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Concept# Four-force

Summary

In the special theory of relativity, four-force is a four-vector that replaces the classical force.
The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time:
For a particle of constant invariant mass , 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-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.
From the formulae of the previous section it appears that the time component of the four-force is the power expended, , apart from relativistic corrections . This is only true in purely mechanical situations, when heat exchanges vanish or can be neglected.
In the full thermo-mechanical case, not only work, but also heat contributes to the change in energy, which is the time component of the energy–momentum covector. The time component of the four-force includes in this case a heating rate , besides the power . Note that work and heat cannot be meaningfully separated, though, as they both carry inertia. This fact extends also to contact forces, that is, to the stress–energy–momentum tensor.
Therefore, in thermo-mechanical situations the time component of the four-force is not proportional to the power but has a more generic expression, to be given case by case, which represents the supply of internal energy from the combination of work and heat, and which in the Newtonian limit becomes .
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.
In addition, we can formulate force using the concept of coordinate transformations between different coordinate systems. Assume that we know the correct expression for force in a coordinate system at which the particle is momentarily at rest.

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