ζ ) A c ...c k d ...d l = k X i =1 ( L ζ Γ kbc i ) A c ...b...c k d ...d l − l X j − ( L ζ Γ kd j b ) A c ...c k d ...b...d l . (26)The connection Γ kji , the transposed connection Γ kji , and the Rieman-nian connection e Γ kji define the three respective covariant derivatives: ∇ k , ∇ k , and e ∇ k . The covariant derivative defined by the Riemannian connectionis conventionally denoted by a semicolon “ ; a ”.We will assume that these differential operators act on tensors. In ad-dition, we will need the covariant operators that act on densities. For anarbitrary tensor density B ni...j... we introduce the covariant derivative b ∇ n B ni...j... := ∂ n B ni...j... + Γ nlj B ni...l... − Γ nil B nl...j... , (27)which produces a tensor density of the same weight. We denote a similar dif-ferential operation constructed with the help of the Riemannian connectionby ˇ ∇ n B ni...j... := ∂ n B ni...j... + e Γ nlj B ni...l... − e Γ nil B nl...j... , (28)When B ni...j... = √− gB ni...j... , we find b ∇ n B ni...j... = √− g ∗ ∇ n B ni...j... , (29)ˇ ∇ n B ni...j... = √− g e ∇ n B ni...j... , (30)where we introduced a modified covariant derivative ∗ ∇ i := ∇ i + N kik . (31)9 .1.3 Matter variables We will not specialize the discussion of matter to any particular physicalfield. It will be more convenient to describe matter by a generalized field ψ A . The range of the indices A, B, . . . is not important in our study. How-ever, we do need to know the behavior of the matter field under spacetimediffeomorphisms (2): δψ A = − ( ∂ i ξ j ) ( σ AB ) ji ψ B . (32)Here ( σ AB ) j i are the generators of general coordinate transformations thatsatisfy the commutation relations( σ AC ) j i ( σ C B ) lk − ( σ AC ) lk ( σ C B ) j i = ( σ AB ) li δ kj − ( σ AB ) j k δ il . (33)We immediately recognize in (33) the Lie algebra of the general linear group GL (4 , R ). This fact is closely related to the standard gauge-theoretic in-terpretation [25] of metric-affine gravity as the gauge theory of the generalaffine group GA (4 , R ), which is a semidirect product of spacetime translationgroup times GL (4 , R ).The transformation properties (32) determine the form of the covariantand the Lie derivative of a matter field: ∇ k ψ A := ∂ k ψ A − Γ kij ( σ AB ) j i ψ B , (34) L ζ ψ A := ζ k ∂ k ψ A + ( ∂ i ζ j )( σ AB ) j i ψ B . (35)The commutators of these differential operators read( ∇ k ∇ l − ∇ l ∇ k ) ψ A = − R klji ( σ AB ) ij ψ B − T kli ∇ i ψ A , (36)( L ζ ∇ k − ∇ k L ζ ) ψ A = − ( L ζ Γ kji )( σ AB ) ij ψ B . (37) As is well known, symmetries of a Riemannian spacetime are generatedby Killing vector fields. Each such field defines a so-called motion of thespacetime manifold, that is a diffeomorphism which preserves the metric g ij . Suppose ζ i is a Killing vector field. By definition, it satisfies e ∇ i ζ j + e ∇ j ζ i = 0 . (38)By differentiation, we derive from this the second covariant derivative e ∇ i e ∇ j ζ k = e R jkil ζ l . (39)Apply another covariant derivative and antisymmetrize: e ∇ [ n e ∇ i ] e ∇ j ζ k = e ∇ [ n ( e R | jk | i ] l ζ l ) . (40)10fter some algebra, the last equation is recast into ζ n e ∇ n e R ijkl + e R njkl e ∇ i ζ n + e R inkl e ∇ j ζ n + e R ijnl e ∇ k ζ n + e R ijkn e ∇ l ζ n = 0 . (41)The equations (38), (39) and (41) have a geometrical meaning: L ζ g ij = 0 , (42) L ζ e Γ ijk = 0 , (43) L ζ e R ijkl = 0 . (44)That is, the Lie derivatives along the Killing vector field ζ vanish for allRiemannian geometrical objects. Moreover, one can show that the same istrue for all higher covariant derivatives of the Riemannian curvature tensor[31] L ζ (cid:16) e ∇ n . . . e ∇ n N e R ijkl (cid:17) = 0 . (45)Let us generalize the notion of a symmetry to the metric-affine spacetime.We take an ordinary Killing vector field ζ and postulate the vanishing of theLie derivative L ζ N kj i = 0 (46)of the distortion tensor. Combining this with (11) and (43), we find anequivalent formulation L ζ g ij = 0 , (47) L ζ Γ ijk = 0 . (48)We call a vector field that satisfies (47) and (48) a generalized Killing vector of the metric-affine spacetime. By definition, such a ζ generates a diffeo-morphism of the spacetime manifold that is simultaneously an isometry (47)and an isoparallelism (48).Since the Lie derivative along a Killing vector commutes with the covari-ant derivative, L ζ e ∇ i = e ∇ i L ζ , see (26), we conclude from (14) and (44) thatthe generalized Killing vector leaves the non-Riemannian curvature tensorinvariant L ζ R klji = 0 . (49)It is also straightforward to verify that L ζ (cid:0) ∇ n . . . ∇ n N R klji (cid:1) = 0 (50)for any number of covariant derivatives of the curvature.Later we will show that generalized Killing vectors have an importantproperty: they induce conserved quantities on the metric-affine spacetime.11 .2 Dynamics of metric-affine gravity The explicit form of the dynamical equations of the gravitational field isirrelevant for the conservation laws that will form the basis for the deriva-tion of the test body equations of motion. However, for completeness, wediscuss here the field equations of a general metric-affine theory of gravity.The standard understanding of MAG is its interpretation as a gauge theorybased on the general affine group GA (4 , R ), which is a semidirect productof the general linear group GL (4 , R ), and the group of local translations[25]. The corresponding gauge-theoretic formalism generalizes the approachof Sciama and Kibble [32, 33]; for more details about gauge gravity theories,see [20, 21]. In the standard formulation of MAG as a gauge theory [25],the gravitational gauge potentials are identified with the metric, coframe,and the linear connection. The corresponding gravitational field strengthsare then the nonmetricity, the torsion, and the curvature, respectively.Here we use an alternative formulation of MAG, in which gravity is de-scribed by a different set of fundamental field variables, i.e. the independentmetric g ij and connection Γ kij , see [34, 35, 36, 15, 16, 37, 38, 39, 17, 40],for example. It is worthwhile to compare the field equations in the differ-ent formulations of MAG, and in particular, it is necessary to clarify therole and place of the canonical energy-momentum tensor as a source of thegravitational field. Since one does not have the coframe (tetrad) among thefundamental variables, the corresponding field equation is absent. Here wedemonstrate that one can always rearrange the field equations of MAG insuch a way that the canonical energy-momentum tensor is recovered as oneof the sources of the gravitational field.Assuming standard minimal coupling, the total Lagrangian density ofinteracting gravitational and matter fields reads L = V ( g ij , R ijkl , N kij ) + L mat ( g ij , ψ A , ∇ i ψ A ) . (51)In general, the gravitational Lagrangian density V is constructed as a dif-feomorphism invariant function of the curvature, torsion, and nonmetricity.However, in view of the relations (12) and (13), we can limit ourselves toLagrangian functions that depend arbitrarily on the curvature and the dis-tortion tensors. The matter Lagrangian depends on the matter fields ψ A and their covariant derivatives (34).The field equations of metric-affine gravity can be written in severalequivalent ways. The standard form is the set of the so-called “first” and“second” field equations. Using the modified covariant derivative defined by(31), the field equations are given byˇ ∇ n H ink + 12 T mni H mnk − E ki = − T ki , (52)ˇ ∇ l H klij + 12 T mnk H mnij − E kij = S ij k . (53)12ere the generalized gravitational field momenta densities are introduced by H klij := − ∂ V ∂R klij , H kij := − ∂ V ∂T kij , M kij := − ∂ V ∂Q kij , (54)and the gravitational hypermomentum density E kij := − H kij − M kij = − ∂ V ∂N kij . (55)Furthermore, the generalized energy-momentum density of the gravitationalfield is E ki = δ ik V + 12 Q kln M iln + T kln H iln + R klnm H ilnm . (56)The sources of the gravitational field are the canonical energy-momentumtensor and the canonical hypermomentum densities of matter, respectively: T ki := ∂ L mat ∂ ∇ i ψ A ∇ k ψ A − δ ik L mat . (57) S ijk := ∂ L mat ∂ Γ kij = − ∂ L mat ∂ ∇ k ψ A ( σ AB ) j i ψ B . (58)It is straightforward to verify that instead of the first field equation (52),one can use the so-called zeroth field equation which reads2 δ V δg ij = t ij . (59)On the right-hand side, the matter source is now represented by the metricalenergy-momentum density which is defined by t ij := 2 ∂ L mat ∂g ij . (60)The system (52) and (53) is completely equivalent to the system (53) and(59), and it is a matter of convenience which one to solve. In actual metric-affine gravity models the Lagrangian densities are con-structed in terms of the Lagrangian functions: L = √− gL , V = √− gV , L mat = √− gL mat . Accordingly, we find for the gravitational field momenta H klij = √− gH klij , H kij = √− gH kij , M kij = √− gM kij , (61)and E ij = √− gE ij . The gravitational sources (57), (58) and (60) are rewrit-ten in terms of