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arXiv:0707.2819v3 [gr-qc] 30 Mar 2009

Propagation equations for deformable test bodies with microstructure in extended

theories of gravity

Dirk Puetzfeld∗

Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029, 0315 Oslo, Norway

Yuri N. Obukhov†

Institute for Theoretical Physics, University of Cologne, Z¨ ulpicher Straße 77, 50937 K¨ oln, Germany‡

(Dated:March 30, 2009)

We derive the equations of motion in metric-affine gravity by making use of the conservation laws

obtained from Noether’s theorem. The results are given in the form of propagation equations for the

multipole decomposition of the matter sources in metric-affine gravity, i.e., the canonical energy-

momentum current and the hypermomentum current. In particular, the propagation equations

allow for a derivation of the equations of motion of test particles in this generalized gravity theory,

and allow for direct identification of the couplings between the matter currents and the gauge

gravitational field strengths of the theory, namely, the curvature, the torsion, and the nonmetricity.

We demonstrate that the possible non-Riemannian spacetime geometry can only be detected with

the help of the test bodies that are formed of matter with microstructure. Ordinary gravitating

matter, i.e., matter without microscopic internal degrees of freedom, can probe only the Riemannian

spacetime geometry. Thereby, we generalize previous results of general relativity and Poincar´ e gauge

theory.

PACS numbers: 04.25.-g; 04.50.+h; 04.20.Fy; 04.20.Cv

Keywords: Approximation methods; Equations of motion; Alternative theories of gravity; Variational prin-

ciples

I. INTRODUCTION

The relation between the field equations and the equations of motion within nonlinear gravitational theories has

been subject to many works. The intimate link between these equations is one of the features of general relativity

which distinguishes it from many other physical theories. The fact that, in contrast to linear field theories, the

equations of motion need not to be postulated separately, but can be derived from the field equations, has been

investigated shortly after the proposal of the theory. From a conceptual standpoint the derivability of the equations

of motion is a very satisfactory result, since it reduces the number of additional assumptions in the theory.1The

earliest accounts of this feature of general relativity can be found in the works of Weyl [2], Eddington [3], as well

as Einstein and Grommer [1]. Nowadays this is customarily addressed as the “problem of motion” in the context of

general relativity and other nonlinear field theories.2

One may distinguish between two conceptually different methods. Both were employed in the derivation of the

equations of motion within the theory of general relativity. One of them goes back to the works of Einstein et al.

[8, 9] and is based on the vacuum field equations of the theory. Within this method matter is modeled in the form of

singularities of the field and only the exterior of bodies is considered. The second method, usually attributed to Fock

∗Electronic address: dirk.puetzfeld@astro.uio.no; URL: http://www.thp.uni-koeln.de/∼dp

†Electronic address: yo@thp.uni-koeln.de

‡Also at Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia

1The following german quotes are taken from [1] (translation by the authors):

• “[...] Es sieht daher so aus, wie wenn die allgemeine Relativit¨ atstheorie jenen ¨ argerlichen Dualismus bereits siegreich ¨ uberwunden

h¨ atte. [...]”,

“[...] It looks like the general theory of relativity has victoriously overcome this annoying dualism. [...]”.

• “[...] Der hier erzielte Fortschritt liegt aber darin, daß zum ersten Male gezeigt ist, daß eine Feldtheorie eine Theorie des

mechanischen Verhaltens von Diskontinuit¨ aten in sich enthalten kann. [...] ”,

“[...] The progress achieved in this work is that for the first time we have shown that a field theory can contain the theory of the

mechanical behavior of discontinuities. [...]”.

2A historical account of works can also be found in [4, 5, 6, 7].

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[10], makes use of the differential conservation laws of the theory and also allows for a consideration of the interior of

material bodies. In this work we are going to utilize the latter method; i.e., we base our considerations on differential

identities derived from the symmetry of the action via Noether’s theorem.

In addition, we make use of a multipole decomposition of the matter currents. This allows for a systematic study

of the coupling between the matter currents and field strengths of the theory at different orders of approximation.

Multipole methods have been intensively studied in the context of the problem of motion since the early work of

Mathisson [11]. In table I, we provide a corresponding chronological overview.3

TABLE I: Timeline of works which deal with the problem of motion and

multipole approximation schemes.

Year Reference

1923 Weyl [2]

Comment

Mentions the link between the equations of motion (EOM) and the

field equations.

Show that the field equations contain the EOM in GR (for a special

case).

Early investigation regarding the problem of motion, treated as

boundary value problem.

Systematic account of the problem of motion in GR, one of the first

authors who makes use of the δ-function in this context.

Test particle EOM from divergence condition.

