post-Newtonian Equations Of Motion Research Articles

Post-Newtonian Lagrangian of an N -body system with arbitrary mass and spin multipoles

The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrary-structured bodies with all mass and spin multipoles in a scalar-tensor theory of gravity. The multipoles depend on time and evolve in accordance with their own dynamic equations of motion. The Lagrangian is retrieved from the post-Newtonian equations of motion by solving the inverse problem of the Lagrangian mechanics and generalizes a well-known Lagrangian of pole-dipole-quadrupole massive particles to the particles of higher multipolarity. Analytic treatment of the higher-order multipole contributions is important for more rigorous computation of gravitational waveform of inspiralling compact binaries at the latest stage of their orbital evolution before merger when tidal and rotational deformations of stars are no longer small and rapidly change in time. The Lagrangian of an N-body system with arbitrary mass and spin multipoles is instrumental for formulation of the post-Newtonian conservation laws of energy, momenta and the integrals of the center of mass.

On the energy integral for first post-Newtonian approximation

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials \(W^j (j=1,2,3)\), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.

New General Relativistic Contribution to Mercury’s Perihelion Advance

We point out the existence of a new general relativistic contribution to the perihelion advance of Mercury that, while smaller than the contributions arising from the solar quadrupole moment and angular momentum, is 100 times larger than the second-post-Newtonian contribution. It arises in part from relativistic "crossterms" in the post-Newtonian equations of motion between Mercury's interaction with the Sun and with the other planets, and in part from an interaction between Mercury's motion and the gravitomagnetic field of the moving planets. At a few parts in 10^{6} of the leading general relativistic precession of 42.98arcseconds per century, these effects are likely to be detectable by the BepiColombo mission to place and track two orbiters around Mercury, scheduled for launch around 2018.

Flip-flopping binary black holes.

We study binary spinning black holes to display the long term individual spin dynamics. We perform a full numerical simulation starting at an initial proper separation of d≈25M between equal mass holes and evolve them down to merger for nearly 48 orbits, 3 precession cycles, and half of a flip-flop cycle. The simulation lasts for t=20 000M and displays a total change in the orientation of the spin of one of the black holes from an initial alignment with the orbital angular momentum to a complete antialignment after half of a flip-flop cycle. We compare this evolution with an integration of the 3.5 post-Newtonian equations of motion and spin evolution to show that this process continuously flip flops the spin during the lifetime of the binary until merger. We also provide lower order analytic expressions for the maximum flip-flop angle and frequency. We discuss the effects this dynamics may have on spin growth in accreting binaries and on the observational consequences for galactic and supermassive binary black holes.

Second post-Minkowskian metric for a moving Kerr black hole

In this paper, the harmonic metric for a moving Kerr black hole is presented in the second post-Minkowskian approximation. It is further demonstrated that the obtained metric is consistent with the Liénard–Wiechert gravitational potential for a moving and spinning source with an arbitrary constant velocity. Based on the metric, we also give the post-Newtonian equations of motion for photon and massive test particle in the time-dependent gravitational field.

Gauss collocation methods for efficient structure preserving integration of post-Newtonian equations of motion

In this work, we present the hitherto most efficient and accurate method for the numerical integration of post-Newtonian equations of motion. We first transform the Poisson system as given by the post-Newtonian approximation to canonically symplectic form. Then we apply Gauss Runge-Kutta schemes to numerically integrate the resulting equations. This yields a convenient method for the structure preserving long-time integration of post-Newtonian equations of motion. In extensive numerical experiments, this approach turns out to be faster and more accurate (i) than previously proposed structure preserving splitting schemes and (ii) than standard explicit Runge-Kutta methods. We also show our approach to be appropriate for simulations on transitional precession.

Formulation of the Post-Newtonian Equations of Motion of the Restricted Three Body Problem

In the present work the geodesic equation represents the equations of motion of the particles along the geodesics is derived. The deviation of the curved space-time metric tensor from that of the Minkowski tensor is considered as a perturbation. The quantities is expanded in powers of c-2. The equations of motion of the relativistic three body problem in the PN formalism are obtained.

