arXiv:0707.2819v3 [gr-qc] 30 Mar 2009
Propagation equations for deformable test bodies with microstructure in extended
theories of gravity
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
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-
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 , Eddington , as well
as Einstein and Grommer . 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: firstname.lastname@example.org; URL: http://www.thp.uni-koeln.de/∼dp
†Electronic address: email@example.com
‡Also at Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia
1The following german quotes are taken from  (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].
, 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 . 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.
1923 Weyl 
Mentions the link between the equations of motion (EOM) and the
Show that the field equations contain the EOM in GR (for a special
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
Derivation of the EOM utilizing a method in the spirit of .
Review of the problem of motion in GR.
Relationship of EOM and covariance of a field theory.
1927 Einstein and Grommer 
1931 Mathisson [14, 15, 16]
1937 Robertson 
1938 Einstein et al. [8, 9]
1939 Fock 
1940 Papapetrou 
1941 Lanczos 
1949 Infeld and Schild 
1951 Papapetrou 
1953 Papapetrou 
1955 Meister and Papapetrou  EOM and coordinate conditions in GR.
1957 Infeld 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
Fock Systematic slow motion/weak field approximation.
Tulczyjew Test particle EOM via a simplified version of Mathisson’s method.
1960 Infeld and Plebanski Review of the EIH method.
Kerr Approximation of the quasistatic case, review of three approxima-
momentum pseudotensor definition.
1962 Goldberg  Review of the problems connected with the EOM in GR and the
Havas and Goldberg Derive single-pole EOM by using Mathisson’s method.
Tulczyjew and Tulczyjew  Covariant formulation of a multipole method in GR.
1964 Taub Test particle EOM in a coordinate independent manner using Pa-
Dixon Covariant multipole method for extended test particles in GR.
Havas Generalized version of Mathisson’s method in affine spaces.
1969 Madore EOM for extended bodies using a multipole method which differs
from the one of .
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 , see also .
3An extended version of this table, also including works in the post-Newtonian and post-Minkowskian context, can be found in 
1974 Papapetrou Review of the derivation of the EOM of a single-pole test particle
Review of the multipole formalism in GR in the context of extended
Generalization of the Papapatrou equations to Poincar´ e gauge
Multipole method for the derivation of the EOM for extended
1979 Dixon 
1980 Yasskin and Stoeger 
Bailey and Israel 
1987 Damour 
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) . 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 , 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
 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.
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
the field equations of metric-affine gravity take the form
DMαβ− mαβ= σαβ,
DHα− Eα= Σα,
DHαβ− Eαβ= ∆αβ,
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
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
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
Σα = eα⌋Lmat− (eα⌋Dψ) ∧∂Lmat
− (eα⌋ψ) ∧∂Lmat
4Please see appendix A on page 22 for the definitions of the objects in this section and a short summary of our conventions.
Eα = eα⌋L + (eα⌋Tβ) ∧ Hβ+ (eα⌋Rβγ) ∧ Hβγ+1
Eαβ = −ϑα∧ Hβ− Mαβ,
∆αβ = (ℓαβψ) ∧∂Lmat
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
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)σβγ+ (eα⌋Rβγ) ∧ ∆βγ, (13)
III. CONSERVATION LAWS
An up-to-date discussion of the conservation laws within metric-affine gravity can be found in the recent work
. 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.
The Noether theorem for the diffeomorphism invariance of the matter action yields the conservation law of the
∧ ∆γβ. (15)
After we substitute the components from (A5)-(A10), we finally find the tensor form of the conservation law (15):
? Lξ= ξ⌋
Dξ⌋ is the (Riemannian) covariant Lie derivative.
This can be identically rewritten as
where we denoted
TABLE II: Dimensions of the quantities within this work.
gαβ, δαβ, gij,√−g, hα
xi, dxi, ds, δxi, Ya, ϑα, Tα
eα, Γiαβ, Niαβ, Kiαβ, Qiαβ, Qi, Sijk
i, Γαβ, Nαβ, Kαβ, Rαβ, Qαβ, Q, ℓαβ
uα, va, ρab, ψ
h (Planck constant), L, Lmat, Ltot, ∆αβ, ταβ, σαβ, mαβ, Mαβ, Hαβ, Eαβ, ∆ijk, ∆
Hα, Eα, Σα, Ti
ijk, Tijk, tijk,
ijk, tijk, Ykl, τkl, Lkl, Λkl, τklj, Zk, Z
k, tij, Tkgm/s
ij, tij, Pi, Pi, m
∂i, ∇i, ∇v,
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TABLE III: Directory of symbols.
SymbolExplanation Form degree
Differential form Component
Determinant of the metric
Nonmetricity (Weyl 1-form denoted by Q = Qidxi)
lGeneral curvature, Riemannian curvature
Curvature “object” [defined in eq. (18)]
Linear connection, Riemannian (Christoffel) connection
Contortion (antisymmetric part of the distorsion)
Worldline within the worldtube of the test particle
Velocity along the worldline Yaof the particle
Total, gravitational, matter Lagrangian
Symmetric energy-momentum current
Canonical energy-momentum current
n-th integrated moment of the hypermomentum
n-th integrated moment of the canonical energy-mom.
n-th integrated moment of the symmetric energy-mom.
Generalized integrated momentum
Generalized integrated orbital momentum
Antisymmetric part of the gen. int. orbital momentum
Generalized integrated hypermomentum
Dilaton part, i.e. the trace, of the generalized int. hypermomentum
Spin current (antisymmetric part of the hypermomentum current)
Placeholder for the density of a matter current (e.g.? ∆klj,? Tij, or? tkl)
Generalized total 4-momentum [defined in eq. (86)]
Placeholder for a general matter field
Covariant (exterior) derivative, Riemannian covariant (exterior) derivative
Convective covariant derivative (see, e.g., eq. (55))
n → n + 1
n → n + 1
n → n + 1
n → n
Riemannian covariant Lie derivative
Spatial projector (equals the convective part, denoted by(c))ρab
Denotes the convective part of an object
Denotes the density of an object
Denotes integrated version of a density based on upper-index convention
Denotes integrated version of a density based on lower-index convention
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