Swimming in spacetime: the view from a Fermi observer

TL;DR

This paper investigates how extended bodies move in curved spacetime, showing that their trajectories depend on internal motion cycles and center of mass definitions, and can deviate from geodesics as predicted by Wisdom.

Contribution

It clarifies the conditions under which extended bodies deviate from geodesic motion, reconciling Wisdom's results with the Mathisson-Papapetrou-Dixon equations.

Findings

Tripod's motion depends on internal cycle choices.

Center of mass definition affects trajectory.

Deviations from geodesics align with Wisdom's predictions.

Abstract

An extended test body moving in a curved spacetime does not typically follow a geodesic, because of forces that arise from couplings between its multipole moments and the ambient curvature. An illustration of this fact was provided by Wisdom, who showed that the motion of a quasi-rigid body undergoing cyclic changes of shape in a curved spacetime deviates, in general, from a geodesic. Wisdom's analysis, however, was recently challenged on the grounds that the body's motion should be described by the Mathisson-Papapetrou-Dixon equations, and that these predict geodesic motion for the kind of body considered by Wisdom. We attempt to shed some light on this matter by examining the motion of an internally-moving tripod in Schwarzschild spacetime, as viewed by a Fermi observer moving on a timelike geodesic. We find that the description of the motion depends sensitively on a choice of cycle for the tripod's internal motions, but also on a choice of "center of mass" for the tripod; a sensible (though not unique) prescription for this "center of mass" produces a motion that conforms with Wisdom's prediction: the tripod drifts away from the observer, even when they are given identical initial conditions. We suggest pathways of reconciliation between this conclusion and the null result that apparently follows from the Mathisson-Papapetrou-Dixon equations of motion.