Possibly the earliest work utilizing a multipole method in the deriva-

tion of the EOM.

Derivation of the EOM outside of material bodies.

Systematic slow motion approximation.

Gravitational interaction of particles using the multipole method.

Test particle EOM via Gaussian integral transformation.

Derive the geodesic motion of test particles for empty space.

EOM for pole-dipole test particles in GR (see also the later work

[22]).

Derivation of the EOM utilizing a method in the spirit of [10].

Review of the problem of motion in GR.

Relationship of EOM and covariance of a field theory.

1927 Einstein and Grommer [1]

Lanczos [13]

1931 Mathisson [14, 15, 16]

1937 Robertson [17]

Mathisson [11]

1938 Einstein et al. [8, 9]

1939 Fock [10]

1940 Papapetrou [18]

1941 Lanczos [19]

1949 Infeld and Schild [20]

1951 Papapetrou [21]

Papapetrou [23]

1953 Papapetrou [24]

Goldberg [25]

1955 Meister and Papapetrou [26] EOM and coordinate conditions in GR.

1957 Infeld [27]Review of approximation methods, derives EOM using Einstein-

Infeld-Hoffmann (EIH) method, relaxes harmonic coordinate con-

dition, δ-function as source.

1959 Kerr [28, 29]Systematic post-Minkowskian treatment I + II (fast motion

approximation).

Fock [30]Systematic slow motion/weak field approximation.

Tulczyjew [31]Test particle EOM via a simplified version of Mathisson’s method.

1960 Infeld and Plebanski [32]Review of the EIH method.

Kerr [33]Approximation of the quasistatic case, review of three approxima-

tions schemes.

Synge [34]Integralconservation

momentum pseudotensor definition.

1962 Goldberg [5]Review of the problems connected with the EOM in GR and the

EIH method.

Havas and Goldberg [35]Derive single-pole EOM by using Mathisson’s method.

Tulczyjew and Tulczyjew [36] Covariant formulation of a multipole method in GR.

1964 Taub [37]Test particle EOM in a coordinate independent manner using Pa-

papetrou’s method.

Dixon [38]Covariant multipole method for extended test particles in GR.

Havas [39]Generalized version of Mathisson’s method in affine spaces.

1969 Madore [40]EOM for extended bodies using a multipole method which differs

from the one of [21].

1970 Dixon [41, 42]Extended bodies within a multipole formalism.

1973 Liebscher [43, 44]EOM for pole particles in non-Riemannian spaces using the method

in [40], see also [45].

laws,EOMformasscenter,energy-

3An extended version of this table, also including works in the post-Newtonian and post-Minkowskian context, can be found in [12]

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1974 Papapetrou [46] Review of the derivation of the EOM of a single-pole test particle

in GR.

Review of the multipole formalism in GR in the context of extended

bodies.

Generalization of the Papapatrou equations to Poincar´ e gauge

theory.

Multipole method for the derivation of the EOM for extended

bodies.

EOM review.

1979 Dixon [47]

1980 Yasskin and Stoeger [48]

Bailey and Israel [49]

1987 Damour [7]

In this paper, we work out the equations of motion within a multipole formalism for a generalized gravitational

theory known as metric-affine gravity (MAG) [50]. In the theory of general relativity, the mass, or more precisely the

energy-momentum, of matter is the only physical source of the gravitational field. The energy-momentum current

corresponds (via the Noether theorem) to the local translational, or the diffeomorphism, spacetime symmetry. In

MAG, this symmetry is extended to the local affine group that is a semidirect product of translations times the local

linear spacetime symmetry group. Correspondingly, there are additional conserved currents describing microscopic

characteristics of matter that arise as physical sources of the gravitational field. In continuum mechanics [51, 52,

53, 54, 55, 56], such matter is described as a medium with microstructure. In physical terms this means that the

elements of a material continuum have internal degrees of freedom such as spin, dilation, and shear. The three latter

microscopic sources are represented in MAG by the irreducible parts (that correspond to the Lorentz, dilational

and shear-deformational subgroups of the general linear group) of the hypermomentum current. Fluid models with

microstructure were extensively studied within different gravity theories (including MAG), see, e.g., [57, 58, 59, 60, 61].

The metric-affine theory naturally generalizes the Poincar´ e gravity theory [62, 63] in which the mass (energy-

momentum) and spin are the sources of the gravitational field. The geometry that arises on the spacetime manifold is

non-Riemannian, it is known as the Riemann-Cartan geometry with curvature and torsion. In MAG, this geometrical

structure is further extended to the metric-affine spacetime with curvature, torsion, and nonmetricity. The resulting

general scheme of MAG embeds not only Poincar´ e gravity, but also a wide spectrum of gauge gravitational models

based on the conformal, Weyl, de Sitter, and other spacetime symmetry groups (for an overview, see [50], for example).

