Four-force: Definition from Answers.com

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 $\mathbf{U} \,$ is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

$\mathbf{F} = m\mathbf{A} = \left(\gamma {d\gamma \over dt} mc,\gamma\mathbf f\right)$ .

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

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.

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

References

Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-853971-853951-5.

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