History of Topics in Special Relativity/Four-current

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

${\begin{matrix}J^{\alpha }&\underbrace {=\left(c\rho ,j^{1},j^{2},j^{3}\right)=\left(c\rho ,\mathbf {j} \right)} &=\rho _{0}U^{\alpha }\\&(a)&(b)\end{matrix}}\left(\rho _{0}=\rho {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)$

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 $F^{\alpha \beta }$:

${\begin{matrix}\mu _{0}J^{\beta }&=-\square A^{\beta }&=\partial _{\alpha }F^{\alpha \beta }\\&(c)&(d)\end{matrix}}$

and the generally covariant form

$(e)\ J^{\mu }=\partial _{\nu }{\mathcal {D}}^{\mu \nu },\ \left[{\mathcal {D}}^{\mu \nu }\,=\,{\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\sqrt {-g}}\right]$

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:

${\begin{matrix}\rho ,\ \rho \xi ,\ \rho \eta ,\ \rho \zeta \\\hline \rho ^{\prime }={\frac {k}{l^{3}}}\rho (1+\epsilon \xi ),\quad \rho ^{\prime }\xi ^{\prime }={\frac {k}{l^{3}}}\rho (\xi +\epsilon ),\quad \rho ^{\prime }\eta ^{\prime }={\frac {\rho \eta }{l^{3}}},\ \quad \rho ^{\prime }\zeta ^{\prime }={\frac {\rho \zeta }{l^{3}}}&({\text{June}})\\\rho ^{\prime }={\frac {k}{l^{3}}}(\rho +\epsilon \rho \xi ),\quad \rho '\xi ^{\prime }={\frac {k}{l^{3}}}(\rho \xi +\epsilon \rho ),\quad \rho ^{\prime }\eta ^{\prime }={\frac {1}{l^{3}}}\rho \eta ,\quad \rho ^{\prime }\zeta ^{\prime }={\frac {1}{l^{3}}}\rho \zeta '&({\text{July}})\\\left(k={\frac {1}{\sqrt {1-\epsilon ^{2}}}},\ l=1\right)\end{matrix}}$

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

${\begin{matrix}{\frac {d\rho ^{\prime }}{dt^{\prime }}}+\sum {\frac {d\rho ^{\prime }\xi ^{\prime }}{dx^{\prime }}}=0\\D_{1}^{'}={\frac {d\rho ^{\prime }}{dt^{\prime }}}+\sum {\frac {d\rho ^{\prime }\xi ^{\prime }}{dx^{\prime }}}=0,\ D_{1}={\frac {d\rho }{dt}}+\sum {\frac {d\rho \xi }{dx}}=0\end{matrix}}$

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 $\mathbf {V} ,\varrho$ and its continuity equation:

${\begin{matrix}(\xi ,\eta ,\zeta )=\mathbf {V} ,\ (\xi ',\eta ',\zeta ')=\mathbf {V} '\\\hline \varrho '\xi '=\varrho \left(\alpha _{1}\xi +\beta _{1}\eta +\gamma _{1}\zeta -i\delta _{1}\right)\\\dots \\\varrho '=\varrho \left(\alpha _{4}\xi +\beta _{4}\eta +\gamma _{4}\zeta -i\delta _{4}\right)\\\hline {\frac {\partial \varrho '}{\partial t'}}+{\frac {\partial \varrho '\xi '}{\partial x'}}+{\frac {\partial \varrho '\eta '}{\partial y'}}+{\frac {\partial \varrho '\zeta '}{\partial z'}}=0\\(t=iu)\end{matrix}}$

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

${\begin{matrix}\Box \mathbf {J} '_{x}=-4\pi \varrho '\xi '=-4\pi \varrho \left(\alpha _{1}\xi +\beta _{1}\eta +\gamma _{1}\zeta -i\delta _{1}\right),\dots \\\Box \mathbf {J} =-4\pi \varrho \mathbf {V,\ \Box \varphi } =-4\pi \rho ,\ \Box \varphi '=-4\pi \varrho '\end{matrix}}$

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 ${\mathfrak {v}}$ as velocity:

$\left(\varrho _{1},\varrho _{2},\varrho _{3},\varrho _{4}\right)=(\varrho {\mathfrak {v}},\ i\varrho )$

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

$(\sigma )=\left(\sigma _{1},\ \sigma _{2},\ \sigma _{3},\ \sigma _{4}\right)=(i_{x},i_{y},i_{z},\ i\sigma )$

