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.

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• Alternative gravity theories
• Astrophysical studies of gravity
• Spacetime symmetries
• Differential equations