In the special theory of relativity, four-force is a four-vector that replaces the classical force.
Contents
In special relativity
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
Here
and
where
Including thermodynamic interactions
From the formulae of the previous section it appears that the time component of the four-force is the power expended,
If 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
Therefore, in thermo-mechanical situations the time component of the four-force is not proportional to the power
In General Relativity
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. Then we can perform a transformation to another system to get the corresponding expression of force. In special relativity the transformation will be a Lorentz transformation between coordinate systems moving with a relative constant velocity whereas in general relativity it will be a general coordinate transformation.
Consider the four-force
where
In general relativity, the expression for force becomes
with covariant derivative
where
Examples
In special relativity, Lorentz 4-force (4-force acting to charged particle situated in electromagnetic field) can be expressed as:
where