the canonical energy-momentum tensor, the canonical hyper-momentum tensor, and the metrical energy-momentum tensor, respectively: T ki = √− g Σ ki , S ijk = √− g ∆ ijk , t ij = √− gt ij . (62)13he usual spin arises as the antisymmetric part of the hypermomentum, τ ij k := ∆ [ ij ] k , (63)whereas the trace ∆ k = ∆ iik is the dilation current. The symmetric tracelesspart describes the proper hypermomentum [25].As a result, the metric-affine field equations (52) and (53) are recast into ∗ ∇ n H ink + 12 T mni H mnk − E ki = − Σ ki , (64) ∗ ∇ l H klij + 12 T mnk H mnij − E kij = ∆ ijk . (65) In order to give an explicit example of physical matter with microstructure,we recall the hyperfluid model [41]. This is a direct generalization of thegeneral relativistic ideal fluid variational theory [42, 43] and of the spinningfluid model of Weyssenhoff and Raabe [44, 45]. Using the variational princi-ple for the hyperfluid [41], one derives the canonical energy-momentum andhypermomentum tensors:Σ ki = v i P k − p (cid:0) δ ik − v i v k (cid:1) , (66)∆ nmi = v i J mn , (67)where v i is the 4-velocity of the fluid and p is the pressure. Fluid elements arecharacterized by their microstructural properties: the momentum density P k and the intrinsic hypermomentum density J mn . As a first step, we notice that the gravitational (geometrical) and materialvariables can be described together by means of a multiplet, which we denoteby Φ J = ( g ij , Γ kij , ψ A ). We do not specify the range of the multi-index J atthis stage. The matter fields may include, besides the true material variables,also auxiliary fields such as Lagrange multipliers. With the help of thelatter we can impose various constraints on the geometry of the spacetime.Furthermore, we can use the Lagrange multipliers to describe models inwhich the Lagrangian depends on arbitrary-order covariant derivatives ofthe curvature, torsion, and nonmetricity. Then the general action reads I = Z d x L , (68)where the Lagrangian density L = L (Φ J , ∂ i Φ J ) depends arbitrarily on theset of fields Φ J and their first derivatives.Our aim is to derive Noether identities that correspond to general coordi-nate transformations. However, it is more convenient to start with arbitrary14nfinitesimal transformations of the spacetime coordinates and the matterfields. They are given as follows: x i −→ x ′ i ( x ) = x i + δx i , (69)Φ J ( x ) −→ Φ ′ J ( x ′ ) = Φ J ( x ) + δ Φ J ( x ) . (70)Within the present context it is not important whether this is a symmetrytransformation under the action of any specific group. The total variation(70) is a result of the change of the form of the functions and of the changeinduced by the transformation of the spacetime coordinates (69). To distin-guish the two pieces in the field transformation, it is convenient to introducethe substantial variation : δ ( s ) Φ J := Φ ′ J ( x ) − Φ J ( x ) = δ Φ J − δx k ∂ k Φ J . (71)By definition, the substantial variation commutes with the partial derivative, δ ( s ) ∂ i = ∂ i δ ( s ) .We need the total variation of the action: δI = Z (cid:2) d x δ L + δ ( d x ) L (cid:3) . (72)A standard derivation shows that under the action of the transformation(69)-(70) the total variation reads δI = Z d x (cid:20) δ L δ Φ J δ ( s ) Φ J + ∂ i (cid:18) L δx i + ∂ L ∂ ( ∂ i Φ J ) δ ( s ) Φ J (cid:19)(cid:21) . (73)Here the variational derivative is defined, as usual, by δ L δ Φ J := ∂ L ∂ Φ J − ∂ i (cid:18) ∂ L ∂ ( ∂ i Φ J ) (cid:19) . (74) Now we specialize to general coordinate transformations. For infinitesimalchanges of the spacetime coordinates and (matter and gravity) fields (69)and (70) we have x i → x i + δx i , g ij → g ij + δg ij , Γ kij → Γ kij + δ Γ kij ,and ψ A → ψ A + δψ A , where variations are given by (2), (3), (4), and (32).Substituting these variations into (73), and making use of the substantialderivative definition (71), we find δI = − Z d x (cid:20) ξ k Ω k + ( ∂ i ξ k ) Ω ki + ( ∂ ij ξ k ) Ω kij + ( ∂ ijn ξ k ) Ω kijn (cid:21) , (75)15here explicitlyΩ k = δ L δg ij ∂ k g ij + δ L δψ A ∂ k ψ A + ∂ L ∂ Γ lnm ∂ k Γ lnm + ∂ L ∂∂ i Γ lnm ∂ k ∂ i Γ lnm + ∂ i (cid:18) ∂ L ∂∂ i g mn ∂ k g mn + ∂ L ∂∂ i ψ A ∂ k ψ A − δ ik L (cid:19) , (76)Ω ki = 2 δ L δg ij g kj + δ L δψ A ( σ AB ) ki ψ B + ∂ L ∂∂ i ψ A ∂ k ψ A − δ ik L + ∂ L ∂∂ i g mn ∂ k g mn + ∂ j (cid:18) ∂ L ∂∂ j g in g nk + ∂ L ∂∂ j ψ A ( σ AB ) ki ψ B (cid:19) + ∂ L ∂ Γ lij Γ lkj + ∂ L ∂ Γ ilj Γ klj − ∂ L ∂ Γ lj k Γ lji + ∂ L ∂∂ i Γ lnm ∂ k Γ lnm + ∂ L ∂∂ n Γ ilm ∂ n Γ klm + ∂ L ∂∂ n Γ lim ∂ n Γ lkm − ∂ L ∂∂ n Γ lmk ∂ n Γ lmi , (77)Ω kij = ∂ L ∂∂ ( i ψ A ( σ AB ) kj ) ψ B + ∂ L ∂ Γ ( ij ) k + ∂ L ∂∂ ( i Γ j ) lm Γ klm + 2 ∂ L ∂∂ ( i g j ) n g kn + ∂ L ∂∂ ( i Γ | l | j ) m Γ lkm − ∂ L ∂∂ ( i Γ | ln | k Γ lnj ) , (78)Ω kijn = ∂ L ∂∂ ( n Γ ij ) k . (79)If the action is invariant under general coordinate transformations, δI = 0,in view of the arbitrariness of the function ξ i and its derivatives, we find theset of four Noether identities:Ω k = 0 , Ω ki = 0 , Ω kij = 0 , Ω kijn = 0 . (80)General coordinate invariance is a natural consequence of the fact thatthe action (68) and the Lagrangian L are constructed only from covariantobjects. Namely, L = L ( ψ A , ∇ i ψ A , g ij , R klij , N kji ) is a function of the met-ric, the curvature (5), the torsion (6), the matter fields, and their covariantderivatives (34). Denoting ρ ijkl := ∂ L ∂R ijkl , µ ijk := ∂ L ∂N ijk , (81)we find for the derivatives of the Lagrangian ∂ L ∂ Γ ijk = − ∂ L ∂ ∇ i ψ A ( σ AB ) kj ψ B − µ ij k +2 ρ inlk Γ nlj + 2 ρ nij l Γ nkl , (82) ∂ L ∂∂ i Γ jkl = 2 ρ ijkl , (83) ∂ L ∂∂ k g ij = 12 (cid:16) µ ( ki ) j + µ ( kj ) i − µ ( ij ) k (cid:17) . (84)As a result, we straightforwardly verify that Ω kij = 0 and Ω kijn = 0 areindeed satisfied identically. 16sing (82) and (83), we then recast the two remaining Noether identities(76) and (77) intoΩ k = δ L δg ij ∂ k g ij + δ L δψ A ∂ k ψ A + ∂ i (cid:18) ∂ L ∂ ∇ i ψ A ∇ k ψ A − δ ik L (cid:19) + b ∇ j (cid:18) ∂ L ∂ ∇ j ψ A ( σ AB ) mn ψ B (cid:19) Γ knm + ρ ilnm ∂ k R ilnm + ∂ L ∂ ∇ l ψ A ( σ AB ) mn ψ B R lknm + µ lnm ∂ k N lnm + 12 ˇ ∇ i (cid:16) µ ( im ) n + µ ( in ) m − µ ( mn ) i (cid:17) ∂ k g mn , (85)Ω ki = 2 δ L δg ij g kj + δ L δψ A ( σ AB ) ki ψ B + ∂ L ∂ ∇ i ψ A ∇ k ψ A − δ ik L + b ∇ j (cid:18) ∂ L ∂ ∇ j ψ A ( σ AB ) ki ψ B (cid:19) − µ lnk N lni + µ iln N kln + µ lin N lkn + 2 ρ ilnm R klnm + ρ lnim R lnkm − ρ lnmk R lnmi + ˇ ∇ n (cid:16) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:17) g jk = 0 . (86)Notice that the variational derivative (74) with respect to the matterfields can be identically rewritten as δ L δψ A ≡ ∂ L ∂ψ A − b ∇ j (cid:18) ∂ L ∂ ∇ j ψ A (cid:19) , (87)and turns out to be a covariant tensor density. It is also worthwhile tonote, that the variational derivative w.r.t. the metric is explicitly a covariantdensity. This follows from the fact that the Lagrangian depends on g ij notonly directly, but also through the objects Q kij and N kij . Taking this intoaccount, we find δ L δg ij = d L dg ij − ∂ n (cid:18) ∂ L ∂∂ n g ij (cid:19) = ∂ L ∂g ij −
12 ˇ ∇ n (cid:16) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:17) . (88)The Noether identity (86) is a covariant relation. In contrast, (85)is apparently non-covariant. However, this can be easily repaired by re-placing Ω k = 0 with an equivalent covariant Noether identity: Ω ′ k :=Ω k − Γ knm Ω mn = 0. Explicitly, we findΩ ′ k = δ L δψ A ∇ k ψ A + b ∇ i (cid:18) ∂ L ∂ ∇ i ψ A ∇ k ψ A − δ ik L (cid:19) − (cid:18) ∂ L ∂ ∇ i ψ A ∇ l ψ A − δ il L (cid:19) T kil + ∂ L ∂ ∇ l ψ A ( σ AB ) mn ψ B R lknm + (cid:20) − δ L δg ij −
12 ˇ ∇ n (cid:16) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:17)(cid:21) Q kij + µ lnm ∇ k N lnm + ρ ilnm ∇ k R ilnm = 0 . (89)17n-shell, i.e. assuming that the matter fields satisfy the field equations δ L /δψ A = 0, the Noether identities (86) and (89) reduce to the conservationlaws for the energy-momentum and hypermomentum, respectively.Equation (86) contains a relation between the canonical and the metricalenergy-momentum tensor, and the conservation law of the hypermomentum.In the next section we turn to the discussion of models with general non-minimal coupling. The results obtained in the previous section are applicable to any theory inwhich the Lagrangian depends arbitrarily on the matter fields and the grav-itational field strengths. Now we specialize to the class of models describedby an interaction Lagrangian of the form L = F ( g ij , R klij , N kli ) L mat ( ψ A , ∇ i ψ A ) . (90)Here L mat ( ψ A , ∇ i ψ A ) is the ordinary matter Lagrangian. We call F = F ( g ij , R klij , N kli ) the coupling function and assume that it can depend ar-bitrarily on its arguments, i.e., on all covariant gravitational field variablesof MAG. When F = 1 we recover the minimal coupling case. As a preliminary step, let us derive identities which are satisfied for thenonminimal coupling function F = F ( g ij , R klij , N kli ). For this, we applythe above Lagrange-Noether machinery to the auxiliary Lagrangian