Hyperbolic-like elastic scattering of spinning particles by a Schwarzschild black hole

The scattering of spinning test particles by a Schwarzschild black hole is studied. The motion is described according to the Mathisson–Papapetrou–Dixon model for extended bodies in a given gravitational background field. The equatorial plane is taken as the orbital plane, the spin vector being orthogonal to it with constant magnitude. The equations of motion are solved analytically in closed form to first-order in spin and the solution is used to compute corrections to the standard geodesic scattering angle as well as capture cross section by the black hole.     

Nonequatorial circular orbits of spinning particles in the Schwarzschild–de Sitter background

In this paper analytical solutions of the Mathisson–Papapetrou equations that describe nonequatorial circular orbits of a spinning particle in the Schwarzschild–de Sitter background are studied, and the role of the cosmological constant is emphasized. It is shown that generally speaking a highly relativistic velocity of the particle is a necessary condition of motion along this type of orbits, with an exception of orbits locating close to the position of the static equilibrium, where low velocities are possible as well. Depending on the correlation between the spin orientation of the particle and its orbital velocity some of the possible nonequatorial circular orbits exist due to the repulsive action on the particle caused by the spin–gravity coupling and the others are caused by the attractive action. Here values of the energy of the particle on the corresponding orbits are also analyzed.     

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.     

Dixon’s extended bodies and weak gravitational waves

General relativity considers Dixon’s theory as the standard theory to deal with the motion of extended bodies in a given gravitational background. We discuss here the features of the “reaction” of an extended body to the passage of a weak gravitational wave. We find that the body acquires a dipolar moment induced by its quadrupole structure. Furthermore, we derive the “world function” for the weak field limit of a gravitational wave background and use it to estimate the deviation between geodesics and the world lines of structured bodies. Measuring such deviations, due to the existence of cumulative effects, should be favorite with respect to measuring the amplitude of the gravitational wave itself.     

Extended bodies with quadrupole moment interacting with gravitational monopoles: reciprocity relations

An exact solution of Einstein’s equations representing the static gravitational field of a quasi-spherical source endowed with both mass and mass quadrupole moment is considered. It belongs to the Weyl class of solutions and reduces to the Schwarzschild solution when the quadrupole moment vanishes. The geometric properties of timelike circular orbits (including geodesics) in this spacetime are investigated. Moreover, a comparison between geodesic motion in the spacetime of a quasi-spherical source and non-geodesic motion of an extended body also endowed with both mass and mass quadrupole moment as described by Dixon’s model in the gravitational field of a Schwarzschild black hole is discussed. Certain “reciprocity relations” between the source and the particle parameters are obtained, providing a further argument in favor of the acceptability of Dixon’s model for extended bodies in general relativity.     

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.Astrophysics Institute, Kazan Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 117–124, October, 1992.     

On the accuracy of the IWM–CFC approximation in differentially rotating relativistic stars

A circular restricted three-body problem describes the motion of a test particle around two massive bodies in circular orbits. In this system, orbital decay caused by a gravitational radiation reaction between the two primary bodies is considered but the direct effect of gravitational radiation on the test particle is neglected. We adopt distance- and time-scale transformations to Newtonian problems so that systems without orbital decay will not depend on separation between the primaries but systems with orbital decay will depend on this separation. If a regular or chaotic orbit is given in a Newtonian system, the starting separation of the primaries varies according to the corresponding decay system. Thus, insights into the chaotic behaviour of a third body in a decay case are provided. For a large initial separation between the primaries, the chaos that exists in a Newtonian problem may be retained for a long enough time scale of dissipative evolution before the primaries coalesce. The final state of a third body is escape attributed to orbital decay.     

