The general form of the global conservation laws for $N$-body
systems in the first post-Newtonian approximation of general relativity is
considered. Our approach applies to the motion of an isolated
system of $N$ arbitrarily composed and shaped, weakly self-gravitating,
rotating, deformable bodies and uses a framework recently introduced by
Damour, Soffel and Xu (DSX). We succeed in showing that seven of
the first integrals of the system (total mass-energy, total dipole mass
moment and total linear momentum)
can be broken up into a sum of contributions which can be entirely
expressed in terms of the basic quantities entering the DSX
framework:
namely, the relativistic individual multipole moments of the
bodies, the relativistic tidal moments experienced by each body, and the
positions and orientations with respect to the global coordinate
system of the local reference frames attached to each body.
On the other hand, the total angular momentum of the
system does not seem to be expressible in such a form due to the
unavoidable presence of irreducible nonlinear gravitational effects.
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