density L = √− g F . This quantity does not depend on the matter fields, and both(86) and (89) are considerably simplified. In particular, we have δ L δg ij = √− g (cid:18) F g ij + F ij (cid:19) , F ij := δFδg ij . (91)Then we immediately see that (86) and (89) reduce to ∇ k F ≡ (cid:20) − F ij − e ∇ n (cid:18) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:19)(cid:21) Q kij + ρ ilnm ∇ k R ilnm + µ lnm ∇ k N lnm , (92)2 F ki ≡ − ρ ilnm R klnm − ρ lnim R lnkm + ρ lnmk R lnmi − µ lnk N lni + µ iln N kln + µ lin N lkn + e ∇ n (cid:18) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:19) g jk . (93)Here we denoted ρ ijkl := ∂F∂R ijkl , µ ijk := ∂F∂N ijk . (94)18otice that for any tensor density we have relations (30) and (29).The identity (92) is naturally interpreted as a generally covariant gen-eralization of the chain rule for the total derivative of a function of severalvariables. This becomes obvious when we notice that (88) implies (cid:20) − F ij − e ∇ n (cid:18) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:19)(cid:21) Q kij = ∂F∂g ij ∇ k g ij . (95)It should be stressed that (92) and (93) are true identities, they are satisfiedfor any function F ( g ij , R klij , N kli ) irrespectively of the field equations thatcan be derived from the corresponding action. Now we are in a position to derive the conservation laws for the generalnonminimal coupling model (90). Recall the definitions of the canonicalenergy-momentum tensor (57), the canonical hypermomentum tensor (58)and the metrical energy-momentum tensor (60).In view of the product structure of the Lagrangian (90), the derivativesare easily evaluated, and the conservation laws (86) and (89) reduce to − F t ki − ∗ ∇ n (cid:0) F ∆ ikn (cid:1) + F Σ ki + h F ki + 2 ρ ilnm R klnm + ρ lnim R lnkm − ρ lnmk R lnmi + µ lnk N lni − µ iln N kln − µ lin N lkn − e ∇ n (cid:18) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:19) g jk i L mat = 0 , (96) ∗ ∇ i (cid:0) F Σ ki (cid:1) + F h − Σ li T kil + ∆ mnl R klmn + 12 t ij Q kij i + n ρ ilnm ∇ k R ilnm + µ lnm ∇ k N lnm + h − F ij − e ∇ n (cid:16) µ ( ni ) j + µ ( nj ) i − µ ( ij ) n (cid:17)i Q kij o L mat = 0 . (97)After we take into account the identities (92) and (93), the conservationlaws (96) and (97) are brought to the final form: F Σ ki = F t ki + ∗ ∇ n (cid:0) F ∆ ikn (cid:1) , (98) ∗ ∇ i (cid:0) F Σ ki (cid:1) = F (cid:18) Σ li T kil − ∆ mnl R klmn − t ij Q kij (cid:19) − L mat ∇ k F. (99)Lowering the index in (98) and antisymmetrizing, we derive the conservationlaw for the spin F Σ [ ij ] + ∗ ∇ n ( F τ ij n ) + Q nl [ i ∆ lj ] n = 0 . (100)This is a generalization of the usual conservation law of the total angularmomentum for the case of nonminimal coupling.19 .3 Rewriting the conservation laws Using the definition (31) and decomposing the connection into the Rieman-nian and non-Riemannian parts (10), we can recast the conservation law(98) into an equivalent form e ∇ j ( F ∆ ikj ) = F (Σ ki − t ki + N nmi ∆ mkn − N nkm ∆ imn ) . (101)In a similar way we can rewrite the conservation law (99). At first, with thehelp of (12) and (13) we notice that F (cid:18) Σ li T kil − t ij Q kij (cid:19) = F ( t li − Σ li ) N kil + F Σ li N ikl . (102)Then substituting here (101) and making use of (10), (31), and the curvaturedecomposition (14), after some algebra we recast (99) into e ∇ j { F (Σ kj +∆ mnj N kmn ) } = − F ∆ mni ( e R kimn − e ∇ k N imn ) − L mat e ∇ k F. (103)For the minimal coupling case, such a conservation law was derived in [46,47]. The importance of this form of the energy-momentum conservationlaw lies in the clear separation of the Riemannian and non-Riemanniangeometrical variables. As we see, the post-Riemannian geometry enters (103)only in the form of the distortion tensor N kji which is coupled only to thehypermomentum current ∆ mni . This means that, in the minimal coupling case, ordinary matter – i.e. without microstructure, ∆ mni = 0 – does not couple to the non-Riemannian geometry. In contrast, in the nonminimalcoupling case, the derivative of the coupling coupling function F on theright-hand side of (103) may lead to a coupling between non-Riemannianstructures and ordinary matter. Every generalized Killing vector field ζ k generates a conserved current. Thiscan be demonstrated from the analysis of the system (101) and (103) asfollows. Let us contract equation (101) with e ∇ i ζ k and equation (103)with ζ k , then subtract the resulting expressions. Note that the contrac-tion t ki e ∇ i ζ k = 0 vanishes because the first factor is a symmetric tensor andthe second one is skew-symmetric. Then after some algebra we derive e ∇ i ζ I i = F ∆ mni L ζ N imn − L mat L ζ F. (104)Here we associate a current with a Killing vector field via ζ I i := F h ζ k Σ ki − ( ∇ m ζ n )∆ mni i . (105)Note that the transposed connection appears here. The right-hand sideof (104) depends linearly on the Lie derivatives along the Killing vector: L ζ F = ζ k e ∇ k F and L ζ N imn = ζ k e ∇ k N imn + ( e ∇ i ζ k ) N kmn + ( e ∇ m ζ k ) N ikn − ( e ∇ k ζ n ) N imk . (106)20hen ζ k is a generalized Killing vector, we have L ζ N imn = 0 in view of(46). Furthermore, recalling that F = F ( g ij , R klji , N kj i ), we find L ζ F = ∂F∂g ij L ζ g ij + ∂F∂R klj i L ζ R klji + ∂F∂N kj i L ζ N kji = 0 , (107)by making use of (46), (47), and (49).As a result, the right-hand side of (104) vanishes for the generalizedKilling vector field, and we conclude that the induced current (105) is con-served e ∇ i ζ I i = 0 . (108)This generalizes the earlier results reported in [30, 48, 49]. In section 6.1.2 wewill show that there is a conserved quantity constructed from the multipolemoments which is a direct counterpart of the induced current (105). It isworthwhile to give an equivalent form of the latter: ζ I i = F h ζ k (Σ ki + ∆ mni N kmn ) − ( e ∇ m ζ n )∆ mni i . (109) Our results contain the Riemannian theory as a special case. Suppose thatthe torsion and the nonmetricity are absent T ij k = 0, Q kij = 0, hence N ijk =0. Then for usual matter without microstructure (i.e. matter with ∆ mni = 0)the canonical and the metrical energy-momentum tensors coincide, Σ ki = t ki . As a result, the conservation law (99) reduces to e ∇ i t ki = 1 F (cid:0) − L mat δ ik − t ki (cid:1) e ∇ i F. (110)This conservation law for the general nonminimal coupling model was de-rived earlier in [26] without using the Noether theorem, directly from thefield equations . The old result established the conservation law for the casein which F = F ( e R ijkl ) depends arbitrarily on the components of the curva-ture tensor, correcting some erroneous derivations in the literature, see [26]for details.Quite remarkably, (110) generalizes the earlier result to the case in whichthe nonminimal coupling function F is a general scalar function of the cur-vature tensor. In this section we derive “master equations of motion” for a general extendedtest body, which is characterized by a set of currents J Aj . (111) Notice a different conventional sign, as compared to our previous work [26]. J Aj dynam-ical currents. The generalized index (capital Latin letters A, B, . . . ) labelsdifferent components of the currents.As the starting point for the derivation of the equations of motion forgeneralized multipole moments, we consider the following conservation law: e ∇ j J Aj = − Λ jBA J Bj − Π A ˙ B K ˙ B . (112)On the right-hand side, we introduce objects that can be called materialcurrents K ˙ A (113)to distinguish them from the dynamical currents J Aj . The number of com-ponents of the dynamical and material currents is different, hence we use adifferent index with dot ˙ A, ˙ B, . . . , the range of which does not coincide withthat of
