Four-force

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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:

F = {dp \over d\tau}.

Since p = mU \, where m is the invariant mass and U \, is the four-velocity, we can relate the four-force with the four-acceleration as like Newton's second law:

F = mA = \left(\gamma \dot \gamma mc,\gamma\mathbf f\right).

Here, m is the invariant mass and \mathbf f=m\left(\dot\gamma\mathbf u+\gamma\mathbf{\dot u}\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

See also: four-vector, four-velocity, four-acceleration, four-momentum.

[edit] References

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