Extended bodies in general relativity do not necessarily follow geodesics; instead, they can experience accelerations. These accelerations depend on an object's angular momentum, as well as its higher-order multipole moments, such as quadrupole and octupole moments. Since these moments can evolve differently for various bodies, objects with identical initial conditions can fall differently, a phenomenon known as the non-universality of free-fall. A new study has specifically explored how octupole moments contribute to this behavior.

The researchers demonstrated that, in arbitrary vacuum spacetimes, only the trace-free component of an octupole moment can influence an object's motion. This implies that at least 16 out of 40 octupole components decouple from the laws of motion. Two decompositions for trace-free octupole moments were obtained, one using a timelike frame vector and the other a null tetrad, applying them to both generic Newtonian spacetimes and fully-relativistic vacuum spacetimes of Petrov type D.

In Newtonian spacetimes, mass moments exhibit their ordinary Newtonian effects, while momentum moments determine a body's hidden momentum—the misalignment between its momentum and its velocity. For Petrov type D spacetimes, such as those generated by Kerr black holes, the study revealed that octupole moments can generate torques that are impossible with quadrupole moments. This suggests that octupole moments have qualitatively different and more complex effects on the dynamics of objects in strong gravitational fields.