Generalized Kerr spacetime with an arbitrary mass quadrupole moment: geometric properties vs particle motion

TL;DR

This paper analyzes an exact solution to Einstein's equations that generalizes Kerr spacetime by including an arbitrary mass quadrupole moment, exploring its geometric properties and particle motion near the source.

Contribution

It provides a detailed analysis of a generalized Kerr solution with arbitrary quadrupole moment, including its geometric features and particle dynamics.

Findings

Presence of causality violations near the source

Existence of directional singularities and repulsive effects

Geodesic and accelerated motions studied on the equatorial plane

Abstract

An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with arbitrary mass quadrupole moment and is specified by three parameters, the mass $M$, the angular momentum per unit mass $a$ and the quadrupole parameter $q$. It reduces to the Kerr spacetime in the limiting case $q=0$ and to the Erez-Rosen spacetime when the specific angular momentum $a$ vanishes. The geometrical properties of such a solution are investigated. Causality violations, directional singularities and repulsive effects occur in the region close to the source. Geodesic motion and accelerated motion are studied on the equatorial plane which, due to the reflection symmetry property of the solution, turns out to be also a geodesic plane.