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:

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

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.

See also

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

References

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