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Equations of Motion for Test Particles in a Poincar Gauge Field.

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Annalen der Physik. 7. Folge, Band 43', Heft 1/2, 1986, S. 76-82
J. A. Barth, Leipzig
Equations of Motion for Test Particles
in a Poincar6 Gauge Field
Akademie der Wissenschaften der DDR, Zentralinstitut fur Astrophysik, Postsdam-Babelsberg
A b s t r a c t . We consider the equations of motion for test particles with internal structure in a
Poincarb gauge field by a general Lagrangian approach. The break down of the Poincarb symmetry
is also considered with respect t o the equations of motion.
Bewegungsgleichungen fur Probeteilchen in einem Poincarh-Eichfeld
I n h a l t s u b e r s i c h t . Mit Hilfe eines Lagrange-Prinzips leiten wir die Bewegungsgleichungen fur
Probeteilchen mit innerer Struktur in einem PoincarB-Eichfeld ab. Es werden Folgerungen fur die
Bewegungsgleiehungen betrachtet, die sich aus einer Brechung der PoincarB-Symmetrie zur LorentzSymmetrie ergeben.
1. Introduction
The equations of motion for monopole singularities in General Relativity are identical with the equations for the geodesics of the space-time under consideration. It is
well known that this law of motion follows from the field equations [l].On the other
hand, the equations for the geodesics can be derived from a Lagrange principle
6 $ I/gmnxmxn dz = 0.
It seems to be quite natural t o ask wether the equations of motion for point particles
corresponding t o more complex singularities (e.g. pole-dipole singularities) can be derived
from a generalized geodesic principle or not. This question can be answered affirmatively.
One has to introduce additional (internal) variables and to postulate a minimal coupling
with the help of the affinity of the space-time manifold [2]. I n [3] we have extended the
formalism to incorporate also the motion of gauge charged particles and arrived at
generalized Wong equations [4]. We remark that in both cases the form of the Lagrangian was not specified, so that the results are quite general. The equations of motion
are determined by the fact that a generalized geodesic principle is assumed. The special
form of the equations of motion depends essentially on the kind of the internal variables
(their transformation properties) and on the kind of the external field (the affinity used
for the minimal coupling).
I n the last years there is a permanent interest in gauge theoretical aspects of gravitational theories (for the role of the local Lorentz group in theories of gravitation see e.g.
[5-71 and for a general discussion of various gauge theories of gravitation see [S]). I n
this paper we will derive equations of motion for point test particles with internal structure in a Poincarb gauge field with the help of a Lagrange principle. We won't specify
the special form of the Lagrangian so that the approach is sufficiently general to incorporate several cases considered in the literature. To make the notation clear we recall
some relations from general gauge theories (for details see e.g. [S-131).
Particles in a Poinear6 Gauge Field
For a given gauge theory we have a gauge group G (of order N ) and particle fields
which transform according to a n-dimensional representation of G, e.g.
-uATAUp1P, 6plN = U ' T S , ~ ~ ,
A = l , 2 ,..., N , a , B = 1 , 2 ,..., n .
Here the T&, are the infinitesimal generators of the representation.'The gauge potentials
are given by a connection in the principal fiber bundel with structure group G and spacetime as base manifold. Let the components of the connection one-form-the gauge
potential-be T&.Then the components of the curvature two-form - the field
strength - are given by
A - r A
F m n - n,m
- r;,n+ ciDr:r:.
Here the CABn ( A , B, D = 1, 2 , . .., N ) are the structure constants of G with respect
t o a given parametrization of the group manifold. The field strength transforms as a
(0, 1)-entity under the adjoint representation [3].
The equations of motion for a test particle in a non-abelian gauge field are given by
the Wong equations [4]. I n [3] we have shown the following: if there is a particle Lagrangian depending on the metric tensor gmn of the space time, the velocity xm of the particle,
internal variables pl, 1" transforming according to a given representation of a gauge
group and their absolute derivatives, the entities
are the absolute derivatives corresponding t,o the representation) fulfil the genp,
eralized Wong-equations.
