History of Topics in Special Relativity/Four-current

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Overview

The w:Four-current is the four-dimensional analogue of the w:electric current density

$\begin{array}{ccc}{J}^{\alpha }& \underbrace{=(c\rho ,j^1,j^2,j^3)=(c\rho ,𝐣)}& ={\rho }_0{U}^{\alpha }\\ & (a)& (b)\end{array}({\rho }_0=\rho \sqrt{1-\frac{v^2}{c^2}})$

where c is the w:speed of light, $U^{\alpha }$ the four-velocity, ρ is the w:charge density, ${\rho }_0$ the rest charge density , and j the conventional w:current density. Alternatively, it can be defined in terms of the inhomogeneous Maxwell equations as the negative product of the D'Alembert operator and the electromagnetic potential $A^{\beta }$, or the four-divergence of the electromagnetic tensor $F^{\alpha \beta }$:

$\begin{array}{ccc}{\mu }_0{J}^{\beta }& =-◻A^{\beta }& ={\partial }_{\alpha }{F}^{\alpha \beta }\\ & (c)& (d)\end{array}$

and the generally covariant form

$(e)J^{\mu }={\partial }_{\nu }{𝒟}^{\mu \nu },[{𝒟}^{\mu \nu }=\frac{1}{{\mu }_0}{g}^{\mu \alpha }{F}_{\alpha \beta }{g}^{\beta \nu }\sqrt{-g}]$

The Lorentz transformation of the four-potential components was given by #Poincaré (1905/6) and #Marcolongo (1906). It was explicitly formulated in modern form by #Minkowski (1907/15) and reformulated in different notations by #Born (1909), #Bateman (1909/10), #Ignatowski (1910), #Sommerfeld (1910), #Lewis (1910), Wilson/Lewis (1912), #Von Laue (1911), #Silberstein (1911). The generally covariant form was first given by #Kottler (1912) and #Einstein (1913).

Historical notation

Poincaré (1905/6)

w:Henri Poincaré (June 1905; July 1905, published 1906) showed that the four quantities related to charge density $\rho$ are connected by a Lorentz transformation:

$\begin{array}{c}\rho ,\rho \xi ,\rho \eta ,\rho \zeta \\ {\rho }^{\prime }=\frac{k}{l^3}\rho (1+ϵ\xi ),{\rho }^{\prime }{\xi }^{\prime }=\frac{k}{l^3}\rho (\xi +ϵ),{\rho }^{\prime }{\eta }^{\prime }=\frac{\rho \eta }{l^3},{\rho }^{\prime }{\zeta }^{\prime }=\frac{\rho \zeta }{l^3}& (\text{June})\\ {\rho }^{\prime }=\frac{k}{l^3}(\rho +ϵ\rho \xi ),{\rho }^{\prime }{\xi }^{\prime }=\frac{k}{l^3}(\rho \xi +ϵ\rho ),{\rho }^{\prime }{\eta }^{\prime }=\frac{1}{l^3}\rho \eta ,{\rho }^{\prime }{\zeta }^{\prime }=\frac{1}{l^3}\rho {\zeta }^{\prime }& (\text{July})\\ (k=\frac{1}{\sqrt{1-{ϵ}^2}},l=1)\end{array}$

and in his July paper he further stated the continuity equation and the invariance of Jacobian D:

$\begin{array}{c}\frac{d{\rho }^{\prime }}{d{t}^{\prime }}+\sum \frac{d{\rho }^{\prime }{\xi }^{\prime }}{d{x}^{\prime }}=0\\ D_1^{\text{'}}=\frac{d{\rho }^{\prime }}{d{t}^{\prime }}+\sum \frac{d{\rho }^{\prime }{\xi }^{\prime }}{d{x}^{\prime }}=0,D_1=\frac{d\rho }{dt}+\sum \frac{d\rho \xi }{dx}=0\end{array}$

Even though Poincaré didn't directly use four-vector notation in those cases, his quantities are the components of four-current (a).

