History of Topics in Special Relativity/Four-force (electromagnetism)

Overview
The electromagnetic four-force or covariant Lorentz force $$K^{\mu}$$ can be expressed as


 * (a) the rate of change in the four-momentum $$P^{\mu}=mU^{\mu}$$ of a particle with respect to the particle's proper time $$\tau$$,


 * (b) function of three-force $$\mathbf{f}$$


 * (c) assuming constant mass as the product of invariant mass m and four-acceleration $$A^{\mu}$$


 * (d) using the Lorentz force per unit charge $$\mathbf{f}=q\left\{ \mathbf{E}+\mathbf{v}\times\mathbf{B}\right\} $$


 * (e) the product of the electromagnetic tensor $$F^{\alpha\beta}$$ with the four-velocity $$U^{\mu}$$ and charge q


 * (f) by integrating the four-force density $$D^{\mu}$$ with respect to rest unit volume $$V_{0}=V\gamma$$

The corresponding four-force density $$D^{\mu}$$ is defined as


 * (a1) the rate of change of four-momentum density $$M^{\mu}=\mu_{0}U^{\mu}$$ with rest mass density $$\mu_{0}=\mu/\gamma$$ and four-velocity $$U^{\mu}$$,


 * (b1) function of three-force density $$\mathbf{d}$$


 * (c1) assuming constant mass the product of rest mass density $$\mu_{0}$$ and four-acceleration $$A^{\mu}$$


 * (d1) using the Lorentz force density $$\mathbf{d}=\rho\left\{ \mathbf{E}+\mathbf{v}\times\mathbf{B}\right\} =\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}$$ with $$\rho$$ as charge density,


 * (e1) as the product of electromagnetic tensor $$F^{\alpha\beta}$$ with four-current $$J^{\mu}$$ or with four-velocity $$U^{\mu}$$ using rest charge density $$\rho_{0}=\rho/\gamma$$,


 * (f1) as the negative four-divergence of the electromagnetic energy-momentum tensor $$T^{\alpha\beta}$$ (compare with History of Topics in Special Relativity/Stress-energy tensor (electromagnetic)):


 * $$\begin{matrix}\begin{matrix}K^{\mu} & =\frac{\mathrm{d}P^{\mu}}{\mathrm{d}\tau} & =\gamma\left(\frac{1}{c}\mathbf{f}\cdot\mathbf{v},\ \mathbf{f}\right) & =mA^{\mu} & =\gamma q\left(\frac{1}{c}\mathbf{v\cdot E},\ \mathbf{E}+\mathbf{v}\times\mathbf{B}\right) & =qF_{\alpha\beta}U^{\beta} & =\int D^{\mu}\,dV_{0}\\

& (a) & (b) & (c) & (d) & (e) & (f)\\ D^{\mu} & =\frac{\mathrm{d}M^{\mu}}{\mathrm{d}\tau} & =\left(\frac{1}{c}\mathbf{d\cdot v},\ \mathbf{d}\right) & =\mu_{0}A^{\mu} & =\rho\left(\frac{1}{c}\mathbf{v\cdot E},\ \mathbf{E}+\mathbf{v}\times\mathbf{B}\right) & \underbrace{=F_{\alpha\beta}J^{\beta}=\rho_{0}F_{\alpha\beta}U^{\beta}} & =-\partial_{\alpha}T^{\alpha\beta}\\ & (a1) & (b1) & (c1) & (d1) & (e1) & (f1) \end{matrix}\\ \left(\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\ \mathbf{f}=\int\mathbf{d}\,dV\right) \end{matrix}$$

Poincaré (1905/6)
Henri Poincaré (July 1905, published January 1906) defined the Lorentz force and its Lorentz transformation:


 * $$\begin{matrix}X=\rho f+\rho(\eta\gamma-\zeta\beta)\dots\\

X'=\rho'f'+\rho'(\eta'\gamma'-\zeta'\beta')\dots\\ \begin{cases} X^{\prime}=k(X+\epsilon T),\quad & T^{\prime}=k(T+\epsilon X),\\ Y^{\prime}=Y, & Z^{\prime}=Z; \end{cases} \end{matrix}$$ with $$T=\Sigma X\xi$$ and $$k=\frac{1}{\sqrt{1-\epsilon^{2}}}$$.

in which (X,Y,Z,T) represent the components of four-force density (b1, d1) because


 * $$\epsilon=\frac{v}{c},\ \left(X,\ Y,\ Z\right)=\rho\left\{ \mathbf{E}+\mathbf{v}\times\mathbf{B}\right\} =\mathbf{d},\ T=\Sigma X\xi=\mathbf{d}\cdot\mathbf{v}=\mathbf{v\cdot E}$$.

