scholarly journals Equations of Motion for Two-Body Problem According to an Observer Inside the Gravitational Field

2011 ◽  
Vol 9 (2) ◽  
pp. 115-135
Author(s):  
Kostadin Trenčevski ◽  
Emilija Celakoska
Keyword(s):  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

The two-body problem: an effective-one-body approach

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Body Problem ◽  
Equations Of Motion ◽  
Reaction Force ◽  
Kerr Black Hole ◽  
Radiation Reaction ◽  
Spherically Symmetric ◽  
Geodesic Motion ◽  
Two Body Problem ◽  
Symmetric Spacetime ◽  
Radiation Reaction Force

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.

scholarly journals Relativistic Reduction of Astrometric Observations

1986 ◽  
Vol 109 ◽  
pp. 19-42 ◽  
Author(s):  
V. A. Brumberg
Keyword(s):  
Body Problem ◽  
Equations Of Motion ◽  
Satellite Observations ◽  
List Type ◽  
Type Number ◽  
General Technique ◽  
Two Body Problem ◽  
The One ◽  
Relative Measurements ◽  
Relativistic Reduction

The nonuniqueness of the quasi-Galilean coordinates of general relativity leads to the emergence of unmeasurable coordinate-dependent quantities in astronomical practice. One may offer three possible ways to overcome the related difficulties: 1.developing theoretical conclusions only in terms of measurable quantities2.using arbitrary coordinates and developing an unambiguous procedure for comparing measurable and calculated quantities3.agreement to utilize one and only one coordinate system.In this paper we prefer the second way. After formulating the heliocentric planetary and geocentric satellite equations of motion, the general technique for relativistic reduction in astrometry and geodynamics is developed. Specific algorithms for the reduction of absolute and relative measurements are derived for the one- and the two- body problem. For illustration, the relativistic reduction of stellar parallaxes, Doppler satellite observations, navigation measurements with the aid of satellites and radiointerferometric measurements are presented in detail.

scholarly journals The planar two-body problem for spheroids and disks

2021 ◽  
Vol 133 (6) ◽  
Author(s):  
Margrethe Wold ◽  
John T. Conway
Keyword(s):  
Gravitational Potential ◽  
Body Problem ◽  
Equations Of Motion ◽  
Surface Integral ◽  
Conservation Of Energy ◽  
Two Body Problem ◽  
Truncation Errors ◽  
Thin Disks ◽  
Solid Bodies ◽  
Two Bodies

AbstractWe outline a new method suggested by Conway (CMDA 125:161–194, 2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not involve series expansions with subsequent truncation errors. In order to test the method, we solve the equations of motion in an inertial frame and run simulations with two spheroids and two infinitely thin disks, restricted to torque-free planar motion. The resulting trajectories display precession patterns typical for non-Keplerian potentials. We follow the conservation of energy and orbital angular momentum and also investigate how the spheroid model approaches the two cases where the surface integral can be solved analytically, i.e., for point masses and infinitely thin disks.

scholarly journals Classical Yang-Mills observables from amplitudes

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Leonardo de la Cruz ◽  
Ben Maybee ◽  
Donal O’Connell ◽  
Alasdair Ross
Keyword(s):  
General Relativity ◽  
Body Problem ◽  
Equations Of Motion ◽  
Colour Change ◽  
Classical Dynamics ◽  
Particle Scattering ◽  
Leading Order ◽  
Yang Mills ◽  
Two Body Problem ◽  
Double Copy

Abstract The double copy suggests that the basis of the dynamics of general relativity is Yang-Mills theory. Motivated by the importance of the relativistic two-body problem, we study the classical dynamics of colour-charged particle scattering from the perspective of amplitudes, rather than equations of motion. We explain how to compute the change of colour, and the radiation of colour, during a classical collision. We apply our formalism at next-to-leading order for the colour change and at leading order for colour radiation.

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

1974 ◽  
Vol 64 ◽  
pp. 102-102
Author(s):  
Arnold Rosenblum
Keyword(s):  
Gravitational Radiation ◽  
Body Problem ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Covariant Approach ◽  
Two Body Problem ◽  
General Relativistic ◽  
Present Understanding ◽  
Relativistic Equations

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.

scholarly journals Binary collisions in popovici’s photogravitational model

2002 ◽  
pp. 9-16 ◽  
Author(s):  
V. Mioc ◽  
C. Blaga
Keyword(s):  
Black Hole ◽  
Body Problem ◽  
Equations Of Motion ◽  
Gravitational Attraction ◽  
Two Body Problem ◽  
Binary Collisions ◽  
Combined Action ◽  
Collision Manifold ◽  
S Model ◽  
Hole Effect

The dynamics of bodies under the combined action of the gravitational attraction and the radiative repelling force has large and deep implications in astronomy. In the 1920s, the Romanian astronomer Constantin Popovici proposed a modified photogravitational law (considered by other scientists too). This paper deals with the collisions of the two-body problem associated with Popovici?s model. Resorting to McGehee-type transformations of the second kind, we obtain regular equations of motion and define the collision manifold. The flow on this boundary manifold is wholly described. This allows to point out some important qualitative features of the collisional motion: existence of the black-hole effect, gradientlikeness of the flow on the collision manifold, regularizability of collisions under certain conditions. Some questions, coming from the comparison of Levi-Civita?s regularizing transformations and McGehee?s ones, are formulated.

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