Poincaré (1905/6) edit w:Henri Poincaré (June 1905[R 1] ; July 1905, published 1906[R 2] ) showed that the four quantities related to charge density ρ {\displaystyle \rho } are connected by a Lorentz transformation:
ρ , ρ ξ , ρ η , ρ ζ ρ ′ = k l 3 ρ ( 1 + ϵ ξ ) , ρ ′ ξ ′ = k l 3 ρ ( ξ + ϵ ) , ρ ′ η ′ = ρ η l 3 , ρ ′ ζ ′ = ρ ζ l 3 ( June ) ρ ′ = k l 3 ( ρ + ϵ ρ ξ ) , ρ ′ ξ ′ = k l 3 ( ρ ξ + ϵ ρ ) , ρ ′ η ′ = 1 l 3 ρ η , ρ ′ ζ ′ = 1 l 3 ρ ζ ′ ( July ) ( k = 1 1 − ϵ 2 , l = 1 ) {\displaystyle {\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 :[R 3]
d ρ ′ d t ′ + ∑ d ρ ′ ξ ′ d x ′ = 0 D 1 ′ = d ρ ′ d t ′ + ∑ d ρ ′ ξ ′ d x ′ = 0 , D 1 = d ρ d t + ∑ d ρ ξ d x = 0 {\displaystyle {\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) edit Following Poincaré, w:Roberto Marcolongo defined the general Lorentz transformation α , β , γ , δ {\displaystyle \alpha ,\beta ,\gamma ,\delta } of the components of the four independent variables V , ϱ {\displaystyle \mathbf {V} ,\varrho } and its continuity equation:[R 4]
( ξ , η , ζ ) = V , ( ξ ′ , η ′ , ζ ′ ) = V ′ ϱ ′ ξ ′ = ϱ ( α 1 ξ + β 1 η + γ 1 ζ − i δ 1 ) … ϱ ′ = ϱ ( α 4 ξ + β 4 η + γ 4 ζ − i δ 4 ) ∂ ϱ ′ ∂ t ′ + ∂ ϱ ′ ξ ′ ∂ x ′ + ∂ ϱ ′ η ′ ∂ y ′ + ∂ ϱ ′ ζ ′ ∂ z ′ = 0 ( t = i u ) {\displaystyle {\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 J , φ {\displaystyle \mathbf {J} ,\varphi } of the four-potential
◻ J x ′ = − 4 π ϱ ′ ξ ′ = − 4 π ϱ ( α 1 ξ + β 1 η + γ 1 ζ − i δ 1 ) , … ◻ J = − 4 π ϱ V , ◻ φ = − 4 π ρ , ◻ φ ′ = − 4 π ϱ ′ {\displaystyle {\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) edit 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 ϱ {\displaystyle \varrho } as charge density and v {\displaystyle {\mathfrak {v}}} as velocity:[R 5]
( ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 ) = ( ϱ v , i ϱ ) {\displaystyle \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 i {\displaystyle \mathbf {i} } as current and σ {\displaystyle \sigma } as charge density:[R 6]
( σ ) = ( σ 1 , σ 2 , σ 3 , σ 4 ) = ( i x , i y , i z , i σ ) {\displaystyle (\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[R 7]
( ϱ w x , ϱ w y , ϱ w z , i ϱ ) ⇒ ( ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 ) ϱ 3 ′ = x 3 cos i ψ + ϱ 4 sin i ψ , ϱ 4 ′ = − ϱ 3 sin i ψ + ϱ 4 cos i ψ , ϱ 1 ′ = ϱ 1 , ϱ 2 ′ = ϱ 2 ϱ ′ w z ′ ′ = ϱ ( w z − q 1 − q 2 ) , ϱ ′ = ϱ ( − q w z + 1 1 − q 2 ) , ϱ ′ w x ′ ′ = ϱ w x , ϱ ′ w y ′ ′ = ϱ w y , − ( ϱ 1 2 + ϱ 2 2 + ϱ 3 2 + ϱ 4 2 ) = ϱ 2 ( 1 − w x 2 − w y 2 − w z 2 ) = ϱ 2 ( 1 − w 2 ) {\displaystyle {\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” s {\displaystyle {\mathfrak {s}}} which becomes s = σ E {\displaystyle {\mathfrak {s}}=\sigma {\mathfrak {E}}} in isotropic media:[R 8]
( s x , s y , s z , i ϱ ) ⇒ ( s 1 , s 2 , s 3 , s 4 ) {\displaystyle {\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) edit Following Minkowski, w:Max Born (1909) defined the “space-time vector of first kind” (four-vector) and its continuity equation[R 9]
( ϱ w x c , ϱ w y c , ϱ w z c , i ϱ ) ⇒ ( ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 ) ∂ ϱ w x ∂ x + ∂ ϱ w y ∂ y + ∂ ϱ w z ∂ z + ∂ ϱ ∂ t = 0 {\displaystyle {\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 Φ α {\displaystyle \Phi _{\alpha }} :
∂ ∂ x α ∑ β = 1 4 ∂ Φ β ∂ x β − ∑ β = 1 4 ∂ 2 Φ α ∂ x β 2 = ϱ α ( ∑ β = 1 4 ∂ Φ β ∂ x β = 0 ) {\displaystyle {\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
ϱ 1 = ϱ ∗ c ⋅ ∂ x ∂ τ , ϱ 2 = ϱ ∗ c ⋅ ∂ y ∂ τ , ϱ 3 = ϱ ∗ c ⋅ ∂ z ∂ τ , ϱ 4 = i ϱ ∗ ∂ t ∂ τ ( ϱ ∗ = ϱ 1 − w 2 c 2 = − ( ϱ 1 2 + ϱ 2 2 + ϱ 3 2 + ϱ 4 2 ) , d τ = d t 1 − w 2 c 2 ) ϱ α = i ϱ ∗ ∂ x α ∂ ξ 4 ( ξ 4 = i c τ ) {\displaystyle {\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) edit 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 ( ρ w x , ρ w y , ρ w z , ρ ) {\displaystyle \left(\rho w_{x},\rho w_{y},\rho w_{z},\rho \right)} [R 10]
ρ w x = β ( ρ ′ w ′ − v ρ ′ ) , ρ w y = ρ ′ w y ′ , ρ w z = ρ ′ w z ′ , − ρ = β ( v ρ ′ w x ′ − ρ ′ ) , [ β = 1 1 − v 2 ] {\displaystyle \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:[R 11]
1 λ 2 [ ρ w x d x + ρ w y d y + ρ w z d z − ρ d t ] ρ 2 λ 2 ( 1 − w 2 ) d x d y d z d t ρ [ A x w x + A y w y + A z w z − Φ ] d x d y d z d t {\displaystyle {\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 λ 2 = 1 {\displaystyle \lambda ^{2}=1} in relativity.
