four-force four-vector proper time energy momentum
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:





F

ν


=



d

P

ν




d
τ



=
m

A

ν




{\displaystyle F_{\nu }={\frac {dP_{\nu }}{d\tau }}=mA_{\nu }}


