General Relativistic Equations Of Motion Research Articles

Attempt to obtain the general relativistic planet's motion by special relativity techniques

AbstractIt is attempted to derive the general relativistic equation of motion for a planet and its solution solely by the special relativity techniques. The motion of a planet relative to the Sun and that of the Sun to the planet are solved independently in a special relativistic framework using the perturbation theory in celestial mechanics. The solution reveals a nature of the structure of the spacetime under the gravitation of the Sun, and then its effect on the planet's motion is examined. When the motion thus examined are compared with the one obtained by the general relativity theory in post‐Newtonian approximation, both are different concerning the mean motion and the radius of the orbit but exactly the same as for the perihelion precession.

An application of symplectic integration for general relativistic planetary orbitography subject to non-gravitational forces

Spacecraft propagation tools describe the motion of near-Earth objects and interplanetary probes using Newton’s theory of gravity supplemented with the approximate general relativistic n-body Einstein–Infeld–Hoffmann equations of motion. With respect to the general theory of relativity and the long-standing recommendations of the International Astronomical Union for astrometry, celestial mechanics and metrology, we believe modern orbitography software is now reaching its limits in terms of complexity. In this paper, we present the first results of a prototype software titled General Relativistic Accelerometer-based Propagation Environment (GRAPE). We describe the motion of interplanetary probes and spacecraft using extended general relativistic equations of motion which account for non-gravitational forces using end-user supplied accelerometer data or approximate dynamical models. We exploit the unique general relativistic quadratic invariant associated with the orthogonality between four-velocity and acceleration and simulate the perturbed orbits for Molniya, Parker Solar Probe and Mercury Planetary Orbiter-like test particles subject to a radiation-like four-force. The accuracy of the numerical procedure is maintained using a 5-stage, 10^mathrm{th}-order structure-preserving Gauss collocation symplectic integration scheme. GRAPE preserves the norm of the tangent vector to the test particle worldline at the order of 10^{-32}.

Lambda Perturbations of Keplerian Orbits in the Expanding Universe

To estimate the influence of “dark energy” on Keplerian orbits, we solve the general-relativistic equations of motion of a test particle in the field of a pointlike mass embedded in the cosmological background formed by the cosmological constant with realistic cosmological Robertson–Walker asymptotics at infinity. It is found that under certain relations between three crucial parameters of the problem—the initial radius of the orbit, the Schwarzschild and de Sitter radii—a specific secular perturbation caused by $$\Lambda$$ -term becomes significant, i.e., can reach the rate of the standard Hubble flow. This fact is interesting both by itself and may have important consequences for the long-term dynamics of planets and stellar binary systems.

Three-dimensional general relativistic Poynting-Robertson effect: Radial radiation field

In this paper we investigate the three-dimensional (3D) motion of a test particle in a stationary, axially symmetric spacetime around a central compact object, under the influence of a radiation field. To this aim we extend the two-dimensional (2D) version of the Poynting-Robertson effect in General Relativity (GR) that was developed in previous studies. The radiation flux is modeled by photons which travel along null geodesics in the 3D space of a Kerr background and are purely radial with respect to the zero angular momentum observer (ZAMO) frames. The 3D general relativistic equations of motion that we derive are consistent with the classical (i.e. non-GR) description of the Poynting-Robertson effect in 3D. The resulting dynamical system admits a critical hypersurface, on which radiation force balances gravity. Selected test particle orbits are calculated and displayed, and their properties described. It is found that test particles approaching the critical hypersurface at a finite latitude and with non-zero angular moment are subject to a latitudinal drift and asymptotically reach a circular orbit on the equator of the critical hypersurface, where they remain at rest with respect to the ZAMO. On the contrary, test particles that have lost all their angular momentum by the time they reach the critical hypersurface do not experience this latitudinal drift and stay at rest with respects to the ZAMO at fixed non-zero latitude.

Orbital dynamics of the gravitational field of stringy black holes

In this paper, we study the orbital dynamics of the gravitational field of stringy black holes by analyzing the effective potential and the phase plane diagram. By solving the equation of Lagrangian, the general relativistic equations of motion in the gravitational field of stringy black holes are given. It is easy to find that the motion of test particles depends on the energy and angular momentum of the test particles. Using the phase plane analysis method and combining the conditions of the stability, we discuss different types of the test particles' orbits in the gravitational field of stringy black holes. We get the innermost stable circular orbit which occurs at rmin= 5.47422 and when the angular momentum b ≤ 4.3887 the test particles will fall into the black hole.

Dynamics of a radiating thick shell

The starting point in this work is that an expanding shell is accelerated outward by radiation from the remnant star and slowed by ram pressure and accretion as it blows into the interstellar medium. The general relativistic equations of motion for such a shell is formulated. It is reduced to the standard Ostriker Gunn equations, in the non-relativistic limit. Israel formalism is used to describe the development of shell of dust which interacts only gravitationally.

