Multipole moments in stationary spacetimes

TL;DR

This paper reviews the Geroch--Hansen multipole moments in stationary asymptotically flat vacuum spacetimes, clarifies their uniqueness, and discusses their behavior under conformal transformations, enhancing understanding of gravitational field characterization.

Contribution

The paper revises and proves a refined uniqueness theorem for Geroch--Hansen multipole moments, addressing gaps in the original formulation and clarifying their invariance properties.

Findings

Established a revised uniqueness result for multipole moments.

Clarified the behavior of moments under conformal transformations.

Filled gaps in the original Geroch--Hansen approach.

Abstract

Multipole moments in general relativity serve as a powerful tool for characterising the gravitational field. In this paper, we review the construction of the Geroch--Hansen multipole moments for stationary asymptotically flat vacuum spacetimes. A particular focus is placed on the well-definedness of these moments, which hinges on the uniqueness of the one-point conformal completion in Geroch's asymptotic flatness definition. Based on Geroch's approach, we formulate and prove a revised uniqueness result, thereby filling in some gaps in the original approach. Uniqueness holds up to certain conformal transformations, and we discuss how the multipole moments behave under such transformations.