34 research outputs found
Mathisson-Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds
A new representation, which does not contain the third-order derivatives of
the coordinates, of the exact Mathisson-Papapetrou-Dixon equations, describing
the motion of a spinning test particle, is obtained under the assumption of the
Mathisson-Pirani condition in a Kerr background. For this purpose the integrals
of energy and angular momentum of the spinning particle as well as a
differential relationship following from the Mathisson-Papapetrou-Dixon
equations are used. The form of these equations is adapted for their computer
integration with the aim to investigate the influence of the spin-curvature
interaction on the particle's behavior in the gravitational field without
restrictions on its velocity and spin orientation. Some numerical examples for
a Schwarzschild background are presented.Comment: 21 pages, 11 figures. arXiv admin note: substantial text overlap with
arXiv:1105.240
Highly relativistic spinning particle in the Schwarzschild field: Circular and other orbits
The Mathisson-Papapetrou equations in the Schwarzschild background both at
Mathisson-Pirani and Tulczyjew-Dixon supplementary condition are considered.
The region of existence of highly relativistic circular orbits of a spinning
particle in this background and dependence of the particle's orbital velocity
on its spin and radial coordinate are investigated. It is shown that in
contrast to the highly relativistic circular orbits of a spinless particle,
which exist only for , , the corresponding
orbits of a spinning particle are allowed in a wider space region, and the
dimension of this region significantly depends on the supplementary condition.
At the Mathisson-Pirani condition new numerical results which describe some
typical cases of non-circular highly relativistic orbits of a spinning particle
starting from are presented.Comment: 10 pages, 11 figure
Ultrarelativistic circular orbits of spinning particles in a Schwarzschild field
Ultrarelativistic circular orbits of spinning particles in a Schwarzschild
field described by the Mathisson-Papapetrou equations are considered. The
preliminary estimates of the possible synchrotron electromagnetic radiation of
highly relativistic protons and electrons on these orbits in the gravitational
field of a black hole are presentedComment: 9 page
Mathisson's helical motions for a spinning particle --- are they unphysical?
It has been asserted in the literature that Mathisson's helical motions are
unphysical, with the argument that their radius can be arbitrarily large. We
revisit Mathisson's helical motions of a free spinning particle, and observe
that such statement is unfounded. Their radius is finite and confined to the
disk of centroids. We argue that the helical motions are perfectly valid and
physically equivalent descriptions of the motion of a spinning body, the
difference between them being the choice of the representative point of the
particle, thus a gauge choice. We discuss the kinematical explanation of these
motions, and we dynamically interpret them through the concept of hidden
momentum. We also show that, contrary to previous claims, the frequency of the
helical motions coincides, even in the relativistic limit, with the
zitterbewegung frequency of the Dirac equation for the electron
Spinning particles in Schwarzschild-de Sitter space-time
After considering the reference case of the motion of spinning test bodies in
the equatorial plane of the Schwarzschild space-time, we generalize the results
to the case of the motion of a spinning particle in the equatorial plane of the
Schwarzschild-de Sitter space-time. Specifically, we obtain the loci of turning
points of the particle in this plane. We show that the cosmological constant
affect the particle motion when the particle distance from the black hole is of
the order of the inverse square root of the cosmological constant.Comment: 8 pages, 5 eps figures, submitted to Gen.Rel.Gra
Spacetime dynamics of spinning particles - exact electromagnetic analogies
We compare the rigorous equations describing the motion of spinning test
particles in gravitational and electromagnetic fields, and show that if the
Mathisson-Pirani spin condition holds then exact gravito-electromagnetic
analogies emerge. These analogies provide a familiar formalism to treat
gravitational problems, as well as a means for comparing the two interactions.