Final spins from the merger of precessing binary black holes

The inspiral of binary black holes is governed by gravitational radiation reaction at binary separations r < 1000 M, yet it is too computationally expensive to begin numerical-relativity simulations with initial separations r > 10 M. Fortunately, binary evolution between these separations is well described by post-Newtonian equations of motion. We examine how this post-Newtonian evolution affects the distribution of spin orientations at separations r ~ 10 M where numerical-relativity simulations typically begin. Although isotropic spin distributions at r ~ 1000 M remain isotropic at r ~ 10 M, distributions that are initially partially aligned with the orbital angular momentum can be significantly distorted during the post-Newtonian inspiral. Spin precession tends to align (anti-align) the binary black hole spins with each other if the spin of the more massive black hole is initially partially aligned (anti-aligned) with the orbital angular momentum, thus increasing (decreasing) the average final spin. Spin precession is stronger for comparable-mass binaries, and could produce significant spin alignment before merger for both supermassive and stellar-mass black hole binaries. We also point out that precession induces an intrinsic accuracy limitation (< 0.03 in the dimensionless spin magnitude, < 20 degrees in the direction) in predicting the final spin resulting from the merger of widely separated binaries.

Measuring black hole spin in OJ287

We model the binary black hole system OJ287 as a spinning primary and a non-spinning secondary. It is assumed that the primary has an accretion disk which is impacted by the secondary at specific times. These times are identified as major outbursts in the light curve of OJ287. This identification allows an exact solution of the orbit, with very tight error limits. Nine outbursts from both the historical photographic records as well as from recent photometric measurements have been used as fixed points of the solution: 1913, 1947, 1957, 1973, 1983, 1984, 1995, 2005 and 2007 outbursts. This allows the determination of eight parameters of the orbit. Most interesting of these are the primary mass of $${1.84\cdot 10^{10}M_\odot}$$ , the secondary mass $${1.40\cdot 10^{8} M_\odot}$$ , major axis precession rate 39°.1 per period, and the eccentricity of the orbit 0.70. The dimensionless spin parameter is 0.28 ± 0.01 (1 sigma). The last parameter will be more tightly constrained in 2015 when the next outburst is due. The outburst should begin on 15 December 2015 if the spin value is in the middle of this range, on 3 January 2016 if the spin is 0.25, and on 26 November 2015 if the spin is 0.31. We have also tested the possibility that the quadrupole term in the post-Newtonian equations of motion does not exactly follow Einstein’s theory: a parameter q is introduced as one of the 8 parameters. Its value is within 30% (1 sigma) of the Einstein’s value q = 1. This supports the no-hair theorem of black holes within the achievable precision. We have also measured the loss of orbital energy due to gravitational waves. The loss rate is found to agree with Einstein’s value with the accuracy of 2% (1 sigma). There is a possibility of improving the accuracy of both quantities using the exact timing of the outburst on 21 July 2019. Because of closeness of OJ287 to the Sun (8–12°), the observations would be best carried out by a telescope in space.

Summary of sessions B1/B2 and B2: relativistic astrophysics and numerical relativity

The numerical relativity session at GR18 was dominated by physics results on binary black hole mergers. Several groups can now simulate these from a time when the post-Newtonian equations of motion are still applicable, through several orbits and the merger to the ringdown phase, obtaining plausible gravitational waves at infinity, and showing some evidence of convergence with resolution. The results of different groups roughly agree. This new-won confidence has been used by these groups to begin mapping out the (finite dimensional) initial data space of the problem, with a particular focus on the effect of black hole spins, and the acceleration by gravitational wave recoil to hundreds of km s−1 of the final merged black hole. Other work was presented on a variety of topics, such as evolutions with matter, extreme mass ratio inspirals and technical issues such as gauge choices.

Equations of motion for binary systems with relativistic quadrupole-quadrupole moments interaction

By making use of the theme developed by Damour, Soffel and Xu, in terms of Maple-computer algebra system, the first post-Newtonian equations of motion for binary systems with monopole, spin and quadrupole interaction are for the first time derived in an explicit and complete form. The equations of motion are expressed in the local coordinate system as well as in a global coordinate system. The new terms containing quadrupole-quadrupole moments might be of some importance for a precise description of coalescing neutron star binaries.

Post-Newtonian Equations of Motion in the Flat Universe

Using the (3 + 1) formalism, we derive the post-Newtonian (PN) equations of motion in a flat unvierse. To derive the equations of motion, we must carefully consider two points, one being the choice of the density in the Newtonian order (ρN) and the other the choice of the gauge condition. In choosing ρN, we require that the density fluctuation ρN - ρ0 agrees with a gauge invariant quantity in the linear approximation theory. As a gauge condition, we propose the cosmological post-Newtonian (CPN) slice condition with the pseudo transverse-traceless gauge condition, by which the evolution of the geometric variables derived in the PN approximation in the early stage of universe agrees with that of the gauge invariant quantities in the linear approximation. In the derived equations of motion, the force is calculated from six potentials which satisfy the Poisson equations. Hence, our formalism can be easily applied to numerical simulations in which the standard technique (e.g., particle-mesh method) is used. We apply the PN formula to the one-dimensional (1D) Zel'dovich solution to demonstrate that our strategy works well, and also to determine the effect of the PN forces on the evolution of the large-scale structure. It is found that the behavior of the density fluctuation and metric quantities in the early stage obtained by the present formalism agrees with that of the gauge invariant quantities in the linear approximation, although they do not always agree within the previous formalism due to the appearance of spurious gauge modes. We also discuss the evolution of the non-linear density fluctuation with very large scale, which may be affected by the PN correction in the last stage of the evolution.

