Page 1
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].
Page 2
2
[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]
Page 3
3
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.
Page 4
4
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.
Page 5
5
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)
Page 24
24
TABLE II: Dimensions of the quantities within this work.
Dimension (SI)Symbol
Geometrical quantities
1
m
m−1
m−2
m4
gαβ, δαβ, gij,√−g, hα
xi, dxi, ds, δxi, Ya, ϑα, Tα
eα, Γiαβ, Niαβ, Kiαβ, Qiαβ, Qi, Sijk
Rijαβ, fij
η
i, Γαβ, Nαβ, Kαβ, Rαβ, Qαβ, Q, ℓαβ
Matter quantities
1
kgm2/s
uα, va, ρab, ψ
h (Planck constant), L, Lmat, Ltot, ∆αβ, ταβ, σαβ, mαβ, Mαβ, Hαβ, Eαβ, ∆ijk, ∆
T
Hα, Eα, Σα, Ti
∆αβi
Tαi, Lmat
ijk, Tijk, tijk,
ijk, tijk, Ykl, τkl, Lkl, Λkl, τklj, Zk, Z
k, tij, Tkgm/s
kg/(m s)
kg/(m2s)
ij, tij, Pi, Pi, m
Operators
1d, D
m−1
∂i, ∇i, ∇v,
{ }
? Lξ
[1] A. Einstein and J. Grommer. Allgemeine Relativit¨ atstheorie und Bewegungsgesetz. Sitzungsb. Preuss. Akad. Wiss., page 2,
1927.
[2] H. Weyl. Raum-Zeit-Materie. Springer-Verlag, Berlin, 1923.
[3] A. S. Eddington. The mathematical theory of relativity. Cambridge University Press, London, 1924.
[4] A. E. Scheidegger. Gravitational motion. Rev. Mod. Phys., 25:451, 1953.
[5] J. N. Goldberg. The equations of motion. Gravitation: An introduction to current research, edited by L. Witten, Wiley,
New York, page 102, 1962.
[6] P. Havas. The early history of the “problem of motion” in General Relativity. Einstein Studies, 1:234, 1986.
[7] T. Damour. The problem of motion in Newtonian and Einsteinian gravity. 300 Years of Gravitation, Cambridge University
Press, edited by S.W. Hawking and W. Israel, page 128, 1987.
[8] A. Einstein, L. Infeld, and B. Hoffmann. The gravitational equations and the problem of motion. Ann. Math., 39:65, 1938.
[9] A. Einstein, L. Infeld, and B. Hoffmann. Appendices to ‘The gravitational equations and the problem of motion’. Hand-
written supplement (IAS Library), 1938.
[10] V. A. Fock. Sur le mouvement des masses finies d’apr` es la th´ eorie de gravitation einsteinienne. J. Phys. (Moscow), 1:81,
1939.
[11] M. Mathisson. Neue Mechanik materieller Systeme. Acta Phys. Pol., 6:163, 1937.
[12] D. Puetzfeld. The cosmological post-Newtonian equations of hydrodynamics in General Relativity. Unpublished, 2007.
[13] C. Lanczos. Zur Dynamik der allgemeinen Relativit¨ atstheorie. Z. Phys., 44:773, 1927.
[14] M. Mathisson. Die Beharrungsgesetze in der allgemeinen Relativit¨ atstheorie. Z. Phys., 67:270, 1931.
[15] M. Mathisson. Die Mechanik des Materieteilchens in der allgemeinen Relativit¨ atstheorie. Z. Phys., 67:826, 1931.
[16] M. Mathisson. Bewegungsproblem der Feldphysik und Elektronenkonstanten. Z. Phys., 69:389, 1931.
[17] H. P. Robertson. Test corpuscles in general relativity. Proc. Edn. Math. Soc., 5:63, 1937.
[18] A. Papapetrou. Gravitationswirkungen zwischen Pol-Dipol Teilchen. Z. Phys., 116:298, 1940.
