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

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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 $P^{\mu }=m{U}^{\mu }$ of a particle with respect to the particle's w:proper time $\tau$,

(b) function of three-force $𝐟$

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

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

(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 $𝐝$

(c1) assuming constant mass the product of rest mass density ${\mu }_0$ and four-acceleration $A^{\mu }$

(d1) using the Lorentz force density $𝐝=\rho \{𝐄+𝐯\times 𝐁\}=\rho 𝐄+𝐉\times 𝐁$ 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)):

---

$\begin{array}{c}\begin{array}{ccccccc}{K}^{\mu }& =\frac{\mathrm{d}{P}^{\mu }}{\mathrm{d}\tau }& =\gamma (\frac{1}{c}𝐟\cdot 𝐯,𝐟)& =m{A}^{\mu }& =\gamma q(\frac{1}{c}𝐯\mathbf{\cdot }𝐄,𝐄+𝐯\times 𝐁)& =q{F}_{\alpha \beta }{U}^{\beta }& =\int D^{\mu }d{V}_0\\ & (a)& (b)& (c)& (d)& (e)& (f)\\ D^{\mu }& =\frac{\mathrm{d}{M}^{\mu }}{\mathrm{d}\tau }& =(\frac{1}{c}𝐝\mathbf{\cdot }𝐯,𝐝)& ={\mu }_0{A}^{\mu }& =\rho (\frac{1}{c}𝐯\mathbf{\cdot }𝐄,𝐄+𝐯\times 𝐁)& \underbrace{=F_{\alpha \beta }{J}^{\beta }={\rho }_0{F}_{\alpha \beta }{U}^{\beta }}& =-{\partial }_{\alpha }{T}^{\alpha \beta }\\ & (a1)& (b1)& (c1)& (d1)& (e1)& (f1)\end{array}\\ (\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},𝐟=\int 𝐝dV)\end{array}$

Historical notation

Poincaré (1905/6)

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

$\begin{array}{c}X=\rho f+\rho (\eta \gamma -\zeta \beta )\dots \\ X^{\prime }={\rho }^{\prime }{f}^{\prime }+{\rho }^{\prime }({\eta }^{\prime }{\gamma }^{\prime }-{\zeta }^{\prime }{\beta }^{\prime })\dots \\ \{\begin{array}{cc}{X}^{\prime }=k(X+ϵT),& T^{\prime }=k(T+ϵX),\\ Y^{\prime }=Y,& Z^{\prime }=Z;\end{array}\end{array}$ with $T=\mathrm{\Sigma }X\xi$ and $k=\frac{1}{\sqrt{1-{ϵ}^2}}$.

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

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

Additionally, he explicitly obtained the four-force per unit charge by setting $\frac{X_1}{X}=\frac{Y_1}{Y}=\frac{Z_1}{Z}=\frac{T_1}{T}=\frac{1}{\rho }$:

$(k_0{X}_1,k_0{Y}_1,k_0{Z}_1,k_0{T}_1)$ with $T_1=\mathrm{\Sigma }{X}_1\xi$ and $k_0=\frac{1}{\sqrt{1-{ϵ}^2}}$

equivalent to (b, d) and obtained its Lorentz transformation by multiplying the transformation of (X,Y,Z,T) by

$\frac{\rho }{{\rho }^{\prime }}=\frac{1}{k(1+\xi ϵ)}=\frac{\delta t}{\delta t^{\prime }}$.