Additionally, he explicitly obtained the four-force per unit charge by setting $$\scriptstyle \frac{X_{1}}{X}=\frac{Y_{1}}{Y}=\frac{Z_{1}}{Z}=\frac{T_{1}}{T}=\frac{1}{\rho}$$:


 * $$\left(k_{0}X_{1},\quad k_{0}Y_{1},\quad k_{0}Z_{1},\quad k_{0}T_{1}\right)$$ with $$T_{1}=\Sigma X_{1}\xi$$ and $$k_{0}=\tfrac{1}{\sqrt{1-\epsilon^{2}}}$$

equivalent to (b, d) and obtained its Lorentz transformation by multiplying the transformation of (X,Y,Z,T) by
 * $$\frac{\rho}{\rho^{\prime}}=\frac{1}{k(1+\xi\epsilon)}=\frac{\delta t}{\delta t^{\prime}}$$.

Minkowski (1907/8)
Hermann Minkowski employed matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product from the outset. In a lecture held in November 1907, published 1915, Minkowski defined the four-force density X,Y,Z,iA as the product of the electromagnetic tensor and the four-current with $$\varrho$$ as charge density and $$\mathfrak{v}$$ as velocity:


 * $$\begin{matrix}X,Y,Z,iA\\

\hline X_{j}=\varrho_{1}\psi_{j1}+\varrho_{2}\psi_{j2}+\varrho_{3}\psi_{j3}+\varrho_{4}\psi_{j4}\\ A=X\mathfrak{v}_{x}+Y\mathfrak{v}_{y}+Z\mathfrak{v}_{z}\\ \left\{ \begin{align}\psi_{23},\ \psi_{31},\ \psi_{12} & \Rightarrow\mathfrak{H}_{x},\mathfrak{H}_{y},\mathfrak{H}_{z}\\ \psi_{14},\ \psi_{24},\ \psi_{34} & \Rightarrow-i\mathfrak{E}_{x},-i\mathfrak{E}_{y},-i\mathfrak{E}_{z}\\ & \psi_{jj}=0 \end{align} \right\} \end{matrix}$$

equivalent to (b1, d1, e1) because $$\epsilon=\frac{v}{c},\ \left(X,Y,Z\right)=\mathbf{d},\ A=\mathbf{d}\mathbf{v}=\mathbf{vE}$$.

In another lecture from December 1907, he used the symbols $$\mathfrak{e},\mathfrak{m}$$ (i.e. $$\mathbf{E},\mathbf{B}$$) and $$\mathfrak{E},\mathfrak{M}$$ (i.e. $$\mathbf{D},\mathbf{H}$$) which he represented in the form of “vectors of second kind”, i.e. the electromagnetic tensor f and its dual $$f^{*}$$, from which he derived the electric rest force $$\Phi$$ and magnetric rest force $$\Psi$$ as the product with four-velocity w, which in turn can be used to express F and f and four-conductivity:


 * $$\begin{matrix}\begin{matrix}\Phi=-wF\\

\left(\Phi_{1},\Phi_{2},\Phi_{3}\right)=\frac{\mathfrak{E}+[\mathfrak{wM}]}{\sqrt{1-\mathfrak{w}^{2}}},\ \Phi_{4}=\frac{i(\mathfrak{wE})}{\sqrt{1-\mathfrak{w}^{2}}} & \Psi=iwf^{*},\\ \hline \left.{\scriptstyle \begin{array}{ccccccccr} \Phi_{1} & = & &  & w_{2}F_{12} & + & w_{3}F_{13} & + & w_{4}F_{14},\\ \Phi_{2} & = & w_{1}F_{21} & &  & + & w_{3}F_{23} & + & w_{4}F_{24},\\ \Phi_{3} & = & w_{1}F_{31} & + & w_{2}F_{32} & &  & + & w_{4}F_{34},\\ \Phi_{4} & = & w_{1}F_{41} & + & w_{2}F_{42} & + & w_{3}F_{43} & &. \end{array}}\right| & {\scriptstyle \begin{array}{cclcccccr} \Psi_{1} & = & -i( & & w_{2}f_{34} & + & w_{3}f_{42} & + & w_{4}f_{23}),\\ \Psi_{2} & = & -i(w_{1}f_{43} & &  & + & w_{3}f_{14} & + & w_{4}f_{31}),\\ \Psi_{3} & = & -i(w_{1}f_{24} & + & w_{2}f_{41} & &  & + & w_{4}f_{12}),\\ \Psi_{4} & = & -i(w_{1}f_{32} & + & w_{2}f_{13} & + & w_{3}f_{21} & & ). \end{array}} \end{matrix}\\ \hline wF=-\Phi,\quad wF^{*}=-i\mu\Psi,\quad wf=-\varepsilon\Phi,\quad wf^{*}=-i\Psi\\ F=[w,\Phi]+i\mu[w,\Psi]^{*},\\ f=\varepsilon[w,\Phi]+i[w,\Psi]^{*},\\ s+(w\overline{s})w=-\sigma wF. \end{matrix}$$

which becomes the covariant Lorentz force (d,e) when multiplied with the charge. He gave the general expression of four-force density as