Post-Newtonian Equations of Motion in the Flat Universe

Post-Newtonian Equations of Motion in the Flat Universe

Corrections of the finite relativistic contributions to the synodic month period Earth-Moon range oscillations: Agreement between the geocentric and the solar-system barycentric inertial-frame calculations.

We have corrected our calculation of the finite general relativistic contribution to the synodic month period Earth-Moon range oscillation by including previously overlooked terms in the Moon's post-Newtonian equation of motion: the corrected result $x(t)\ensuremath{\simeq}(\frac{3{g}_{S}{r}^{2}}{{c}^{2}})cos(\ensuremath{\omega}\ensuremath{-}\ensuremath{\Omega})t$ agrees with the Shahid-Saless calculation which was performed in the geocentric frame. It is also pointed out that at the level of a few millimeters synodic month period amplitude, the Moon's orbit is polarized by the solar radiation pressure force on the Moon.

Post-Newtonian Equations of Motion in Flat Cosmological Models

The post-Newtonian equations of motion for particles in an expanding universe are derived in spatially flat models with cosmological constant. This system of equations is an extended version of the previous one which was derived in the Einstein-de Sitter model. The applicability condition is also examined and some errors in the previous expressions are indicated and corrected

Extension of the parametrized post-Newtonian formalism to the case of a quasi-Lorentz frame of reference

Thus, without changing the conditions for metric theories of gravitation to be conservative, the post-newtonian parmater τ occurs explicitly in the experession (35) for the Lagrange function of a system of two extended bodies, and it cannot be eliminated from this expression by any redefinition of the remaining post-Newtonian parameters γ, β, ξ. This means that the parameter τ plays an important role in the post-Newtonian equations of motion of a binary system and must lead to experimentally observable post-Newtonian effects. This offers hope that after a sufficient number of post-Newtonian experiments has been made it will be possible to measure the parameter τ, like the other parameters.

Post-Newtonian Equations of Motion in an Expanding Universe

Post-Newtonian Equations of Motion in an Expanding Universe

The apsidal motion of a binary star in the nonsymmetric gravitation theory

The two-body post-Newtonian equations of motion in the nonsymmetric gravitational theory are reduced to the one-body form in the centre-of-mass coordinates. The periastron shift formula is calculated and an additional negative contribution is found that can have important consequences for apsidal motion in close binary systems.

The two-body problem in Rosen's bimetric theory of gravitation

This is a continuation of a previous paper, in which the field equations in successive approximations and the post-Newtonian equations of motion in Rosen's theory of gravitation were derived. In this paper the energy integral and the center of mass for an insular system with an arbitrary structure are obtained in the post-Newtonian approximation. A many-body system is considered, and in the extreme case of point bodies (particles) the center-of-mass coordinates are found to be identical with the Einsteinian ones. The two-body problem is considered. For a system of two identical neutron stars of mass 1.3M⊙ (a possible model of the Hulse-Taylor binary pulsar system) the trajectory and the perihelion precession are calculated. It is found that the expressions obtained depend on the gravitational self-energy of the stars. The relations deduced from Rosen's bimetric gravitation in the case of small velocities and weak fields are compared with those of general relativity.

Equations of motion in Rosen's bimetric theory of gravitation

The paper contains an investigation of Rosen's bimetric theory of gravitation in the case of slow velocities and weak fields. Newtonian and post-Newtonian approximations are obtained. The post-Newtonian equation of motion is integrated for an insular system of spherical bodies that move translationally at large mutual distances. It appears that the post-Newtonian law of motion obtained in this way contains terms that depend on the self-energy of the test body (a self-influence phenomenon). It is proved that also in the Einsteinian gravitation this influence is present, but it can be canceled out from the post-Newtonian law of motion if one takes into account the de Donder conditions. The self-influence discovered here seems to be a general gravitation phenomenon, which usually appears in theories of gravitation in the post-Newtonian approximation.