In the special theory of relativity, fourforce is a fourvector that replaces the classical force.
$$\backslash mathbf\{F\}\; =\; \{\backslash mathrm\{d\}\backslash mathbf\{P\}\; \backslash over\; \backslash mathrm\{d\}\backslash tau\}.$$
For a particle of constant invariant mass $m\; >\; 0$, $\backslash mathbf\{P\}\; =\; m\backslash mathbf\{U\}$ where $\backslash mathbf\{U\}=\backslash gamma(c,\backslash mathbf\{u\})$ is the fourvelocity, so we can relate the fourforce with the fouracceleration $\backslash mathbf\{A\}$ as in Newton's second law:
$$\backslash mathbf\{F\}\; =\; m\backslash mathbf\{A\}\; =\; \backslash left(\backslash gamma\; \{\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}\; \backslash over\; c\},\backslash gamma\{\backslash mathbf\; f\}\backslash right).$$
Here
$$\{\backslash mathbf\; f\}=\{\backslash mathrm\{d\}\; \backslash over\; \backslash mathrm\{d\}t\}\; \backslash left(\backslash gamma\; m\; \{\backslash mathbf\; u\}\; \backslash right)=\{\backslash mathrm\{d\}\backslash mathbf\{p\}\; \backslash over\; \backslash mathrm\{d\}t\}$$
and
$$\{\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}\}=\{\backslash mathrm\{d\}\; \backslash over\; \backslash mathrm\{d\}t\}\; \backslash left(\backslash gamma\; mc^2\; \backslash right)=\{\backslash mathrm\{d\}E\; \backslash over\; \backslash mathrm\{d\}t\}\; .$$
where $\backslash mathbf\{u\}$, $\backslash mathbf\{p\}$ and $\backslash mathbf\{f\}$ are 3space vectors describing the velocity, the momentum of the particle and the force acting on it respectively.
In the full thermomechanical case, not only work, but also heat contributes to the change in energy, which is the time component of the fourmomentum. The time component of the fourforce includes in this case a heating rate $h$, besides the power $\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}$. 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.C. A. Truesdell, R. A. Toupin: The Classical Field Theories (in S. Flügge (ed.): Encyclopedia of Physics, Vol. III1, Springer 1960). §§152–154 and 288–289.
Therefore, in thermomechanical situations the time component of the fourforce is not proportional to the power $\backslash mathbf\{f\}\backslash cdot\backslash mathbf\{u\}$ 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 $h\; +\; \backslash mathbf\{f\}\; \backslash cdot\; \backslash mathbf\{u\}$.
$$F^\backslash lambda\; :=\; \backslash frac\{DP^\backslash lambda\; \}\{d\backslash tau\}\; =\; \backslash frac\{dP^\backslash lambda\; \}\{d\backslash tau\; \}\; +\; \backslash Gamma^\backslash lambda\; \{\}\_\{\backslash mu\; \backslash nu\}U^\backslash mu\; P^\backslash nu$$
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 fourforce $F^\backslash mu=(F^0,\; \backslash mathbf\{F\})$ acting on a particle of mass $m$ which is momentarily at rest in a coordinate system. The relativistic force $f^\backslash mu$ in another coordinate system moving with constant velocity $v$, relative to the other one, is obtained using a Lorentz transformation:
$$\backslash begin\{align\}$$
\mathbf{f} &= \mathbf{F} + (\gamma  1) \mathbf{v} {\mathbf{v}\cdot\mathbf{F} \over v^2}, \\ f^0 &= \gamma \boldsymbol{\beta}\cdot\mathbf{F} = \boldsymbol{\beta}\cdot\mathbf{f}.\end{align}
where $\backslash boldsymbol\{\backslash beta\}\; =\; \backslash mathbf\{v\}/c$.
In general relativity, the expression for force becomes
$$f^\backslash mu\; =\; m\; \{DU^\backslash mu\backslash over\; d\backslash tau\}$$
with covariant derivative $D/d\backslash tau$. The equation of motion becomes
$$m\; \{d^2\; x^\backslash mu\backslash over\; d\backslash tau^2\}\; =\; f^\backslash mu\; \; m\; \backslash Gamma^\backslash mu\_\{\backslash nu\backslash lambda\}\; \{dx^\backslash nu\; \backslash over\; d\backslash tau\}\; \{dx^\backslash lambda\; \backslash over\; d\backslash tau\},$$
where $\backslash Gamma^\backslash mu\_\{\backslash nu\backslash lambda\}$ is the Christoffel symbol. If there is no external force, this becomes the equation for in the curved spacetime. The second term in the above equation, plays the role of a gravitational force. If $f^\backslash alpha\_f$ is the correct expression for force in a freely falling frame $\backslash xi^\backslash alpha$, we can use then the equivalence principle to write the fourforce in an arbitrary coordinate $x^\backslash mu$:
$$f^\backslash mu\; =\; \{\backslash partial\; x^\backslash mu\; \backslash over\; \backslash partial\backslash xi^\backslash alpha\}\; f^\backslash alpha\_f.$$
where