 * $$\begin{matrix}K+(w\overline{K})w\\

\hline K=\text{lor }S=-sF+N\\ \hline \begin{align}K_{1} & =\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial X_{t}}{\partial t}=\varrho\mathfrak{E}_{x}+\mathfrak{s}_{y}\mathfrak{M}_{z}-\mathfrak{s}_{z}\mathfrak{M}_{y}-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\varepsilon}{\partial x}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial x}+\frac{\varepsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial x}\right)\\ K_{2} & =\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial Y_{t}}{\partial t}=\varrho\mathfrak{E}_{y}+\mathfrak{s}_{z}\mathfrak{M}_{x}-\mathfrak{s}_{x}\mathfrak{M}_{z}-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\varepsilon}{\partial y}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial y}+\frac{\varepsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial y}\right)\\ K_{3} & =\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial Z_{t}}{\partial t}=\varrho\mathfrak{E}_{z}+\mathfrak{s}_{x}\mathfrak{M}_{y}-\mathfrak{s}_{y}\mathfrak{M}_{x}-\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\varepsilon}{\partial z}-\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial z}+\frac{\varepsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial z}\right)\\ \frac{1}{i}K_{4} & =-\frac{\partial T_{x}}{\partial x}-\frac{\partial T_{y}}{\partial y}-\frac{\partial T_{z}}{\partial z}-\frac{\partial T_{t}}{\partial t}=\mathfrak{s}_{x}\mathfrak{E}_{x}+\mathfrak{s}_{y}\mathfrak{E}_{y}+\mathfrak{s}_{z}\mathfrak{E}_{z}+\frac{1}{2}\Phi\overline{\Phi}\frac{\partial\varepsilon}{\partial t}+\frac{1}{2}\Psi\overline{\Psi}\frac{\partial\mu}{\partial t}-\frac{\varepsilon\mu-1}{\sqrt{1-\mathfrak{w}^{2}}}\left(\mathfrak{W}\frac{\partial\mathfrak{w}}{\partial t}\right) \end{align} \\ \left[\text{lor }=\left|\frac{\partial}{\partial x_{1}},\ \frac{\partial}{\partial x_{2}},\ \frac{\partial}{\partial x_{3}},\ \frac{\partial}{\partial x_{4}}\right|\right] \end{matrix}$$

equivalent to (f).

In his lecture “space and time” (1908, published 1909), Minkowski defined the “moving force vector” as


 * $$\begin{matrix}\dot{t}X,\ \dot{t}Y,\ \dot{t}Z,\ \dot{t}T\\

T=\frac{1}{c^{2}}\left(\frac{\dot{x}}{\dot{t}}X,\ \frac{\dot{y}}{\dot{t}}Y,\ \frac{\dot{z}}{\dot{t}}Z\right) \end{matrix}$$

equivalent to (b) because $$\dot{x},\ \dot{y},\ \dot{z},\ \dot{t}$$ is the four-velocity $$\gamma[\mathbf{u},c]$$ and $$\mathbf{f}=[X,Y,Z]$$.

Born (1909)
Max Born (1909) summarized Minkowski's work using index notation, defining the equations of motion in terms of rest mass density $$\mu$$ and rest charge density $$\varrho_{0}^{\ast}$$


 * $$\begin{matrix}ci\mu\frac{\partial^{2}x_{\alpha}}{\partial\xi_{4}^{2}}=\frac{\varrho_{0}^{\ast}}{c}\sum_{\beta=1}^{4}f_{\alpha\beta}\frac{\partial x_{\beta}}{\partial\xi_{4}}\\

\hline \begin{align}\mu\frac{\partial^{2}x}{\partial\tau^{2}} & =\varrho_{0}^{\ast}\left\{ \mathfrak{E}_{x}\frac{\partial t}{\partial\tau}+\frac{1}{c}\left(\frac{\partial y}{\partial\tau}\mathfrak{M}_{z}-\frac{\partial z}{\partial\tau}\mathfrak{M}_{y}\right)\right\} \\ \mu\frac{\partial^{2}y}{\partial\tau^{2}} & =\varrho_{0}^{\ast}\left\{ \mathfrak{E}_{y}\frac{\partial t}{\partial\tau}+\frac{1}{c}\left(\frac{\partial z}{\partial\tau}\mathfrak{M}_{x}-\frac{\partial x}{\partial\tau}\mathfrak{M}_{z}\right)\right\} \\ \mu\frac{\partial^{2}z}{\partial\tau^{2}} & =\varrho_{0}^{\ast}\left\{ \mathfrak{E}_{z}\frac{\partial t}{\partial\tau}+\frac{1}{c}\left(\frac{\partial x}{\partial\tau}\mathfrak{M}_{y}-\frac{\partial y}{\partial\tau}\mathfrak{M}_{z}\right)\right\} \\ \mu\frac{\partial^{2}t}{\partial\tau^{2}} & =\frac{1}{c^{2}}\varrho_{0}^{\ast}\left\{ \mathfrak{E}_{x}\frac{\partial x}{\partial\tau}+\mathfrak{E}_{y}\frac{\partial y}{\partial\tau}+\mathfrak{E}_{z}\frac{\partial z}{\partial\tau}\right\} \end{align} \end{matrix}$$

equivalent to (c1, d1, e1).

