Do forces compose/superpose in special relativity?

Jano L.

The proof is bit long so I will not write all the steps in their full extent here. You can find a good start in Heitler's book, 2nd ed., Chap. I, sec. I. He derives the Poynting theorem and the Maxwell stress tensor theorem in this way: he integrates the density of force ##\rho\mathbf E + \mathbf j\times \mathbf B## and density of power ##\mathbf j\cdot \mathbf E## to obtain total force and power that the body receives.

Assume part of the fluid inside a volume region ##V##. Heitler's equations 22a, 22b are, in changed notation

$$
\int_V \partial_t \bigg( g^k_{matter} + g^k_{field} \bigg) d^3x = \int_V \partial_s M^{ks} d^3 x,
$$

$$
\int_V \partial_t \bigg( \epsilon_{matter} + \epsilon_{field} \bigg) d^3x = -\int_V \partial_s S^{s} d^3 x,
$$

where ##g^k##-s are densities of momentum, ##\epsilon##-s densities of energy, ##M^{ks}## is the Maxwell stress tensor and ##S^s## the Poynting vector. If the boundary terms in RHS are negligible, these equations express conservation of momentum and energy within the volume ##V##.

To see that these equations are relativistic, take the RHS to the left side and express everything as one volume integral:

$$
\int_V \partial_t \bigg( g^k_{matter} + g^k_{field}\bigg) -\partial_s M^{ks} d^3x = 0,
$$
$$
\int_V \partial_t \bigg( \epsilon_{matter} + \epsilon_{field} \bigg) + \partial_s S^{s} d^3x = 0.
$$

These equations can be written as

$$
\int_V \partial_\mu \bigg( T_{\mathrm{field}}^{\rho\mu} + T_{\mathrm{matter}}^{\rho\mu}\bigg) d^3x = 0
$$

using the tensors


$$
T_{\mathrm{field}}^{\rho\mu} = {{F^{\rho\nu}} F^{\mu}_{~~\,\nu}} - \frac{1}{4} \delta^{\rho\mu} F^{\sigma\nu} F_{\sigma\nu}
=
\left(
\begin{matrix}
\epsilon_{field}& ~\mathbf S \\
\mathbf S & - \mathbf M\\
\end{matrix}
\right),
$$


$$
T_{\mathrm{matter}}^{\rho\mu} = \rho_0 u^\rho u^\mu,
$$

and the continuity of mass flow ##\partial_\mu (\rho_0 u^\mu) = 0##. Here ##\rho_0## is the rest mass density and ##u^\mu## is the local four-velocity of the fluid ##dx^\mu/d\tau##.


Finally, since the volume ##V## is arbitrary, we obtain

$$
\partial_\mu \bigg( T_{\mathrm{field}}^{\rho\mu} + T_{\mathrm{matter}}^{\rho\mu}\bigg) = 0,
$$

which is exact, tensor equation.

To summarize, model of fluid whose motion is acted upon by EM forces, described by the above force and power density, leads to conservation law that is the same in every inertial frame. This is an evidence that the concept of force works well in relativity.

Can you please explain why did you claim that the above expression for force density is not meaningful in relativistic theory?