Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem

We derive the tensor-virial theorem and the angular momentum integral for a system of bodies of finite dimensions in the post-Newtonian approximation of general relativity. For this purpose we first examine the Newtonian case. Then we prove that, if tidal interactions between the bodies are neglected, it is possible in the post-Newtonian approximation and for each body of the system to define a conserved mass and a corresponding center of mass, to provide expressions for the vanishing of the post-Newtonian self-linear momentum, self-force and self-torque, and also to write down an appropriate tensor-virial equation. This, in turn, enables us to write down the tensor-virial theorem and the angular momentum integral for the many-body system, which, when expressed in terms of the above masses and centers of masses, reduce to the corresponding expressions valid for a system of point masses, in the limiting case when the dimensions of the bodies tend to zero.

Tensor-virial equations for post-Newtonian relativistic stellar dynamics

Synge's approximation procedure is applied to calculate the spin precession of either body in a binary system consisting of two rotating, spherical, rigid bodies of comparable mass and radius. The calculations are valid for the case in which the mass-radius ratio of each body, as well as the ratio of the radius of either body to the distance between their centers, is small. The results agree with those of earlier authors, who use different techniques, except for a term that arises from the effect of the rotation on the stress within the bodies. This term is similar in form to the quadrupole term of Barker and O'Connell, which they obtain when they allow the bodies to become distorted under the influence of the rotation.

Spin precession in the relativistic two-body problem.

We derive the most general equations of motion in the post-Newtonian approximation of general relativity for a bounded system of extended bodies with arbitrary internal structure and internal motions, and we explore in detail the conditions under which the motion of the bodies deviates from the geodesic motion.

Relativistic equations of motion of extended bodies

We present the results of a systematic study of the dynamics of realistic binary systems in the post-Newtonian approximation (PNA) of general relativity. We propose definitions valid in the PNA for the self-angular-momenta of the binary's members, as well as for the angular momentum of their relative orbital motion, and we examine under which conditions they can be considered as constant in the PNA. This enables us to define to the same approximation the plane relative orbital motion. Then we find the form of the differential equations of motion from an integration of which we prove that in the PNA the relative motion is a processing ellipse composed of a basic orbit and a correction, both of which are of post-Newtonian character. Moreover, using the polar equation of the above ellipse we define the elements of the post-Newtonian, relative, basic orbit, we generalize to the PNA the three well-known laws of classical celestial mechanics of Kepler, and we derive the precessional motion of the relative orbit's pericenter. Finally, we compare our method with other methods existing in the literature, and we expose its theoretical and conceptual differences with them.

N. Spyrou

Papers

The Center of Mass in the Post-Newtonian Approximation of General Relativity