Frank (1909)
Philipp Frank (1909) discussed “electromagnetic mechanics” by defining four-force (X,Y,Z,T) as follows:


 * $$\begin{matrix}m\frac{d}{d\sigma}\frac{dt}{d\sigma}=T,\ m\frac{d^{2}x}{d\sigma^{2}}=X,\ \frac{d^{2}y}{d\sigma^{2}}=Y,\ \frac{d^{2}z}{d\sigma^{2}}=Z\\

\hline \begin{align}T & =m\frac{d}{d\sigma}\frac{dt}{d\sigma}=m\frac{1}{\left(1-w^{2}\right)^{2}}\left(w_{x}\frac{dw_{x}}{dt}+w_{y}\frac{dw_{y}}{dt}+w_{z}\frac{dw_{z}}{dt}\right)\\ \frac{d^{2}x}{d\sigma^{2}} & =\frac{1}{1-w^{2}}\frac{dw_{x}}{dt}+\frac{w_{x}}{\left(1-w^{2}\right)^{2}}\left(w_{x}\frac{dw_{x}}{dt}+w_{y}\frac{dw_{y}}{dt}+w_{z}\frac{dw_{z}}{dt}\right)\\ & {\rm etc}. \end{align} \end{matrix}$$

corresponding to (b, c).

Abraham (1909/10)
While Minkowski used the four-force density as $$K+(K\overline{w})w$$ with $$K=\text{lor }S$$, Max Abraham (1909) directly used $$K$$ as four-force density, and expressed it in terms of “momentum equations” and an “energy equation” using momentum density $$\mathfrak{g}$$, energy density $$\psi$$, Poynting vector $$\mathfrak{S}$$, Joule heat Q


 * $$\begin{matrix}\begin{align}\mathfrak{K}_{x} & =\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{x}}{\partial t} & \mathfrak{K}_{x} & =\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial X_{t}}{\partial l}\\

\mathfrak{K}_{y} & =\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{y}}{\partial t}\quad\Rightarrow & \mathfrak{K}_{y} & =\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial Y_{t}}{\partial l}\\ \mathfrak{K}_{z} & =\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{z}}{\partial t} & \mathfrak{K}_{z} & =\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial Z_{t}}{\partial l}\\ \mathfrak{wK}+Q & =-\mathrm{div}\,\mathfrak{S}-\frac{\partial\psi}{\partial t} & \mathfrak{K}_{t} & =-\frac{\partial T_{x}}{\partial x}-\frac{\partial T_{y}}{\partial y}-\frac{\partial T_{z}}{\partial z}-\frac{\partial T_{t}}{\partial l} \end{align} \\ \hline \left(\mathrm{div}\,\mathfrak{S}=\frac{\partial\mathfrak{S}_{x}}{\partial x}+\frac{\partial\mathfrak{S}_{y}}{\partial y}+\frac{\partial\mathfrak{S}_{z}}{\partial z},\ \mathfrak{g}=\frac{\mathfrak{S}}{c^{2}}\right) \end{matrix}$$

or alternatively by introducing “relative stresses” and the “relative energy flux”:


 * $$\begin{matrix}\begin{align}\mathfrak{K}_{x} & =\frac{\partial X_{x}^{\prime}}{\partial x}+\frac{\partial X_{y}^{\prime}}{\partial y}+\frac{\partial X_{z}^{\prime}}{\partial z}-\frac{\partial\mathfrak{g}_{x}}{\partial t}\\

\mathfrak{K}_{y} & =\frac{\partial Y_{x}^{\prime}}{\partial x}+\frac{\partial Y_{y}^{\prime}}{\partial y}+\frac{\partial Y_{z}^{\prime}}{\partial z}-\frac{\partial\mathfrak{g}_{y}}{\partial t}\\ \mathfrak{K}_{z} & =\frac{\partial Z_{x}^{\prime}}{\partial x}+\frac{\partial Z_{y}^{\prime}}{\partial y}+\frac{\partial Z_{z}^{\prime}}{\partial z}-\frac{\partial\mathfrak{g}_{z}}{\partial t}\\ \mathfrak{wK}+Q & =-\mathrm{div}\,\left\{ \mathfrak{S}-\mathfrak{w}\psi\right\} -\frac{\partial\psi}{\partial t} \end{align} \\ \hline \left(\begin{align}X_{x}^{\prime} & =X_{x}+\mathfrak{w}_{x}\mathfrak{g}_{x} & X_{y}^{\prime} & =X_{y}+\mathfrak{w}_{y}\mathfrak{g}_{x} & X_{z}^{\prime} & =X_{z}+\mathfrak{w}_{z}\mathfrak{g}_{x}\\ Y_{x}^{\prime} & =Y_{x}+\mathfrak{w}_{x}\mathfrak{g}_{x} & Y_{y}^{\prime} & =Y_{y}+\mathfrak{w}_{y}\mathfrak{g}_{x} & Y_{z}^{\prime} & =Y_{z}+\mathfrak{w}_{z}\mathfrak{g}_{x}\\ Z_{x}^{\prime} & =Z_{x}+\mathfrak{w}_{x}\mathfrak{g}_{x} & Z_{y}^{\prime} & =Z_{y}+\mathfrak{w}_{y}\mathfrak{g}_{x} & Z_{z}^{\prime} & =Z_{z}+\mathfrak{w}_{z}\mathfrak{g}_{x} \end{align} \right) \end{matrix}$$

