When does a sphere fall like a point particle? Quadrupole universality and Weyl-driven hexadecapole deviations in vacuum general relativity

Abstract

We ask when a spinless spherical extended test body in vacuum general relativity moves as its point-particle counterpart. In Newtonian gravity, harmonicity of the external potential gives an all-order cancellation: in source-free regions all spherical multipole forces beyond the monopole vanish. Using Dixon's covariant multipole formalism, with spherical symmetry defined as $O(3)$ invariance in the Tulczyjew-Dixon momentum rest space, we show that the relativistic analogue holds through quadrupole order in any Ricci-flat spacetime. At this order the torque vector vanishes, the force reduces to Ricci contractions, and the representative worldline is geodesic; the spherical octupole is forbidden by symmetry. This universality, however, is not an all-order effacement principle. At hexadecapole (16-pole) order the torque vector still vanishes, so the spinless sector remains dynamically consistent, but curvature-squared terms generate Weyl-driven forces that can survive in vacuum. In Schwarzschild spacetime we compute the resulting force for radial infall and the invariant leading correction to the infall proper time. We also show that periodic modulations of the hexadecapole moments act as an internal drive: Melnikov's method gives transverse homoclinic splitting and local chaotic layers near the geodesic separatrix for generic driving frequencies. The analysis is restricted to the small-body regime of Dixon's finite multipole expansion.