[19] C. Lanczos. The dynamics of a particle in General Relativity. Phys. Rev., 59:813, 1941.
[20] L. Infeld and A. Schild. On the motion of test particles in General Relativity. Rev. Mod. Phys., 21:408, 1949.
[21] A. Papapetrou. Spinning test-particles in General Relativity. I. Proc. Roy. Soc. London Ser. A: Math. Phys. Sci., 209:248,
1951.
[22] A. Papapetrou and W. Urich. Das Pol-Dipol-Teilchen im Gravitationsfeld und elektromagnetischen Feld. Z. Naturforsch.,
10A:109, 1955.
[23] A. Papapetrou. Equations of motion in General Relativity. Proc. Phys. Soc. London A, 64:57, 1951.
[24] A. Papapetrou. Das Problem der Bewegung in der allgemeinen Relativit¨ atstheorie. Fortschr. Phys., 1:29, 1953.
Page 25
25
TABLE III: Directory of symbols.
SymbolExplanation Form degree
Differential form Component
notation
Geometrical quantities
gαβ
gab
g
Metric
Determinant of the metric
Volume form
Coframe
Torsion
Vector basis
Nonmetricity (Weyl 1-form denoted by Q = Qidxi)
0
0
4
1
2
0
1
η
ϑα
Tα
eα
Qαβ
Sijk
Qijk
Rαβ,
{}
Rα
β
Rijkl,
? Rijkl
Nijk
Kijk
Ya
ua
{ }
Rijk
lGeneral curvature, Riemannian curvature
Curvature “object” [defined in eq. (18)]
2
0
Γαβ,
Nαβ
Kαβ
{ }
Γ αβ
Γijk,
{ }
Γ ij
k
Linear connection, Riemannian (Christoffel) connection
Distorsion
Contortion (antisymmetric part of the distorsion)
Worldline within the worldtube of the test particle
Velocity along the worldline Yaof the particle
1
1
1
0
0
Matter quantities
Ltot,L,Lmat
σαβ
Σα
∆αβ
Total, gravitational, matter Lagrangian
Symmetric energy-momentum current
Canonical energy-momentum current
Hypermomentum 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)]
4
4
3
3
0
0
0
0
0
0
0
0
3
0
0
0
tij
Tij
∆ijk
∆
T
tb1···bnij
Pi
L
Λ
Y
Z, Z
τijk
JA
ψ
Pi
b1···bnijk
b1···bnij
ab
ab
ab
k
ταβ
Placeholder for a general matter field
Operators
{ }
D
D,
∇i,
∇v
,i
{ }
∇i
Covariant (exterior) derivative, Riemannian covariant (exterior) derivative
Convective covariant derivative (see, e.g., eq. (55))
Exterior/partial derivative
n → n + 1
n → n + 1
n → n + 1
n → n
0
d
{ }
? Lξ
Riemannian covariant Lie derivative
Spatial projector (equals the convective part, denoted by(c))ρab
Accents
“(c)”
“?”
“
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
Tilde
Overline
Underline
“”
”
Page 26
26
[25] J. N. Goldberg. Strong conservation laws and equations of motion in covariant field theories. Phys. Rev., 89:263, 1953.
[26] H. J. Meister and A. Papapetrou. Die Bewegungsgleichungen in der allgemeinen Relativit¨ atstheorie und die Koordinatenbe-
dingung. Bull. Acad. Pol. Sci., 3:163, 1955.
[27] L. Infeld. Equations of motion in general relativity theory and the action principle. Rev. Mod. Phys., 29:398, 1957.
[28] R. P. Kerr. The Lorentz-covariant approximation method in General Relativity I. Nuovo Cimento, 13:469, 1959.
[29] R. P. Kerr. The Lorentz-covariant approximation method in General Relativity II - Second approximation. Nuovo Cimento,
13:492, 1959.
