Four-force ($K^{\mu }=d{p}^{\mu }/d\tau$)

The four-force is the proper time derivative of four-momentum. For constant rest mass and coordinate three-force $\mathbf{F}=d\mathbf{p}/dt$, its components are $K^{\mu }=\gamma (\mathbf{F}\cdot \mathbf{v}/c,\mathbf{F})$ in the convention $p^{\mu }=(E/c,\mathbf{p})$. Thus it packages the power transfer and three-force into one Lorentz-covariant vector.

A four-force preserving the particle's rest mass. For a massive particle, $P=mU$ and $U^2=c^2$ in the mostly-minus convention imply $F\cdot U=c^{2}dm/d\tau$. Thus constant mass is equivalent to orthogonality of the force and four-velocity. Energy transfer in a moving frame is allowed even though rest mass does not change.

A nonzero four-force cannot have all components constant in an inertial frame along an entire constant-mass trajectory. If $U=U_0+F\tau /m$, preserving $U^2=c^2$ for all proper times requires both $U_0\cdot F=0$ and $F^2=0$. The orthogonal complement of timelike $U_0$ is negative definite, forcing $F=0$. A constant laboratory three-force or a four-force of constant invariant magnitude is a different, consistent condition.