all equivalent to (f1).

In a subsequent paper (1909) he formulated these relations as follows


 * $$\begin{matrix}\begin{align}\mathfrak{K}_{x} & =\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{x}}{\partial t} & \mathfrak{K}_{x} & =\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}+\frac{\partial X_{u}}{\partial u}\\

\mathfrak{K}_{y} & =\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{y}}{\partial t}\quad\Rightarrow & \mathfrak{K}_{y} & =\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}+\frac{\partial Y_{u}}{\partial u}\\ \mathfrak{K}_{z} & =\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{z}}{\partial t} & \mathfrak{K}_{z} & =\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}+\frac{\partial Z_{u}}{\partial u}\\ Q+(\mathfrak{wK}) & =-\frac{\partial\mathfrak{S}_{x}}{\partial x}-\frac{\partial\mathfrak{S}_{y}}{\partial y}-\frac{\partial\mathfrak{S}_{z}}{\partial z}-\frac{\partial\psi}{\partial t} & \mathfrak{K}_{u} & =\frac{\partial U_{x}}{\partial x}+\frac{\partial U_{y}}{\partial y}+\frac{\partial U_{z}}{\partial z}+\frac{\partial U_{u}}{\partial u} \end{align} \\ \hline \left(u=ict\right) \end{matrix}$$

In addition, Abraham gave arguments in favor of his choice to use $$K=\text{lor }S$$ directly as four-force density: He argued that Minkowski's definition of force as the product of constant rest mass with four-acceleration together with a complementary force is disadvantageous at non-adiabatic motion, because mass-energy equivalence according to which mass depends on its energy content would suggest a variable rest mass $$m_{0}$$, and showed that $$m_{0}$$ is compatible with the general definition of force density $$\mathfrak{K}$$ over volume dv as the rate of momentum change without the need of complementary components


 * $$\frac{d}{dt}\left\{ \frac{m_{0}c\mathfrak{q}}{\sqrt{1-\mathfrak{q}^{2}}}\right\} =\mathfrak{K}\,dv$$.

As he showed in 1910, this implies that the equations of motion in terms of four-force density $$\mathfrak{K}$$ and rest mass density $$\nu$$ and four-velocity assume the form:


 * $$\begin{align}\frac{d}{d\tau}\left(\nu\frac{dx}{d\tau}\right) & =\mathfrak{K}_{x}\\

\frac{d}{d\tau}\left(\nu\frac{dy}{d\tau}\right) & =\mathfrak{K}_{y}\\ \frac{d}{d\tau}\left(\nu\frac{dz}{d\tau}\right) & =\mathfrak{K}_{z}\\ \frac{d}{d\tau}\left(\nu\frac{du}{d\tau}\right) & =\mathfrak{K}_{u}\ (u=ict) \end{align} $$

This is equivalent to (a1) defining four-force density as the rate of change of four-momentum density.

Bateman (1909/10)
The first discussion in an English language paper of four-force in terms of integral forms (even though in the broader context of spherical wave transformations), was given by Harry Bateman in a paper read 1909 and published 1910, who defined the following invariants (with $$\lambda^{2}=1$$ in relativity):


 * $$\begin{matrix}\frac{\rho}{\lambda^{2}}\left[\left(E_{x}-w_{z}H_{y}+w_{y}H_{z}\right)dy\ dz\ dt+\left(E_{y}-w_{x}H_{z}+w_{z}H_{x}\right)dz\ dx\ dt\right.\\

\left.+\left(E_{z}-w_{y}H_{z}+w_{x}H_{y}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\\ \\ \frac{\rho}{\lambda^{4}}\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.\\ \left.+\left(E_{z}+w_{z}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right],\\ \\ \rho\ dx\ dy\ dz\ dt\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.\\ \left.+\left(E_{z}+w_{x}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right]. \end{matrix}$$

equivalent to (e) because it can be seen as the product of charge, four-velocity and the electromagnetic tensor, which is the definition of the covariant Lorentz force.

