The static two body problem

1974 ◽  
Vol 75 (2) ◽  
pp. 249-260 ◽  
Author(s):  
H. Müller Zum Hagen
Keyword(s):  
Perfect Fluid ◽  
Physical Meaning ◽  
Body Problem ◽  
Space Time ◽  
Equilibrium Conditions ◽  
Two Body Problem ◽  
Special Cases ◽  
Static Solutions ◽  
Two Bodies ◽  
Extended Bodies

AbstractThe problem of whether there exist static solutions of Einstein's equations for two extended bodies surrounded by vacuum is investigated. Equilibrium conditions for the general static situation are derived and their physical meaning discussed. Special cases of the problem are solved by using these equilibrium conditions. In one of these cases, the essential assumptions are that the space time is axisymmetric and that the matter of which the two bodies are composed consists of perfect fluid. Solu tions are also obtained in which these restrictions are weakened.

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.

PRECESSION AND CHAOS IN THE CLASSICAL TWO-BODY PROBLEM IN A SPHERICAL UNIVERSE

2008 ◽  
Vol 18 (02) ◽  
pp. 455-464 ◽  
Author(s):  
JOHN F. LINDNER ◽  
MARTHA I. ROSEBERRY ◽  
DANIEL E. SHAI ◽  
NICHOLAS J. HARMON ◽  
KATHERINE D. OLAKSEN
Keyword(s):  
Body Problem ◽  
Initial Conditions ◽  
Flat Space ◽  
Three Body Problem ◽  
Two Body Problem ◽  
Extreme Sensitivity ◽  
Three Body ◽  
Spherical Curvature ◽  
Spherical Universe ◽  
Two Bodies

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

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.

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 GENERALIZED W3-STRINGS FROM FREE FIELDS

1995 ◽  
Vol 10 (06) ◽  
pp. 515-524 ◽  
Author(s):  
J. M. FIGUEROA-O'FARRILL ◽  
C. M. HULL ◽  
L. PALACIOS ◽  
E. RAMOS
Keyword(s):  
Bosonic Strings ◽  
Free Field ◽  
Space Time ◽  
General Construction ◽  
Special Cases ◽  
Free Fields ◽  
Rule Out ◽  
General Conditions ◽  
String Theories

The conventional quantization of w3-strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W3-string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W3-algebra. We study W3-like strings with a direct space-time interpretation — that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w3-algebra. We give the general conditions for the existence of W3-like strings, and comment on how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations.

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