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

History of 4-Vectors :
History of Four-velocity
History of Four-acceleration
History of Four-momentum
History of Four-current
History of Four-potential
History of Four-force (electromagnetism)
History of Four-force (mechanics) :
History of Topics in Special Relativity :
* Multiple chapter topics
* History of Lorentz transformations
* History of relativistic four vectors
* History of relativistic tensors
* Single page topics
* History of twin paradox
* History of time dilation
* History of three-acceleration transformation
* Müller's history of relativity

## Overview

The electromagnetic w:four-force or covariant w:Lorentz force $K^{\mu }$ can be expressed as
    (a) the rate of change in the w:four-momentum $P^{\mu }=mU^{\mu }$ of a particle with respect to the particle's w: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)):
    ${\begin{matrix}{\begin{matrix}K^{\mu }&={\frac {\mathrm {d} P^{\mu }}{\mathrm {d} \tau }}&=\gamma \left({\frac {1}{c}}\mathbf {f} \cdot \mathbf {v} ,\ \mathbf {f} \right)&=mA^{\mu }&=\gamma q\left({\frac {1}{c}}\mathbf {v\cdot E} ,\ \mathbf {E} +\mathbf {v} \times \mathbf {B} \right)&=qF_{\alpha \beta }U^{\beta }&=\int D^{\mu }\,dV_{0}\\&(a)&(b)&(c)&(d)&(e)&(f)\\D^{\mu }&={\frac {\mathrm {d} M^{\mu }}{\mathrm {d} \tau }}&=\left({\frac {1}{c}}\mathbf {d\cdot v} ,\ \mathbf {d} \right)&=\mu _{0}A^{\mu }&=\rho \left({\frac {1}{c}}\mathbf {v\cdot E} ,\ \mathbf {E} +\mathbf {v} \times \mathbf {B} \right)&\underbrace {=F_{\alpha \beta }J^{\beta }=\rho _{0}F_{\alpha \beta }U^{\beta }} &=-\partial _{\alpha }T^{\alpha \beta }\\&(a1)&(b1)&(c1)&(d1)&(e1)&(f1)\end{matrix}}\\\left(\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ \mathbf {f} =\int \mathbf {d} \,dV\right)\end{matrix}}$

## Historical notation
### Poincaré (1905/6)

w: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)

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

w: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 $\varrho _{0}^{\ast }$
    ${\begin{matrix}ci\mu {\frac {\partial ^{2}x_{\alpha }}{\partial \xi _{4}^{2}}}={\frac {\varrho _{0}^{\ast }}{c}}\sum _{\beta =1}^{4}f_{\alpha \beta }{\frac {\partial x_{\beta }}{\partial \xi _{4}}}\\\hline {\begin{aligned}\mu {\frac {\partial ^{2}x}{\partial \tau ^{2}}}&=\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{x}{\frac {\partial t}{\partial \tau }}+{\frac {1}{c}}\left({\frac {\partial y}{\partial \tau }}{\mathfrak {M}}_{z}-{\frac {\partial z}{\partial \tau }}{\mathfrak {M}}_{y}\right)\right\}\\\mu {\frac {\partial ^{2}y}{\partial \tau ^{2}}}&=\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{y}{\frac {\partial t}{\partial \tau }}+{\frac {1}{c}}\left({\frac {\partial z}{\partial \tau }}{\mathfrak {M}}_{x}-{\frac {\partial x}{\partial \tau }}{\mathfrak {M}}_{z}\right)\right\}\\\mu {\frac {\partial ^{2}z}{\partial \tau ^{2}}}&=\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{z}{\frac {\partial t}{\partial \tau }}+{\frac {1}{c}}\left({\frac {\partial x}{\partial \tau }}{\mathfrak {M}}_{y}-{\frac {\partial y}{\partial \tau }}{\mathfrak {M}}_{z}\right)\right\}\\\mu {\frac {\partial ^{2}t}{\partial \tau ^{2}}}&={\frac {1}{c^{2}}}\varrho _{0}^{\ast }\left\{{\mathfrak {E}}_{x}{\frac {\partial x}{\partial \tau }}+{\mathfrak {E}}_{y}{\frac {\partial y}{\partial \tau }}+{\mathfrak {E}}_{z}{\frac {\partial z}{\partial \tau }}\right\}\end{aligned}}\end{matrix}}$
equivalent to (c1, d1, e1).