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

${\begin{matrix}\left(\varrho \,{\mathfrak {w}}_{x},\ \varrho \,{\mathfrak {w}}_{y},\ \varrho \,{\mathfrak {w}}_{z},\ i\varrho \right)\Rightarrow \left(\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}\right)\\\hline \varrho '_{3}=x_{3}\cos \ i\psi +\varrho _{4}\sin \ i\psi ,\quad \varrho '_{4}=-\varrho _{3}\sin \ i\psi +\varrho _{4}\cos \ i\psi ,\quad \varrho '_{1}=\varrho _{1},\quad \varrho '_{2}=\varrho _{2}\\\varrho '{\mathfrak {w}}'_{z'}=\varrho \left({\frac {{\mathfrak {w}}_{z}-q}{\sqrt {1-q^{2}}}}\right),\quad \varrho '=\varrho \left({\frac {-q\,{\mathfrak {w}}_{z}+1}{\sqrt {1-q^{2}}}}\right),\quad \varrho '{\mathfrak {w}}'_{x'}=\varrho \,{\mathfrak {w}}_{x},\quad \varrho '{\mathfrak {w}}'_{y'}=\varrho \,{\mathfrak {w}}_{y},\\\hline -(\varrho _{1}^{2}+\varrho _{2}^{2}+\varrho _{3}^{2}+\varrho _{4}^{2})=\varrho ^{2}(1-{\mathfrak {w}}_{x}^{2}-{\mathfrak {w}}_{y}^{2}-{\mathfrak {w}}_{z}^{2})=\varrho ^{2}(1-{\mathfrak {w}}^{2})\end{matrix}}$

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

${\begin{matrix}\left({\mathfrak {s}}_{x},\ {\mathfrak {s}}_{y},\ {\mathfrak {s}}_{z},\ i\varrho \right)\Rightarrow \left(s_{1},\ s_{2},\ s_{3},\ s_{4}\right)\end{matrix}}$

Born (1909)

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

${\begin{matrix}\left({\frac {\varrho w_{x}}{c}},\ {\frac {\varrho w_{y}}{c}},\ {\frac {\varrho w_{z}}{c}},\ i\varrho \right)\Rightarrow \left(\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}\right)\\{\frac {\partial \varrho w_{x}}{\partial x}}+{\frac {\partial \varrho w_{y}}{\partial y}}+{\frac {\partial \varrho w_{z}}{\partial z}}+{\frac {\partial \varrho }{\partial t}}=0\end{matrix}}$

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

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

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

${\begin{matrix}\varrho _{1}={\frac {\varrho ^{\ast }}{c}}\cdot {\frac {\partial x}{\partial \tau }},\ \varrho _{2}={\frac {\varrho ^{\ast }}{c}}\cdot {\frac {\partial y}{\partial \tau }},\ \varrho _{3}={\frac {\varrho ^{\ast }}{c}}\cdot {\frac {\partial z}{\partial \tau }},\ \varrho _{4}=i\varrho ^{\ast }{\frac {\partial t}{\partial \tau }}\\\left(\varrho ^{\ast }=\varrho {\sqrt {1-{\frac {w^{2}}{c^{2}}}}}={\sqrt {-\left(\varrho _{1}^{2}+\varrho _{2}^{2}+\varrho _{3}^{2}+\varrho _{4}^{2}\right)}},\ d\tau =dt{\sqrt {1-{\frac {w^{2}}{c^{2}}}}}\right)\\\varrho _{\alpha }=i\varrho ^{\ast }{\frac {\partial x_{\alpha }}{\partial \xi _{4}}}\quad \left(\xi _{4}=ic\tau \right)\end{matrix}}$

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 $\left(\rho w_{x},\rho w_{y},\rho w_{z},\rho \right)$

$\rho w_{x}=\beta (\rho 'w'-v\rho '),\ \rho w_{y}=\rho 'w'_{y},\ \rho w_{z}=\rho 'w'_{z},\ -\rho =\beta (v\rho 'w'_{x}-\rho '),\ \left[\beta ={\frac {1}{\sqrt {1-v^{2}}}}\right]$

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

${\begin{matrix}{\frac {1}{\lambda ^{2}}}\left[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt\right]\\{\frac {\rho ^{2}}{\lambda ^{2}}}\left(1-w^{2}\right)dx\ dy\ dz\ dt\\\rho \left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi \right]dx\ dy\ dz\ dt\end{matrix}}$

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 ${\mathfrak {v}}$:

${\begin{matrix}\left(\varrho {\mathfrak {v}},\ \varrho \right)\\\hline \left[\varrho {\sqrt {1-n{\mathfrak {v}}^{2}}}=\varrho '{\sqrt {1-n{\mathfrak {v}}^{\prime 2}}}=\varrho _{0}\right]\end{matrix}}$

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):

${\begin{matrix}P_{x}=\varrho {\frac {{\mathfrak {v}}_{x}}{c}},\ P_{y}=\varrho {\frac {{\mathfrak {v}}_{y}}{c}},\ P_{z}=\varrho {\frac {{\mathfrak {v}}_{z}}{c}},\ P_{l}=i\varrho \\\hline \beta ^{2}={\frac {1}{c^{2}}}\left({\mathfrak {v}}_{x}^{2}+{\mathfrak {v}}_{y}^{2}+{\mathfrak {v}}_{z}^{2}\right)\quad \Rightarrow \quad \left|P\right|=i\varrho {\sqrt {1-\beta ^{2}}}\\{}[l=ict]\end{matrix}}$

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

${\begin{matrix}{\begin{aligned}P&={\mathfrak {Div}}\mathrm {Rot} \ \Phi ={\mathfrak {Div}}\ f\\-P&=\square \Phi ,\ (\mathrm {Div} \ \Phi =0)\\\mathrm {Div} \ P&=0\end{aligned}}\\\left[{\begin{aligned}\mathrm {Rot} &={\text{exterior product}}\\\mathrm {Div} &={\text{divergence four-vector}}\\{\mathfrak {Div}}&={\text{divergence six-vector}}\\\square &={\text{D'Alembert operator}}\end{aligned}}\right]\end{matrix}}$

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

$(P\Phi )$

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

$(Pf)={\mathfrak {F}}$

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”:

${\begin{aligned}\mathbf {q} &={\frac {\varrho }{c}}\mathbf {v} +i\varrho \mathbf {k} _{4}\\&={\frac {\varrho }{c}}v_{1}\mathbf {k} _{1}+{\frac {\varrho }{c}}v_{2}\mathbf {k} _{2}+{\frac {\varrho }{c}}v_{3}\mathbf {k} _{3}+i\varrho \mathbf {k} _{4}\end{aligned}}$

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

${\begin{matrix}{\begin{aligned}\lozenge \lozenge \times \mathbf {m} &=\mathbf {q} \\\lozenge \mathbf {M} &=\mathbf {q} \\\lozenge ^{2}\mathbf {m} &=-\mathbf {q} \end{aligned}}\\{\begin{aligned}\left({\frac {\partial H_{12}}{\partial x_{2}}}+{\frac {\partial H_{13}}{\partial x_{3}}}+{\frac {\partial E_{14}}{\partial x_{4}}}\right)\mathbf {k} _{1}&={\frac {\varrho }{c}}v_{1}\mathbf {k} _{1}\\\left({\frac {\partial H_{21}}{\partial x_{1}}}+{\frac {\partial H_{23}}{\partial x_{3}}}+{\frac {\partial E_{24}}{\partial x_{4}}}\right)\mathbf {k} _{2}&={\frac {\varrho }{c}}v_{2}\mathbf {k} _{2}\\\left({\frac {\partial H_{31}}{\partial x_{1}}}+{\frac {\partial H_{32}}{\partial x_{2}}}+{\frac {\partial E_{34}}{\partial x_{4}}}\right)\mathbf {k} _{3}&={\frac {\varrho }{c}}v_{3}\mathbf {k} _{3}\\\left({\frac {\partial H_{41}}{\partial x_{1}}}+{\frac {\partial H_{42}}{\partial x_{2}}}+{\frac {\partial E_{43}}{\partial x_{4}}}\right)\mathbf {k} _{4}&={\frac {\varrho }{c}}i\mathbf {k} _{4}\end{aligned}}\\\left[{\begin{matrix}\lozenge =\mathbf {k} _{1}{\frac {\partial }{\partial x_{1}}}+\mathbf {k} _{2}{\frac {\partial }{\partial x_{2}}}+\mathbf {k} _{3}{\frac {\partial }{\partial x_{3}}}+\mathbf {k} _{4}{\frac {\partial }{\partial x_{4}}}\\\lozenge ^{2}={\frac {\partial ^{2}}{\partial x_{1}}}+{\frac {\partial ^{2}}{\partial x_{2}}}+{\frac {\partial ^{2}}{\partial x_{3}}}+{\frac {\partial ^{2}}{\partial x_{4}}}\end{matrix}}\right]\end{matrix}}$

equivalent to (c,d).