[30] V. A. Fock. The theory of space time and gravitation. Pergamon Press, New York (Orig. 1955), 1959.
[31] W. Tulczyjew. Motion of multipole particles in general relativity theory. Acta Phys. Pol., 18:393, 1959.
[32] L. Infeld and J. Plebanski. Motion and Relativity. Pergamon Press, New York, 1960.
[33] R. P. Kerr. On the quasi-static approximation in General Relativity. Nuovo Cimento, 16:26, 1960.
[34] J. L. Synge. Relativity: The general theory. North-Holland, Amsterdam, 1960.
[35] P. Havas and J. N. Goldberg. Lorentz-invariant equations of motion of point masses in the general theory of relativity.
Phys. Rev., 128:398, 1962.
[36] B. Tulczyjew and W. Tulczyjew. On multipole formalism in General Relativity. Recent Developments in General Relativity,
Polish Scientific Publishers, Warsaw, page 465, 1962.
[37] A. H. Taub. Motion of test bodies in General Relativity. J. Math. Phys. (N.Y.), 5:112, 1964.
[38] W. G. Dixon. A covariant multipole formalism for extended test bodies in General Relativity. Nuovo Cimento, 34:317,
1964.
[39] P. Havas. The connection between conservation laws and laws of motion in affine spaces. J. Math. Phys. (N.Y.), 5:373,
1964.
[40] J. Madore. The equations of motion of an extended body in General Relativity. Ann. Inst. Henri Poincar´ e., A11:221,
1969.
[41] W. G. Dixon. Dynamics of extended bodies in General Relativity. I. Momentum and angular momentum. Proc. R. Soc.
A, 314:499, 1970.
[42] W. G. Dixon. Dynamics of extended bodies in General Relativity. II. Moments of the charge-current vector. Proc. R. Soc.
A, 319:509, 1970.
[43] D. E. Liebscher. Generalized equations of motion I: The equivalence principle and non-Riemannian space-times. Ann.
Phys. (Leipzig), 485:309, 1973.
[44] D. E. Liebscher. Generalized equations of motion II: The integration of generalized systems of dynamical equations. Ann.
Phys. (Leipzig), 485:321, 1973.
[45] D. E. Liebscher. The motion of test-particles in non-Riemannian space-time. Cosmology and Gravitation: Spin, torsion,
rotation, and supergravity, edited by P.G. Bergmann and V. De Sabbata, NATO Advanced Study Institutes, Plenum Press,
New York, 58:125, 1979.
[46] A. Papapetrou. Lectures on General Relativity. Reidel, Dordrecht, 1974.
[47] W. G. Dixon. Extended bodies in General Relativity: Their description and motion. Isolated gravitating systems in General
Relativity, Proceedings of the International School of Physics “Enrico Fermi,” Course LXVII, edited by J. Ehlers, North
Holland, Amsterdam, page 156, 1979.
[48] P. B. Yasskin and W. R. Stoeger. Propagation equations for test bodies with spin and rotation in theories of gravity with
torsion. Phys. Rev. D, 21:2081, 1980.
[49] I. Bailey and W. Israel. Relativistic dynamics of extended bodies and polarized media: An eccentric approach. Ann. Phys.
(N.Y.), 130:188, 1980.
[50] F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman. Metric-affine gauge theory of gravity: Field equations, Noether
identities, world spinors, and breaking of dilation invariance. Phys. Rep., 258:1, 1995.
[51] E. Cosserat and F. Cosserat. Th´ eorie des corps d´ eformables. Hermann, Paris, 1909.
[52] J. Weyssenhoff and A. Raabe. Relativistic dynamics of spin-fluids and spin-particles. Acta Phys. Pol., 9:7, 1947.
[53] E. Kr¨ oner. Kontinuumstheorie der Versetzungen und Eigenspannungen. Ergebnisse der angewandten Mathematik, edited
by L. Collatz and F. L¨ osch, Springer-Verlag, Berlin, 5:1 – 179, 1958.