Marcolongo (1906)

Following Poincaré, w:Roberto Marcolongo defined the general Lorentz transformation $\alpha ,\beta ,\gamma ,\delta$ of the components of the four independent variables $𝐕,\varrho$ and its continuity equation:

$\begin{array}{c}(\xi ,\eta ,\zeta )=𝐕,({\xi }^{\prime },{\eta }^{\prime },{\zeta }^{\prime })={𝐕}^{\prime }\\ {\varrho }^{\prime }{\xi }^{\prime }=\varrho ({\alpha }_1\xi +{\beta }_1\eta +{\gamma }_1\zeta -i{\delta }_1)\\ \dots \\ {\varrho }^{\prime }=\varrho ({\alpha }_4\xi +{\beta }_4\eta +{\gamma }_4\zeta -i{\delta }_4)\\ \frac{\partial {\varrho }^{\prime }}{\partial t^{\prime }}+\frac{\partial {\varrho }^{\prime }{\xi }^{\prime }}{\partial x^{\prime }}+\frac{\partial {\varrho }^{\prime }{\eta }^{\prime }}{\partial y^{\prime }}+\frac{\partial {\varrho }^{\prime }{\zeta }^{\prime }}{\partial z^{\prime }}=0\\ (t=iu)\end{array}$

equivalent to the components of four-current (a), and pointed out its relation to the components $𝐉,\phi$ of the four-potential

$\begin{array}{c}◻𝐉{\text{'}}_x=-4\pi {\varrho }^{\prime }{\xi }^{\prime }=-4\pi \varrho ({\alpha }_1\xi +{\beta }_1\eta +{\gamma }_1\zeta -i{\delta }_1),\dots \\ ◻𝐉=-4\pi \varrho 𝐕,\mathit{◻}\mathit{\phi }=-4\pi \rho ,◻{\phi }^{\prime }=-4\pi {\varrho }^{\prime }\end{array}$

equivalent to the components of Maxwell's equations (b).

Minkowski (1907/15)

w:Hermann Minkowski from the outset employed vector and matrix representation of four-vectors and six-vectors (i.e. antisymmetric second rank tensors) and their product. In a lecture held in November 1907, published 1915, Minkowski defined the four-current in vacuum with $\varrho$ as charge density and $𝔳$ as velocity:

$({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4)=(\varrho 𝔳,i\varrho )$

equivalent to (a), and the electric four-current in matter with $𝐢$ as current and $\sigma$ as charge density:

$(\sigma )=({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4)=(i_x,i_y,i_z,i\sigma )$

In another lecture from December 1907, Minkowski defined the “space-time vector current” and its Lorentz transformation

$\begin{array}{c}(\varrho {𝔴}_x,\varrho {𝔴}_y,\varrho {𝔴}_z,i\varrho )\Rightarrow ({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4)\\ \varrho {\text{'}}_3=x_3\cos i\psi +{\varrho }_4\sin i\psi ,\varrho {\text{'}}_4=-{\varrho }_3\sin i\psi +{\varrho }_4\cos i\psi ,\varrho {\text{'}}_1={\varrho }_1,\varrho {\text{'}}_2={\varrho }_2\\ {\varrho }^{\prime }𝔴{\text{'}}_{z^{\prime }}=\varrho \left(\frac{{𝔴}_z-q}{\sqrt{1-q^2}}\right),{\varrho }^{\prime }=\varrho \left(\frac{-q{𝔴}_z+1}{\sqrt{1-q^2}}\right),{\varrho }^{\prime }𝔴{\text{'}}_{x^{\prime }}=\varrho {𝔴}_x,{\varrho }^{\prime }𝔴{\text{'}}_{y^{\prime }}=\varrho {𝔴}_y,\\ -({\varrho }_1^2+{\varrho }_2^2+{\varrho }_3^2+{\varrho }_4^2)={\varrho }^2(1-{𝔴}_x^2-{𝔴}_y^2-{𝔴}_z^2)={\varrho }^2(1-{𝔴}^2)\end{array}$

equivalent to (a). In moving media and dielectrics, Minkowski more generally used the current density vector “electric current” $𝔰$ which becomes $𝔰=\sigma 𝔈$ in isotropic media:

$\begin{array}{c}({𝔰}_x,{𝔰}_y,{𝔰}_z,i\varrho )\Rightarrow (s_1,s_2,s_3,s_4)\end{array}$

Born (1909)

Following Minkowski, w:Max Born (1909) defined the “space-time vector of first kind” (four-vector) and its continuity equation

$\begin{array}{c}(\frac{\varrho w_x}{c},\frac{\varrho w_y}{c},\frac{\varrho w_z}{c},i\varrho )\Rightarrow ({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4)\\ \frac{\partial \varrho w_x}{\partial x}+\frac{\partial \varrho w_y}{\partial y}+\frac{\partial \varrho w_z}{\partial z}+\frac{\partial \varrho }{\partial t}=0\end{array}$

equivalent to (a), and pointed out its relation to Maxwell's equations as the product of the D'Alembert operator with the electromagnetic potential ${\mathrm{\Phi }}_{\alpha }$:

$\frac{\partial }{\partial x_{\alpha }}\sum _{\beta =1}^4\frac{\partial {\mathrm{\Phi }}_{\beta }}{\partial x_{\beta }}-\sum _{\beta =1}^4\frac{{\partial }^2{\mathrm{\Phi }}_{\alpha }}{\partial x_{\beta }^2}={\varrho }_{\alpha }(\sum _{\beta =1}^4\frac{\partial {\mathrm{\Phi }}_{\beta }}{\partial x_{\beta }}=0)$

equivalent to (c). He also expressed the four-current in terms of rest charge density and four-velocity

$\begin{array}{c}{\varrho }_1=\frac{{\varrho }^{\ast }}{c}\cdot \frac{\partial x}{\partial \tau },{\varrho }_2=\frac{{\varrho }^{\ast }}{c}\cdot \frac{\partial y}{\partial \tau },{\varrho }_3=\frac{{\varrho }^{\ast }}{c}\cdot \frac{\partial z}{\partial \tau },{\varrho }_4=i{\varrho }^{\ast }\frac{\partial t}{\partial \tau }\\ ({\varrho }^{\ast }=\varrho \sqrt{1-\frac{w^2}{c^2}}=\sqrt{-({\varrho }_1^2+{\varrho }_2^2+{\varrho }_3^2+{\varrho }_4^2)},d\tau =dt\sqrt{1-\frac{w^2}{c^2}})\\ {\varrho }_{\alpha }=i{\varrho }^{\ast }\frac{\partial x_{\alpha }}{\partial {\xi }_4}({\xi }_4=ic\tau )\end{array}$

equivalent to (b).

Bateman (1909/10)

A discussion of four-current 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 Lorentz transformations of its components $(\rho w_x,\rho w_y,\rho w_z,\rho )$

$\rho w_x=\beta ({\rho }^{\prime }{w}^{\prime }-v{\rho }^{\prime }),\rho w_y={\rho }^{\prime }w{\text{'}}_y,\rho w_z={\rho }^{\prime }w{\text{'}}_z,-\rho =\beta (v{\rho }^{\prime }w{\text{'}}_x-{\rho }^{\prime }),[\beta =\frac{1}{\sqrt{1-v^2}}]$

forming the following invariant relations together with the differential four-position and four-potential:

$\begin{array}{c}\frac{1}{{\lambda }^2}[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt]\\ \frac{{\rho }^2}{{\lambda }^2}(1-w^2)dxdydzdt\\ \rho [A_x{w}_x+A_y{w}_y+A_z{w}_z-\mathrm{\Phi }]dxdydzdt\end{array}$

with ${\lambda }^2=1$ in relativity.

Ignatowski (1910)

w:Wladimir Ignatowski (1910) defined the “vector of first kind” using charge density $\varrho$ and three-velocity $𝔳$:

$\begin{array}{c}(\varrho 𝔳,\varrho )\\ [\varrho \sqrt{1-n{𝔳}^2}={\varrho }^{\prime }\sqrt{1-n{𝔳}^{\prime 2}}={\varrho }_0]\end{array}$

equivalent to four-current (a).

