Editing Four-force (section)

== Including thermodynamic interactions ==
From the formulae of the previous section it appears that the time component of the four-force is the power expended, <math>\mathbf{f}\cdot\mathbf{u}</math>, apart from relativistic corrections <math>\gamma/c</math>. 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 (thermodynamics)|work]], but also [[Heat (thermodynamics)|heat]] contributes to the change in energy, which is the time component of the [[four-momentum|energy–momentum covector]]. The time component of the four-force includes in this case a heating rate <math>h</math>, besides the power <math>\mathbf{f}\cdot\mathbf{u}</math>.<ref name=grotetal1966>{{cite journal|last1=Grot|first1=Richard A.|last2=Eringen|first2=A. Cemal| title=Relativistic continuum mechanics: Part I – Mechanics and thermodynamics|date=1966|journal=Int. J. Engng Sci.| volume=4|issue=6|pages=611–638, 664|doi=10.1016/0020-7225(66)90008-5}}</ref> Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.<ref name=eckart1940>{{cite journal| last1=Eckart|first1=Carl| title=The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid|date=1940|journal=Phys. Rev.| volume=58| issue=10| pages=919–924| doi=10.1103/PhysRev.58.919| bibcode=1940PhRv...58..919E}}</ref> This fact extends also to contact forces, that is, to the [[stress–energy tensor|stress–energy–momentum tensor]].<ref name=truesdelletal1960>C. 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.</ref><ref name=eckart1940 />

Therefore, in thermo-mechanical situations the time component of the four-force is ''not'' proportional to the power <math>\mathbf{f}\cdot\mathbf{u}</math> 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,<ref name=eckart1940 /><ref name=grotetal1966 /><ref>{{cite journal| last1=Maugin|first1=Gérard A.|title=On the covariant equations of the relativistic electrodynamics of continua. I. General equations|date=1978|journal=J. Math. Phys.| volume=19| issue=5| pages=1198–1205| doi=10.1063/1.523785| bibcode=1978JMP....19.1198M}}</ref><ref name=truesdelletal1960 /> and which in the Newtonian limit becomes <math>h + \mathbf{f} \cdot \mathbf{u}</math>.