Mathisson-Papapetrou Equations Research Articles

Highly relativistic circular orbits of spinning particle in the Kerr field

The Mathisson-Papapetrou equations in Kerr's background are considered. The region of existence of highly relativistic planar circular orbits of a spinning particle in this background and dependence of the particle's Lorentz $\gamma$-factor on its spin and radial coordinate are investigated. It is shown that in contrast to the highly relativistic circular orbits of a spinless particle the corresponding orbits of a spinning particle are allowed in much wider space region. Some of these orbits show the significant attractive action of the spin-gravity coupling on a particle and others are caused by the significant repulsive action. Numerical estimates for electrons, protons and neutrinos in the gravitational field of black holes are presented.

Test particle motion in modified gravity theories

We derive the equations of motion of an electrically neutral test particle for modified gravity theories for which the covariant divergence of the ordinary matter energy-momentum tensor does not vanish (i.e., ${\ensuremath{\nabla}}_{\ensuremath{\mu}}{T}^{\ensuremath{\mu}\ensuremath{\nu}}\ensuremath{\ne}0$). In fact, we generalize the Mathisson-Papapetrou equations by deriving a general form for the equations of motion of a test particle. Furthermore, using the generalized Mathisson-Papapetrou equations, we investigate the equations of motion of a pole-dipole (spinning) particle in the context of modified gravity.

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.

Republication of: The mechanics of matter particles in general relativity

This is an English translation of the first of two papers by Myron Mathisson, first published in German in 1931 and 1937, in which he presented the correct formulation of equations of motion of spinning bodies in general relativity (today known as the Mathisson–Papapetrou equations). The papers have been selected by the Editors of General Relativity and Gravitation for republication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note and Mathisson’s brief biography, both written by Andrzej Trautman.

Republication of: New mechanics of material systems

This is an English translation of the second of two papers by Myron Mathisson, first published in German in 1931 and 1937, in which he presented the correct formulation of equations of motion of spinning bodies in general relativity (today known as the Mathisson–Papapetrou equations). The papers have been selected by the Editors of General Relativity and Gravitation for republication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note and Mathisson’s brief biography, both written by Andrzej Trautman.

Tangent Euler Top in General Relativity

We define the tangent Euler top in General Relativity through a constrained Lagrangian on the orthonormal frame bundle. The corresponding motions are studied to various degrees of approximation, the lowest of which is shown to yield the Mathisson-Papapetrou equations.

Quasi-Maxwell interpretation of the spin–curvature coupling

We write the Mathisson-Papapetrou equations of motion for a spinning particle in a stationary spacetime using the quasi-Maxwell formalism and give an interpretation of the coupling between spin and curvature. The formalism is then used to compute equilibrium positions for spinning particles in the NUT spacetime.

Mathisson–Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation

We present a simple method of deriving the semiclassical equations of motion for a spinning particle in a gravitational field. We investigate the cases of classical, rotating particles, i.e. the so-called pole–dipole particles, as well as particles with an additional intrinsic spin. We show that, starting with a simple Lagrangian, one can derive equations for the spin evolution and momentum propagation in the framework of metric theories of gravity (general relativity) and in theories based on a Riemann–Cartan geometry (Poincaré gauge theory), without explicitly referring to matter current densities (spin and stress energy). Our results agree with those derived from the multipole expansion of the current densities by the conventional Papapetrou method and from the WKB analysis for elementary particles.

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 presented.

Spinning particles in the vacuum C metric

The motion of a spinning test particle given by the Mathisson–Papapetrou equations is studied on an exterior vacuum C metric background spacetime describing the accelerated motion of a spherically symmetric gravitational source. We consider circular orbits of the particle around the direction of acceleration of the source. The symmetries of this configuration lead to the reduction of the differential equations of motion to algebraic relations. The spin supplementary conditions as well as the coupling between the spin of the particle and the acceleration of the source are discussed.

Spinning test particles and clock effect in Kerr spacetime

We study the motion of spinning test particles in Kerr spacetime using the Mathisson–Papapetrou equations; we impose different supplementary conditions from the well-known conditions of Corinaldesi–Papapetrou, Pirani and Tulczyjew, and analyse their physical implications in order to decide which is the most natural to use. We find that if the particle's centre-of-mass world line, namely the one chosen for multipole reduction, is a spatially circular orbit (sustained by the tidal forces due to the spin), then the generalized momentum P of the test particle is also tangent to a spatially circular orbit intersecting the centre-of-mass line at a point. There exists one such orbit for each point of the centre-of-mass line where they intersect; although fictitious, these orbits are essential to define the properties of the spinning particle along its physical motion. In the small spin limit, the particle orbit is almost a geodesic and the difference in its angular velocity with respect to the geodesic value can be of arbitrary sign, corresponding to the possible spin-up and -down alignments along the z-axis. We also find that the choice of the supplementary conditions leads to clock effects of substantially different magnitude. In fact, for co- and counter-rotating particles having the same spin magnitude and orientation, the gravitomagnetic clock effect induced by the background metric can be magnified or inhibited and even suppressed by the contribution of the spin of individual particles. Quite surprisingly, this contribution can itself be made vanishingly small, leading to a clock effect indistiguishable from that of non-spinning particles. The results of our analysis can be observationally tested.

