Extended bodies with quadrupole moment interacting with gravitational monopoles: reciprocity relations

TL;DR

This paper derives an exact solution for the gravitational field of a quasi-spherical mass with quadrupole moment, analyzing orbital dynamics and reciprocity relations to validate Dixon's model for extended bodies in general relativity.

Contribution

It introduces a new exact solution for a mass-quadrupole source and explores reciprocity relations between source and particle parameters in gravitational fields.

Findings

Geometric analysis of timelike circular orbits in the new spacetime

Comparison of geodesic and non-geodesic motion of extended bodies

Reciprocity relations supporting Dixon's model

Abstract

An exact solution of Einstein's equations representing the static gravitational field of a quasi-spherical source endowed with both mass and mass quadrupole moment is considered. It belongs to the Weyl class of solutions and reduces to the Schwarzschild solution when the quadrupole moment vanishes. The geometric properties of timelike circular orbits (including geodesics) in this spacetime are investigated. Moreover, a comparison between geodesic motion in the spacetime of a quasi-spherical source and non-geodesic motion of an extended body also endowed with both mass and mass quadrupole moment as described by Dixon's model in the gravitational field of a Schwarzschild black hole is discussed. Certain "reciprocity relations" between the source and the particle parameters are obtained, providing a further argument in favor of the acceptability of Dixon's model for extended bodies in general relativity.