  Massive gravity in the weak field limit is described by the Fierz-Pauli
theory with 5 degrees of freedom in four dimensions. In this theory, we
calculate the gravitomagnetic effects (potential energy) between two
point-like, spinning sources that also orbit around each other in the limit
where the spins and the velocities are small. Spin-spin, spin-orbit and
orbit-orbit interactions in massive gravity theory have rather remarkable,
discrete differences from their counterparts in General Relativity. Our
computation is applicable for large distances, for example, for interaction
between galaxies or galaxy clusters where massive gravity is expected to play a
role. We also extend the computations to quadratic gravity theories in four
dimensions and find the lowest order gravitomagnetic effects and show that at
small separations quadratic gravity behaves differently than General
Relativity.