### Frank (1909)

w:Philipp Frank (1909) discussed “electromagnetic mechanics” by defining four-force (X,Y,Z,T) as follows:
    ${\begin{matrix}m{\frac {d}{d\sigma }}{\frac {dt}{d\sigma }}=T,\ m{\frac {d^{2}x}{d\sigma ^{2}}}=X,\ {\frac {d^{2}y}{d\sigma ^{2}}}=Y,\ {\frac {d^{2}z}{d\sigma ^{2}}}=Z\\\hline {\begin{aligned}T&=m{\frac {d}{d\sigma }}{\frac {dt}{d\sigma }}=m{\frac {1}{\left(1-w^{2}\right)^{2}}}\left(w_{x}{\frac {dw_{x}}{dt}}+w_{y}{\frac {dw_{y}}{dt}}+w_{z}{\frac {dw_{z}}{dt}}\right)\\{\frac {d^{2}x}{d\sigma ^{2}}}&={\frac {1}{1-w^{2}}}{\frac {dw_{x}}{dt}}+{\frac {w_{x}}{\left(1-w^{2}\right)^{2}}}\left(w_{x}{\frac {dw_{x}}{dt}}+w_{y}{\frac {dw_{y}}{dt}}+w_{z}{\frac {dw_{z}}{dt}}\right)\\&{\rm {etc}}.\end{aligned}}\end{matrix}}$
corresponding to (b, c).

### Abraham (1909/10)

While Minkowski used the four-force density as $K+(K{\overline {w}})w$ with $K={\text{lor }}S$, w: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
    ${\begin{matrix}{\begin{aligned}{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}}&{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial X_{t}}{\partial l}}\\{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}}\quad \Rightarrow &{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial Y_{t}}{\partial l}}\\{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}&{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial Z_{t}}{\partial l}}\\{\mathfrak {wK}}+Q&=-\mathrm {div} \,{\mathfrak {S}}-{\frac {\partial \psi }{\partial t}}&{\mathfrak {K}}_{t}&=-{\frac {\partial T_{x}}{\partial x}}-{\frac {\partial T_{y}}{\partial y}}-{\frac {\partial T_{z}}{\partial z}}-{\frac {\partial T_{t}}{\partial l}}\end{aligned}}\\\hline \left(\mathrm {div} \,{\mathfrak {S}}={\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}},\ {\mathfrak {g}}={\frac {\mathfrak {S}}{c^{2}}}\right)\end{matrix}}$
or alternatively by introducing “relative stresses” and the “relative energy flux”:
    ${\begin{matrix}{\begin{aligned}{\mathfrak {K}}_{x}&={\frac {\partial X_{x}^{\prime }}{\partial x}}+{\frac {\partial X_{y}^{\prime }}{\partial y}}+{\frac {\partial X_{z}^{\prime }}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}}\\{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}^{\prime }}{\partial x}}+{\frac {\partial Y_{y}^{\prime }}{\partial y}}+{\frac {\partial Y_{z}^{\prime }}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}}\\{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}^{\prime }}{\partial x}}+{\frac {\partial Z_{y}^{\prime }}{\partial y}}+{\frac {\partial Z_{z}^{\prime }}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}\\{\mathfrak {wK}}+Q&=-\mathrm {div} \,\left\{{\mathfrak {S}}-{\mathfrak {w}}\psi \right\}-{\frac {\partial \psi }{\partial t}}\end{aligned}}\\\hline \left({\begin{aligned}X_{x}^{\prime }&=X_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x}&X_{y}^{\prime }&=X_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x}&X_{z}^{\prime }&=X_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x}\\Y_{x}^{\prime }&=Y_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x}&Y_{y}^{\prime }&=Y_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x}&Y_{z}^{\prime }&=Y_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x}\\Z_{x}^{\prime }&=Z_{x}+{\mathfrak {w}}_{x}{\mathfrak {g}}_{x}&Z_{y}^{\prime }&=Z_{y}+{\mathfrak {w}}_{y}{\mathfrak {g}}_{x}&Z_{z}^{\prime }&=Z_{z}+{\mathfrak {w}}_{z}{\mathfrak {g}}_{x}\end{aligned}}\right)\end{matrix}}$
all equivalent to (f1).
In a subsequent paper (1909) he formulated these relations as follows
    ${\begin{matrix}{\begin{aligned}{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}}&{\mathfrak {K}}_{x}&={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}+{\frac {\partial X_{u}}{\partial u}}\\{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}}\quad \Rightarrow &{\mathfrak {K}}_{y}&={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}+{\frac {\partial Y_{u}}{\partial u}}\\{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}&{\mathfrak {K}}_{z}&={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}+{\frac {\partial Z_{u}}{\partial u}}\\Q+({\mathfrak {wK}})&=-{\frac {\partial {\mathfrak {S}}_{x}}{\partial x}}-{\frac {\partial {\mathfrak {S}}_{y}}{\partial y}}-{\frac {\partial {\mathfrak {S}}_{z}}{\partial z}}-{\frac {\partial \psi }{\partial t}}&{\mathfrak {K}}_{u}&={\frac {\partial U_{x}}{\partial x}}+{\frac {\partial U_{y}}{\partial y}}+{\frac {\partial U_{z}}{\partial z}}+{\frac {\partial U_{u}}{\partial u}}\end{aligned}}\\\hline \left(u=ict\right)\end{matrix}}$
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:
    ${\begin{aligned}{\frac {d}{d\tau }}\left(\nu {\frac {dx}{d\tau }}\right)&={\mathfrak {K}}_{x}\\{\frac {d}{d\tau }}\left(\nu {\frac {dy}{d\tau }}\right)&={\mathfrak {K}}_{y}\\{\frac {d}{d\tau }}\left(\nu {\frac {dz}{d\tau }}\right)&={\mathfrak {K}}_{z}\\{\frac {d}{d\tau }}\left(\nu {\frac {du}{d\tau }}\right)&={\mathfrak {K}}_{u}\ (u=ict)\end{aligned}}$
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 w:spherical wave transformations), was given by w: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)