Unraveling gravity beyond Einstein with extended test bodies

The motion of test bodies in gravity is tightly linked to the conservation laws. This well-known fact in the context of General Relativity is also valid for gravitational theories which go beyond Einstein?s theory. Here we derive the equations of motion for test bodies for a very large class of gravitational theories with a general nonminimal coupling to matter. These equations form the basis for future systematic tests of alternative gravity theories. Our treatment is covariant and generalizes the classic Mathisson–Papapetrou–Dixon result for spinning (extended) test bodies. The equations of motion for structureless test bodies turn out to be surprisingly simple, despite the very general nature of the theories considered.     

Republication of: New mechanics of material systems

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.     

Motion and gravitational radiation of a binary system consisting of an oscillating and rotating coplanar dusty disk and a point-like object

A binary system composed of an oscillating and rotating coplanar dusty disk and a point mass is considered. The conservative dynamics is treated on the Newtonian level. The effects of gravitational radiation reaction and wave emission are studied to leading quadrupole order. The related waveforms are given. The dynamical evolution of the system is determined semi-analytically exploiting the Hamiltonian equations of motion which comprise the effects both of the Newtonian tidal interaction and the radiation reaction on the motion of the binary system in elliptic orbits. Tidal resonance effects between orbital and oscillatory motions are considered in the presence of radiation damping.     

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. Zh. éksp. Teor. Fiz. 113, 1537–1557 (May 1998)     

Deformable gyroscope in a non-euclidean space classical,non-relativistic theory

The classical mechanics of structured test particles in a manifold with affine connection is studied. Gyroscopic rotations and homogeneous deformations are taken into account as internal degrees of freedom. Hence, in addition to the orbital motion of the centre of mass, the body undergoes affine rotations about the centre (“affinely-rigid body”). Configurations of particles are described mathematically by linear frames in an underlying manifold (physical space).Symmetries of the theory are discussed and in some special cases the equations of motion are derived. The orbital motion is found to be influenced by internal degrees of freedom which are dynamically coupled to the geometry of the manifold. For example, in a Riemann-Cartan space ([4], [8], [9], [25]), internal degrees of freedom interact with curvature and torsion tensors. Imposing then some holonomic constraints (orthonormal frames only), one gets the theory of test rigid bodies in a curved space with torsion. In a Riemannian case (no torsion) such a theory seems to coincide with non-relativistic (although spatially non-Euclidean) limit of theories studied by Dixon, Künzle, Tulczyjew and Papapetrou [5], [13], [26], [15].As for all standard techniques and notations of differential geometry, we follow mainly Kobayashi-Nomizu [12] and Sternberg [23].     

Gravitational Waves from a Pseudo-Newtonian Kerr Field with Halos

A close relation between gravitational waveforms and the types of trajectories in a superposed field between a pseudo-Newtonian Kerr black hole and quadrupolar halos is shown in detail. The gravitational waveforms emitted from circular, KAM tori and chaotic orbits must be periodic, quasiperiodic and stochastic, respectively. The chaotic motion can maximally enhance both the amplitudes and the energy emission rates of the waves.     

Relativistic orbital perturbations in a weak gravitational field

With the general third-order equations of motion for a test particle, Synge's third-order orbital equations at great distance in the weak gravitational field generated by a massive body are derived. The body has an axis of symmetry around which is rotating steadily. The results found for the advance of perihelion using first integrals of motion for the general equations show that the effect due to the inner stress of the body can be derived for orbits with inclination with respect to the equator of the body. Then, by means of the variation of the parameters method, we obtain with the equations at great distance the corresponding perturbations on the elements of such orbits in the field considered. These perturbations result to be of second order with regard to the mass of the body (the basis of the approximation).     

