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:

<math>F^\mu= {dp^\mu \over d\tau}</math>.

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

<math>F^\mu = mA^\mu = \left(\gamma \dot \gamma mc,\gamma\mathbf f\right)</math>.

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

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.

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