Spinning test particles in general relativity: Nongeodesic motion
in the Reissner-Nordstro
¨
m spacetime
Donato Bini
Istituto per Applicazioni della Matematica, C.N.R., I – 80131 Napoli, Italy
and International Center for Relativistic Astrophysics, University of Rome, I – 00185 Roma, Italy
Gianluca Gemelli
Istituto Matematico, Universita ` di Roma ‘‘La Sapienza,’’ P.le A. Moro 2, I-00185 Roma, Italy
and International Center for Relativistic Astrophysics, University of Rome, I – 00185 Roma, Italy
Remo Ruffini
Istituto di Fisica, Universita ` di Roma ‘‘La Sapienza,’’ P.le A. Moro 2, I-00185 Roma, Italy
and International Center for Relativistic Astrophysics, University of Rome, I – 00185 Roma, Italy
Received 30 October 1998; published 24 February 2000
The dynamics of a charged spinning test particle in general relativity is studied in the context of gravito-
electromagnetism. Various families of test observers and supplementary conditions are examined. The spin-
gravity-electromagnetism coupling is investigated for motion in the background of a Reissner-Nordstro ¨m black
hole both in the exact spacetime and in the weak-field approximation. Results are compared with those of the
theory.
PACS numbers: 04.20.Cv
I. INTRODUCTION
In this paper we consider a charged massive test particle
in the Dixon-Souriau model 1–9, which is a first order
cutoff in the multipole expansion of the equations of motion
of a small extended body. The model includes spin-
electromagnetism and spin-gravity coupling terms, but it re-
quires the assumption of additional supplementary condi-
tions to be completed. For uncharged particles it reduces to
the well-known Papapetrou model.
Here we briefly review the model and the different possi-
bilities for the supplementary conditions, with their corre-
sponding meaning in terms of the center-of-mass world line
definition. We then provide the splitting of the equations of
motion with respect to a frame associated with a generic
four-velocity field u, and then specialize the results for cer-
tain preferred families of observers.
As an example, we consider the Reissner-Nordstro
¨
m
background and study the evolution of a particle with respect
to the family of static observers. We thus incorporate into a
single framework the exact generalization of earlier results
belonging to the purely electromagnetic case 10–12 and to
the purely gravitational case 11,13,14. We also discuss the
weak field slow motion approximation and somewhat com-
plete and generalize the picture given in the literature.
II. THE PAPAPETROU EQUATIONS OF MOTION
In general relativity, an extended body is described by its
associated energy-momentum tensor. A small body can be
studied by a multipole expansion method enabling it to be
equivalently described by a set of multipole energy-
momentum moments defined along a central line 1–3,15.
An analogous scheme is obtained by considering a singular
energy-momentum tensor distribution defined along a single
curve 6–8. The cutoff at successive multipole orders de-
fines a hierarchy of elementary multipole particles see, e.g.,
5,15–17. The first cutoff yields a point particle or single
pole governed by the geodesic equation of motion.
The next cutoff leads to the dipole ‘‘spinning’’ particle
which interests us here. The equations of motion for such a
particle were first derived in the purely gravitational case by
Papapetrou 15 as
D
d
U
p
=
1
2
R
S
U
,
D
d
U
S
= p
U
- p
U
, 1
where R
is the Riemann tensor, p
is the generalized
momentum vector, S
is a antisymmetric spin tensor, U
=DX / d
U
is the unit tangent vector ( U
U
=-1) of the
‘‘center line’’ l
U
used to make the multipole reduction, and
where X =X (
U
) is the center point whose world line is l
U
.
The fields S, U, and p are defined only along l
U
. Units are
chosen here so that the speed of light in empty space satisfies
c =1.
It is well known that the number of independent equations
in Eq. 1 is less than that of the unknown quantities; three
additional scalar supplementary conditions SC are needed
for the scheme to be completed. Once a suitable choice has
been made, l
U
, p, and S can in principle be determined by
the complete set of equations.
The various supplementary conditions which are consid-
ered in the literature are all of the form u
ˆ
S
=0 for some
timelike unit vector u
ˆ
along the world line l
U
. According to
the special relativistic analogy 18, p. 161, this is equiva-
PHYSICAL REVIEW D, VOLUME 61, 064013
0556-2821/2000/616/06401310/$15.00 ©2000 The American Physical Society 61 064013-1