Sommerfeld (1910)

In influential papers on 4D vector calculus in relativity, w:Arnold Sommerfeld defined the four-current P, which he called four-density (Viererdichte):

$\begin{array}{c}{P}_x=\varrho \frac{{𝔳}_x}{c},P_y=\varrho \frac{{𝔳}_y}{c},P_z=\varrho \frac{{𝔳}_z}{c},P_l=i\varrho \\ {\beta }^2=\frac{1}{c^2}({𝔳}_x^2+{𝔳}_y^2+{𝔳}_z^2)\Rightarrow \left|P\right|=i\varrho \sqrt{1-{\beta }^2}\\ [l=ict]\end{array}$

equivalent to (a). In the second paper he pointed out its relation to four-potential $\mathrm{\Phi }$ and the electromagnetic tensor (six-vector) f together with the continuity condition:

$\begin{array}{c}\begin{array}{cc}P& =𝔇𝔦𝔳\mathrm{R}\mathrm{o}\mathrm{t}\mathrm{\Phi }=𝔇𝔦𝔳f\\ -P& =◻\mathrm{\Phi },(\mathrm{D}\mathrm{i}\mathrm{v}\mathrm{\Phi }=0)\\ \mathrm{D}\mathrm{i}\mathrm{v}P& =0\end{array}\\ \left[\begin{array}{cc}\mathrm{R}\mathrm{o}\mathrm{t}& =\text{exterior product}\\ \mathrm{D}\mathrm{i}\mathrm{v}& =\text{divergence four-vector}\\ 𝔇𝔦𝔳& =\text{divergence six-vector}\\ ◻& =\text{D'Alembert operator}\end{array}\right]\end{array}$

equivalent to Maxwell's equations (c). The scalar product with the four-potential

$(P\mathrm{\Phi })$

he called “electro-kinetic potential” whereas the vector product with the electromagnetic tensor

$(Pf)=𝔉$

he called the electrodynamic force (four-force density).

Lewis (1910), Wilson/Lewis (1912)

w:Gilbert Newton Lewis (1910) devised an alternative 4D vector calculus based on w:Dyadics which, however, never gained widespread support. The four-current is a “1-vector”:

$\begin{array}{cc}𝐪& =\frac{\varrho }{c}𝐯+i\varrho {𝐤}_4\\ & =\frac{\varrho }{c}{v}_1{𝐤}_1+\frac{\varrho }{c}{v}_2{𝐤}_2+\frac{\varrho }{c}{v}_3{𝐤}_3+i\varrho {𝐤}_4\end{array}$

equivalent to (a) and its relation to the four-potential $𝐦$ and electromagnetic tensor $𝐌$:

$\begin{array}{c}\begin{array}{cc}◊◊\times 𝐦& =𝐪\\ ◊𝐌& =𝐪\\ {◊}^2𝐦& =-𝐪\end{array}\\ \begin{array}{cc}(\frac{\partial H_{12}}{\partial x_2}+\frac{\partial H_{13}}{\partial x_3}+\frac{\partial E_{14}}{\partial x_4}){𝐤}_1& =\frac{\varrho }{c}{v}_1{𝐤}_1\\ (\frac{\partial H_{21}}{\partial x_1}+\frac{\partial H_{23}}{\partial x_3}+\frac{\partial E_{24}}{\partial x_4}){𝐤}_2& =\frac{\varrho }{c}{v}_2{𝐤}_2\\ (\frac{\partial H_{31}}{\partial x_1}+\frac{\partial H_{32}}{\partial x_2}+\frac{\partial E_{34}}{\partial x_4}){𝐤}_3& =\frac{\varrho }{c}{v}_3{𝐤}_3\\ (\frac{\partial H_{41}}{\partial x_1}+\frac{\partial H_{42}}{\partial x_2}+\frac{\partial E_{43}}{\partial x_4}){𝐤}_4& =\frac{\varrho }{c}i{𝐤}_4\end{array}\\ \left[\begin{array}{c}◊={𝐤}_1\frac{\partial }{\partial x_1}+{𝐤}_2\frac{\partial }{\partial x_2}+{𝐤}_3\frac{\partial }{\partial x_3}+{𝐤}_4\frac{\partial }{\partial x_4}\\ {◊}^2=\frac{{\partial }^2}{\partial x_1}+\frac{{\partial }^2}{\partial x_2}+\frac{{\partial }^2}{\partial x_3}+\frac{{\partial }^2}{\partial x_4}\end{array}\right]\end{array}$

equivalent to (c,d).