w: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, w: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, w: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
    ${\begin{matrix}{\begin{aligned}F&=[P{\mathfrak {M}}]\\&=-\varDelta iv\,T\end{aligned}}\\\hline[P{\mathfrak {M}}]_{x}={\mathfrak {F}}_{x},\ [P{\mathfrak {M}}]_{y}={\mathfrak {F}}_{y},\ [P{\mathfrak {M}}]_{z}={\mathfrak {F}}_{z},\ [P{\mathfrak {M}}]_{l}={\frac {i\varrho }{c}}({\mathfrak {qE}})={\frac {i}{c}}({\mathfrak {qF}})\\\left[{\mathfrak {F}}=\varrho ({\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {qH}}]),\ \varDelta iv={\text{divergence six-vector}},\ l=ict\right]\end{matrix}}$
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 $\delta V:{\sqrt {c^{2}-q^{2}}}$:
    $K\Rightarrow {\begin{aligned}K_{x}&={\frac {{\mathfrak {K}}_{x}}{\sqrt {c^{2}-q^{2}}}},&K_{y}&={\frac {{\mathfrak {K}}_{y}}{\sqrt {c^{2}-q^{2}}}},\\K_{z}&={\frac {{\mathfrak {K}}_{z}}{\sqrt {c^{2}-q^{2}}}},&K_{l}&={\frac {i}{c}}{\frac {({\mathfrak {qK}})}{\sqrt {c^{2}-q^{2}}}},\end{aligned}}\Rightarrow {\frac {d(mY)}{d\tau }}\Rightarrow m{\frac {dY}{d\tau }}$