Ignatowski (1910)
Wladimir Ignatowski (1910) formulated the four-force in terms of the Lorentz force as well as the equation of motion, and following Abraham (1909) he assumed variable rest mass in the case of non-adiabatic motion


 * $$\begin{matrix}\left(\frac{\mathfrak{K}}{\sqrt{1-n\mathfrak{v}^{2}}},\ \frac{n\mathfrak{vK}}{\sqrt{1-n\mathfrak{v}^{2}}}\right)\\

\frac{d\frac{m_{0}\mathfrak{v}}{\sqrt{1-n\mathfrak{v}^{2}}}}{dt}=\mathfrak{K}\quad\Rightarrow\quad\left(\frac{\mathfrak{K}}{\sqrt{1-n\mathfrak{v}^{2}}},\ \frac{n\mathfrak{vK}}{\sqrt{1-n\mathfrak{v}^{2}}}+\frac{dm_{0}}{dt}\right)\\ \hline \begin{matrix}\mathfrak{K}=e\mathfrak{E}+[e\mathfrak{vH}]\\ \mathfrak{K}'=e'\mathfrak{E}'+[e'\mathfrak{v'H}'] \end{matrix}\quad m_{0}\frac{d\frac{\mathfrak{v}}{\sqrt{1-n\mathfrak{v}^{2}}}}{dt}=\mathfrak{K} \end{matrix}{\scriptstyle \left[n=\frac{1}{c^{2}}\right]}$$

equivalent to (b, d). He also derived the four-force density in terms of the density of three-force or Lorentz force $$\mathfrak{K}_{1}$$ by integrating three-force $$\mathfrak{K}$$ with respect to volume:


 * $$\begin{matrix}\left(\mathfrak{K}_{1},\ n\mathfrak{K}_{1}\mathfrak{v}\right)\\

\hline \begin{matrix}\mathfrak{K}_{1}=\mathfrak{E}\varrho+[\varrho\mathfrak{vH}]\\ \mathfrak{K}'_{1}=\mathfrak{E}'\varrho+[\varrho'\mathfrak{v'H}'] \end{matrix}\left[\mathfrak{K}\ dv=\mathfrak{K}_{1},\ n=\frac{1}{c^{2}}\right] \end{matrix}$$

equivalent to (b1, d1).

Sommerfeld (1910)
In an influential paper, Arnold Sommerfeld translated Minkowski's matrix formalism into a four-dimensional vector formalism involving four-vectors and six-vectors. He defined the four-force density as the covariant product of a six-vector denoted as “field vector” f (now known as electromagnetic tensor) and the four-current P, which he related to Lorentz force density $$\mathfrak{F}$$:


 * $$\begin{matrix}[Pf]\\

\hline \begin{array}{ll} \mathfrak{F}_{x}=\left(Pf_{x}\right) & =\varrho\left(\frac{\mathfrak{v}_{x}}{c}f_{xx}+\frac{\mathfrak{v}_{y}}{c}f_{xy}+\frac{\mathfrak{v}_{z}}{c}f_{xz}+if_{xl}\right)\\ & =\varrho\left(\frac{\mathfrak{v}_{y}}{c}\mathfrak{H}_{z}-\frac{\mathfrak{v}_{z}}{c}\mathfrak{H}_{y}+\mathfrak{E}_{x}\right)\\ & \dots\\ \mathfrak{F}_{l}=\left(Pf_{l}\right) & =\varrho\left(\frac{\mathfrak{v}_{x}}{c}f_{lx}+\frac{\mathfrak{v}_{y}}{c}f_{ly}+\frac{\mathfrak{v}_{z}}{c}f_{lz}+if_{ll}\right)\\ & =\frac{i\varrho}{c}\left(\mathfrak{v}_{x}\mathfrak{E}_{x}+\mathfrak{v}_{y}\mathfrak{E}_{y}+\mathfrak{v}_{z}\mathfrak{E}_{z}\right) \end{array}\\ \hline \mathfrak{F}_{j}=\varrho\left(\mathfrak{E}+\frac{1}{c}[\mathfrak{vH}]\right),\ \mathfrak{F}_{l}=\frac{i\varrho}{c}(\mathfrak{Ev})\\ (j=x,y,z;\ l=ict) \end{matrix}$$

equivalent to (b1, d1, e1).

