Proper force (${\mathbf{f}}_0=d\mathbf{p}/dt\text{in the instantaneous rest frame}$)

The three-force measured in a particle's instantaneous rest frame. For a pure force, ${\mathbf{f}}_0=m_0{\mathbf{a}}_0$, where ${\mathbf{a}}_0$ is the rest-frame acceleration. In that frame the four-force has zero temporal component and spatial part ${\mathbf{f}}_0$; the four-force itself is a different object.