# 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:

$\mathbf{F} = {d\mathbf{P} \over d\tau}$.

For a particle of constant invariant mass m > 0, $\mathbf{P} = m\mathbf{U} \,$ where $\mathbf{U}=\gamma(c,\mathbf{u}) \,$ is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

$\mathbf{F} = m\mathbf{A} = \left(\gamma {\mathbf{f}\cdot\mathbf{u} \over c},\gamma{\mathbf f}\right)$.

Here

${\mathbf f}={d \over dt} \left(\gamma m {\mathbf u} \right)={d\mathbf{p} \over dt}$

and

${\mathbf{f}\cdot\mathbf{u}}={d \over dt} \left(\gamma mc^2 \right)={dE \over dt}$.

where $\mathbf{u}$, $\mathbf{p}$ and $\mathbf{f}$ 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.

$F^\lambda := \frac{DP^\lambda }{d\tau} = \frac{dP^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu P^\nu$

## Examples

In special relativity, Lorentz 4-force (4-force acting to charged particle situated in electromagnetic field) can be expressed as:

$F_\mu = qE_{\mu\nu}U^\nu$, where $E_{\mu\nu}$ - electromagnetic tensor, $U^\nu$ - 4-velocity, $q$ - electric charge.