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:

:$mathbf\{F\}\; =\; \{dmathbf\{P\}\; over\; d\; au\}$.

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

:$mathbf\{F\}\; =\; mmathbf\{A\}\; =\; left(gamma\; \{dgamma\; over\; dt\}\; mc,gammamathbf\; f\; ight)$.

Here $mathbf\; f=mleft(\{dgamma\; over\; dt\}\; mathbf\; u+gamma\{d\; mathbf\{u\}\; over\; dt\}\; ight)$.

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.

:$F^lambda\; :=\; frac\{DP^lambda\; \}\{d\; au\}\; =\; frac\{dP^lambda\; \}\{d\; au\; \}\; +\; Gamma^lambda\; \{\}\_\{mu\; u\}U^mu\; P^\; u$

See also

* four-vector
* four-velocity
* four-acceleration
* four-momentum

References

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