A, B, . . . . At this stage we do not specify the ranges of both typesof indices, this will be done for the particular examples which we analyzelater. As usual, Einstein’s summation rule over repeated indices is assumedfor the generalized indices as well as for coordinate indices.Both sets of currents J Aj and K ˙ A are constructed from the variablesthat describe the structure and the properties of matter inside the body. Incontrast, the objects Λ jBA , Π A ˙ B , (114)do not depend on the matter, but they are functions of the external classicalfields which act on the body and thereby determine its motion. The list ofsuch external fields include the electromagnetic, gravitational and scalarfields.We will now derive the equations of motion of a test body by utilizingthe covariant expansion method of Synge [7]. For this we need the followingauxiliary formula for the absolute derivative of the integral of an arbitrarybitensor density B x y = B x y ( x, y ) (the latter is a tensorial function oftwo spacetime points): Dds Z Σ( s ) B x y d Σ x = Z Σ( s ) e ∇ x B x y w x d Σ x + Z Σ( s ) v y e ∇ y B x y d Σ x . (115)Here v y := dx y /ds , s is the proper time, Dds = v i e ∇ i , and the integral isperformed over a spatial hypersurface. Note that in our notation the pointto which the index of a bitensor belongs can be directly read from the indexitself; e.g., y n denotes indices at the point y . Furthermore, we will nowassociate the point y with the world-line of the test body under consider-ation. Here σ denotes Synge’s [7] world-function and σ y its first covariantderivative; g yx is the parallel propagator for vectors. For objects with morecomplicated tensorial properties the parallel propagator is straightforwardly22eneralized to G Y X and G ˙ Y ˙ X . We will need these generalized propagatorsto deal with the dynamical and material currents J Aj and K ˙ A . More detailsare collected in the appendix on our conventions.After these preliminaries, we introduce integrated moments for the twotypes of currents via (for n = 0 , , . . . ) j y ··· y n Y = ( − n Z Σ( τ ) σ y · · · σ y n G Y X J X x ′′ d Σ x ′′ , (116) i y ...y n Y y ′ = ( − n Z Σ( τ ) σ y · · · σ y n G Y X g y ′ x ′ J X x ′ w x ′′ d Σ x ′′ , (117) m y ...y n ˙ Y = ( − n Z Σ( τ ) σ y · · · σ y n G ˙ Y ˙ X K ˙ X w x ′′ d Σ x ′′ . (118)Here the densities are constructed from the currents: J X x = √− gJ X x and K ˙ X = √− gK ˙ X . Integrating (112) and making use of (115), we findthe following “master equation of motion” for the generalized multipolemoments: Dds j y ··· y n Y = − n v ( y j y ...y n ) Y + n i ( y ...y n − | Y | y n ) − γ Y Y ′ y ′′ y n +1 (cid:16) i y ...y n Y ′ y ′′ + j y ...y n Y ′ v y ′′ (cid:17) − Λ y ′ Y ′′ Y i y ...y n Y ′′ y ′ − Λ y ′ Y ′′ Y ; y n +1 i y ...y n +1 Y ′′ y ′ − Π Y ˙ Y ′ m y ...y n ˙ Y ′ − Π Y ˙ Y ′ ; y n +1 m y ...y n +1 ˙ Y ′ + ∞ X k =2 k ! h − ( − k n α ( y y ′ y n +1 ...y n + k i y ...y n ) y n +1 ...y n + k Y y ′ +( − k n v y ′ β ( y y ′ y n +1 ...y n + k j y ...y n ) y n +1 ...y n + k Y +( − k γ Y Y ′ y ′′ y n +1 ...y n + k (cid:16) i y ...y n + k Y ′ y ′′ + j y ...y n + k Y ′ v y ′′ (cid:17) − Λ y ′ Y ′′ Y ; y n +1 ...y n + k i y ...y n + k Y ′′ y ′ − Π Y ˙ Y ′ ; y n +1 ...y n + k m y ...y n + k ˙ Y ′ i . (119) To see how the general formalism works, let us consider the motion of elec-trically charged extended test bodies under the influence of the electromag-netic field in flat Minkowski spacetime. This problem was earlier analyzedby means of a different approach in [50].In this case, it is convenient to recast the set of dynamical currents intothe form of a column J Aj = (cid:18) J j Σ kj (cid:19) , (120)23here J j is the electric current and Σ kj is the energy-momentum tensor.Physically, the structure of the dynamical current is crystal clear: the mat-ter elements of an extended body are characterized by the two types of“charges”, the electrical charge (the upper component) and the mass (thelower component).The generalized conservation law comprises two components of differenttensor dimension: e ∇ j (cid:18) J j Σ kj (cid:19) = (cid:18) − F kj J j (cid:19) , (121)where the lower component of the right-hand side describes the usual Lorentzforce.Accordingly, we indeed recover for the dynamical current (120) the con-servation law in the form (112), where K ˙ B = 0 andΛ jBA = (cid:18) F j k (cid:19) . (122)The generalized moments (116)-(118) have the same column structure, re-flecting the two physical charges of matter: j y ··· y n Y = (cid:18) j y ··· y n p y ··· y n y (cid:19) , (123) i y ··· y n Y y ′ = (cid:18) i y ··· y n y ′ k y ··· y n y y ′ (cid:19) , (124)whereas m y ...y n ˙ Y = 0.As a result, the master equation (119) reduces to the coupled system ofthe two sets of equations for the moments: Dds j y ··· y n = − n v ( y j y ...y n ) + n i ( y ...y n ) , (125) Dds p y ··· y n y = − n v ( y p y ...y n ) y + n k ( y ...y n − | y | y n ) − F y ′ y i y ...y n y ′ − ∞ X k =1 k ! F y ′ y ; y n +1 ...y n + k i y ...y n + k y ′ . (126)These equations should be compared to those of [50]. We are now in a position to derive the equations of motion for extendedtest bodies in metric-affine gravity. As a preliminary step, we rewrite theconservation laws (98) and (99) as [22]: e ∇ j ∆ ikj = − U jmnik ∆ mnj + Σ ki − t ki , (127) e ∇ j Σ kj = − V j nk Σ nj − R kjmn ∆ mnj − Q kjn t nj − A k L mat . (128)24ere we denoted A k := e ∇ k log F , and U jmnik := A j δ im δ kn − N jmi δ kn + N j kn δ im , (129) V jnk := A j δ kn + N kjn . (130)Introducing the dynamical current J Aj = (cid:18) ∆ ikj Σ kj (cid:19) , (131)and the material current K ˙ A = (cid:18) t ik L mat (cid:19) , (132)we then recast the system (127) and (128) into the generic conservation law(112), where we now haveΛ jBA = (cid:18) U ji ′ k ′ ik − δ ij δ kk ′ R kji ′ k ′ V jk ′ k (cid:19) , (133)Π A ˙ B = (cid:18) δ ii ′ δ kk ′ Q ki ′ k ′ A k (cid:19) . (134)Like in the previous example of an electrically charged body, the matterelements in metric-affine gravity are also characterized by two “charges”: thecanonical hypermomentum (upper component) and the canonical energy-momentum (lower component). This is reflected in the column structure ofthe dynamical current (131). The material current (132) takes into accountthe metrical energy-momentum and the matter Lagrangian related to thenonminimal coupling. The multi-index A = { ik, k } , whereas ˙ A = { ik, } .Accordingly, the generalized propagator reads G Y X = (cid:18) g y x g y x g y x (cid:19) , (135)and we easily construct the expansion coefficients of its derivatives from thecorresponding expansions of the derivatives of vector propagator g yx : γ Y Y y ...y k +2 = (cid:18) γ { y ˜ y }{ y ′ y ′′ } y ...y k +2 γ y y ′ y ...y k +2 (cid:19) , (136)where we denoted γ { y ˜ y }{ y ′ y ′′ } y ...y k +2 = γ y y ′ y ...y k +2 δ ˜ yy ′′ + γ ˜ yy ′′ y ...y k +2 δ y y ′ . (137)In particular, for the first expansion coefficient ( k = 1), we find γ { y ˜ y }{ y ′ y ′′ } y y = 12 (cid:16) e R y y ′ y y δ ˜ yy ′′ + e R ˜ yy ′′ y y δ y y ′ (cid:17) , (138) γ y y ′ y y = 12 e R y y ′ y y . (139)25or completeness, let us write down also another generalized propagator G ˙ Y ˙ X = (cid:18) g y x g y x
00 1 (cid:19) . (140)The last step is to write the generalized moments (116)-(118) in termsof their components: j y ··· y n Y = (cid:18) h y ··· y n y ′ y ′′ p y ··· y n y ′ (cid:19) , (141) i y ...y n Y y = (cid:18) q y ··· y n y ′ y ′′ y k y ··· y n y ′ y (cid:19) , (142) m y ...y n ˙ Y = (cid:18) µ y ··· y n y ′ y ′′ ξ y ··· y n (cid:19) . (143)For the two most important moments, “ h ” stands for the hypermomentum,whereas “ p ” stands for the momentum. Finally, substituting all the aboveinto the “master equation” (119), we obtain the system of multipolar equa-tions of motion for extended test bodies in metric-affine gravity:
Dds h y ...y n y a y b = − n v ( y h y ...y n ) y a y b + n q ( y ...y n − | y a y b | y n ) + k y ...y n y b y a − µ y ...y n y a y b − e R y a y ′ y ′′ y n +1 (cid:16) q y ...y n +1 y ′ y b y ′′ + v y ′′ h y ...y n +1 y ′ y b (cid:17) − e R y b y ′ y ′′ y n +1 (cid:16) q y ...y n +1 y a y ′ y ′′ + v y ′′ h y ...y n +1 y a y ′ (cid:17) − U y y ′ y ′′ y a y b q y ...y n y ′ y ′′ y − U y y ′ y ′′ y a y b ; y n +1 q y ...y n +1 y ′ y ′′ y + ∞ X k =2 k ! " ( − k v y ′ n β ( y y ′ y n +1 ...y n + k h y ...y n ) y n +1 ...y n + k y a y b +( − k γ y a y ′ y ′′ y n +1 ...y n + k (cid:16) q y ...y n + k y ′ y b y ′′ + v y ′′ h y ...y n + k y ′ y b (cid:17) +( − k γ y b y ′ y ′′ y n +1 ...y n + k (cid:16) q y ...y n + k y a y ′ y ′′ + v y ′′ h y ...y n + k y a y ′ (cid:17) − ( − k n α ( y y ′ y n +1 ...y n + k q y ...y n ) y n +1 ...y n + k y a y b y ′ − U y y ′ y ′′ y a y b ; y n +1 ...y n + k q y ...y n + k y ′ y ′′ y , (144) Note that in order to facilitate the comparison with our previous work [51], we providein appendix the explicit form of integrated conservation laws (127) and (128), as well asthe generalized integrated moments (141) – (143) in the notation used in [51]. ds p y ...y n y a = − n v ( y p y ...y n ) y a + n k ( y ...y n − | y a | y n ) − A y a ξ y ...y n − A y a ; y n +1 ξ y ...y n +1 − V y ′′ y ′ y a k y ...y n y ′ y ′′ − V y ′′ y ′ y a ; y n +1 k y ...y n +1 y ′ y ′′ − e R y a y ′ y ′′ y n +1 (cid:16) k y ...y n +1 y ′ y ′′ + v y ′′ p y ...y n +1 y ′ (cid:17) − R y a y y ′ y ′′ q y ...y n y ′ y ′′ y − R y a y y ′ y ′′ ; y n +1 q y ...y n +1 y ′ y ′′ y − Q y a y ′′ y ′ µ y ...y n y ′ y ′′ − Q y a y ′′ y ′ ; y n +1 µ y ...y n +1 y ′ y ′′ + ∞ X k =2 k ! " ( − k n v y ′ β ( y y ′ y n +1 ...y n + k p y ...y n ) y n +1 ...y n + k y a +( − k γ y a y ′ y ′′ y n +1 ...y n + k (cid:16) k y ...y n + k y ′ y ′′ + v y ′′ p y ...y n + k y ′ (cid:17) − ( − k n α ( y y ′ y n +1 ...y n + k k y ...y n ) y n +1 ...y n + k y a y ′ − R y a y y ′ y ′′ ; y n +1 ...y n + k q y ...y n + k y ′ y ′′ y − V y ′′ y ′ y a ; y n +1 ...y n + k k y ...y n + k y ′ y ′′ − Q y a y ′′ y ′ ; y n +1 ...y n + k µ y ...y n + k y ′ y ′′ − A y a ; y n +1 ...y n + k ξ y ...y n + k . (145) The general equations of motion (144) and (145) are valid to any multipolarorder. In the following sections we focus on some special cases, in particularwe work out the two lowest multipolar orders of approximation, and considerthe explicit form of the equations of motion in special geometries.