In 1912, Lewis and w:Edwin Bidwell Wilson used only real coordinates, writing the above expressions as

$\begin{array}{c}\begin{array}{cc}◊\cdot 𝐌& =4\pi 𝐪\\ {◊}^2𝐦& =-4\pi 𝐪\end{array}\\ \left[\begin{array}{c}◊={𝐤}_1\frac{\partial }{\partial x_1}+{𝐤}_2\frac{\partial }{\partial x_2}+{𝐤}_3\frac{\partial }{\partial x_3}-{𝐤}_4\frac{\partial }{\partial x_4}\\ {◊}^2=\frac{{\partial }^2}{\partial x_1}+\frac{{\partial }^2}{\partial x_2}+\frac{{\partial }^2}{\partial x_3}-\frac{{\partial }^2}{\partial x_4}\end{array}\right]\end{array}$

equivalent to (c,d).

Von Laue (1911)

In the first textbook on relativity in 1911, w:Max von Laue elaborated on Sommerfeld's methods and explicitly used the term “four-current” (Viererstrom) of density $\varrho$ in relation to four-potential $\mathrm{\Phi }$ and electromagnetic tensor $𝔐$:

$\begin{array}{c}P\Rightarrow (P_x=\frac{\varrho {𝔮}_x}{c},P_y=\frac{\varrho {𝔮}_y}{c},P_z=\frac{\varrho {𝔮}_z}{c},P_l=i\varrho )\\ \begin{array}{cc}P& =𝞓iv(𝔐)\\ -P& =◻\mathrm{\Phi }(Div\mathrm{\Phi }=0)\\ Div(P)& =0\end{array}\\ \left[\begin{array}{cc}\Re 𝔬𝔱& =\text{exterior product}\\ Div& =\text{divergence four-vector}\\ 𝞓iv& =\text{divergence six-vector}\\ ◻& =\text{D'Alembert operator}\end{array}\right]\end{array}$

equivalent to (a,c,d). He went on to define four-force density F as vector-product with $𝔐$, four-convection K and four-conduction $\mathrm{\Lambda }$ using four-velocity Y:

$\begin{array}{c}F=[P𝔐],(PF)=(P[P𝔐])=0\\ K=-(YP)Y\\ \mathrm{\Lambda }=P+(YP)Y\end{array}$,

Silberstein (1911)

w:Ludwik Silberstein devised an alternative 4D calculus based on w:Biquaternions which, however, never gained widespread support. He defined the “current-quaternion” (i.e. four-current) C and its relation to the “electromagnetic bivector” (i.e. field tensor) $𝐅$ and “potential-quaternion” (i.e. four-potential) $\mathrm{\Phi }$

$\begin{array}{c}\begin{array}{cc}\mathrm{C}& =\rho (\iota +\frac{1}{c}𝐩)\\ & =\iota \rho \frac{dq}{dl}\\ \mathrm{C}& =\mathrm{D}𝐅=-◻\mathrm{\Phi }\\ \mathrm{S}{\mathrm{D}}_c\mathrm{C}& =0\end{array}\\ [\mathrm{D}=\frac{\partial }{\partial l}-\mathrm{\nabla },\mathrm{D}{\mathrm{D}}_c=◻=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}+\frac{{\partial }^2}{\partial l^2}]\end{array}$

Kottler (1912)

w:Friedrich Kottler defined the four-current ${𝐏}^{(\alpha )}$ and its relation to four-velocity $V^{(\alpha )}$, four-potential ${\mathrm{\Phi }}_{\alpha }$, four-force $F_{\alpha }$, electromagnetic field-tensor $F_{\alpha \beta }$, stress-energy tensor $S_{\alpha \beta }$:

