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:

.

For a particle of constant invariant mass m > 0, where is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

.

Here

and

.

where , and 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.

Examples

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

, where - electromagnetic tensor, - 4-velocity, - electric charge.

See also

References

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