Monge array

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:

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

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:

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

Here

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

and

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

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.

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

Examples

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

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

See also

• four-vector
• four-velocity
• four-acceleration
• four-momentum

References

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