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 $𝔳$ as velocity:

$\begin{array}{c}X,Y,Z,iA\\ X_j={\varrho }_1{\psi }_{j1}+{\varrho }_2{\psi }_{j2}+{\varrho }_3{\psi }_{j3}+{\varrho }_4{\psi }_{j4}\\ A=X{𝔳}_x+Y{𝔳}_y+Z{𝔳}_z\\ \left\{\begin{array}{cc}{\psi }_{23},{\psi }_{31},{\psi }_{12}& \Rightarrow {ℌ}_x,{ℌ}_y,{ℌ}_z\\ {\psi }_{14},{\psi }_{24},{\psi }_{34}& \Rightarrow -i{𝔈}_x,-i{𝔈}_y,-i{𝔈}_z\\ & {\psi }_{jj}=0\end{array}\right\}\end{array}$

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

In another lecture from December 1907, he used the symbols $𝔢,𝔪$ (i.e. $𝐄,𝐁$) and $𝔈,𝔐$ (i.e. $𝐃,𝐇$) 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 $\mathrm{\Phi }$ and magnetric rest force $\mathrm{\Psi }$ as the product with four-velocity w, which in turn can be used to express F and f and four-conductivity:

$\begin{array}{c}\begin{array}{c}\mathrm{\Phi }=-wF\\ ({\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3)=\frac{𝔈+[𝔴𝔐]}{\sqrt{1-{𝔴}^2}},{\mathrm{\Phi }}_4=\frac{i(𝔴𝔈)}{\sqrt{1-{𝔴}^2}}& \mathrm{\Psi }=iw{f}^{*},\\ \begin{array}{ccccccccc}{\mathrm{\Phi }}_1& =& & & w_2{F}_{12}& +& w_3{F}_{13}& +& w_4{F}_{14},\\ {\mathrm{\Phi }}_2& =& w_1{F}_{21}& & & +& w_3{F}_{23}& +& w_4{F}_{24},\\ {\mathrm{\Phi }}_3& =& w_1{F}_{31}& +& w_2{F}_{32}& & & +& w_4{F}_{34},\\ {\mathrm{\Phi }}_4& =& w_1{F}_{41}& +& w_2{F}_{42}& +& w_3{F}_{43}& & .\end{array}|& \begin{array}{ccccccccc}{\mathrm{\Psi }}_1& =& -i(& & w_2{f}_{34}& +& w_3{f}_{42}& +& w_4{f}_{23}),\\ {\mathrm{\Psi }}_2& =& -i(w_1{f}_{43}& & & +& w_3{f}_{14}& +& w_4{f}_{31}),\\ {\mathrm{\Psi }}_3& =& -i(w_1{f}_{24}& +& w_2{f}_{41}& & & +& w_4{f}_{12}),\\ {\mathrm{\Psi }}_4& =& -i(w_1{f}_{32}& +& w_2{f}_{13}& +& w_3{f}_{21}& & ).\end{array}\end{array}\\ wF=-\mathrm{\Phi },w{F}^{*}=-i\mu \mathrm{\Psi },wf=-\epsilon \mathrm{\Phi },w{f}^{*}=-i\mathrm{\Psi }\\ F=[w,\mathrm{\Phi }]+i\mu [w,\mathrm{\Psi }{]}^{*},\\ f=\epsilon [w,\mathrm{\Phi }]+i[w,\mathrm{\Psi }{]}^{*},\\ s+(w\bar{s})w=-\sigma wF.\end{array}$

