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).

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