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 $r=1.5 r_g(1+\delta)$, $0<\delta \ll 1$, 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 $r>1.5 r_g$ 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