[54] C. Truesdell and R. A. Toupin. The classical field theories. Handbuch der Physik, edited by S. Fl¨ ugge, Springer-Verlag,
Berlin, III/1:226, 1960.
[55] R. D. Mindlin. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 16:51, 1964.
[56] G. Capriz. Continua with microstructure. Springer Tracts in Natural Philosophy, Springer-Verlag, Berlin, 1989.
[57] Y. N. Obukhov and V. A. Korotky. The Weyssenhoff fluid in Einstein-Cartan theory. Class. Quantum Grav., 4:1633, 1987.
[58] W. Kopczy´ nski. Lagrangian dynamics of particles and fluids with intrinsic spin in Einstein-Cartan space-time. Phys. Rev.
D, 34:352, 1986.
[59] Y. N. Obukhov and O. B. Piskareva. Spinning fluid in general relativity. Class. Quantum Grav., 6:L15, 1989.
[60] W. Kopczy´ nski. Variational principles for gravity and fluids. Ann. Phys. (N.Y.), 203:308, 1990.
[61] Y. N. Obukhov and R. Tresguerres. Hyperfluid - a model of classical matter with hypermomentum. Phys. Lett. A, 184:17,
1993.
[62] Y. N. Obukhov, V. N. Ponomariev, and V. V. Zhytnikov. Quadratic Poincar´ e gauge theory of gravity: A comparison with
the General Relativity Theory. Gen. Relativ. Gravit., 21:1107, 1989.
[63] Y. N. Obukhov. Poincar´ e gauge theory: Selected topics. Int. J. Geom. Meth. Mod. Phys., 3:95, 2006.
[64] F. Gronwald and F. W. Hehl. On the gauge aspects of gravity. Proceedings of the International School of Cosmology and
Gravitation: 14th Course, Erice, Italy, edited by P.G. Bergmann et al. (World Scientific, Singapore), page 148, 1996.
Page 27
27
[65] W. Adamowicz. Plane waves in gauge theories of gravitation. Gen. Relativ. Gravit., 12:677, 1980.
[66] F. M¨ uller-Hoissen and J. Nitsch. Teleparallelism - a viable theory of gravity? Phys. Rev. D, 28:718, 1983.
[67] M. Q. Chen, D. C. Chern, R. R. Hsu, and W. B. Yeung. Plane-fronted torsion waves in a gravitational gauge theory with
a quadratic Lagrangian. Phys. Rev. D, 28:2094, 1983.
[68] R. Sippel and H. Goenner. Symmetry classes of pp-waves. Gen. Relativ. Gravit., 18:1229, 1986.
[69] P. Singh and J. B. Griffiths. A new class of exact solutions of the vacuum quadratic Poincar´ e gauge field theory. Gen.
Relativ. Gravit., 22:947, 1990.
[70] A. V. Zhytnikov. Wavelike exact solutions of R + R2+ Q2gravity. J. Math. Phys. (N.Y.), 35:6001, 1994.
[71] O. V. Babourova, B. N. Frolov, and E. A. Klimova. Plane torsion waves in quadratic gravitational theories in Riemann-
Cartan space. Class. Quantum Grav., 16:1149, 1999.
[72] R. W. Tucker and C. Wang. Black holes with Weyl charge and non-Riemannian waves. Class. Quantum Grav., 12:2587,
1995.
[73] A. Garc´ ıa, C. L¨ ammerzahl, A. Mac´ ıas, E. W. Mielke, and J. Socorro. Colliding waves in metric-affine gravity. Phys. Rev.
D, 57:3457, 1998.
[74] A. Garc´ ıa, A. Mac´ ıas, D. Puetzfeld, and J. Socorro. Plane-fronted waves in metric-affine gravity. Phys. Rev. D, 62:044021,
2000.