Pm = IAFAmnX",
= 0.
P, is a momentum vector as defined in the next chapter. D/dr and @at' are the a,bsolute
derivatives constructed with the Christoffel affinity and the gauge potentials respectively.
I n general we have for a n entity 1" say, transforming as in (1)
and for an entity I A transforming under the adjoint representation of the gauge group
= uBC:AI,
(The notations in [3] are different from those used in this paper).
Let us now consider the Poincar6 group as the internal gauge group (for various
approachs t o Poincar6 gauge theories see f.e. [lo, 13-17]). The Poincar6 group is the
semi-direct product of the Lorentz group and the translational group R4.We label
the ten group parameters by wAB, uA ( A , B = 1, 2 , 3, 4,uAB= -uBA), the gauge
potentials by
FAmand the corresponding components of the curvature form by
Ann. Physik Leipzig 43 (1986) 1/2
FARmn,FAm,. The structure constants of the PoincarA group are given by [13]
(TAC,,= qCRTABm).
The curvature transforms according to the adjoint representation,
that is
We see that the rotational part of the curvature transfornis as a Lorentz tensor.
The second part of the Wong equations-the propagation law for the gauge charges
I,,, I,-is given for the Poincari: group as an internal gauge group by
The translational part transforms as a covariant Lorentz vector.
2. Equations of Motion
We assume that the internal structure of the part'icle is described by variables which
transform as Poincar6 matter fields (see e.g. [lo]). That means: in addition to the matter
field variables q2,A" transforming under Poincari: transformations as under the representation of the Lorentz subgroup
we have an internal vector Y* transforming as
- w4.
( u ,uA
~ are
~ the parameters of the Poincar6 group).
Particles in a Poincarh Gauge Field
The absolute derivatives of these quantities have the form
a” = 1” = ;i.+ yxgn~w,
= @“- ybJnCpgsn,
YA =
T A BagrA ” n
FA + r*,,y%m
+ rAmsm.
To derive the equations of motion we use a Lagrangian approach. Besides the internal
variables and their absolute derivatives the geometry of space-time and the external
variables (the curve) are introduced in the Lagrangian by the components of the canonical one-form h i and the velocity component,s xm. The velocity components transform
under the Poincari: transformations as scalars whereas the tetrads transform as a Lorent’z vector
6hAm= -uABhBm
We start with an unspecified Lagrangian
L = L(xm,hAm,A”, A”,
$jJ, p*).
It is assumed that L is homogeneous in the derivatives and that it behaves as a scalar
both under coordinate transformations and under Poincarit transformations.
According to the second Noether theorem we have because of the assumed Poincari:
gauge invariance the following identity
+ Y(q-Jp)
- Y(I’”) T A R D C Y C ! 7
I A B = I A Y c ~ j BC IBYC?jAG.
The y’s are the Lagrangian expressions corresponding to the variables written in t>he
The Euler-Lagrange equat’ions for the internal variables have the following form
A* denotes that in the derivative only the explicite occurrence of the variable is considered not that in a absolute derivative. Using these equations and the Noether identity the other part of the equations of motion corresponding to the variations of the
curve can be written as follows
3. Discussion
Let us now discuss the equations of motion by considering special cases for the
Lagrangian. At first we assume that the Poincar6 vect'or Y A doesn't occur in the Lagrangian, that is, we come back to a Lorentz invariant t'heory. Then we have
0 , I A B= 0 .
Two subcases are of int'erest.
a) The Lagrangian depends on h& only via the invariant (qAsh$hfdixk). e.g. in
the form of a kinematical term
Lo = m fqABhfhfxixk.
Then we have
and the equations of motion can be written as
- -#AgFABmnXn
The absolute derivative Dldz is built with the Christoffel affinity belonging to the
and the derivative @d-c is constructed with the Lorente part of the Poincar6 affinity.
Comparing with (4a, b), we see that the equations (24a, b) are just the Wong equations
for a Lorentz charged particle. Here the Lorentz group plays the role of an internal gauge
group. The soldering of the internal space to spacetime via h i is not taken into account.