Spinning test particles and clock effect in Schwarzschild spacetime

We study the behaviour of spinning test particles in the Schwarzschild spacetime. Using Mathisson–Papapetrou equations of motion, we confine our attention to spatially circular orbits and search for observable effects which could eventually discriminate among the standard supplementary conditions, namely the Corinaldesi–Papapetrou, Pirani and Tulczyjew. We find that if the world line chosen for the multipole reduction and whose unit tangent we denote as U is a circular orbit then the generalized momentum P of the spinning test particle is also tangent to a circular orbit even though P and U are not parallel four-vectors. These orbits are shown to exist because the spin-induced tidal forces provide the required acceleration irrespective of the supplementary conditions we select. Of course, in the limit of a small spin, the particle's orbit is close to being a circular geodesic and the (small) deviation of the angular velocities from the geodesic values can be of an arbitrary sign, corresponding to the possible spin-up and spin-down alignment to the z-axis. When two spinning particles orbit around a gravitating source in opposite directions, they make one loop with respect to a given static observer with different arrival times. This difference is termed the clock effect. We find that a nonzero gravitomagnetic clock effect appears for oppositely orbiting spin-up or spin-down particles even in the Schwarzschild spacetime. This allows us to establish a formal analogy with the case of (spin-less) geodesics on the equatorial plane of the Kerr spacetime. This result can be verified experimentally.

Spinning test particles in a Kerr field - II

The motion of small spinning free test bodies is usually treated within the 'pole-dipole' approximation, which - in general relativity - leads to Mathisson-Papapetrou (MP) equations. These have to be supplemented by three side constraints in order to provide a unique solution. Several different 'spin conditions' have been proposed and used in the literature, each leading to different worldlines. In a previous paper, we integrated the MP equations with the p σ S μσ = 0 condition numerically in Kerr space-time and illustrated the effect of the spin-curvature interaction by comparing the trajectories obtained for various spin magnitudes. Here we also consider other spin conditions and clarify their interrelations analytically as well as numerically on particular trajectories. The notion of a 'minimal worldtube' is introduced in order to judge the individual supplementary conditions and to expose the limitations of the pole-dipole approximation.

Gravitational ultrarelativistic interaction of classical particles in the context of unification of interactions

The response of the ultrarelativistic particle with spin in a Schwarzschild field to the gravitomagnetic components as measured by the comoving observer is investigated. The dependence of the particle's spin–orbit acceleration on the Lorentz γ-factor and the spin orientation is studied. The concrete circular ultrarelativistic orbit of radius r = 3m is considered as a partial solution of the Mathisson–Papapetrou equations and as the corresponding high-energy quantum state of the Dirac particle. Numerical estimates for protons and electrons near black holes are given. A tendency of gravitational and electromagnetic interactions to approach in quantitative terms at ultrarelativistic velocities is discussed.

On the comparison of solutions of the Dirac and Mathisson–Papapetrou equations in a Schwarzschild field

On the comparison of solutions of the Dirac and Mathisson–Papapetrou equations in a Schwarzschild field

Spinning test particles in a Kerr field -- I

Mathisson—Papapetrou equations are solved numerically to obtain trajectories of spinning test particles in (the meridional section of) the Kerr space—time. The supplementary conditions pSμσ=0 are used to close the system of equations. The results show that in principle a spin-curvature interaction may lead to considerable deviations from geodesic motion, although in astrophysical situations of interest probably no large spin effects can be expected for values of spin consistent with a pole—dipole test-particle approximation. However, a significant cumulative effect may occur, e.g. in the inspiral of a spinning particle on to a rotating compact body, that would modify gravitational waves generated by such a system. A thorough literature review is included in the paper.

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 correct

SPINOR MATTER IN A GRAVITATIONAL FIELD : COVARIANT EQUATIONS A LA HEISENBERG

A fundamental tenet of general relativity is geodesic motion of point particles. For extended objects, however, tidal forces make the trajectories deviate from geodesic form. In fact Mathisson, Papapetrou, and others have found that even in the limit of very small size there exists a residual curvature-spin force. Another important physical case is that of field theory. Here the ray (WKB) approximation may be used to obtain the equation of motion. In this article I consider an alternative procedure, the proper time translation operator formalism, to obtain the covariant Heisenberg equations for the quantum velocity, momentum, and angular momentum operators for the case of spinor fields. I review the flat spacetime results for Dirac particles in Yang-Mills fields, where we recover the Lorentz force. For curved spacetime I find that the geodesic equation is modified by an additional term involving the spin tensor, and the parallel transport equation for the momentum is modified by an additional term involving the curvature tensor. This curvature term is the “Lorentz force” of the gravitational field. The main result of this article is that these equations are exactly the (symmetrized) Mathisson-Papapetrou equations for the quantum operators. Extension of these results to the case of spin-one fields may be possible by use of the KDP formalism.

The spherically-symmetrical trajectories of spin particles in the Schwarzschild field

The motion of spin particles in the Schwarzschild field is examined in this paper. It is shown that Mathisson-Papapetrou equations under additional conditions\(\mathop L\limits_j {\text{ }}S^{\alpha \beta } {\text{ = }}0\), where\(\mathop L\limits_j\) is the Lie derivative of the Killing vector ξj, permit only radial motion, motion in the equatorial plane and in the equilibrium points. All these types of motion are considered more in detail.

Conditions for the appearance of gravitational ultrarelativistic spin-orbital interaction

A qualitative analysis of the energy and momentum integrals for the Mathisson-Papapetrou equations in a Schwarzschild field is employed to determine the conditions under which all solutions of the exact Mathisson-Papapetrou equations corresponding to fixed initial coordinate and velocity values differ significantly from the solution of the abbreviated equations normally used for the same initial data. Numerical computer calculations provide additional indication of a difference in the world lines of the exact Mathisson-Papapetrou equations from solutions of the geodesic equations at ultrarelativistic velocity values. For high energy particles entering into the composition of cosmic rays one can expect appearance of gravitational ultrarelativistic spin-orbital interaction upon motion of such particles in the field of a neutron star.