Laue (1911-13)
In the first textbook on relativity, Max von Laue (1911) followed Abraham in defining the four-force density (Viererkraft) F as the product of a six-vector denoted as “field vector” $$\mathfrak{M}$$ (now known as electromagnetic tensor) and the four-current related to Lorentz force density $$\mathfrak{F}$$, and alternatively as the divergence of the electromagnetic stress-energy tensor T, which included some printing errors corrected in the 1913 edition


 * $$\begin{matrix}\begin{align}F & =[P\mathfrak{M}]\\

& =-\varDelta iv\,T \end{align} \\ \hline {}[P\mathfrak{M}]_{x}=\mathfrak{F}_{x},\ [P\mathfrak{M}]_{y}=\mathfrak{F}_{y},\ [P\mathfrak{M}]_{z}=\mathfrak{F}_{z},\ [P\mathfrak{M}]_{l}=\frac{i\varrho}{c}(\mathfrak{qE})=\frac{i}{c}(\mathfrak{qF})\\ \left[\mathfrak{F}=\varrho(\mathfrak{E}+\frac{1}{c}[\mathfrak{qH}]),\ \varDelta iv=\text{divergence six-vector},\ l=ict\right] \end{matrix}$$

equivalent to (b1, d1, e1, f1).

In 1913, Laue also showed that four-force density F and three force density $$\mathfrak{F}$$ can be used to derive the “Minkowskian force vector” (i.e. four-force) K and three-force $$\mathfrak{K}$$ per unit charge by defining the volume $$\delta V:\sqrt{c^{2}-q^{2}}$$:


 * $$K\Rightarrow\begin{align}K_{x} & =\frac{\mathfrak{K}_{x}}{\sqrt{c^{2}-q^{2}}}, & K_{y} & =\frac{\mathfrak{K}_{y}}{\sqrt{c^{2}-q^{2}}},\\

K_{z} & =\frac{\mathfrak{K}_{z}}{\sqrt{c^{2}-q^{2}}}, & K_{l} & =\frac{i}{c}\frac{(\mathfrak{qK})}{\sqrt{c^{2}-q^{2}}}, \end{align} \Rightarrow\frac{d(mY)}{d\tau}\Rightarrow m\frac{dY}{d\tau}$$

Silberstein (1911-14)
Ludwik Silberstein in 1911 and published 1912, devised an alternative 4D calculus based on Biquaternions which, however, never gained widespread support. He defined the "force-quaternion" (i.e. four-force density) as the electrical part $$\mathrm{P}_{e}$$ of P, which he related to the “current-quaternion” (i.e. four-current) C and the “electromagnetic bivector” (i.e. Weber vector) $$\mathbf{F}$$}


 * $$\begin{matrix}\mathrm{P}=\mathrm{C}\mathbf{F}=\mathrm{D}[\mathbf{F}\cdot\mathbf{F}]\\

\hline \begin{align}\mathrm{P} & =\rho\left\{ \iota\mathbf{M}+\mathbf{E}+\frac{1}{c}p\mathbf{M}-\frac{\iota}{c}p\mathbf{E}\right\} \\ & =\mathrm{P}_{e}+\iota\mathrm{P}_{m}\\ \mathrm{P}_{e} & =\rho\left\{ \frac{\iota}{c}(p\mathbf{E})+\mathbf{E}+\frac{1}{c}\mathrm{V}p\mathbf{M}\right\} \\ \mathrm{P}_{m} & =\rho\left\{ \frac{\iota}{c}(p\mathbf{M})+\mathbf{M}+\frac{1}{c}\mathrm{V}p\mathbf{E}\right\} \end{align} \\ \left[\mathrm{D}=\frac{\partial}{\partial l}-\nabla\right] \end{matrix}$$

equivalent to (d1,e1,f1).

In his textbook on quaternionic special relativity written 1914, he defined four-force density using stress-energy tensor $$\mathfrak{S}$$ like Abraham and Laue, as well as in terms of Minkowski's alternative force definition $$F_{\mathrm{Mnk}}$$ using four-velocity Y:


 * $$\begin{matrix}\begin{align}F & =\rho\left\{ \frac{\iota}{c}(p\mathbf{E})+\mathbf{E}+\frac{1}{c}\mathrm{V}p\mathbf{M}\right\} \\

& =\frac{\iota}{c}(\mathbf{Pp})+\mathbf{P}\\ & =-\frac{1}{c}\mathbf{R}[D]\mathbf{L}\\ & =-\mathrm{lor}\mathfrak{S}\\ F_{\mathrm{Mnk}} & =\frac{1}{2}\left[F+\frac{1}{c^{2}}YF_{c}Y\right] \end{align} \\ \left[\mathbf{P}=\rho\left\{ \mathbf{E}+\frac{1}{c}\mathrm{V}\mathbf{p}\mathbf{M}\right\} ,\ D=\frac{\partial}{\partial l}-\nabla,\ {\rm lor}=\left|\frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z},\ \frac{\partial}{i\partial t}\right|\right] \end{matrix}$$

equivalent to (d1,e1,f1).

Lewis and Wilson (1912)
Gilbert Newton Lewis and Edwin Bidwell Wilson explicitly defined a four-vector called “extended momentum” (i.e. four-momentum) $$m_{0}\mathbf{w}$$, deriving the “extended force” (i.e. four force) using $$\mathbf{c}$$ as four-acceleration:


 * $$m_{0}\mathbf{c}=\frac{dm_{0}\mathbf{w}}{ds}=\frac{dmv}{ds}\mathbf{k}_{1}+\frac{dm}{ds}\mathbf{k}_{4}=\frac{1}{\sqrt{1-v^{2}}}\left(\frac{dmv}{dt}\mathbf{k}_{1}+\frac{dm}{dt}\mathbf{k}_{4}\right)$$

equivalent to (a, c).