### Silberstein (1911-14)

w:Ludwik Silberstein in 1911 and published 1912, devised an alternative 4D calculus based on w: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. w:Weber vector) $\mathbf {F}$}
    ${\begin{matrix}\mathrm {P} =\mathrm {C} \mathbf {F} =\mathrm {D} [\mathbf {F} \cdot \mathbf {F} ]\\\hline {\begin{aligned}\mathrm {P} &=\rho \left\{\iota \mathbf {M} +\mathbf {E} +{\frac {1}{c}}p\mathbf {M} -{\frac {\iota }{c}}p\mathbf {E} \right\}\\&=\mathrm {P} _{e}+\iota \mathrm {P} _{m}\\\mathrm {P} _{e}&=\rho \left\{{\frac {\iota }{c}}(p\mathbf {E} )+\mathbf {E} +{\frac {1}{c}}\mathrm {V} p\mathbf {M} \right\}\\\mathrm {P} _{m}&=\rho \left\{{\frac {\iota }{c}}(p\mathbf {M} )+\mathbf {M} +{\frac {1}{c}}\mathrm {V} p\mathbf {E} \right\}\end{aligned}}\\\left[\mathrm {D} ={\frac {\partial }{\partial l}}-\nabla \right]\end{matrix}}$
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:
    ${\begin{matrix}{\begin{aligned}F&=\rho \left\{{\frac {\iota }{c}}(p\mathbf {E} )+\mathbf {E} +{\frac {1}{c}}\mathrm {V} p\mathbf {M} \right\}\\&={\frac {\iota }{c}}(\mathbf {Pp} )+\mathbf {P} \\&=-{\frac {1}{c}}\mathbf {R} [D]\mathbf {L} \\&=-\mathrm {lor} {\mathfrak {S}}\\F_{\mathrm {Mnk} }&={\frac {1}{2}}\left[F+{\frac {1}{c^{2}}}YF_{c}Y\right]\end{aligned}}\\\left[\mathbf {P} =\rho \left\{\mathbf {E} +{\frac {1}{c}}\mathrm {V} \mathbf {p} \mathbf {M} \right\},\ D={\frac {\partial }{\partial l}}-\nabla ,\ {\rm {lor}}=\left|{\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}},\ {\frac {\partial }{i\partial t}}\right|\right]\end{matrix}}$
equivalent to (d1,e1,f1).

### Lewis and Wilson (1912)

w:Gilbert Newton Lewis and w: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, w: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, w: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 $\left(T_{\mu \nu }\right)$:
    ${\begin{matrix}\left({\mathfrak {F}}_{\mu \nu }\right)\left({\mathfrak {J}}_{\nu }\right)=\left(K_{\mu }\right)\\\hline {\begin{aligned}K_{1}&=\rho \left({\mathfrak {e}}_{x}+{\frac {{\mathfrak {q}}_{y}}{c}}{\mathfrak {h}}_{z}-{\frac {{\mathfrak {q}}_{z}}{c}}{\mathfrak {h}}_{y}\right)\\K_{2}&=\rho \left({\mathfrak {e}}_{y}+{\frac {{\mathfrak {q}}_{z}}{c}}{\mathfrak {h}}_{x}-{\frac {{\mathfrak {q}}_{x}}{c}}{\mathfrak {h}}_{z}\right)\\K_{3}&=\rho \left({\mathfrak {e}}_{z}+{\frac {{\mathfrak {q}}_{x}}{c}}{\mathfrak {h}}_{y}-{\frac {{\mathfrak {q}}_{y}}{c}}{\mathfrak {h}}_{x}\right)\\K_{4}&={\frac {i}{c}}\rho \left({\mathfrak {q}}_{x}{\mathfrak {e}}_{x}+{\mathfrak {q}}_{y}{\mathfrak {e}}_{y}+{\mathfrak {q}}_{z}{\mathfrak {e}}_{z}\right)\end{aligned}}\\\hline \left(K_{\mu }\right)=-\left({\frac {\partial }{\partial x_{\nu }}}\right)\left(T_{\mu \nu }\right)\\\hline {\begin{aligned}K_{1}&=-{\frac {\partial p_{xx}}{\partial x_{1}}}-{\frac {\partial p_{xy}}{\partial x_{2}}}-{\frac {\partial p_{xz}}{\partial x_{3}}}-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{x}}{\partial x_{4}}}\\&\dots \\&\dots \\K_{4}&=-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{x}}{\partial x_{1}}}-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{y}}{\partial x_{2}}}-{\frac {\partial {\frac {i}{c}}\mathbf {s} _{z}}{\partial x_{3}}}-{\frac {\partial (-w)}{\partial x_{4}}}\end{aligned}}\end{matrix}}$
equivalent to (d1,e1,f1).