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

$\begin{array}{c}K+(w\bar{K})w\\ K=\text{lor }S=-sF+N\\ \begin{array}{cc}{K}_1& =\frac{\partial X_x}{\partial x}+\frac{\partial X_y}{\partial y}+\frac{\partial X_z}{\partial z}-\frac{\partial X_t}{\partial t}=\varrho {𝔈}_x+{𝔰}_y{𝔐}_z-{𝔰}_z{𝔐}_y-\frac{1}{2}\mathrm{\Phi }\bar{\mathrm{\Phi }}\frac{\partial \epsilon }{\partial x}-\frac{1}{2}\mathrm{\Psi }\bar{\mathrm{\Psi }}\frac{\partial \mu }{\partial x}+\frac{\epsilon \mu -1}{\sqrt{1-{𝔴}^2}}\left(𝔚\frac{\partial 𝔴}{\partial x}\right)\\ K_2& =\frac{\partial Y_x}{\partial x}+\frac{\partial Y_y}{\partial y}+\frac{\partial Y_z}{\partial z}-\frac{\partial Y_t}{\partial t}=\varrho {𝔈}_y+{𝔰}_z{𝔐}_x-{𝔰}_x{𝔐}_z-\frac{1}{2}\mathrm{\Phi }\bar{\mathrm{\Phi }}\frac{\partial \epsilon }{\partial y}-\frac{1}{2}\mathrm{\Psi }\bar{\mathrm{\Psi }}\frac{\partial \mu }{\partial y}+\frac{\epsilon \mu -1}{\sqrt{1-{𝔴}^2}}\left(𝔚\frac{\partial 𝔴}{\partial y}\right)\\ K_3& =\frac{\partial Z_x}{\partial x}+\frac{\partial Z_y}{\partial y}+\frac{\partial Z_z}{\partial z}-\frac{\partial Z_t}{\partial t}=\varrho {𝔈}_z+{𝔰}_x{𝔐}_y-{𝔰}_y{𝔐}_x-\frac{1}{2}\mathrm{\Phi }\bar{\mathrm{\Phi }}\frac{\partial \epsilon }{\partial z}-\frac{1}{2}\mathrm{\Psi }\bar{\mathrm{\Psi }}\frac{\partial \mu }{\partial z}+\frac{\epsilon \mu -1}{\sqrt{1-{𝔴}^2}}\left(𝔚\frac{\partial 𝔴}{\partial z}\right)\\ \frac{1}{i}{K}_4& =-\frac{\partial T_x}{\partial x}-\frac{\partial T_y}{\partial y}-\frac{\partial T_z}{\partial z}-\frac{\partial T_t}{\partial t}={𝔰}_x{𝔈}_x+{𝔰}_y{𝔈}_y+{𝔰}_z{𝔈}_z+\frac{1}{2}\mathrm{\Phi }\bar{\mathrm{\Phi }}\frac{\partial \epsilon }{\partial t}+\frac{1}{2}\mathrm{\Psi }\bar{\mathrm{\Psi }}\frac{\partial \mu }{\partial t}-\frac{\epsilon \mu -1}{\sqrt{1-{𝔴}^2}}\left(𝔚\frac{\partial 𝔴}{\partial t}\right)\end{array}\\ [\text{lor }=|\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4}|]\end{array}$

equivalent to (f).

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

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

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

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 ${\varrho }_0^{\ast }$

$\begin{array}{c}ci\mu \frac{{\partial }^2{x}_{\alpha }}{\partial {\xi }_4^2}=\frac{{\varrho }_0^{\ast }}{c}\sum _{\beta =1}^4{f}_{\alpha \beta }\frac{\partial x_{\beta }}{\partial {\xi }_4}\\ \begin{array}{cc}\mu \frac{{\partial }^{2}x}{\partial {\tau }^2}& ={\varrho }_0^{\ast }\{{𝔈}_x\frac{\partial t}{\partial \tau }+\frac{1}{c}(\frac{\partial y}{\partial \tau }{𝔐}_z-\frac{\partial z}{\partial \tau }{𝔐}_y)\}\\ \mu \frac{{\partial }^{2}y}{\partial {\tau }^2}& ={\varrho }_0^{\ast }\{{𝔈}_y\frac{\partial t}{\partial \tau }+\frac{1}{c}(\frac{\partial z}{\partial \tau }{𝔐}_x-\frac{\partial x}{\partial \tau }{𝔐}_z)\}\\ \mu \frac{{\partial }^{2}z}{\partial {\tau }^2}& ={\varrho }_0^{\ast }\{{𝔈}_z\frac{\partial t}{\partial \tau }+\frac{1}{c}(\frac{\partial x}{\partial \tau }{𝔐}_y-\frac{\partial y}{\partial \tau }{𝔐}_z)\}\\ \mu \frac{{\partial }^{2}t}{\partial {\tau }^2}& =\frac{1}{c^2}{\varrho }_0^{\ast }\{{𝔈}_x\frac{\partial x}{\partial \tau }+{𝔈}_y\frac{\partial y}{\partial \tau }+{𝔈}_z\frac{\partial z}{\partial \tau }\}\end{array}\end{array}$

equivalent to (c1, d1, e1).