This fact makes the analysis of the equations of motion in MAG especially interesting, with possible direct physical

applications for all the gravitational models mentioned.

The energy-momentum current and the hypermomentum current (spin + dilaton + shear charge) are the sources

of the gravitational field in MAG. Accordingly, test bodies that are formed of matter with microstructure have

two kinds of physical properties which determine their dynamics in a curved spacetime. The properties of the first

type have microscopic origin; they arise directly from the fact that the elements of a medium have internal degrees of

freedom (microstructure). The properties of the second type are essentially macroscopic; they arise from the collective

dynamics of matter elements characterized by mass (energy) and momentum. More exact definitions will be given

later, but the qualitative picture is as follows. The averaging of the microscopic hypermomentum current yields the

integrated spin, dilaton, and shear charge of a test body. In addition, the averaging of the energy-momentum and of

its multipole moments gives rise to the orbital integrated momenta. In Poincar´ e gravity, there is only one relevant

first moment, namely, the orbital angular momentum. It describes the behavior of a test particle as a rigid body,

i.e., its rotation. In metric-affine gravity, one finds, in addition, the orbital moments that describe deformations of

body. These are the orbital dilation momentum (that describes isotropic volume expansion) and the orbital shear

momentum (that determines the anisotropic deformations with fixed volume). The three together (orbital angular

momentum, orbital dilation momentum, and orbital shear momentum) comprise the generalized integrated orbital

momentum. In this paper, we compare the gravitational interaction of the integrated hypermomentum to that of the

integrated orbital momentum of a rotating and deformable test body. Thereby, we generalize the previous analysis

[48] in which the effects of the integrated spin were compared to the effects of the orbital angular momentum of a

rotating rigid test body.

The paper is organized as follows. In section II we recall some basic facts about the gravity theory under consid-

eration, namely, metric-affine gravity. This is followed by a discussion of the conservation laws within this theory in

section III which form the basis for the derivation of the equations of motion. We then work out the explicit form

of the propagation equations in sections IV and V. In section VI we provide some relations between the different

definitions of momenta within the multipole formalism. We discuss our findings in section VII and present an outlook

on the open questions within this field. Our notation and conventions are summarized in appendix A. A table with

the dimensions of all quantities appearing throughout the work can be found in appendix B.

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II. METRIC-AFFINE GRAVITY

Metric-affine gravity represents a gauge-theoretical formulation of a theory of gravitation which is based on the

general affine group A(4,R), i.e., the semidirect product of the four-dimensional translation group R4and the general

linear group GL(4,R). For a review of the theory see [50, 64], and references therein. In such a theory, besides

the usual “weak” Newton-Einstein–type gravity, described by the metric of spacetime, additional “strong” gravity

pieces will arise that are supposed to be mediated by additional degrees of freedom related to the independent linear

connection Γαβ. Alternatively, the strong gravity pieces can also be expressed in terms4of the nonmetricity Qαβand

the torsion Tα. The propagating modes related to the new degrees of freedom are expected to manifest themselves

in the non-Riemannian pieces of the curvature Rαβ. The existence of such modes certainly depends on the choice

of the dynamical scheme, or in technical terms, on the choice of the Lagrangian. The simplest generalization of the

linear Hilbert-Einstein Lagrangian leads to a model with contact interaction. However, quadratic Yang-Mills–type

Lagrangians describe a wide spectrum of non-Riemannian propagating gravitational modes. This is revealed, for

example, by studies of generalized gravitational waves in models with torsion [65, 66, 67, 68, 69, 70, 71] and in models

with torsion and nonmetricity [72, 73, 74, 75, 76, 77, 78, 79, 80].

In a Lagrangian framework one usually considers the geometrical “potentials” (metric gαβ, coframe 1-form ϑα,

connection 1-form Γαβ) to be minimally coupled to matter fields, collectively called ψ, such that the total Lagrangian,

i.e., the geometrical and the matter part, is given by

Ltot= L?gαβ,ϑα,Qαβ,Tα,Rαβ?+ Lmat(gαβ,ϑα,ψ,Dψ).

Here D = d+ℓαβΓαβ, with ℓαβdenoting the generators of the linear transformations (namely, δψ = εβαℓαβψ, where

εβαare the infinitesimal parameters). With the following general definitions for the gauge field momenta

(1)

Mαβ:= −2

∂L

∂Qαβ,Hα:= −∂L

∂Tα,Hαβ:= −

∂L

∂Rαβ,(2)

the field equations of metric-affine gravity take the form

(δ/δgαβ)

(δ/δϑα)

?δ/δΓαβ?

