Multipolar equations of motion for extended test bodies in General Relativity

TL;DR

This paper derives the equations of motion for extended test bodies in General Relativity using a multipolar approximation up to quadrupolar order, emphasizing the canonical form of energy-momentum density.

Contribution

It introduces a systematic derivation of multipolar equations of motion in Einstein's theory, including explicit construction of canonical energy-momentum forms.

Findings

Derived equations of motion up to quadrupolar order

Constructed explicit canonical form of energy-momentum density

Provided a framework for comparing multipolar approximation schemes

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.