Multipole moments do not uniquely characterise spacetimes beyond general relativity

Abstract

Spacetimes in general relativity can be uniquely decomposed into a set of multipole moments. Given the usefulness of moments in the categorisation 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.

Figures (1)

• Figure 1: Depiction of equality between the critical impact parameters for the RN and JNW spacetimes, $b^{\rm RN}_{\rm cr} = b^{\rm JNW}_{\rm cr}$, as a function of RN electric monopole ($P_{0}$) and JNW scalar monopole ($\Phi_0$). Units such that $M_0=1$ are used.