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

${\begin{matrix}{\begin{aligned}\lozenge \cdot \mathbf {M} &=4\pi \mathbf {q} \\\lozenge ^{2}\mathbf {m} &=-4\pi \mathbf {q} \end{aligned}}\\\left[{\begin{matrix}\lozenge =\mathbf {k} _{1}{\frac {\partial }{\partial x_{1}}}+\mathbf {k} _{2}{\frac {\partial }{\partial x_{2}}}+\mathbf {k} _{3}{\frac {\partial }{\partial x_{3}}}-\mathbf {k} _{4}{\frac {\partial }{\partial x_{4}}}\\\lozenge ^{2}={\frac {\partial ^{2}}{\partial x_{1}}}+{\frac {\partial ^{2}}{\partial x_{2}}}+{\frac {\partial ^{2}}{\partial x_{3}}}-{\frac {\partial ^{2}}{\partial x_{4}}}\end{matrix}}\right]\end{matrix}}$

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 $\Phi$ and electromagnetic tensor ${\mathfrak {M}}$:

${\begin{matrix}P\Rightarrow \left(P_{x}={\frac {\varrho {\mathfrak {q}}_{x}}{c}},\ P_{y}={\frac {\varrho {\mathfrak {q}}_{y}}{c}},\ P_{z}={\frac {\varrho {\mathfrak {q}}_{z}}{c}},\ P_{l}=i\varrho \right)\\\hline {\begin{aligned}P&=\varDelta iv\ ({\mathfrak {M}})\\-P&=\square \Phi \ (Div\ \Phi =0)\\Div\ (P)&=0\end{aligned}}\\\left[{\begin{aligned}{\mathfrak {Rot}}&={\text{exterior product}}\\Div&={\text{divergence four-vector}}\\\varDelta iv&={\text{divergence six-vector}}\\\square &={\text{D'Alembert operator}}\end{aligned}}\right]\end{matrix}}$

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

${\begin{matrix}F=[P{\mathfrak {M}}],\ (PF)=(P[P{\mathfrak {M}}])=0\\K=-(YP)Y\\\Lambda =P+(YP)Y\end{matrix}}$,

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) $\mathbf {F}$ and “potential-quaternion” (i.e. four-potential) $\Phi$

${\begin{matrix}{\begin{aligned}\mathrm {C} &=\rho \left(\iota +{\frac {1}{c}}\mathbf {p} \right)\\&=\iota \rho {\frac {dq}{dl}}\\\mathrm {C} &=\mathrm {D} \mathbf {F} =-\Box \Phi \\\mathrm {S} \mathrm {D} _{c}\mathrm {C} &=0\end{aligned}}\\\left[\mathrm {D} ={\frac {\partial }{\partial l}}-\nabla ,\ \mathrm {D} \mathrm {D} _{c}=\Box ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {\partial ^{2}}{\partial l^{2}}}\right]\end{matrix}}$

Kottler (1912)

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

${\begin{matrix}P^{(1)}=\rho {\frac {{\mathfrak {v}}_{x}}{c}}=i\rho _{0}V^{(1)},\ P^{(2)}=\rho {\frac {{\mathfrak {v}}_{y}}{c}}=i\rho _{0}V^{(2)},\ P^{(3)}=\rho {\frac {{\mathfrak {v}}_{z}}{c}}=i\rho _{0}V^{(3)},\ P^{(4)}=i\rho =i\rho _{0}V^{(4)}\\\hline \sum _{h=1}^{4}{\frac {\partial F_{gh}}{\partial x^{(h)}}}=\mathbf {P} ^{(g)},\ \Box \Phi _{\alpha }=-\mathbf {P} ^{(\alpha )}\\F_{\alpha }(y)=\sum _{\beta }{\frac {F_{\alpha \beta }(y)\mathbf {P} ^{(\beta )}(y)}{\sqrt {1-{\mathfrak {w}}^{2}/c^{2}}}}\\\left[{\underset {\beta }{\sum }}F_{\alpha \beta }(y)\mathbf {P} ^{(\beta )}(y)={\underset {\beta }{\sum }}F_{\alpha \beta }(y){\underset {\gamma }{\sum }}{\frac {\partial }{\partial y^{(\gamma )}}}F_{\beta \gamma }(y)={\underset {\beta }{\sum }}{\frac {\partial }{\partial y^{(\beta )}}}S_{\alpha \beta },\ \rho _{0}=\rho {\sqrt {1-{\mathfrak {v}}^{2}/c^{2}}}\right]\end{matrix}}$

