Conservation laws for systems of extended bodies in the first post-Newtonian approximation.

TL;DR

This paper derives global conservation laws for complex N-body systems in the first post-Newtonian approximation of general relativity, expressing key integrals in terms of individual relativistic multipole moments and tidal effects.

Contribution

It extends the DSX framework to show how seven key conserved quantities can be decomposed into contributions from individual bodies in a relativistic setting.

Findings

Seven integrals (mass-energy, dipole moment, linear momentum) expressed in terms of multipole and tidal moments.

Total angular momentum not fully expressible due to nonlinear gravitational effects.

Framework applicable to weakly self-gravitating, deformable, rotating bodies in N-body systems.

Abstract

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.