Multipole moments do not uniquely characterize spacetimes beyond general relativity

TL;DR

This paper demonstrates that multipole moments, used to characterize spacetimes, are not unique identifiers beyond general relativity, affecting interpretations of black-hole shadows and universal relations.

Contribution

It reveals that in theories beyond Einstein's gravity, different spacetimes can share identical multipole moments, challenging their use as unique identifiers.

Findings

Different objects can have the same Geroch-Hansen moments.

Two metrics can match yet have different moments.

Implications affect black-hole shadow analysis and universal relations.

Abstract

Spacetimes in general relativity can be uniquely decomposed into a set of multipole moments. Given the usefulness of moments in the categorization of radiation patterns, tidal deformations, and other phenomena associated with compact objects, a number of studies have explored their construction in beyond-Einstein theories of gravity. It is shown here that uniqueness does not necessarily extend across theories: by comparing a few static and spherically-symmetric solutions in different theories, we find that two distinct objects can possess the same Geroch-Hansen moments. Moreover, two metrics can match and yet take different moments. Implications of this result are explored in the context of black-hole shadows and ``universal'' relations hinging on moment computations.