(matter)

DMαβ− mαβ= σαβ,

DHα− Eα= Σα,

DHαβ− Eαβ= ∆αβ,

δL

δψ= 0.

(3)

(4)

(5)

(6)

On the right-hand side (rhs) of the field equations we have the physical sources: the metrical energy-momentum σαβ,

the canonical energy-momentum Σα, and the canonical hypermomentum ∆αβcurrents of the matter fields

σαβ:= 2δLmat

δgαβ

,Σα:=δLmat

δϑα,∆αβ:=δLmat

δΓαβ.(7)

On the left-hand side (lhs) there are typical Yang-Mills–like terms governing the gauge gravitational fields, and the

corresponding terms that describe the currents of the gauge fields themselves that arise due to the nonlinearity of

the theory. The metrical energy-momentum, the canonical energy-momentum, and the canonical hypermomentum

currents of the gauge gravitational fields are introduced by

mαβ:= 2∂L

∂gαβ,Eα:=

∂L

∂ϑα,Eαβ:=

∂L

∂Γαβ. (8)

MAG has a wide gauge symmetry group. With the help of the Noether theorems for the diffeomorphism symmetry and

for the local linear symmetry, one can verify that [provided the matter field equations (6) are fulfilled] the following

identities hold:

Σα = eα⌋Lmat− (eα⌋Dψ) ∧∂Lmat

∂Dψ

− (eα⌋ψ) ∧∂Lmat

∂ψ

, (9)

4Please see appendix A on page 22 for the definitions of the objects in this section and a short summary of our conventions.

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Eα = eα⌋L + (eα⌋Tβ) ∧ Hβ+ (eα⌋Rβγ) ∧ Hβγ+1

Eαβ = −ϑα∧ Hβ− Mαβ,

∆αβ = (ℓαβψ) ∧∂Lmat

?eα⌋Tβ?∧ Σβ−1

D∆αβ = gβγσαγ− ϑα∧ Σβ.

The gauge symmetry and the corresponding Noether identities play an essential role in MAG. The most important

result is as follows: It can be shown that, by means of (10)-(14), the field equation (3) is redundant. It is a consequence

of the two other MAG field equations (4) and (5) and of the Noether identities. The explanation is straightforward:

One can use the local linear transformations of the frames to “gauge away” the metric gαβby making it equal to the

constant Minkowski metric diag(1,−1,−1,−1) everywhere on the spacetime manifold. After doing this, equation (3)

is trivially solved, and one needs to solve only the remaining equations (4) and (5) to determine the coframe ϑαand

connection Γαβ.

There are many nontrivial exact solutions for different MAG models ranging from black holes, gravitational waves,

to cosmological models known in the literature. Nearly all of the corresponding references can be found in the works

[50, 81, 82, 83].

2(eα⌋Qβγ)Mβγ,(10)

(11)

∂Dψ,

(12)

DΣα =

2(eα⌋Qβγ)σβγ+ (eα⌋Rβγ) ∧ ∆βγ,(13)

(14)

III.CONSERVATION LAWS

An up-to-date discussion of the conservation laws within metric-affine gravity can be found in the recent work

[84]. In the following sections IIIA-IIIC we recall the conservation laws for the canonical energy-momentum and

hypermomentum. These conservation laws serve as starting point for our subsequent derivation of the propagation

equations for the multipole moments of the matter currents. In IIIC we make contact with Poincar´ e gauge theory,

which represents the special case of metric-affine gravity for which the distorsion, i.e., the difference between the full

and the metric-compatible connection, reduces to the antisymmetric contortion, and the hypermomentum reduces to

the spin current.

A.Energy-momentum conservation

The Noether theorem for the diffeomorphism invariance of the matter action yields the conservation law of the

energy-momentum current

?

{ }

D

?Σα− ∆γβeα⌋Nγβ?≡eα⌋

{ }

Rγβ−

{ }

? LαNγβ

?

∧ ∆γβ.(15)

Here

After we substitute the components from (A5)-(A10), we finally find the tensor form of the conservation law (15):

{ }

? Lξ= ξ⌋

{}

D +

{}

Dξ⌋ is the (Riemannian) covariant Lie derivative.

{ }

∇j

?Tij− Nikl∆klj?=?{}

Rijkl−

{}

∇iNjkl

?∆klj.(16)

This can be identically rewritten as

{}

∇jTij=?Rijkl∆klj+ Nikl

{}

∇j∆klj,(17)

where we denoted

?Rijkl:=

{ }

Rijkl−

{ }

∇iNjkl+

{}

∇jNikl. (18)