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:
For a particle of constant invariant mass m > 0, where is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:
where , and are 3-vectors describing the velocity and the momentum of the particle and the force acting on it respectively.
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.
In special relativity, Lorentz 4-force (4-force acting to charged particle situated in electromagnetic field) can be expressed as:
- , where - electromagnetic tensor, - 4-velocity, - electric charge.