From (144) and (145), we can derive the general pole-dipole equations ofmotion. The relevant moments to be kept at this order of approximationare: p a , p ab , h ab , q abc , k ab , k abc , µ ab , µ abc , ξ a , and ξ . Since all objects are nowevaluated on the world-line, we switch back to the usual tensor notation.For n = 1 and n = 0, eq. (144) yields0 = k acb − µ abc + q bca − v a h bc , (146) Dds h ab = k ba − µ ab − U cdeab q dec . (147)27urthermore for n = 2 , , k ( a | c | b ) − v ( a p b ) c , (148) Dds p ab = k ba − v a p b − A b ξ a − V dcb k acd − Q bdc µ acd , (149) Dds p a = − V cba k bc − R adbc q bcd − Q acb µ bc − A a ξ − e R acdb (cid:16) k bcd + v d p bc (cid:17) − V dca ; b k bcd − Q adc ; b µ bcd − A a ; b ξ b . (150) Rewriting the equations of motion
Let us decompose (146) and (147)into symmetric and skew-symmetric parts: µ abc = k a ( bc ) + q ( bc ) a − v a h ( bc ) , (151)0 = − k a [ bc ] + q [ bc ] a − v a h [ bc ] , (152) µ ab = − Dds h ( ab ) + k ( ab ) − U cde ( ab ) q dec , (153) Dds h [ ab ] = − k [ ab ] − U cde [ ab ] q dec . (154)As a result, we can express the moments symmetric in the two last indices µ ab = µ ( ab ) and µ cab = µ c ( ab ) (in general, this is possible also for an arbitraryorder µ c ...c n ab = µ c ...c n ( ab ) ) in terms of the other moments.Let us denote the skew-symmetric part s ab := h [ ab ] , as this simplifiesgreatly the subsequent manipulations and the comparison with [51].The system of the two equations (148) and (152) can be resolved in termsof the 3rd rank k -moment. The result reads explicitly k abc = v a p cb + v c (cid:16) p [ ab ] − s ab (cid:17) + v b (cid:16) p [ ac ] − s ac (cid:17) + v a (cid:16) p [ bc ] − s bc (cid:17) + q [ ab ] c + q [ ac ] b + q [ bc ] a . (155)This yields some useful relations: k a [ bc ] = − v a s bc + q [ bc ] a , (156) k [ ab ] c = v [ a p | c | b ] + v c (cid:16) p [ ab ] − s ab (cid:17) + q [ ab ] c . (157)The next step is to use the equations (151), (153) together with (155)and to substitute the µ -moments and k -moments into (147) and (149)-(150).This yields a system that depends only on the p, h, q, and ξ moments.Let us start with the analysis of (150). The latter contains the combina-tion k [ b | c | d ] + v [ d p b ] c where the skew symmetry is imposed by the contractionwith the Riemann curvature tensor, which is antisymmetric in the two lastindices. Making use of (155), we derive k [ a | c | b ] + v [ b p a ] c = κ abc + κ acb − κ bca , (158)28here we introduced the abbreviation κ abc := v c (cid:16) p [ ab ] − s ab (cid:17) + q [ ab ] c . (159)Note that by construction κ abc = κ [ ab ] c .Then by making use of the Ricci identity we find − e R acdb (cid:16) k bcd + v d p bc (cid:17) = e R abcd h q [ cd ] b + v b (cid:16) p [ cd ] − s cd (cid:17)i . (160)Substituting k bc from (149) and µ bc from (153), we find after some algebra − V cba k bc − Q acb µ bc = − A b Dp ba ds − N acd Dh cd ds − (cid:16) p a + N acd h cd (cid:17) v b A b − A a A b ξ b − k bac A b A c + ( N anb N dcn − N acn N dnb ) q cbd . (161)Further simplification is achieved by noticing that v b A b = DAds , k bac A b A c = p ca A c DAds , (162)where we used (148) and recalled that A b = A ; b .Analogously, taking k b [ cd ] from (156) and µ bcd from (151), we derive − V dca ; b k bcd − Q adc ; b µ bcd = − A b ; c k cab + N acd ; b q cdb − N acd ; b v b h cd . (163)We can again use A b = A ; b and (148) to simplify − A b ; c k cab = − p ba DA b ds . (164)After these preliminary calculations, we substitute (160)-(164) into (150)to recast the latter into Dds (cid:16)
F p a + F N acd h cd + p ba b ∇ b F (cid:17) = F e R abcd v b (cid:16) p [ cd ] − s cd (cid:17) − F q cbd h R adcb − e R adcb − N acb ; d − N anb N dcn + N acn N dnb i − F A a (cid:16) ξ + ξ b A b (cid:17) − F ξ b A a ; b . (165)Finally, combining (147) and (149) to eliminate k ba we derive the equa-tion Dds (cid:16) p ab − h ab (cid:17) = µ ab − v a (cid:16) p b + N bcd h cd (cid:17) + q cda N bcd − q cbd N dca + q acd N dbc − ξ a A b + ( q abc − k abc ) A c . (166)Following [51], we introduce the total orbital and the total spin angularmoments L ab := 2 p [ ab ] , S ab := − h [ ab ] , (167)29nd define the generalized total energy-momentum 4-vector and the gener-alized total angular momentum by P a := F ( p a + N acd h cd ) + p ba b ∇ b F, (168) J ab := F ( L ab + S ab ) . (169)Then, taking into account the identity (14), which with the help of theraising and lowering of indices can be recast into e ∇ a N dcb = − R adcb + e R adcb + N acb ; d + N anb N dcn − N dnb N acn , (170)we rewrite the pole-dipole equations of motion (165) and (166) in the finalform D P a ds = 12 e R abcd v b J cd + F q cbd e ∇ a N dcb − ξ e ∇ a F − ξ b e ∇ b e ∇ a F, (171) D J ab ds = − v [ a P b ] + 2 F ( q cd [ a N b ] cd + q c [ a | d | N dcb ] + q [ a | cd | N db ] c ) − ξ [ a e ∇ b ] F. (172)The last equation arises as the skew-symmetric part of (166), whereas thesymmetric part of the latter is a non-dynamical relation that determines the µ ab moment µ ab = D Υ ab ds + 1 F v ( a (cid:16) P b ) + J b ) c A c (cid:17) + ξ ( a A b ) − q cd ( a N b ) cd + q c ( a | d | N dcb ) − q ( a | cd | N db ) c + ( q [ ac ] b + q [ bc ] a − q ( ab ) c ) A c . (173)Here the symmetric moment of the total hypermomentum is introduced viaΥ ab := p ( ab ) − h ( ab ) . (174) The equations of motion for the multipole moments are derived from theconservation laws of the energy-momentum and the hypermomentum cur-rents Σ ki and ∆ mni . In section 4.4 we demonstrated that every generalizedKilling vector induces a conserved current constructed from Σ ki and ∆ mni .Quite remarkably, there is a direct counterpart of such an induced currentbuilt from the multipole moments.Let ζ k be a generalized Killing vector, and let us contract equation (171)with ζ a and equation (172) with e ∇ a ζ b , and then take the sum. This yields Dds (cid:18) P a ζ a + 12 J ab e ∇ a ζ b (cid:19) = F q cbd L ζ N dcb − ξ L ζ F − ξ a e ∇ a L ζ F. (175)30n the right-hand side, the Lie derivatives of the distortion tensor and ofthe coupling function both vanish in view of (46) and (107).Consequently, we conclude that for every generalized Killing vector fieldthe quantity P a ζ a + 12 J ab e ∇ a ζ b = const (176)is conserved along the trajectory of an extended body.We thus observe a complete consistency between (104), (105), (109) and(176), (175). Let us look more carefully at how the post-Riemannian pieces of the grav-itational field couple to extended test bodies. At first, we notice that thegeneralized energy-momentum vector (168) contains the term N acd h cd thatdescribes the direct interaction of the distortion (torsion plus nonmetricity)with the intrinsic dipole moment of the hypermomentum. Decomposingthe latter into the skew-symmetric (spin) part and the symmetric (properhypermomentum + dilation) part, we find N acd h cd = − N a [ cd ] S cd − Q acd h ( cd ) . (177)Here we made use of (13). This is well consistent with the gauge-theoreticstructure of metric-affine gravity. The second term shows that the intrinsicproper hypermomentum and the dilation moment couple to the nonmetric-ity, whereas the first term displays the typical spin-torsion coupling.Similar observations can be made for the coupling of higher momentswhich appear on the right-hand sides of (171) and (172) - and thus determinethe force and torque acting on an extended body due to the post-Riemanniangravitational field. In order to see this, let us introduce the decomposition − q abc = d q abc + s q cab (178)into the two pieces d q abc := 12 (cid:16) q [ ac ] b + q [ bc ] a − q ( ab ) c (cid:17) , (179) s q abc := 12 (cid:16) q [ ab ] c + q [ ac ] b − q [ bc ] a (cid:17) . (180)The overscript “ d ” and “ s ” notation shows the relevance of these objects tothe dilation plus proper hypermomentum and to the spin, respectively. Byconstruction, we have the following algebraic properties d q [ ab ] c ≡ , s q ( ab ) c ≡ . (181)31aking use of the decomposition (178) and of the explicit structure ofthe distortion (11), we then recast the equations of motion (171) and (172)into D P a ds = 12 e R abcd v b J cd + F s q cbd e ∇ a T cbd + F d q cbd e ∇ a Q dcb − ξ e ∇ a F − ξ b e ∇ b e ∇ a F, (182) D J ab ds = − v [ a P b ] + 2 F ( s q cd [ a T cdb ] + 2 s q [ a | cd | T b ] cd )+ 2 F ( d q cd [ a Q b ] cd + 2 d q [ a | dc | Q cdb ] ) − ξ [ a e ∇ b ] F. (183)Now we clearly see the fine structure of the coupling of extended bodies tothe post-Riemannian geometry. The first lines in the equations of motiondescribe the usual Mathisson-Papapetrou force and torque. They depend onthe Riemannian geometry only. A body with the nontrivial moment (180)is affected by the torsion field, whereas the nontrivial moment (179) feelsthe nonmetricity. This explains the different physical meaning of the highermoments (179) and (180). In addition, the last lines in (182) and (183)describe contributions due to the nonminimal coupling. At the monopolar order we have nontrivial moments p a , k ab , µ ab and ξ . Thenontrivial equations of motion then arise from (144) for n = 0 and from(145) for n = 1 , n = 0:0 = k ba − µ ab , (184)0 = k ba − v a p b , (185) Dp a ds = − V cba k bc − Q acb µ bc − A a ξ. (186)The first two equations (184) and (185) yield k [ ab ] = 0 , v [ a p b ] = 0 , (187)and substituting (184), (185) and (187) into (186) we find D ( F p a ) ds = − ξ e ∇ a F. (188)From (187) we have p a = M v a with the mass M := v a p a , and this allows torecast (188) into the final form M Dv a ds = − ξ ( g ab − v a v b ) e ∇ b FF . (189)32ence, in general the motion of nonminimally coupled monopole test bod-ies is nongeodetic. Furthermore, the general monopole equation of mo-tion (189) reveals an interesting feature of theories with nonminimal cou-pling. There is an “indirect” coupling, i.e. through the coupling function F ( g ij , R ijkl , T ij k , Q ij k ), of post-Riemannian spacetime features to structure-less test bodies. In Weyl-Cartan spacetime the nonmetricity reads Q kij = Q k g ij , where Q k is the Weyl covector. Hence the distortion is given by N kji = K kji + 12 (cid:0) Q i g kj − Q k δ ij − Q j δ ik (cid:1) . (190)The contortion tensor is constructed from the torsion, K kji = −
12 ( T kji + T ikj + T ijk ) . (191)As a result, the generalized momentum (168) in Weyl-Cartan spacetimetakes the form: P a = F p a − F (cid:16) K acd S cd − Q b S ba + Q a D (cid:17) + p ba e ∇ b F. (192)Here we introduced the intrinsic dilation moment D := g ab h ab .Substituting the distortion (190) into (171) and (172), we find the pole-dipole equations of motion in Weyl-Cartan spacetime: D P a ds = 12 e R abcd v b J cd + F s q cbd e ∇ a T cbd + Z b e ∇ a Q b − ξ e ∇ a F − ξ b e ∇ b e ∇ a F, (193) D J ab ds = − v [ a P b ] + 2 F ( s q cd [ a T cdb ] + 2 s q [ a | cd | T b ] cd )+2 F Z [ a Q b ] − ξ [ a e ∇ b ] F. (194)Here we introduced the trace of the modified moment (179) Z a := g bc d q bca = 12 g bc (cid:16) q bac − q bca − q abc (cid:17) . (195)It is coupled to the Weyl nonmetricity. Weyl spacetime [52] is obtained as a special case of the results above forvanishing torsion. Hence the contortion is trivial K abc = 0 . (196)Taking this into account, the generalized momentum (192) and the equationsof motion (193) and (194) are simplified even further.It is interesting to note that besides a direct coupling of the dilationmoment to the Weyl nonmetricity on the right-hand sides of (193) and (194),there is also a nontrivial coupling of the spin to the nonmetricity in (192).33 .1.7 Riemann-Cartan spacetime Another special case is obtained when the Weyl vector vanishes Q a = 0.Equations (192)-(194) then reproduce in a covariant way the findings ofYasskin and Stoeger [53] when the coupling is minimal ( F = 1). For non-minimal coupling we recover our earlier results in [51]. The geometrical arena of scalar-tensor theories is the Riemannian space-time, hence N kji = 0 which means that both the torsion T ijk = 0 and thenonmetricity Q kij = 0 vanish.Scalar-tensor theories have a long history and they belong to the moststraightforward generalizations of Einstein’s general relativity theory. In theso-called Brans-Dicke theory [54, 55, 56, 57, 58] a scalar field is introducedas a variable “gravitational coupling constant” (which is thus more correctlycalled a “gravitational coupling function”). Similar formalisms were devel-oped earlier by Jordan [59, 60], Thiry [61] and their collaborators usingthe 5-dimensional Kaluza-Klein approach. An overview of the history anddevelopments of scalar-tensor theories can be found in [62, 63, 64, 65].Surprisingly little attention was paid to the equations of motion of ex-tended test bodies in scalar-tensor theories. Some early discussions can befound in [56, 66, 67], and in [68] the dynamics of compact bodies was thor-oughly studied in the framework of the post-Newtonian formalism. We consider the class of scalar-tensor theories along the lines of [68] wherethe action I = R d x J L is constructed on the manifold with the spacetimemetric J g ij . The Lagrangian density J L = J √− g J L has the following form J L = 12 κ (cid:18) − F e R ( J g ) + J g ij J γ AB ∂ i ϕ A ∂ j ϕ B − J U (cid:19) + L m ( ψ, ∂ψ, J g ij ) . (197)This action is an extension of standard Brans-Dicke theory [69] to the casewhen we have a multiplet of scalar fields ϕ A (capital Latin indices A, B, C =1 , . . . , N label the components of the multiplet). Here κ = 8 πG/c denotesEinstein’s gravitational constant and in general we have several functions ofscalar fields, F = F ( ϕ A ) , J U = J U ( ϕ A ) , J γ AB = J γ AB ( ϕ A ) . (198)The Lagrangian L m ( ψ, ∂ψ, J g ij ) depends on the matter fields ψ and the grav-itational field. 34he metric J g ij determines angles and intervals in the Jordan referenceframe . The Riemannian curvature scalar e R ( J g ) is constructed from the Jor-dan metric. With the help of the conformal transformation J g ij −→ g ij = F J g ij (199)we obtain a different metric on the spacetime manifold. This is called anEinstein reference frame. In the
Einstein reference frame the Lagrangian density in the scalar-tensortheory reads L = √− gL with L = 12 κ (cid:16) − e R + g ij γ AB ∂ i ϕ A ∂ j ϕ B − U (cid:17) + 1 F L mat ( ψ, ∂ψ, F − g ij ) . (200)Here the scalar curvature e R ( g ) is constructed from the Einstein metric g ij ,and γ AB = 1 F ( J γ AB + 6 F ,A F ,B ) , U = 1 F J U . (201)The metrical energy-momentum tensor for the matter Lagrangian L mat is constructed as usual via (60). From the Noether theorem we find that itsatisfies the generalized conservation law e ∇ i t ij = 1 F (4 t ij − g ij t ) ∂ i F. (202)Here t := g ij t ij . The derivation is given in [23].To begin with, we recast (202) into an equivalent form e ∇ i t ij = − A i (cid:0) Ξ ij + t ij (cid:1) (203)by introducing A i := ∂ i log F − , Ξ ij := − g ij t/ . (204) We now derive the equations of motion for extended test bodies in scalar-tensor gravity by making use of the master equations obtained for the generalcase of MAG. In scalar-tensor theory, the hypermomentum is zero ∆ ikj = 0.Following the general scheme, we introduce the dynamical current which isa special case of the MAG current: J Aj = (cid:18) kj (cid:19) . (205)Taking into account the structure of the conservation laws (203) and (204),we define the material current K ˙ A = (cid:18) t ik − t/ (cid:19) , (206)35e then recast the system (203) into the generic conservation law, where wenow have Λ jBA = (cid:18) − δ ij δ kk ′ V jk ′ k (cid:19) , (207)Π A ˙ B = (cid:18) δ ii ′ δ kk ′ A k (cid:19) . (208)In accordance with (203) and (204) we now have V jnk = A j δ kn , A i = ∂ i log F − . (209)In view of (205)-(206) the generalized moments read j y ··· y n Y = (cid:18) p y ··· y n y ′ (cid:19) , (210) i y ...y n Y y = (cid:18) k y ··· y n y ′ y (cid:19) , (211) m y ...y n ˙ Y = (cid:18) µ y ··· y n y ′ y ′′ ξ y ··· y n (cid:19) . (212)It is worthwhile to notice that although we formally use a different notationfor the upper and lower components of the moment (212), they are notindependent. Recalling (206), we have the obvious relation ξ y ...y n = − g y ′ y ′′ µ y ...y n y ′ y ′′ , (213)which we will take into account when rewriting the equations of motion.Now we can make use of the general MAG equations of motion (119).The first equation is a degenerate version of (144) since we have h y ...y n y a y b =0 and q y ...y n − y a y b y n = 0, and we are left with