# 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:

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

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

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

Here

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

and

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

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

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

## Examples

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

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