Analytical effective one-body formalism for extreme-mass-ratio inspirals with eccentric orbits

Extreme-mass-ratio inspirals(EMRIs) are among the most important sources for future spaceborne gravitational wave detectors. In this kind of system, compact objects usually orbit around central supermassive black holes on complicated trajectories. Usually, these trajectories are approximated as the geodesics of Kerr space-times, and orbital evolution is simulated with the help of the adiabatic approximation. However, this approach omits the influence of the compact object on its background. In this paper, using the effective one-body formalism, we analytically calculate the trajectory of a nonspinning compact object around a massive Kerr black hole in an equatorial eccentric orbit(omitting the orbital inclination) and express the fundamental orbital frequencies in explicit forms. Our formalism includes the first-order corrections for the mass ratio in the conservative orbital motion. Furthermore, we insert the mass-ratio-related terms into the first post-Newtonian energy fluxes. By calculating the gravitational waves using the Teukolsky equations, we quantitatively reveal the influence of the mass of the compact object on the data analysis. We find that the shrinking of geodesic motion by taking small objects as test particles may not be appropriate for the detection of EMRIs.     

Fourth-order orbital equations in stationary weak gravitational fields

The equations of motion in fourth approximation for gravitational bodies are used to obtain orbital equations, first integrals, differential equations for the corresponding trajectories, and fourth-order contributions to the orbital motions in stationary weak gravitational fields.     

Spinor Matter in a Gravitational Field: Covariant Equations à 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.     

Gravitational radiation from n-body systems

An expression for the quadrupole moment of any two-body system with structure is derived from a “paralel axes” theorem. Within the weak-field limit of the theory of general relativity, expressions for the gravitational radiation flux of energy and angular momentum from two particles or two spherically symmetric bodies in arbitrary plane motion arising from any type of forces are consequently obtained in terms of time derivatives of the relative coordinates of the system. An estimate of the gravitational flux from any plane motion follows. In particular, the flux from systems with Keplerian and straight-line motion are deduced as special cases. For the general problem of a two-body system with intrinsic quadrupole moment (due to deviation from spherical symmetry), it is found that in addition to the flux from the orbital and the spin motion there is another source of flux—the interaction flux. This is shown explicitly in two special cases—the system of a particle moving in the plane of symmetry of a Jacobi ellipsoid, and that of two spinning rigid rods in plane circular motion with parallel spin and orbital angular momentum. The interaction flux is regarded as the result of interaction of the bodies with gravitational waves. An outline of the method for the calculation of gravitational radiation flux from an n-body system is given. For a three-body system—an astrophysically interesting situation—this is worked out in detail. It is seen that the presence of an unsuspected third body can, by virtue of the interaction power term, increase the generation of gravitational waves significantly.     

Chaos and dynamics of spinning particles in Kerr spacetime

We study chaos dynamics of spinning particles in Kerr spacetime of rotating black holes use the Papapetrou equations by numerical integration. Because of spin, this system exists many chaos solutions, and exhibits some exceptional dynamic character. We investigate the relations between the orbits chaos and the spin magnitude S, pericenter, polar angle and Kerr rotation parameter a by means of a kind of brand new Fast Lyapulov Indicator (FLI) which is defined in general relativity. The classical definition of Lyapulov exponent (LE) perhaps fails in curve spacetime. And we emphasize that the Poincaré sections cannot be used to detect chaos for this case. Via calculations, some new interesting conclusions are found: though chaos is easier to emerge with bigger S, but not always depends on S monotonically; the Kerr parameter a has a contrary action on the chaos occurrence. Furthermore, the spin of particles can destroy the symmetry of the orbits about the equatorial plane. And for some special initial conditions, the orbits have equilibrium points.     

旋转致密双星的引力波特征

研究了轨道和旋转效果到2.5阶后牛顿旋转致密双星拉格朗日动力学与引力波的关系, 分析了有序和混沌轨道的引力波特征.发现当加速度不考虑辐射项时, 有序双星系统辐射的引力波具有周期或拟周期的特征, 而混沌双星系统辐射的引力波却具有明显的混沌特征.当加速度含有辐射项贡献时, 双星必会出现并合现象.此时, 原保守有序双星系统需较长时间才能完成并合过程, 引力波形在双星并合前仍保留拟周期的基本特点;然而, 原保守混沌双星系统仅在较短时间内就会并合, 但因并合时间太短, 无法获取足够的动力学信息导致引力波形的特征不易分辨.