the algebraic relation k y ...y n y b y a = µ y ...y n y a y b . (214)for all n . Hence we find ξ y ...y n = − g y ′ y ′′ k y ...y n y ′ y ′′ , (215)and the second equation (145) then gives rise to the equations of motion in36calar-tensor theory Dds p y ...y n y a = − n v ( y p y ...y n ) y a + n k ( y ...y n − | y a | y n ) − A y a ξ y ...y n − A y a ; y n +1 ξ y ...y n +1 − V y ′′ y ′ y a k y ...y n y ′ y ′′ − V y ′′ y ′ y a ; y n +1 k y ...y n +1 y ′ y ′′ − e R y a y ′ y ′′ y n +1 (cid:16) k y ...y n +1 y ′ y ′′ + v y ′′ p y ...y n +1 y ′ (cid:17) + ∞ X k =2 k ! " − ( − k n α ( y y ′ y n +1 ...y n + k k y ...y n ) y n +1 ...y n + k y a y ′ +( − k n v y ′ β ( y y ′ y n +1 ...y n + k p y ...y n ) y n +1 ...y n + k y a +( − k γ y a y ′ y ′′ y n +1 ...y n + k (cid:16) k y ...y n + k y ′ y ′′ + v y ′′ p y ...y n + k y ′ (cid:17) − V y ′′ y ′ y a ; y n +1 ...y n + k k y ...y n + k y ′ y ′′ − A y a ; y n +1 ...y n + k ξ y ...y n + k . (216)The system (216) is valid up to any multipole order. In the following wespecialize it to the dipole and the monopole case. For n = 2 , , k ( a | c | b ) − v ( a p b ) c , (217) Dds p ab = k ba − v a p b − A b ξ a − V dcb k acd , (218) Dds p a = − V cba k bc − V dca ; b k bcd − A a ; b ξ b − A a ξ − e R acdb (cid:16) k bcd + v d p bc (cid:17) . (219)Taking into account that k a [ bc ] = 0, we resolve (217) in a standard way tofind explicitly k abc = v a p cb + v c p [ ab ] + v b p [ ac ] + v a p [ bc ] . (220)In view of (215), we have in addition ξ a = − g bc k abc , (221) ξ = − g ab k ab . (222)Then repeating the same algebra as we did in MAG, we recast the system(218) and (219) into D P a ds = 12 e R abcd v b J cd − ξ e ∇ a F − − ξ b e ∇ b e ∇ a F − , (223) D J ab ds = − v [ a P b ] − ξ [ a e ∇ b ] F − . (224)37ere, following [51], we have the total orbital angular moment L ab := 2 p [ ab ] .Whereas the generalized total energy-momentum 4-vector and the general-ized total angular momentum are introduced by P a := F − p a + p ba e ∇ b F − , (225) J ab := F − L ab . (226) At the monopolar order we find from eq. (216) for n = 1 , n = 0:0 = k ba − v a p b , (227) Dp a ds = − V cba k bc − A a ξ. (228)Making use of k [ ab ] = 0, the first equation yields v [ a p b ] = 0, hence we have p a = M v a (229)with the mass M := v a p a . Substituting (209), (227) and (229) into (228)we find D ( F − M v a ) ds = − ξ e ∇ a F − . (230)Contracting this with v a , we derive D ( F − M ) ds = − ξv a e ∇ a F − , (231)we write (230) in the final form M Dv a ds = − ξ ( g ab − v a v b ) e ∇ b F − F − . (232)Combining (222) with (227) we find ξ = − v a p a − M . (233)Substituting this into (232), we obtain Dv a ds = − ( g ab − v a v b ) e ∇ b FF . (234)Quite remarkably, we thus find that the dynamics of an extended test bodyin the monopole approximation is independent of the body’s mass. In caseof a trivial coupling function F , equation (234) reproduces the well knowngeneral relativistic result.Interestingly, the mass of a body is not constant. Substituting (233) into(231) we can solve the resulting differential equation to find explicitly thedependence of mass on the scalar function: M = F M with M =const.38able 1: Overview: MAG equations of motion.Lagrangian and conservation laws L = V ( g ij , R ijkl , N kij ) + F ( g ij , R klij , N kli ) L mat ( ψ A , ∇ i ψ A ) e ∇ j ( F ∆ ikj ) = F (Σ ki − t ki + N nmi ∆ mkn − N nkm ∆ imn ) e ∇ j { F (Σ kj + ∆ mnj N kmn ) } = − F ∆ mni ( e R kimn − e ∇ k N imn ) − L mat e ∇ k F Equations of motion (any order)See equations (144) and (145).Equations of motion (pole-dipole order) D P a ds = e R abcd v b J cd + F q cbd e ∇ a N dcb − ξ e ∇ a F − ξ b e ∇ b e ∇ a F D J ab ds = − v [ a P b ] + 2 F ( q cd [ a N b ] cd + q c [ a | d | N dcb ] + q [ a | cd | N db ] c ) − ξ [ a e ∇ b ] FL ab = 2 p [ ab ] S ab = − h [ ab ] P a = F ( p a + N acd h cd ) + p ba b ∇ b F J ab = F ( L ab + S ab )Equations of motion (monopole order) M Dv a ds = − ξ ( g ab − v a v b ) e ∇ b FF M = p a v a L = κ (cid:16) − e R + g ij γ AB ∂ i ϕ A ∂ j ϕ B − U (cid:17) + F L mat ( ψ, ∂ψ, F − g ij ) F = F ( ϕ A ) U = U ( ϕ A ) γ AB = γ AB ( ϕ A ) e ∇ i t ij = F (4 t ij − g ij t ) ∂ i F Equations of motion (any order)See equation (216).Equations of motion (pole-dipole order) D P a ds = e R abcd v b J cd − ξ e ∇ a F − − ξ b e ∇ b e ∇ a F − D J ab ds = − v [ a P b ] − ξ [ a e ∇ b ] F − L ab = 2 p [ ab ] P a = F − p a + p ba e ∇ b F − J ab = F − L ab Equations of motion (monopole order) Dv a ds = − ( g ab − v a v b ) e ∇ b FF Conclusions
We have presented a general multipolar framework of covariant test bodyequations of motion for standard metric-affine gravity, as well as its exten-sions with nonminimal coupling between matter and gravity. Our resultscover gauge theories of gravity (based on spacetime symmetry groups), andvarious so-called f ( R ) models (and their generalizations), as well as scalar-tensor gravity.Our results unify and extend a whole set of works [70, 71, 72, 53, 73, 74,75, 26, 76, 51, 77, 78]. In particular they can be viewed as a completion ofthe program initiated in [73], in which a noncovariant Papapetrou [2] typeof approach was used. The general equations of motion (144) and (145)cover all of the previously reported cases. As demonstrated explicitly, themaster equation (119) allows for a quick adoption to any physical theory, assoon as the conservation laws and (multi-)current structure is fixed. Table1 contains an overview of our main results in the context of metric-affinegravity. In particular, the explicit equations of motion at the pole-dipole andmonopole order are given. We stress that these equations hold for the mostgeneral case, i.e. including a general nonminimal coupling between matterand gravity. The results for standard (minimal) MAG are easily recoveredby choosing a trivial coupling function F .Our analysis reveals how the new geometrical structures in generalizedtheories of gravity couple to matter, which in turn should be used for thedesign of experimental tests of gravity beyond the Einsteinian (purely Rie-mannian) geometrical picture. In the case of minimal coupling, we have oncemore confirmed – now in a very general context – that only matter matterwith microstructure (such as the intrinsic hypermomentum, including spin,dilation and shear charges) allows for the detection of post-Riemannianstructures. However, in gravitational theories with nonminimal coupling,there seems to be a loophole which may proof to be interesting for possibleexperiments; i.e. there is an indirect coupling of new geometrical quanti-ties to regular matter via the nonminimal coupling function F . This maybe exploited to devise new strategies to detect possible post-Riemannianspacetime features in future experiments.In addition to the results in MAG, we have explicitly worked out thetest body equations of motion for a very general class of scalar-tensor grav-itational theories. Table 2 contains an overview of our main results in thecontext of this theory class. Again the equations of motion at the pole-dipoleand monopole order for a general coupling function F , which now dependson the scalar degrees of freedom, are explicitly given.We hope that our covariant unified multipolar framework sheds morelight on the systematic test of gravitational theories by means of extendedand microstructured test bodies. We would like to conclude with a statementby Einstein [79] who stressed that“[...] the question whether this continuum has a Euclidean,Riemannian, or any other structure is a question of physics41roper which must be answered by experience, and not a ques-tion of a convention to be chosen on grounds of mere expediency.” We would like to thank A. Trautman (University of Warsaw), W.G. Dixon(University of Cambridge), J. Madore (University of Paris South), and W.Tulczyjew (INFN Napoli) for sharing their insights into gravitational mul-tipole formalisms and discussing their pioneering works with us. Further-more, we would like to thank F.W. Hehl (University of Cologne) for fruitfuldiscussion on gauge gravity models, in particular on Metric-Affine Grav-ity (MAG). D.P. was supported by the Deutsche Forschungsgemeinschaft(DFG) through the grants LA-905/8-1/2 and SFB 1128/1 (geo-Q).
