Multipolar equations of motion for extended test bodies in General Relativity

TL;DR

This work advances the problem of extended-body motion in General Relativity by applying Tulczyjew's multipolar expansion up to quadrupole order, deriving a fully covariant set of monopole, dipole, and quadrupole equations without imposing a spin supplementary condition a priori. A transparent canonicalization of the energy-momentum density is developed, and the quadrupole sector is mapped onto Dixon's framework to enable cross-method comparisons. The results reveal spin-curvature and higher-multipole couplings, show that many conserved quantities are not guaranteed at quadrupole order without extra conditions, and provide a foundation for incorporating quadrupole effects into perturbative approaches relevant to gravitational-wave physics. Overall, the paper delivers a comprehensive, covariant formalism that can inform self-gravitating-body models and higher-order relativistic corrections in gravitational-wave astronomy.

Abstract

We derive the equations of motion of an extended test body in the context of Einstein's theory of gravitation. The equations of motion are obtained via a multipolar approximation method and are given up to the quadrupolar order. Special emphasis is put on the explicit construction of the so-called canonical form of the energy-momentum density. The set of gravitational multipolar moments and the corresponding equations of motion allow for a systematic comparison to competing multipolar approximation schemes.