Fundamental differences are manifest in the symmetries and time projections of
the electromagnetic and gravitational tidal tensors. The physical consequences
of the symmetries of the tidal tensors are explored comparing the following
analogous setups: magnetic dipoles in the field of non-spinning/spinning
charges, and gyroscopes in the Schwarzschild, Kerr, and Kerr-de Sitter
spacetimes. The implications of the time projections of the tidal tensors are
illustrated by the work done on the particle in various frames; in particular,
a reciprocity is found to exist: in a frame comoving with the particle, the
electromagnetic (but not the gravitational) field does work on it, causing a
variation of its proper mass; conversely, for "static observers," a stationary
gravitomagnetic (but not a magnetic) field does work on the particle, and the
associated potential energy is seen to embody the Hawking-Wald spin-spin
interaction energy. The issue of hidden momentum, and its counterintuitive
dynamical implications, is also analyzed. Finally, a number of issues regarding
the electromagnetic interaction and the physical meaning of Dixon's equations
are clarified.Comment: 32+11 pages, 5 figures. Edited and further improved version, with new
Section C.2 unveiling analogies for arbitrary spin conditions, and new Sec.
3.2.3 in the Supplement making connection to the post-Newtonian
approximation; former Sec. III.B.4 and Appendix C moved to the (reshuffled)
Supplement; references updated. The Supplement is provided in ancillary file.
Matches the final published versio
Gravitational ultrarelativistic spin-orbit interaction and the weak equivalence principle
It is shown that the gravitational ultrarelativistic spin-orbit interaction
violates the weak equivalence principle in the traditional sense. This fact is
a direct consequence of the Mathisson-Papapetrou equations in the frame of
reference comoving with a spinning test particle. The widely held assumption
that the deviation of a spinning test body from a geodesic trajectory is caused
by tidal forces is not correctComment: 12 page
Stability of circular orbits of spinning particles in Schwarzschild-like space-times
Circular orbits of spinning test particles and their stability in
Schwarzschild-like backgrounds are investigated. For these space-times the
equations of motion admit solutions representing circular orbits with particles
spins being constant and normal to the plane of orbits. For the de Sitter
background the orbits are always stable with particle velocity and momentum
being co-linear along them. The world-line deviation equations for particles of
the same spin-to-mass ratios are solved and the resulting deviation vectors are
used to study the stability of orbits. It is shown that the orbits are stable
against radial perturbations. The general criterion for stability against
normal perturbations is obtained. Explicit calculations are performed in the
case of the Schwarzschild space-time leading to the conclusion that the orbits
are stable.Comment: eps figures, submitted to General Relativity and Gravitatio
Center of mass, spin supplementary conditions, and the momentum of spinning particles
We discuss the problem of defining the center of mass in general relativity
and the so-called spin supplementary condition. The different spin conditions
in the literature, their physical significance, and the momentum-velocity
relation for each of them are analyzed in depth. The reason for the
non-parallelism between the velocity and the momentum, and the concept of
"hidden momentum", are dissected. It is argued that the different solutions
allowed by the different spin conditions are equally valid descriptions for the
motion of a given test body, and their equivalence is shown to dipole order in
curved spacetime. These different descriptions are compared in simple examples.Comment: 45 pages, 7 figures. Some minor improvements, typos fixed, signs in
some expressions corrected. Matches the published version. Published as part
of the book "Equations of Motion in Relativistic Gravity", D. Puetzfeld et
al. (eds.), Fundamental Theories of Physics 179, Springer, 201
Spin-Spin Interactions in Gauge Theory of Gravity, Violation of Weak Equivalence Principle and New Classical Test of General Relativity
For a long time, it is generally believed that spin-spin interactions can
only exist in a theory where Lorentz symmetry is gauged, and a theory with
spin-spin interactions is not perturbatively renormalizable. But this is not
true. By studying the motion of a spinning particle in gravitational field, it
is found that there exist spin-spin interactions in gauge theory of gravity.
Its mechanism is that a spinning particle will generate gravitomagnetic field
in space-time, and this gravitomagnetic field will interact with the spin of
another particle, which will cause spin-spin interactions. So, spin-spin
interactions are transmitted by gravitational field. The form of spin-spin
interactions in post Newtonian approximations is deduced. This result can also
be deduced from the Papapetrou equation. This kind of interactions will not
affect the renormalizability of the theory. The spin-spin interactions will
violate the weak equivalence principle, and the violation effects are
detectable. An experiment is proposed to detect the effects of the violation of
the weak equivalence principle.Comment: 17 pages, no figur