Papapetrou Equations Research Articles

Dynamics of spinning test particles in Kerr spacetime

We investigate the dynamics of relativistic spinning test particles in the spacetime of a rotating black hole using the Papapetrou equations. We use the method of Lyapunov exponents to determine whether the orbits exhibit sensitive dependence on initial conditions, a signature of chaos. In the case of maximally spinning equal-mass binaries (a limiting case that violates the test-particle approximation) we find unambiguous positive Lyapunov exponents that come in pairs $\ifmmode\pm\else\textpm\fi{}\ensuremath{\lambda},$ a characteristic of Hamiltonian dynamical systems. We find no evidence for nonvanishing Lyapunov exponents for physically realistic spin parameters, which suggests that chaos may not manifest itself in the gravitational radiation of extreme mass-ratio binary black-hole inspirals (as detectable, for example, by LISA, the Laser Interferometer Space Antenna).

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.

Equations of motion of spinning relativistic particle in external fields

We consider the motion of a spinning relativistic particle in external electromagnetic and gravitational fields, to first order in the external field, but to an arbitrary order in spin. The noncovariant spin formalism is crucial for the correct description of the influence of the spin on the particle trajectory. We show that the true coordinate of a relativistic spinning particle is its naive, common coordinate $\r$. Concrete calculations are performed up to second order in spin included. A simple derivation is presented for the gravitational spin-orbit and spin-spin interactions of a relativistic particle. We discuss the gravimagnetic moment (GM), a specific spin effect in general relativity. It is shown that for the Kerr black hole the gravimagnetic ratio, i.e., the coefficient at the GM, equals unity (just as for the charged Kerr hole the gyromagnetic ratio equals two). The equations of motion obtained for relativistic spinning particle in external gravitational field differ essentially from the Papapetrou equations.

Equations of motion of a spinning relativistic particle in external fields

We consider the motion of a spinning relativistic particle in external electromagnetic and gravitational fields to first order in the external field but to arbitrary order in the spin. The influence of the spin on the particle trajectory is properly accounted for by describing the spin noncovariantly. Specific calculations are performed through second order in the spin. A simple derivation is presented for the gravitational spin-orbit and spin-spin interactions of a relativistic particle. We discuss the gravimagnetic moment (GM), a particular spin effect in general relativity. We show that for a Kerr black hole the gravimagnetic ratio, i.e., the coefficient of the GM, equals unity (just as the gyromagnetic ratio equals 2 for a charged Kerr hole). The equations of motion obtained for a spinning relativistic particle in an external gravitational field differ substantially from the Papapetrou equations.

THE PAPAPETROU EQUATIONS AND MOTION OF A TEST BODY IN GRAVITATIONAL FIELD

The Papapetrou equations for a dipole particle are criticized in favor of the evolution equations obtained by Dixon and Madore. It is concluded that an additional condition (Pirani, Dixon, Corinaldesi, etc.) which permits the calculation of a unique world line of a particle is a substitute for data of a particle structure. Equations of motion of an extended test body in terms of multipoles are presented.

The multipole particle and the Papapetrou equation

The conclusion is made that the supplementary conditions of Pirani, Dixon, or others, which close the system of dipole equations of the general theory of relativity for a test particle and as a result make it possible to uniquely calculate the motion of such a particle, in essence determine the internal structure of the particle. The conclusion is made against the background of the solution of the problem of the multipole expansion of the tensor density along a line and a multipole formulation of the equations of motion of a body. It is shown that the Papapetrou equations can be employed to calculate the behavior of particles only if information contained in the dipole equations is used.

Dynamical equations of spinning particles. Feynman's proof

We discuss the extension of Feynman's derivation of the equation of motion to the case of spinning particles. We show that a spinning particle interacts only with the electromagnetic and gravitational fields. In the absence of the electromagnetic interactions, we rederive Papapetrou's equations for spinning particles in the background of the conformal gravity. We also find that the effect of spin coupled to non-constant electromagnetic fields leads to further corrections to the Lorentz force equations. Some discussions of these results are given at the end.

Induced rotation in the Papapetrou equations

We obtain an integral form of the Papapetrou equations, which describes the motion of an extended body in an external gravitational field. Using the Fock method, we calculate an explicit form for the components of the spin tensor and derive relativistic equations of rotational motion in the Schwarzschild space V4. We show that the spin of the body becomes proportional not only to the angular velocity but also to the angular momentum. Thus, induced rotation also follows from the Papapetrou equations.

The Influence of Supplementary Conditions for Papapetrou Equations on the Lyapunov Stability of Motion

AbstractThe Lyapunov stability of circular motions of spinning test bodies in the Schwarzschild field has been considered. For Papapetrou equations the generalized supplementary conditions S0iu0 = kSijuj = k const, i, j = 1, 2, 3, have been chosen. Four of the first integrals of motion have been indicated. With the aid of Lyapunov functions the domains of stable motions have been found.