General Relativistic Equation of Motion for a Photon Moving Round a Time Varying Spherical Distribution of Mass

Abstract In this article, Schwarzschild metric is extended to obtain a generalized metric for the gravitational field exterior to time varying spherical distributions of mass. Einstein’s equation for photon moving round a time varying spherical distribution of mass is derived. The second-order differential equation obtained is a modification of the equation of motion in Schwarzschild field. It introduces a unique dependence of the motion of the photon in this field on Newton’s scalar potential exterior to time varying spherical bodies.

Covariant equations of motion for test bodies in gravitational theories with general nonminimal coupling

We present a covariant derivation of the equations of motion for test bodies for a wide class of gravitational theories with nonminimal coupling, encompassing a general interaction via the complete set of 9 parity-even curvature invariants. The equations of motion for spinning test bodies in such theories are explicitly derived by means of Synge's expansion technique. Our findings generalize previous results in the literature and allow for a direct comparison to the general relativistic equations of motion of pole-dipole test bodies.

Relativistic Rotation: A Comparison of Theories

Alternative theories of relativistic rotation considered viable as of 2004 are compared in the light of experiments reported in 2005. En route, the contentious issue of simultaneity choice in rotation is resolved by showing that only one simultaneity choice, the one possessing continuous time, gives rise, via the general relativistic equation of motion, to the correct Newtonian limit Coriolis acceleration. In addition, the widely dispersed argument purporting Lorentz contraction in rotation and the concomitant curved surface of a rotating disk is analyzed and argued to be lacking for more than one reason. It is posited that not by theoretical arguments, but only via experiment can we know whether such effect exists in rotation or not. The Coriolis/simultaneity correlation, and the results of the 2005 experiments, support the Selleri theory as being closest to the truth, though it is incomplete in a more general applicability sense, because it does not provide a global metric. Two alternatives, a modified Klauber approach and a Selleri–Klauber hybrid, are presented which are consistent with recent experiment and have a global metric, thereby making them applicable to rotation problems of all types.

On the Structure of Line‐driven Winds Near Black Holes

A general physical mechanism of the formation of line-driven winds at the vicinity of strong gravitational field sources is investigated in the frame of General Relativity. We argue that gravitational redshifting should be taken into account to model such outflows. The generalization of the Sobolev approximation in the frame of General Relativity is presented. We consider all processes in the metric of a nonrotating (Schwarzschild) black hole. The radiation force that is due to absorbtion of the radiation flux in lines is derived. It is demonstrated that if gravitational redshifting is taken into account, the radiation force becomes a function of the local velocity gradient (as in the standard line-driven wind theory) and the gradient of $g_{00}$. We derive a general relativistic equation of motion describing such flow. A solution of the equation of motion is obtained and confronted with that obtained from the Castor, Abbott & Klein (CAK) theory. It is shown that the proposed mechanism could have an important contribution to the formation of line-driven outflows from compact objects.

Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics

A calculation of perihelion precession is presented that utilizes a phase-plane analysis of the general relativistic equations of motion. The equations of motion are reviewed in addition to the phase-plane analysis required for the calculation. “Exact” phase planes for orbital dynamics in the Schwarzschild geometry are discussed, and bifurcations are identified as a dimensionless parameter involving the angular momentum is varied.

Dynamics of massive shells ejected in a supernova explosion

We set up the general relativistic equations of motion of a shell that models the matter ejected in supernova explosions. The non-relativistic Ostriker-Gunn equation is obtained properly. It is shown that in general the equation of motion admits a first integral and the evolution of the shell is reduced to a dynamical system of four equations.

Invariant Hamiltonians for relativistic particles

The invariant Hamiltonian of a relativistic free particle can be derived via a strong consistency constraint. Invariant mass appears as a constant of the motion, and minimal coupling to electromagnetic and scalar fields arises naturally from the constraint form. The Hamiltonian formulation yields a simple derivation of the general relativistic equations of motion. The corresponding Lagrangian formulation is briefly discussed.

