Four-force

In the special theory of relativity four-force is a four-vector that replaces the classical force; the four-force is the four-vector defined as the change in four-momentum over the particle's own time:

$F = {dp \over d\tau}$ .

Since $p = mU \,$ where m is the invariant mass and $U \,$ is the four-velocity, we can relate the four-force with the four-acceleration as like Newton's second law:

$F = mA = \left(\gamma \dot \gamma mc,\gamma\mathbf f\right)$ .

Here, m is the invariant mass and $\mathbf f=m\left(\dot\gamma\mathbf u+\gamma\mathbf{\dot u}\right)$ .

In general relativity the relation between four-force, and four-acceleration remains the same, but the elements of the four-force are related to the elements of the four-momentum through a covariant derivative with respect to proper time.

$F^\lambda := \frac{Dp^\lambda }{d\tau} = \frac{dp^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu p^\nu$