Monge array

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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=dPdτ.

For a particle of constant invariant mass m > 0, P=mU where U=γ(c,u) is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

F=mA=(γfuc,γf).

Here

f=ddt(γmu)=dpdt

and

fu=ddt(γmc2)=dEdt.

where u, p and 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.

Fλ:=DPλdτ=dPλdτ+ΓλμνUμPν

Examples

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

Fμ=qEμνUν, where Eμν - electromagnetic tensor, Uν - 4-velocity, q - electric charge.

See also

References

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