The generalized Poynting-Robertson effect

view Abstract Citations (11) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The generalized Poynting-Robertson effect. Carroll, Douglas L. Abstract The special relativistic equations of motion for a particle and the general relativistic equations of motion for a fluid in an arbitrary radiation field are formulated. The Poynting-Robertson forces are manifested in some of the velocity-dependent terms obtained. Newtonian approximations to the equations of motion are solved for a particle in orbit about a spherically symmetric source of radiation, where the angular size of the source as seen from the orbit is arbitrary. Situations are considered in which the absorption cross section of the particle is independent of frequency, as well as when the particle is an atom absorbing in a spectral line. In both instances, it is found that the finite size of the source of radiation leads to greater Poynting-Robertson drags than a point source of the same luminosity. The Newtonian equations are solved for a particle moving radially outward from the source. Limits for the velocities of high-speed particles resulting from Poynting-Robertson drags are obtained. The fluid equations are discussed briefly in their relation to theoretical treatments of relativistic jets and accretion disks. Publication: The Astrophysical Journal Pub Date: January 1990 DOI: 10.1086/168265 Bibcode: 1990ApJ...348..588C Keywords: Computational Astrophysics; Equations Of Motion; Poynting-Robertson Effect; Relativistic Effects; Accretion Disks; Hydrodynamics; Optical Thickness; Radiative Transfer; White Dwarf Stars; Astrophysics full text sources ADS | data products SIMBAD (1)

Relativistic effects on an Earth‐orbiting satellite in the Barycenter Coordinate System

The general relativistic equations of motion and range measurement relations for an earth‐orbiting satellite such as LAGEOS were investigated in the barycenter coordinate system. In comparison with the Newtonian equations, modifications must be made to the satellite acceleration, the light time measurement, station clock times, and station position coordinates. Modifications to the measurement computation are also necessary in order to continue to operate formally in an earth‐centered coordinate system. Limited comparisons of data reduction results using Newtonian equations with results using the Einsteinian formulation show that estimated baselines between LAGEOS tracking stations are affected at the several centimeter level. In three of four cases, rms of fit was slightly improved using the relativistic formulation.

Light rays in gravitating, refractive media

The field-to-particle method of H. P. Robertson is applied to the general-relativistic Maxwell equations in order to obtain the general-relativistic equation of motion for a photon in a refractive medium. For the special case of an uncharged, refractive, spherically symmetric mass, the exact first-order differential equation for the light-ray path is given.

Speculation on traceless particles with charge and electromagnetic moments

The general-relativistic equations of motion and subsidiary conditions are found for particles with spin whose energy-momentum tensor has zero trace (normally considered to be therefore massless) but which may have comoving internal charge currents. The procedure is an extension of that used in the pole-dipole model of traceless particles without charge currents.

Relativistic conductive additional terms in Navier-Stokes equations

The general-relativistic equation of motion for a continuum and the corresponding mass balance equation are approximately treated for nonrelativistic velocities and weak gravitational fields, but taking into account the conductive momentum density caused by the irreversible effects. We find a new force term and a modification of the mass balance equation, neither of which can be neglected if irreversible processes are involved, (e.g., nuclear processes occurring in plasmas [1]).

A New General Covariant Approach to the General Relativistic Two-Body Problem

A new general covariant approach to the general relativistic equations of motion is presented. It is stressed that our present understanding of the development of binaries due to general relativistic effects and of the power emitted by these systems in the form of gravitational radiation is highly unsatisfactory.

Lorentz-Invariant Equations of Motion of Point Masses in the General Theory of Relativity

After a general discussion of the problem of motion in the general theory of relativity a simple derivation of the law of motion is given for single poles of the gravitational field, which is based on a method originally developed by Mathisson. This law follows from the covariant conservation law for the matter energy-momentum tensor alone, without reference to any field equations, and takes the form of a geodesic of the (unknown) metric. Expanding this metric in terms of a power series in a parameter gamma and using the Minkowski proper time to parametrize the world lines of the particles, the (Lorentz-invariant) form of the approximate laws of motion follows. A method is developed to obtain the equations of motion (including the explicit form of the metric in terms of the particle variables) from Einsiein's field equations. A systematic linearization procedure leads to a series of second-order differential equations for the metric; the nth order approximation of the equations of motion, as well as the explicit form of the matter tensor in (n + l)st order, is obtained as an integrability condition on the (n + l)st order opproximation for the metric. No coordinate conditions are required to obtain themore » general form of the equations of motion; they are needed only to reduce the approximation equations to wave equations and thus to allow their explicit integration in terms of retarded or symmetric potentials. In developing the approximation method it is shown that consistency requires that any set of approximate equations is solved up to'' rather than in'' nth order; this implies that the form of the lower-order metric be maintained, but the motion corresponding to the nth order solutions rather than to lower order ones. In particular, the equations for the first- order metric imply zero-order equations of motion which restrict the particles to zero acceleration; the equations for the second-order metric imply first-order equation of motion involving the first-order metric, but without the previous restriction. In the retarded case the equations of motion contain retarded interactions and radiation terms of the form familiar from electrodynamics; no such terms appear in the symmetric case. The equations of the symmetric case are derivable from a Fokker-type variational principle. The relation of the results obtained to work on Lorentz-invariant equations by other autnors is discussed. In Appendix I a discussion of alternative derivations is prescnted; Appendix II contains remarks on Wheeler-Feynman type considerations for general relativistic equations of motion. (auth)« less