The spinning bodies motion stability by lyapunov in an axially symmetric gravitational field

The stability by Lyapunov of the circular motion of spinning test bodies in a Lense-Thirring field has been investigated. For Papapetrou's equations the generalized supplementary conditionsS oi U o =kS ij U j ,k=const.,i, j = 1, 2, 3 have been chosen. Four of the first integrals of motion have been indicated. The existence, stability, and instability regions have been found. It is shown that the positions of stability and instability ranges depend considerably upon the type of supplementary conditions.

The equations of motion of a test particle with spin and self-gravity

view Abstract Citations (10) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The equations of motion of a test particle with spin and self-gravity Noonan, T. W. Abstract Papapetrou's equations of motion of a spinning particle in general relativity are extended to the case of a nonsymmetric stress-energy tensor in order to include the test particle's self-gravity. There are derivations of the spin tensor and the momentum tensor and their equations of motion, including gravitational radiation reaction. There is a treatment of a test particle with self-gravity, charge, and electromagnetic moment. The dynamical role of the test particle's self-gravity is illustrated in the weak-field approximation. The principal conclusion is that the test particle's Newtonian gravitational potential energy contributes to the test particle's inertial mass (under electromagnetic forces) in the same way as do other forms of mass-energy (rest mass, internal energy, and kinetic energy). Publication: The Astrophysical Journal Pub Date: April 1985 DOI: 10.1086/163081 Bibcode: 1985ApJ...291..422N Keywords: Equations Of Motion; Gravitation Theory; Particle Motion; Particle Spin; Relativity; Electromagnetism; Gravitational Waves; Momentum; Stress Tensors; Transformations (Mathematics); Weak Interactions (Field Theory); Physics (General) full text sources ADS |

Propagation equations for test bodies with spin and rotation in theories of gravity with torsion

We generalize the Papapetrou equations by deriving propagation equations for the energy-momentum and angular momentum of a test body which has both elementary-particle spin and macroscopic rotation and which is moving in background metric and torsion fields. Our results show that the torsion couples to spin but not to rotation. Thus a rotating test body with no net spin will ignore the torsion and move according to the usual Papapetrou equations. Hence the standard tests of gravity are insensitive to a torsion field. We propose experiments (although still infeasible) to compare the motion of a spin-polarized body with the motion of a rotating body. If the spin and rotation precess differently, the theory of gravity cannot be a metric theory but may be a torsion theory.

Reference systems for the isolated observer and the dynamics of test particles

A method of deriving the equations of motion for rotating test particles with the aid of reference systems for an isolated observer is considered. It is shown that the form of the equations depends upon the manner in which the reference system for the isolated observer are determined (calibrated). Reference systems, which in the case under consideration lead to the Papapetrou equations, were also determined. These systems were used to derive the equations of motion for particles for which the quadrupole moment of their energy-momentum tensor was taken into account.

Motion of a spinning test particle in Vaidya's radiating metric

The motion of a spinning test particle in Vaidya's gravitational field is considered in the framework of Papapetrou's equations of motion. Use is made of the supplementary condition S/sup ..mu..//sup u/ = 0, where u is the retarded Schwarzschild time coordinate. We derive the equations for the dynamical variables, and consider the conservation laws, that follow from the equations of motion. Particular cases of motion are also discussed and additional first integrals corresponding to these cases are found. Some of the new extra integrals are related to the Casimir operators of the Poincare group. It is found that under special conditions on the spin tensor components the particle follows a geodesic. Motion of the spinning test particle in the Schwarzschild field is considered as one of the particular cases.

Mean motion of Dirac electrons in a gravitational field

A sequence of Foldy-Wouthuysen transformations applied to the Dirac equation coupled to a background gravitational field is used to obtain evolution equations for the mean position, mean momentum, and mean spin operators. These equations are compared with the analogous Papapetrou equations for a classical gravitationally coupled pole-dipole particle.

Two examples of massive scattering using twistor Hamiltonians

A twistor Hamiltonian formulation is given for describing the scattering of a massive particle with spin by a plane impulsive gravitational wave and of a massive particle with spin and charge by a plane impulsive electromagnetic wave. As a Taylor series in the spin of the particle, the expressions obtained agree with the Papapetrou equations and the magnetic dipole force law, respectively, to first order in the spin.

On the equivalence of the Einstein−Mayer and Einstein−Cartan theories for describing a spinning medium

It is shown that the five−dimensional formalism of the Einstein−Mayer theory of 1932 can be given a new interpretation and used to describe a continuous medium with intrinsic angular momentum. The equations of motion of a spinning particle are derived from the generalized field equations and are seen to be essentially the same as the Mathisson−Papapetrou equations. Finally the correspondence of the new approach with the theories based on Cartan’s torsion tensor

�ber die Bewegungsgleichungen eines makroskopischen rotierenden Teilchens gem�� der allgemeinen Relativit�tstheorie

It is shown that under special covariant constraints the Papapetrou equations of motion of a spinning particle (rotating planet) moving in a spherically symmetric field (Schwarzschild field) give additional terms in the expressions for the advance of perihelion and the deflection angle of particles.