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

@misc{harte2026octupole,
  title = {{Octupole moments and the non-universality of free-fall in general relativity}},
  author = {Abraham I. Harte and Paul Ramond},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.21314},
  eprint = {2607.21314},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.21314},
}

TY  - UNPB
TI  - Octupole moments and the non-universality of free-fall in general relativity
AU  - Abraham I. Harte
AU  - Paul Ramond
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2607.21314
UR  - https://arxiv.org/abs/2607.21314
AB  - Extended bodies in general relativity do not necessarily fall along geodesics. Instead, a body can experience accelerations that depend on its angular momentum, as well as on its quadrupole, octupole, and higher-order moments. Moreover, due to the fact that one object's multipole moments can evolve differently from those of another, different bodies can 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. We also 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 static Newtonian spacetimes and in fully-relativistic vacuum spacetimes that are of Petrov type D (such as Kerr). In Newtonian spacetimes, the mass moments are shown to have their ordinary Newtonian effects. The momentum moments instead determine a body's hidden momentum---the misalignment between its momentum and its velocity. In Petrov type D spacetimes, we show that some torques that are impossible with quadrupole moments are generically possible with octupole moments. Octupole moments can thus have qualitatively-different effects from quadrupole moments.
ER  -