where , and are
3-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.
Including thermodynamic interactions
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 .[1] Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.[2] This fact extends also to contact forces, that is, to the
stress–energy–momentum tensor.[3][2]
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,[2][1][4][3] and which in the Newtonian limit becomes .
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.[5] 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 acting on a particle of mass which is momentarily at rest in a coordinate system. The relativistic force in another coordinate system moving with constant velocity , relative to the other one, is obtained using a Lorentz transformation:
where is the
Christoffel symbol. If there is no external force, this becomes the equation for
geodesics in the
curved space-time. The second term in the above equation, plays the role of a gravitational force. If is the correct expression for force in a freely falling frame , we can use then the
equivalence principle to write the four-force in an arbitrary coordinate :
Examples
In special relativity,
Lorentz four-force (four-force acting on a charged particle situated in an electromagnetic field) can be expressed as:
^
abGrot, Richard A.; Eringen, A. Cemal (1966). "Relativistic continuum mechanics: Part I – Mechanics and thermodynamics". Int. J. Engng Sci. 4 (6): 611–638, 664.
doi:
10.1016/0020-7225(66)90008-5.
^
abC. A. Truesdell, R. A. Toupin: The Classical Field Theories (in S. Flügge (ed.): Encyclopedia of Physics, Vol. III-1, Springer 1960). §§152–154 and 288–289.
^Maugin, Gérard A. (1978). "On the covariant equations of the relativistic electrodynamics of continua. I. General equations". J. Math. Phys. 19 (5): 1198–1205.
Bibcode:
1978JMP....19.1198M.
doi:
10.1063/1.523785.