# Extended bodies moving on geodesic trajectories

## TL;DR

This paper examines conditions under which extended bodies with spin and quadrupole moments follow geodesic paths in curved spacetimes, focusing on the Mathisson-Papapetrou-Dixon equations and specific spin supplementary conditions.

## Contribution

It provides a detailed analysis of geodesic motion for extended bodies with pole-dipole and quadrupole moments using the Ohashi-Kyrian-Semerak condition, clarifying when such bodies follow geodesics.

## Key findings

- Pole-dipole bodies can follow geodesics under certain conditions.
- The Ohashi-Kyrian-Semerak condition simplifies the analysis by eliminating hidden momentum.
- Specific trajectories in Kerr and Schwarzschild spacetimes are examined.

## Abstract

This work investigates whether an extended test body obeying the Mathisson-Papapetrou- Dixon equations under the Ohashi-Kyrian-Semerak spin supplementary condition can follow geodesic trajectories in curved spacetimes. In particular, we explore what are the requirements under which pole-dipole and pole-dipole-quadrupole approximated bodies moving in the Schwarzschild or Kerr spacetimes can follow equatorial geodesic trajectories. We do this exploration thoroughly in the pole-dipole case, while we focus just on particular trajectories in the pole-dipole-quadrupole case. Using the Ohashi-Kyrian-Semerak spin supplementary condition to fix the center of the mass of a pole-dipole body has the advantage that the hidden momentum is eliminated. This allows the four-velocity to be parallel to the four-momentum, which provides a convenient framework for our investigation. We discuss how this feature can be recovered at a pole-dipole-quadrupole approximation and what are the consequences.