Kottler (1912)
While formulating electrodynamics in a generally covariant way, Friedrich Kottler expressed the “Minkowski force” $$F_{\alpha}$$ in terms of the electromagnetic field-tensor $$F_{\alpha\beta}$$, four-current $$\mathbf{P}^{(\beta)}$$, stress-energy tensor $$S_{\alpha\beta}$$:


 * $$\begin{matrix}F_{\alpha}(y)=\sum_{\beta}\frac{F_{\alpha\beta}(y)\mathbf{P}^{(\beta)}(y)}{\sqrt{1-\mathfrak{w}^{2}/c^{2}}}\\

\left[\underset{\beta}{\sum}F_{\alpha\beta}(y)\mathbf{P}^{(\beta)}(y)=\underset{\beta}{\sum}F_{\alpha\beta}(y)\underset{\gamma}{\sum}\frac{\partial}{\partial y^{(\gamma)}}F_{\beta\gamma}(y)=\underset{\beta}{\sum}\frac{\partial}{\partial y^{(\beta)}}S_{\alpha\beta}\right] \end{matrix}$$

equivalent to (e, f).

In 1914 Kottler derived the equations of motion from the “Nordström tensor” (i.e. dust solution of the stress energy tensor) in terms of rest mass density $$\nu_{0}$$, which he equated to the action of a constant external electromagnetic field in terms of electromagnetic tensor $$F^{(hk)}$$ and charge density $$\varrho$$:


 * $$\sum_{k=1}^{4}\frac{\partial}{\partial x^{(k)}}\left(\nu\frac{dx^{(h)}}{dt}\frac{dx^{(k)}}{dt}\right)=\nu\frac{d^{2}x^{(h)}}{dt^{2}}=K^{(h)}=\frac{\varrho}{c}\sum_{k=1}^{4}F^{(hk)}x^{(k)}$$

equivalent to (f).

Einstein (1912-14)
In an unpublished manuscript on special relativity, written between 1912-1914, Albert Einstein wrote the four-force density $$\left(K_{\mu}\right)$$ in terms of electromagnetic tensor $$\left(\mathfrak{F}_{\mu\nu}\right)$$, four-current $$\left(\mathfrak{J}_{\nu}\right)$$, stress-energy tensor $$\left(T_{\mu\nu}\right)$$:


 * $$\begin{matrix}\left(\mathfrak{F}_{\mu\nu}\right)\left(\mathfrak{J}_{\nu}\right)=\left(K_{\mu}\right)\\

\hline \begin{align}K_{1} & =\rho\left(\mathfrak{e}_{x}+\frac{\mathfrak{q}_{y}}{c}\mathfrak{h}_{z}-\frac{\mathfrak{q}_{z}}{c}\mathfrak{h}_{y}\right)\\ K_{2} & =\rho\left(\mathfrak{e}_{y}+\frac{\mathfrak{q}_{z}}{c}\mathfrak{h}_{x}-\frac{\mathfrak{q}_{x}}{c}\mathfrak{h}_{z}\right)\\ K_{3} & =\rho\left(\mathfrak{e}_{z}+\frac{\mathfrak{q}_{x}}{c}\mathfrak{h}_{y}-\frac{\mathfrak{q}_{y}}{c}\mathfrak{h}_{x}\right)\\ K_{4} & =\frac{i}{c}\rho\left(\mathfrak{q}_{x}\mathfrak{e}_{x}+\mathfrak{q}_{y}\mathfrak{e}_{y}+\mathfrak{q}_{z}\mathfrak{e}_{z}\right) \end{align} \\ \hline \left(K_{\mu}\right)=-\left(\frac{\partial}{\partial x_{\nu}}\right)\left(T_{\mu\nu}\right)\\ \hline \begin{align}K_{1} & =-\frac{\partial p_{xx}}{\partial x_{1}}-\frac{\partial p_{xy}}{\partial x_{2}}-\frac{\partial p_{xz}}{\partial x_{3}}-\frac{\partial\frac{i}{c}\mathbf{s}_{x}}{\partial x_{4}}\\ & \dots\\ & \dots\\ K_{4} & =-\frac{\partial\frac{i}{c}\mathbf{s}_{x}}{\partial x_{1}}-\frac{\partial\frac{i}{c}\mathbf{s}_{y}}{\partial x_{2}}-\frac{\partial\frac{i}{c}\mathbf{s}_{z}}{\partial x_{3}}-\frac{\partial(-w)}{\partial x_{4}} \end{align} \end{matrix}$$

equivalent to (d1,e1,f1).