Frank (1909)

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

$\begin{array}{c}m\frac{d}{d\sigma }\frac{dt}{d\sigma }=T,m\frac{d^{2}x}{d{\sigma }^2}=X,\frac{d^{2}y}{d{\sigma }^2}=Y,\frac{d^{2}z}{d{\sigma }^2}=Z\\ \begin{array}{cc}T& =m\frac{d}{d\sigma }\frac{dt}{d\sigma }=m\frac{1}{{(1-w^2)}^2}(w_x\frac{d{w}_x}{dt}+w_y\frac{d{w}_y}{dt}+w_z\frac{d{w}_z}{dt})\\ \frac{d^{2}x}{d{\sigma }^2}& =\frac{1}{1-w^2}\frac{d{w}_x}{dt}+\frac{w_x}{{(1-w^2)}^2}(w_x\frac{d{w}_x}{dt}+w_y\frac{d{w}_y}{dt}+w_z\frac{d{w}_z}{dt})\\ & \mathrm{e}\mathrm{t}\mathrm{c}.\end{array}\end{array}$

corresponding to (b, c).

Abraham (1909/10)

While Minkowski used the four-force density as $K+(K\bar{w})w$ 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 $𝔤$, energy density $\psi$, Poynting vector $𝔖$, Joule heat Q

$\begin{array}{c}\begin{array}{cccc}{𝔎}_x& =\frac{\partial X_x}{\partial x}+\frac{\partial X_y}{\partial y}+\frac{\partial X_z}{\partial z}-\frac{\partial {𝔤}_x}{\partial t}& {𝔎}_x& =\frac{\partial X_x}{\partial x}+\frac{\partial X_y}{\partial y}+\frac{\partial X_z}{\partial z}-\frac{\partial X_t}{\partial l}\\ {𝔎}_y& =\frac{\partial Y_x}{\partial x}+\frac{\partial Y_y}{\partial y}+\frac{\partial Y_z}{\partial z}-\frac{\partial {𝔤}_y}{\partial t}\Rightarrow & {𝔎}_y& =\frac{\partial Y_x}{\partial x}+\frac{\partial Y_y}{\partial y}+\frac{\partial Y_z}{\partial z}-\frac{\partial Y_t}{\partial l}\\ {𝔎}_z& =\frac{\partial Z_x}{\partial x}+\frac{\partial Z_y}{\partial y}+\frac{\partial Z_z}{\partial z}-\frac{\partial {𝔤}_z}{\partial t}& {𝔎}_z& =\frac{\partial Z_x}{\partial x}+\frac{\partial Z_y}{\partial y}+\frac{\partial Z_z}{\partial z}-\frac{\partial Z_t}{\partial l}\\ 𝔴𝔎+Q& =-\mathrm{d}\mathrm{i}\mathrm{v}𝔖-\frac{\partial \psi }{\partial t}& {𝔎}_t& =-\frac{\partial T_x}{\partial x}-\frac{\partial T_y}{\partial y}-\frac{\partial T_z}{\partial z}-\frac{\partial T_t}{\partial l}\end{array}\\ (\mathrm{d}\mathrm{i}\mathrm{v}𝔖=\frac{\partial {𝔖}_x}{\partial x}+\frac{\partial {𝔖}_y}{\partial y}+\frac{\partial {𝔖}_z}{\partial z},𝔤=\frac{𝔖}{c^2})\end{array}$