[75] A. Mac´ ıas, C. L¨ ammerzahl, and A. Garc´ ıa. A class of colliding waves in metric-affine gravity, nonmetricity and torsion
shock waves. J. Math. Phys. (N.Y.), 41:6369, 2000.
[76] D. Puetzfeld. An exact-plane fronted wave solution in metric-affine gravity. Exact solutions and scalar fields in gravity:
Recent developments, edited by A. Mac´ ıas, J. Cervantes-Cota, and C. L¨ ammerzahl (Kluwer, Dordrecht), page 141, 2001.
[77] A. D. King and D. Vassiliev. Torsion waves in metric-affine field theory. Class. Quantum Grav., 18:2317, 2001.
[78] V. Pasic and D. Vassiliev. PP-waves with torsion and metric-affine gravity. Class. Quantum Grav., 22:3961, 2005.
[79] Y. N. Obukhov. Plane waves in metric-affine gravity. Phys. Rev. D, 73:024025, 2006.
[80] P. Baekler, N. Boulanger, and F. W. Hehl. Linear connections with a propagating spin-3 field in gravity. Phys. Rev. D,
74:125009, 2006.
[81] F. W. Hehl and A. Mac´ ıas. Metric-affine gauge theory of gravity II. Exact solutions. Int. J. Mod. Phys. D, 8:399, 1999.
[82] D. Puetzfeld. Status of non-Riemannian cosmology. New Astron. Rev., 49:59, 2005.
[83] P. Baekler and F. W. Hehl. Rotating black holes in metric-affine gravity. Int. J. Mod. Phys. D, 15:635, 2006.
[84] Y. N. Obukhov and G. F. Rubilar. Invariant conserved currents in gravity theories with local Lorentz and diffeomorphism
symmetry. Phys. Rev. D, 74:064002, 2006.
[85] F. P. Chen. Momentum, angular momentum, and equations of motion for test bodies in space-time with torsion. Int. J.
Theor. Phys., 32:373, 1993.
[86] F. W. Hehl. How does one measure torsion of space-time? Phys. Lett. A, 36:225, 1971.
[87] A. Trautman. On the Einstein-Cartan equations III. Bull. Acad. Pol. Sci., s´ er. sci. math. astr. phys., 20:895, 1972.
[88] W. R. Stoeger and P. B. Yasskin. Can a macroscopic gyroscope feel torsion? Gen. Relativ. Gravit., 11:427, 1979.
[89] Y. Ne’eman and F. W. Hehl. Test matter in a spacetime with nonmetricity. Class. Quantum Grav., 14:A251, 1997.
[90] 2007. URL http://einstein.stanford.edu/.
[91] H. Weyl. Gravitation und Elektrizit¨ at. Sitzungsb. Preuss. Akad. Wiss., page 465, 1918.
[92] H. Weyl. Eine neue Erweiterung der Relativit¨ atstheorie. Ann. Phys. (Leipzig), 364:101, 1919.
[93] F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester. General relativity with spin and torsion: Foundations and
prospects. Rev. Mod. Phys., 48:393, 1976.
[94] W. Beiglb¨ ock. The center-of-mass in Einstein’s theory of gravitation. Commun. Math. Phys., 5:106, 1967.
[95] J. Frenkel. Die Elektrodynamik des rotierenden Elektrons. Z. Phys., 37:243, 1926.
[96] E. Corinaldesi and A. Papapetrou. Spinning test-particles in General Relativity. II. Proc. Roy. Soc. London Ser. A: Math.
Phys. Sci., 209:259, 1951.
[97] F. A. E. Pirani. On the physical significance of the Riemann tensor. Acta Phys. Pol., 15:389, 1956.
[98] A. H. Taub. The motion of multipoles in General Relativity.
on General Relativity, Problems of Energy and Gravitational Waves, edited by G. Barb` era, Comitato Nazionale per le
Manifestazioni Celebrative, Florence, page 100, 1965.
Proceedings of the Galileo Galilei Centenary Meeting