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

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

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):

$\varrho _{0}{\frac {dx_{\nu }}{ds}}={\frac {1}{\sqrt {-g}}}\varrho _{0}{\frac {dx_{\nu }}{dt}}$

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

${\begin{matrix}\sum _{\nu }{\frac {\partial }{\partial x_{\nu }}}\left({\sqrt {-g}}\cdot \varphi _{\mu \nu }\right)=\varrho _{0}{\frac {dx_{\mu }}{dt}}\\\hline {\begin{aligned}{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}-{\frac {\partial {\mathfrak {E}}_{x}}{\partial t}}&=u_{x}\\\dots \\\dots \\{\frac {\partial {\mathfrak {E}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {E}}_{y}}{\partial z}}+{\frac {\partial {\mathfrak {E}}_{x}}{\partial z}}&=\varrho \end{aligned}}\\\left[\varrho _{0}{\frac {dx_{\mu }}{dt}}=u_{\mu }\right]\end{matrix}}$

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

Historical sources

• Bateman, H. (1910) [1909], "The Transformation of the Electrodynamical Equations", Proceedings of the London Mathematical Society, 8: 223–264

• The Transformation of the Electrodynamical Equations on English Wikisource

• Born, M. (1909), "Die träge Masse und das Relativitätsprinzip", Annalen der Physik, 333 (3): 571–584
• Einstein, A. & Grossmann, M. (1913), "Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation", Zeitschrift für Mathematik und Physik, 62: 225–261{{citation}}: CS1 maint: multiple names: authors list (link)
• Herglotz, G. (1904), "Über die Berechnung retardierter Potentiale", Gött. Nachr. (6): 549–556
• Ignatowsky, W. v. (1910), "Das Relativitätsprinzip", Archiv der Mathematik und Physik 17: 1-24, 18: 17-40

• Das Relativitätsprinzip on German Wikisource

• Kottler, F. (1912), "Über die Raumzeitlinien der Minkowski'schen Welt", Wiener Sitzungsberichte 2a, 121: 1659–1759

• On the spacetime lines of a Minkowski world on English Wikisource

• Laue, M. v. (1911), Das Relativitätsprinzip, Braunschweig: Vieweg
• Lewis, G. N. (1910), "On Four-Dimensional Vector Analysis, and Its Application in Electrical Theory", Proceedings of the American Academy of Arts and Sciences, 43 (7): 165–181
• Lewis, G. N. & Wilson, E. B. (1912), "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences, 48: 387–507{{citation}}: CS1 maint: multiple names: authors list (link)
• Marcolongo, R. (1906), "Sugli integrali delle equazioni dell'elettrodinamica" (PDF), Atti della Reale Accademia dei Lincei Rendiconti, 15: 344–349
• Minkowski, H. (1915) [1907], "Das Relativitätsprinzip", Annalen der Physik, 352 (15): 927–938

• Das Relativitätsprinzip on German Wikisource

• Minkowski, H. (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111

• Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern on German Wikisource
• The Fundamental Equations for Electromagnetic Processes in Moving Bodies on English Wikisource

• Poincaré, H. (1905), "Sur la dynamique de l'électron", Comptes Rendus, 140: 1504–1508

• Sur la dynamique de l’électron on French Wikisource
• On the Dynamics of the Electron on English Wikisource

• Poincaré, H. (1906) [1905], "Sur la dynamique de l'électron", Rendiconti del Circolo Matematico di Palermo, 21: 129–176

• Sur la dynamique de l’électron on French Wikisource
• On the Dynamics of the Electron on English Wikisource

• Silberstein, L. (1912) [1911], "Quaternionic form of relativity", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23 (137): 790–809, doi:10.1080/14786440508637276
• Sommerfeld, A. (1910a), "Zur Relativitätstheorie I: Vierdimensionale Vektoralgebra", Annalen der Physik, 337 (9): 749–776

• On the Theory of Relativity I: Four-dimensional Vector Algebra on English Wikisource

• Sommerfeld, A. (1910b), "Zur Relativitätstheorie II: Vierdimensionale Vector Analysis", Annalen der Physik, 338 (14): 649–689

• On the Theory of Relativity II: Four-dimensional Vector Analysis on English Wikisource