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

$\begin{array}{c}\begin{array}{cc}{𝔎}_x& =\frac{\partial X_x^{\prime }}{\partial x}+\frac{\partial X_y^{\prime }}{\partial y}+\frac{\partial X_z^{\prime }}{\partial z}-\frac{\partial {𝔤}_x}{\partial t}\\ {𝔎}_y& =\frac{\partial Y_x^{\prime }}{\partial x}+\frac{\partial Y_y^{\prime }}{\partial y}+\frac{\partial Y_z^{\prime }}{\partial z}-\frac{\partial {𝔤}_y}{\partial t}\\ {𝔎}_z& =\frac{\partial Z_x^{\prime }}{\partial x}+\frac{\partial Z_y^{\prime }}{\partial y}+\frac{\partial Z_z^{\prime }}{\partial z}-\frac{\partial {𝔤}_z}{\partial t}\\ 𝔴𝔎+Q& =-\mathrm{d}\mathrm{i}\mathrm{v}\{𝔖-𝔴\psi \}-\frac{\partial \psi }{\partial t}\end{array}\\ \left(\begin{array}{cccccc}{X}_x^{\prime }& =X_x+{𝔴}_x{𝔤}_x& X_y^{\prime }& =X_y+{𝔴}_y{𝔤}_x& X_z^{\prime }& =X_z+{𝔴}_z{𝔤}_x\\ Y_x^{\prime }& =Y_x+{𝔴}_x{𝔤}_x& Y_y^{\prime }& =Y_y+{𝔴}_y{𝔤}_x& Y_z^{\prime }& =Y_z+{𝔴}_z{𝔤}_x\\ Z_x^{\prime }& =Z_x+{𝔴}_x{𝔤}_x& Z_y^{\prime }& =Z_y+{𝔴}_y{𝔤}_x& Z_z^{\prime }& =Z_z+{𝔴}_z{𝔤}_x\end{array}\right)\end{array}$

all equivalent to (f1).

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

$\begin{array}{c}\begin{array}{cccc}{𝔎}_x& =\frac{\partial X_x}{\partial x}+\frac{\partial X_y}{\partial y}+\frac{\partial X_z}{\partial z}-\frac{\partial {𝔤}_x}{\partial t}& {𝔎}_x& =\frac{\partial X_x}{\partial x}+\frac{\partial X_y}{\partial y}+\frac{\partial X_z}{\partial z}+\frac{\partial X_u}{\partial u}\\ {𝔎}_y& =\frac{\partial Y_x}{\partial x}+\frac{\partial Y_y}{\partial y}+\frac{\partial Y_z}{\partial z}-\frac{\partial {𝔤}_y}{\partial t}\Rightarrow & {𝔎}_y& =\frac{\partial Y_x}{\partial x}+\frac{\partial Y_y}{\partial y}+\frac{\partial Y_z}{\partial z}+\frac{\partial Y_u}{\partial u}\\ {𝔎}_z& =\frac{\partial Z_x}{\partial x}+\frac{\partial Z_y}{\partial y}+\frac{\partial Z_z}{\partial z}-\frac{\partial {𝔤}_z}{\partial t}& {𝔎}_z& =\frac{\partial Z_x}{\partial x}+\frac{\partial Z_y}{\partial y}+\frac{\partial Z_z}{\partial z}+\frac{\partial Z_u}{\partial u}\\ Q+(𝔴𝔎)& =-\frac{\partial {𝔖}_x}{\partial x}-\frac{\partial {𝔖}_y}{\partial y}-\frac{\partial {𝔖}_z}{\partial z}-\frac{\partial \psi }{\partial t}& {𝔎}_u& =\frac{\partial U_x}{\partial x}+\frac{\partial U_y}{\partial y}+\frac{\partial U_z}{\partial z}+\frac{\partial U_u}{\partial u}\end{array}\\ (u=ict)\end{array}$

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 $𝔎$ over volume dv as the rate of momentum change without the need of complementary components

$\frac{d}{dt}\left\{\frac{m_{0}c𝔮}{\sqrt{1-{𝔮}^2}}\right\}=𝔎dv$.

