Octupole moments and the non-universality of free-fall in general relativity

Abstract

Extended bodies in general relativity do not necessarily fall along geodesics, but can be accelerated. These accelerations depend on an object's angular momentum, as well as on its quadrupole, octupole, and higher-order moments. However, these multipole moments can evolve differently from one body to another. Different bodies can thus fall differently, even with identical initial data. This paper examines how octupole moments contribute to the non-universality of free-fall in general relativity. We begin by showing that in arbitrary vacuum spacetimes, only the trace-free component of an octupole moment can affect an object's motion. It follows that at least 16 out of 40 octupole components decouple from the laws of motion. Then, we obtain two decompositions for trace-free octupole moments, one in terms of a timelike frame vector and the other in terms of a null tetrad. These decompositions are applied to motion both in generic Newtonian spacetimes and in fully-relativistic vacuum spacetimes that are of Petrov type D. In Newtonian spacetimes, the mass moments are shown to have their ordinary Newtonian effects, while the momentum moments determine a body's hidden momentum---the misalignment between its momentum and its velocity. In Petrov type D spacetimes (such as Kerr), we show that some torques that are impossible with quadrupole moments are possible with octupole moments. Octupole moments can thus have qualitatively-different effects from quadrupole moments.