Extended bodies moving on geodesic trajectories

Abstract

This work investigates whether an extended test body obeying the Mathisson–Papapetrou–Dixon equations under the Ohashi–Kyrian–Semerák 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–Semerák 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.