$\begin{array}{c}{P}^{(1)}=\rho \frac{{𝔳}_x}{c}=i{\rho }_0{V}^{(1)},P^{(2)}=\rho \frac{{𝔳}_y}{c}=i{\rho }_0{V}^{(2)},P^{(3)}=\rho \frac{{𝔳}_z}{c}=i{\rho }_0{V}^{(3)},P^{(4)}=i\rho =i{\rho }_0{V}^{(4)}\\ \sum _{h=1}^4\frac{\partial F_{gh}}{\partial x^{(h)}}={𝐏}^{(g)},◻{\mathrm{\Phi }}_{\alpha }=-{𝐏}^{(\alpha )}\\ F_{\alpha }(y)=\sum _{\beta }\frac{F_{\alpha \beta }(y){𝐏}^{(\beta )}(y)}{\sqrt{1-{𝔴}^2/c^2}}\\ [\sum _{\beta }{F}_{\alpha \beta }(y){𝐏}^{(\beta )}(y)=\sum _{\beta }{F}_{\alpha \beta }(y)\sum _{\gamma }\frac{\partial }{\partial y^{(\gamma )}}{F}_{\beta \gamma }(y)=\sum _{\beta }\frac{\partial }{\partial y^{(\beta )}}{S}_{\alpha \beta },{\rho }_0=\rho \sqrt{1-{𝔳}^2/c^2}]\end{array}$

equivalent to (a,b,c,d) and subsequently was the first to give the generally covariant formulation of Maxwell's equations using metric tensor $c_{\alpha \beta }$

$\begin{array}{c}\sum c^{(1\alpha )}\sum _{\beta ,\gamma }{c}^{(\beta \gamma )}{\mathrm{\Phi }}_{\alpha /\beta \gamma }=-{𝐏}^{(\alpha )}\text{etc}.\\ [\sum _{\beta ,\gamma }{c}^{(\beta \gamma )}{\mathrm{\Phi }}_{\beta /\gamma }=0]\end{array}$

equivalent to (e).

Einstein (1913)

Independently of Kottler (1912), w:Albert Einstein defined the general covariant four-current in the context of his Entwurf theory (a precursor of general relativity):

${\varrho }_0\frac{d{x}_{\nu }}{ds}=\frac{1}{\sqrt{-g}}{\varrho }_0\frac{d{x}_{\nu }}{dt}$

equivalent to (a), and the generally covariant formulation of Maxwell's equations

$\begin{array}{c}\sum _{\nu }\frac{\partial }{\partial x_{\nu }}(\sqrt{-g}\cdot {\phi }_{\mu \nu })={\varrho }_0\frac{d{x}_{\mu }}{dt}\\ \begin{array}{cc}\frac{\partial {ℌ}_x}{\partial y}-\frac{\partial {ℌ}_y}{\partial z}-\frac{\partial {𝔈}_x}{\partial t}& =u_x\\ \dots \\ \dots \\ \frac{\partial {𝔈}_x}{\partial x}+\frac{\partial {𝔈}_y}{\partial z}+\frac{\partial {𝔈}_x}{\partial z}& =\varrho \end{array}\\ [{\varrho }_0\frac{d{x}_{\mu }}{dt}=u_{\mu }]\end{array}$

equivalent to (e) in the case of $g_{\mu \nu }$ being the Minkowski tensor.

Historical sources

• {{#section:History of Topics in Special Relativity/relsource|bate10elec}}
• {{#section:History of Topics in Special Relativity/relsource|born09elek}}
• {{#section:History of Topics in Special Relativity/relsource|einst13ent}}
• {{#section:History of Topics in Special Relativity/relsource|herg04pot}}
• {{#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|laue11prin}}
• {{#section:History of Topics in Special Relativity/relsource|lew10vec}}
• {{#section:History of Topics in Special Relativity/relsource|lewis12non}}
• {{#section:History of Topics in Special Relativity/relsource|marc06elek}}
• {{#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|poinc05a}}
• {{#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|som10alg}}
• {{#section:History of Topics in Special Relativity/relsource|som10ana}}