The equations of motion (24a, b) are in accordance with that derived in [ 181in the framework of a Lorentz gauge theory on & Riemannian space-time.
b) The Lagrangian depends on h i with
gik ==
Particles in a Poincar.5 Gauge Field
This case corresponds t o the Poincarh gauge theories in the sense of HEHLand al.
(see [14, 151 and the discussion in [lo]). For the equations of motion we get in this case
t 221
P, - XkKimPi- 2 SABFABmnXn= 0 ,
8, B = PBXi., - PAX,.
Here P,, X , are the tetrad components P, = hgP,, x, = qAQXC= qAChixmand
K i m is the contortion tensor K f , = F;, - (f,} (see equ. 21). These are the equations
of motion of an spinning test particle in a space-time with torsion (see f.e. [12]). I n
[301 these equations where derived from a special Lagrangian. The additional incorporation of the electromagnetic field (and of Lagrangian mult,ipliers, as in [ZO]) in our general
Lagrangian is rather trivial and results independently of the special choice of the Lagrangian in generalized Bargmann-MicheI-Telegdi equations of motion.
Let us now consider the equations of motion of a pure Poincarb charged particle,
coupled minimaly-only via connection in the absolute derivative-to the Poincarh gauge
field. This case corresponds to a Lagrangian L = L(Xm,h i , FA), where there is no
coupling between external and internal variables. The only invariants t o built the
Lagrangian are then J, = qABhih:xmxlaand J , = YAFBl;rAB.This gives for the
equations of motion.
+I , BFABmnXn
I* - r B A n I B X n = 0 ,
- -P,
f 1, F&Xn = 0,
I,, - r F B n i A F X n - r F A n i F B 2 n
+ qBurnx
D . n I ,- T I A D r : k v B= 0.
( 2 ~ ~ )
These are exactly the Wong equations for Poincarh charged particles.
The translational part of the connection I'i can be split up in the components of the
canonical one-form hf and in a covariant derivative with respect to the rotational part
of the connection of an affine vector field [16, 191:
Ti = h i V,vA.
If the Poincar6 gauge symmetry is broken down to Lorentz symmetry by V,qP = 0
[16] or yA = 0 [lo] then we get for the equations of motion
X n = 0,
I , B - T B D X D I A - ?]AD2"1,
(28 c)
This again are the equations of motion for a spinning test particle in a Space-time with
torsion. The translational part of the Poincare charge gives a contribution t o four momentum which is generally not parallel t o the four velocity. For this part of themomentum
we have a subsidiary condition: it must be parallelpropagated with respect to the Lorentz connection under consideration. It should be mentioned t h a t the pole particle
copdition is not trivial in this model. If one demands I A $; 0, I A B = 0, then it is easely
be seen that I A must be of the form I , = KhiXmqAB,K = const. This is only in
accordance with the equations of motion, if DJdz .Im
= 0 is compatible with B/dz I , = 0.
Then the particle moves along the geodesics of the space-time with metric gj, = qABh?hf.
Ann. Physik Leipzig 43 (1986) 1 / 2
I n this case the four momentum can be considered as the translational charge of a broken
Poincare gauge theory, if one drops the kinematical term of the Lagrangian.
Finally we remark that the generalized geodesic principle by itself defines point
particle models the physical significance of which can be shown by proving agreement
with the resulbs following from the dynamical equations. I n the cases considered here
and in [2] and [3] agreement' is shown independent of the special choice of Lagrangian.
The author is greatly indebted t o Prof. H.-J. TREDEIR.
for stimulating discussions and
advices. He also wishes to thank Dr. U. KASPER
for many helpful discussions.
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Bei der Redaktion eingegangeii am 24. Januar 1985
Anschr. d. Verf.: Dr. HELMUT
Zentralinstitut fur Astrophysik der AdW der DDR
DDR-1503 Potsdam-Babelsberg
Rosa-Luxemburg-Str. 1 7 LL
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