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

$\begin{array}{cc}\frac{d}{d\tau }\left(\nu \frac{dx}{d\tau }\right)& ={𝔎}_x\\ \frac{d}{d\tau }\left(\nu \frac{dy}{d\tau }\right)& ={𝔎}_y\\ \frac{d}{d\tau }\left(\nu \frac{dz}{d\tau }\right)& ={𝔎}_z\\ \frac{d}{d\tau }\left(\nu \frac{du}{d\tau }\right)& ={𝔎}_u(u=ict)\end{array}$

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):

$\begin{array}{c}\frac{\rho }{{\lambda }^2}[(E_x-w_z{H}_y+w_y{H}_z)dydzdt+(E_y-w_x{H}_z+w_z{H}_x)dzdxdt\\ +(E_z-w_y{H}_z+w_x{H}_y)dxdydt-(w_x{E}_x+w_y{E}_y+w_z{E}_z)dxdydz],\\ \\ \frac{\rho }{{\lambda }^4}[(E_x+w_y{H}_z-w_z{H}_y)\delta x+(E_y+w_z{H}_x-w_x{H}_z)\delta y\\ +(E_z+w_z{H}_y-w_y{H}_x)\delta z-(w_x{E}_x+w_y{E}_y+w_z{E}_z)\delta t],\\ \\ \rho dxdydzdt[(E_x+w_y{H}_z-w_z{H}_y)\delta x+(E_y+w_z{H}_x-w_x{H}_z)\delta y\\ +(E_z+w_x{H}_y-w_y{H}_x)\delta z-(w_x{E}_x+w_y{E}_y+w_z{E}_z)\delta t].\end{array}$

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

$\begin{array}{c}(\frac{𝔎}{\sqrt{1-n{𝔳}^2}},\frac{n𝔳𝔎}{\sqrt{1-n{𝔳}^2}})\\ \frac{d\frac{m_0𝔳}{\sqrt{1-n{𝔳}^2}}}{dt}=𝔎\Rightarrow (\frac{𝔎}{\sqrt{1-n{𝔳}^2}},\frac{n𝔳𝔎}{\sqrt{1-n{𝔳}^2}}+\frac{d{m}_0}{dt})\\ \begin{array}{c}𝔎=e𝔈+[e𝔳ℌ]\\ {𝔎}^{\prime }=e^{\prime }{𝔈}^{\prime }+[e^{\prime }{{𝔳}^{\prime }ℌ}^{\prime }]\end{array}{m}_0\frac{d\frac{𝔳}{\sqrt{1-n{𝔳}^2}}}{dt}=𝔎\end{array}[n=\frac{1}{c^2}]$

equivalent to (b, d). He also derived the four-force density in terms of the density of three-force or Lorentz force ${𝔎}_1$ by integrating three-force $𝔎$ with respect to volume:

$\begin{array}{c}({𝔎}_1,n{𝔎}_1𝔳)\\ \begin{array}{c}{𝔎}_1=𝔈\varrho +[\varrho 𝔳ℌ]\\ 𝔎{\text{'}}_1={𝔈}^{\prime }\varrho +[{\varrho }^{\prime }{{𝔳}^{\prime }ℌ}^{\prime }]\end{array}[𝔎dv={𝔎}_1,n=\frac{1}{c^2}]\end{array}$

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 $𝔉$:

$\begin{array}{c}[Pf]\\ \begin{array}{cc}{𝔉}_x=\left(P{f}_x\right)& =\varrho (\frac{{𝔳}_x}{c}{f}_{xx}+\frac{{𝔳}_y}{c}{f}_{xy}+\frac{{𝔳}_z}{c}{f}_{xz}+i{f}_{xl})\\ & =\varrho (\frac{{𝔳}_y}{c}{ℌ}_z-\frac{{𝔳}_z}{c}{ℌ}_y+{𝔈}_x)\\ & \dots \\ {𝔉}_l=\left(P{f}_l\right)& =\varrho (\frac{{𝔳}_x}{c}{f}_{lx}+\frac{{𝔳}_y}{c}{f}_{ly}+\frac{{𝔳}_z}{c}{f}_{lz}+i{f}_{ll})\\ & =\frac{i\varrho }{c}({𝔳}_x{𝔈}_x+{𝔳}_y{𝔈}_y+{𝔳}_z{𝔈}_z)\end{array}\\ {𝔉}_j=\varrho (𝔈+\frac{1}{c}[𝔳ℌ]),{𝔉}_l=\frac{i\varrho }{c}(𝔈𝔳)\\ (j=x,y,z;l=ict)\end{array}$