AppendixA Conventions & Symbols
In the following we summarize our conventions, and collect some frequentlyused formulas. A directory of symbols used throughout the text can befound in tables 3, 4, 5.For an arbitrary k -tensor T a ...a k , the symmetrization and antisymmetriza-tion are defined by T ( a ...a k ) := 1 k ! k ! X I =1 T π I { a ...a k } , (235) T [ a ...a k ] := 1 k ! k ! X I =1 ( − | π I | T π I { a ...a k } , (236)where the sum is taken over all possible permutations (symbolically denotedby π I { a . . . a k } ) of its k indices. As is well-known, the number of such per-mutations is equal to k !. The sign factor depends on whether a permutationis even ( | π | = 0) or odd ( | π | = 1). The number of independent compo-nents of the totally symmetric tensor T ( a ...a k ) of rank k in n dimensions isequal to the binomial coefficient (cid:0) n − kk (cid:1) = ( n − k )! / [ k !( n − T [ a ...a k ] of rank k in n dimensions is equal to the binomial coefficient (cid:0) nk (cid:1) = n ! / [ k !( n − k )!]. For example, for a second rank tensor T ab the sym-metrization yields a tensor T ( ab ) = ( T ab + T ba ) with 10 independent compo-nents, and the antisymmetrization yields another tensor T [ ab ] = ( T ab − T ba )with 6 independent components.In the derivation of the equations of motion we made use of the bitensorformalism, see, e.g., [7, 8, 80] for introductions and references. In particular,the world-function is defined as an integral σ ( x, y ) := ǫ (cid:18) y R x dτ (cid:19) over the42able 3: Directory of symbols.Symbol ExplanationGeometrical quantities x a , s Coordinates, proper time g ab Metric δ ab Kronecker symbol g Determinant of the metric σ World-functionΓ abc
Connection e Γ abc Riemannian connectionΓ abc
Transposed connection R abcd Curvature Q abc Nonmetricity Q a Weyl covector T abc Torsion K abc Contortion N abc Distortion R ab , R Ricci tensor / scalar( σ AB ) j i Generators of general coordinate transformations V, V Gravitational Lagrangian (density)( H ...... , H ...... ) , ( M ... , M ... ) Generalized gravitational field momenta (densities) E abb , E abb Gravitational hypermomentum (density) E ab , E ab Generalized gravitational energy-momentum (den-sity) I General action L, L Total Lagrangian (density)Φ J General set of fields ζ a (Generalized) Killing vector J Aj , J Aj General dynamical currents (densities) J g ij Jordan frame metric J L, J L Total Lagrangian (density) in Jordan frame43able 4: Directory of symbols.Symbol ExplanationMatter quantities ψ A General matter field L mat , L mat Matter Lagrangian (density)Σ ab , T ab Canonical energy-momentum (density) of matter∆ abc , S abc Canonical hypermomentum (density) of matter t ab , t ab Metrical energy-momentum (density) τ abc Spin current∆ a Dilation current v a Four velocity p Pressure (hyperfluid) P a Momentum density (hyperfluid) J ab Intrinsic hypermomentum density (hyperfluid) F , A Coupling function F ab Variation of the coupling function ζ I a Induced conserved current K ˙ A , K ˙ A General material currents (densities) j ... , i ... , m ... , p ... , k ... , Integrated moments h ... , q ... , µ ... , ξ ... F ab Electromagnetic field J a Electric current L ab , S ab Total orbital / spin angular momentum P a , J ab Generalized total momentum / angular momentumΥ ab Total hypermomentum M Generalized testbody mass D Intrinsic dilation moment ϕ A Multiplet of scalar fields κ Einstein’s gravitational coupling constant44able 5: Directory of symbols.Symbol ExplanationAuxiliary quantitiesΩ a , Ω ab , Ω abc , Ω abcd , Ω ′ k Auxiliary variables (Noether identities) ρ abcd , µ abc Partial derivatives of the total Lagrangian L Auxiliary Lagrangian density ρ abcd , µ abc Partial derivatives of the coupling function
A, B, . . . ; ˙ A, ˙ B, . . .
General multi-indicesΛ jBA , Π A ˙ B General functions of external classical fields U abcd , V abc , κ abc Auxiliary variables (MAG equations of motion) A k Derivative of the coupling function α y y ...y n , β y y ...y n , γ y y ...y n Expansion coefficients of the parallel propagator d q abc , s q abc Decomposition pieces of the q-moments Z a Trace of the d q moment γ AB , J γ AB General function of scalar fields (in Jordan frame) U, J U General potential of scalar fields (in Jordan frame)Ξ ab Auxiliary variable (Einstein frame)Φ y ...y n y x , Ψ y ...y n +1 y y x x Auxiliary variables multipole expansionsOperators / accents ∂ a , L ζ (Partial, Lie) derivative ∇ a Covariant derivative e ∇ a , “ ; a ” Riemannian covariant derivative ∇ a Transposed covariant derivative b ∇ a Covariant density derivativeˇ ∇ a Riemannian covariant density derivative ∗ ∇ a Modified covariant density derivative
Dds =“ ˙ ” Total derivative δ, δ ( s ) Variation, substantial variation B ...... (Bi-)Tensor density g y x , G Y X (Generalized) parallel propagator“ J ” Jordan frame quantity“ g ” Riemannian quantity“[ . . . ]” Coincidence limit45eodesic curve connecting the spacetime points x and y , where ǫ = ± σ y := e ∇ y σ , hence we do notmake explicit use of the semicolon in case of the world-function. The parallelpropagator by g yx ( x, y ) allows for the parallel transportation of objects alongthe unique geodesic that links the points x and y . For example, given avector V x at x , the corresponding vector at y is obtained by means of theparallel transport along the geodesic curve as V y = g yx ( x, y ) V x . For moredetails see, e.g., [7, 8] or section 5 in [80]. A compact summary of usefulformulas in the context of the bitensor formalism can also be found in theappendices A and B of [26].We start by stating, without proof, the following useful rule for a bitensor B with arbitrary indices at different points (here just denoted by dots):[ B ... ] ; y = [ B ... ; y ] + [ B ... ; x ] . (237)Here a coincidence limit of a bitensor B ... ( x, y ) is a tensor[ B ... ] = lim x → y B ... ( x, y ) , (238)determined at y . Furthermore, we collect the following useful identities: σ y y x y x = σ y y y x x = σ x x y y y , (239) g x x σ x σ x = 2 σ = g y y σ y σ y , (240)[ σ ] = 0 , [ σ x ] = [ σ y ] = 0 , (241)[ σ x x ] = [ σ y y ] = g y y , [ σ x y ] = [ σ y x ] = − g y y , (242)[ σ x x x ] = [ σ x x y ] = [ σ x y y ] = [ σ y y y ] = 0 , (243)[ g x y ] = δ y y , [ g x y ; x ] = [ g x y ; y ] = 0 , (244)[ g x y ; x x ] = 12 e R y y y y . (245)46 Covariant expansions
Here we briefly summarize the covariant expansions of the second derivativeof the world-function, and the derivative of the parallel propagator: σ y x = g y ′ x (cid:18) − δ y y ′ + ∞ X k =2 k ! α y y ′ y ...y k +1 σ y · · · σ y k +1 (cid:19) , (246) σ y y = δ y y − ∞ X k =2 k ! β y y y ...y k +1 σ y · · · σ y k +1 , (247) g y x ; x = g y ′ x g y ′′ x (cid:18) e R y y ′ y ′′ y σ y + ∞ X k =2 k ! γ y y ′ y ′′ y ...y k +2 σ y · · · σ y k +2 (cid:19) , (248) g y x ; y = g y ′ x (cid:18) e R y y ′ y y σ y + ∞ X k =2 k ! γ y y ′ y y ...y k +2 σ y · · · σ y k +2 (cid:19) . (249) G Y X ; x = G Y ′ X g y ′′ x ∞ X k =1 k ! γ Y Y ′ y ′′ y ...y k +2 σ y · · · σ y k +2 , (250) G Y X ; y = G Y ′ X ∞ X k =1 k ! γ Y Y ′ y y ...y k +2 σ y · · · σ y k +2 . (251)The coefficients α, β, γ in these expansions are polynomials constructed fromthe Riemann curvature tensor and its covariant derivatives. The first coef-ficients read as follows: α y y y y = − e R y ( y y ) y , (252) β y y y y = 23 e R y ( y y ) y , (253) α y y y y y = 12 e ∇ ( y e R y y y ) y , (254) β y y y y y = − e ∇ ( y e R y y y ) y , (255) γ y y y y y = 13 e ∇ ( y e R y | y | y ) y . (256)In addition, we also need the covariant expansion of a vector: A x = g y x ∞ X k =0 ( − k k ! A y ; y ...y k σ y · · · σ y k . (257) C Explicit form
Here we make contact with our notation in [51] to facilitate a direct com-parison to the results in there. 47e introduce the auxiliary variablesΦ y ...y n y x := σ y · · · σ y n g y x , (258)Ψ y ...y n y y ′ x x ′ := σ y · · · σ y n g y x g y ′ x ′ . (259)Their derivativesΨ y ...y n y y ′ x x ′ ; z = n X a =1 σ y · · · σ y a z · · · σ y n g y x g y ′ x ′ + σ y · · · σ y n (cid:16) g y x ; z g y ′ x ′ + g y x g y ′ x ′ ; z (cid:17) , (260)Φ y ...y n y x ; z = n X a =1 σ y · · · σ y a z · · · σ y n g y x + σ y · · · σ y n g y x ; z , (261)can be straightforwardly evaluated by using the expansions from the previ-ous appendix.In terms of (258) and (259) the integrated conservation laws (127) and(128) take the form: Dds Z Ψ y ...y n y y ′ x x ′ S x x ′ x d Σ x = Z Ψ y ...y n y y ′ x x ′ (cid:16) − U x ′′′′ x ′ x ′′ x x ′′′ S x ′′ x ′′′ x ′′′′ + T x ′ x − t x ′ x (cid:17) w x d Σ x + Z Ψ y ...y n y y ′ x x ′ ; x ′′ S x x ′ x ′′ w x d Σ x + Z v y n +1 Ψ y ...y n y y ′ x x ′ ; y n +1 S x x ′ x d Σ x , (262) Dds Z Φ y ...y n y x T x x d Σ x = Z Φ y ...y n y x (cid:16) − V x ′′ x x ′ T x ′ x ′′ − R x x ′′′ x ′ x ′′ S x ′ x ′′ x ′′′ − Q x x ′′ x ′ t x ′ x ′′ − A x L mat (cid:17) w x d Σ x + Z Φ y ...y n y x ; x ′ T x x ′ w x d Σ x + Z v y n +1 Φ y ...y n y x ; y n +1 T x x d Σ x . (263)This form allows for a direct comparison to (29) and (30) in [51]. Explicitly,in terms of (258) and (259) the integrated moments from (141)–(143) are48iven by: p y ...y n y := ( − n Z Σ( τ ) Φ y ...y n y x T x x d Σ x , (264) k y ...y n +1 y y := ( − n Z Σ( τ ) Ψ y ...y n +1 y y x x T x x w x d Σ x , (265) h y ...y n +1 y y := ( − n Z Σ( τ ) Ψ y ...y n +1 y y x x S x x x d Σ x , (266) q y ...y n +2 y y y := ( − n Z Σ( τ ) Ψ y ...y n +2 y y x x g y x S x x x w x d Σ x , (267) µ y ...y n +1 y y := ( − n Z Σ( τ ) Ψ y ...y n +1 y y x x t x x w x d Σ x , (268) ξ y ...y n := ( − n Z Σ( τ ) σ y · · · σ y n L mat w x d Σ x . (269) References [1] M. Mathisson. Neue Mechanik materieller Systeme.
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