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” $𝔐$ (now known as electromagnetic tensor) and the four-current related to Lorentz force density $𝔉$, and alternatively as the divergence of the electromagnetic stress-energy tensor T, which included some printing errors corrected in the 1913 edition

$\begin{array}{c}\begin{array}{cc}F& =[P𝔐]\\ & =-𝞓ivT\end{array}\\ [P𝔐{]}_x={𝔉}_x,[P𝔐{]}_y={𝔉}_y,[P𝔐{]}_z={𝔉}_z,[P𝔐{]}_l=\frac{i\varrho }{c}(𝔮𝔈)=\frac{i}{c}(𝔮𝔉)\\ [𝔉=\varrho (𝔈+\frac{1}{c}[𝔮ℌ]),𝞓iv=\text{divergence six-vector},l=ict]\end{array}$

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

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

$K\Rightarrow \begin{array}{cccc}{K}_x& =\frac{{𝔎}_x}{\sqrt{c^2-q^2}},& K_y& =\frac{{𝔎}_y}{\sqrt{c^2-q^2}},\\ K_z& =\frac{{𝔎}_z}{\sqrt{c^2-q^2}},& K_l& =\frac{i}{c}\frac{(𝔮𝔎)}{\sqrt{c^2-q^2}},\end{array}\Rightarrow \frac{d(mY)}{d\tau }\Rightarrow m\frac{dY}{d\tau }$

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) $𝐅$}

$\begin{array}{c}\mathrm{P}=\mathrm{C}𝐅=\mathrm{D}[𝐅\cdot 𝐅]\\ \begin{array}{cc}\mathrm{P}& =\rho \{\iota 𝐌+𝐄+\frac{1}{c}p𝐌-\frac{\iota }{c}p𝐄\}\\ & ={\mathrm{P}}_e+\iota {\mathrm{P}}_m\\ {\mathrm{P}}_e& =\rho \{\frac{\iota }{c}(p𝐄)+𝐄+\frac{1}{c}\mathrm{V}p𝐌\}\\ {\mathrm{P}}_m& =\rho \{\frac{\iota }{c}(p𝐌)+𝐌+\frac{1}{c}\mathrm{V}p𝐄\}\end{array}\\ [\mathrm{D}=\frac{\partial }{\partial l}-\mathrm{\nabla }]\end{array}$

equivalent to (d1,e1,f1).

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

$\begin{array}{c}\begin{array}{cc}F& =\rho \{\frac{\iota }{c}(p𝐄)+𝐄+\frac{1}{c}\mathrm{V}p𝐌\}\\ & =\frac{\iota }{c}(𝐏𝐩)+𝐏\\ & =-\frac{1}{c}𝐑[D]𝐋\\ & =-\mathrm{l}\mathrm{o}\mathrm{r}𝔖\\ F_{\mathrm{M}\mathrm{n}\mathrm{k}}& =\frac{1}{2}[F+\frac{1}{c^2}Y{F}_{c}Y]\end{array}\\ [𝐏=\rho \{𝐄+\frac{1}{c}\mathrm{V}𝐩𝐌\},D=\frac{\partial }{\partial l}-\mathrm{\nabla },\mathrm{l}\mathrm{o}\mathrm{r}=|\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z},\frac{\partial }{i\partial t}|]\end{array}$

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) $m_0𝐰$, deriving the “extended force” (i.e. four force) using $𝐜$ as four-acceleration:

$m_0𝐜=\frac{d{m}_0𝐰}{ds}=\frac{dmv}{ds}{𝐤}_1+\frac{dm}{ds}{𝐤}_4=\frac{1}{\sqrt{1-v^2}}(\frac{dmv}{dt}{𝐤}_1+\frac{dm}{dt}{𝐤}_4)$

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 ${𝐏}^{(\beta )}$, stress-energy tensor $S_{\alpha \beta }$:

$\begin{array}{c}{F}_{\alpha }(y)=\sum _{\beta }\frac{F_{\alpha \beta }(y){𝐏}^{(\beta )}(y)}{\sqrt{1-{𝔴}^2/c^2}}\\ [\sum _{\beta }{F}_{\alpha \beta }(y){𝐏}^{(\beta )}(y)=\sum _{\beta }{F}_{\alpha \beta }(y)\sum _{\gamma }\frac{\partial }{\partial y^{(\gamma )}}{F}_{\beta \gamma }(y)=\sum _{\beta }\frac{\partial }{\partial y^{(\beta )}}{S}_{\alpha \beta }]\end{array}$

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 $\varrho$:

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

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 $\left({𝔉}_{\mu \nu }\right)$, four-current $\left({𝔍}_{\nu }\right)$, stress-energy tensor $\left(T_{\mu \nu }\right)$:

$\begin{array}{c}\left({𝔉}_{\mu \nu }\right)\left({𝔍}_{\nu }\right)=\left(K_{\mu }\right)\\ \begin{array}{cc}{K}_1& =\rho ({𝔢}_x+\frac{{𝔮}_y}{c}{𝔥}_z-\frac{{𝔮}_z}{c}{𝔥}_y)\\ K_2& =\rho ({𝔢}_y+\frac{{𝔮}_z}{c}{𝔥}_x-\frac{{𝔮}_x}{c}{𝔥}_z)\\ K_3& =\rho ({𝔢}_z+\frac{{𝔮}_x}{c}{𝔥}_y-\frac{{𝔮}_y}{c}{𝔥}_x)\\ K_4& =\frac{i}{c}\rho ({𝔮}_x{𝔢}_x+{𝔮}_y{𝔢}_y+{𝔮}_z{𝔢}_z)\end{array}\\ \left(K_{\mu }\right)=-\left(\frac{\partial }{\partial x_{\nu }}\right)\left(T_{\mu \nu }\right)\\ \begin{array}{cc}{K}_1& =-\frac{\partial p_{xx}}{\partial x_1}-\frac{\partial p_{xy}}{\partial x_2}-\frac{\partial p_{xz}}{\partial x_3}-\frac{\partial \frac{i}{c}{𝐬}_x}{\partial x_4}\\ & \dots \\ & \dots \\ K_4& =-\frac{\partial \frac{i}{c}{𝐬}_x}{\partial x_1}-\frac{\partial \frac{i}{c}{𝐬}_y}{\partial x_2}-\frac{\partial \frac{i}{c}{𝐬}_z}{\partial x_3}-\frac{\partial (-w)}{\partial x_4}\end{array}\end{array}$

equivalent to (d1,e1,f1).

References

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• {{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
• {{#section:History of Topics in Special Relativity/relsource|kott12mink}}
• {{#section:History of Topics in Special Relativity/relsource|kott14bes}}
• {{#section:History of Topics in Special Relativity/relsource|laue11prin}}
• {{#section:History of Topics in Special Relativity/relsource|laue13prin}}
• {{#section:History of Topics in Special Relativity/relsource|lewis12non}}
• {{#section:History of Topics in Special Relativity/relsource|mink07a}}
• {{#section:History of Topics in Special Relativity/relsource|mink07b}}
• {{#section:History of Topics in Special Relativity/relsource|poinc05b}}
• {{#section:History of Topics in Special Relativity/relsource|silber11quat}}
• {{#section:History of Topics in Special Relativity/relsource|silber14quat}}
• {{#section:History of Topics in Special Relativity/relsource|som10alg}}