Ignatowski (1910) edit w:Wladimir Ignatowski (1910) defined the “vector of first kind” using charge density ϱ {\displaystyle \varrho } and three-velocity v {\displaystyle {\mathfrak {v}}} :[R 12]
( ϱ v , ϱ ) [ ϱ 1 − n v 2 = ϱ ′ 1 − n v ′ 2 = ϱ 0 ] {\displaystyle {\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) edit In influential papers on 4D vector calculus in relativity, w:Arnold Sommerfeld defined the four-current P , which he called four-density (Viererdichte):[R 13]
P x = ϱ v x c , P y = ϱ v y c , P z = ϱ v z c , P l = i ϱ β 2 = 1 c 2 ( v x 2 + v y 2 + v z 2 ) ⇒ | P | = i ϱ 1 − β 2 [ l = i c t ] {\displaystyle {\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 Φ {\displaystyle \Phi } and the electromagnetic tensor (six-vector) f together with the continuity condition:[R 14]
P = D i v R o t Φ = D i v f − P = ◻ Φ , ( D i v Φ = 0 ) D i v P = 0 [ R o t = exterior product D i v = divergence four-vector D i v = divergence six-vector ◻ = D'Alembert operator ] {\displaystyle {\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[R 15]
( P Φ ) {\displaystyle (P\Phi )} he called “electro-kinetic potential” whereas the vector product with the electromagnetic tensor[R 16]
( P f ) = F {\displaystyle (Pf)={\mathfrak {F}}} he called the electrodynamic force (four-force density).
Lewis (1910), Wilson/Lewis (1912) edit 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”:[R 17]
q = ϱ c v + i ϱ k 4 = ϱ c v 1 k 1 + ϱ c v 2 k 2 + ϱ c v 3 k 3 + i ϱ k 4 {\displaystyle {\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 m {\displaystyle \mathbf {m} } and electromagnetic tensor M {\displaystyle \mathbf {M} } :
◊ ◊ × m = q ◊ M = q ◊ 2 m = − q ( ∂ H 12 ∂ x 2 + ∂ H 13 ∂ x 3 + ∂ E 14 ∂ x 4 ) k 1 = ϱ c v 1 k 1 ( ∂ H 21 ∂ x 1 + ∂ H 23 ∂ x 3 + ∂ E 24 ∂ x 4 ) k 2 = ϱ c v 2 k 2 ( ∂ H 31 ∂ x 1 + ∂ H 32 ∂ x 2 + ∂ E 34 ∂ x 4 ) k 3 = ϱ c v 3 k 3 ( ∂ H 41 ∂ x 1 + ∂ H 42 ∂ x 2 + ∂ E 43 ∂ x 4 ) k 4 = ϱ c i k 4 [ ◊ = k 1 ∂ ∂ x 1 + k 2 ∂ ∂ x 2 + k 3 ∂ ∂ x 3 + k 4 ∂ ∂ x 4 ◊ 2 = ∂ 2 ∂ x 1 + ∂ 2 ∂ x 2 + ∂ 2 ∂ x 3 + ∂ 2 ∂ x 4 ] {\displaystyle {\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[R 18]
◊ ⋅ M = 4 π q ◊ 2 m = − 4 π q [ ◊ = k 1 ∂ ∂ x 1 + k 2 ∂ ∂ x 2 + k 3 ∂ ∂ x 3 − k 4 ∂ ∂ x 4 ◊ 2 = ∂ 2 ∂ x 1 + ∂ 2 ∂ x 2 + ∂ 2 ∂ x 3 − ∂ 2 ∂ x 4 ] {\displaystyle {\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) edit 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 ϱ {\displaystyle \varrho } in relation to four-potential Φ {\displaystyle \Phi } and electromagnetic tensor M {\displaystyle {\mathfrak {M}}} :[R 19]
P ⇒ ( P x = ϱ q x c , P y = ϱ q y c , P z = ϱ q z c , P l = i ϱ ) P = Δ i v ( M ) − P = ◻ Φ ( D i v Φ = 0 ) D i v ( P ) = 0 [ R o t = exterior product D i v = divergence four-vector Δ i v = divergence six-vector ◻ = D'Alembert operator ] {\displaystyle {\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 M {\displaystyle {\mathfrak {M}}} , four-convection K and four-conduction Λ {\displaystyle \Lambda } using four-velocity Y :[R 20]
F = [ P M ] , ( P F ) = ( P [ P M ] ) = 0 K = − ( Y P ) Y Λ = P + ( Y P ) Y {\displaystyle {\begin{matrix}F=[P{\mathfrak {M}}],\ (PF)=(P[P{\mathfrak {M}}])=0\\K=-(YP)Y\\\Lambda =P+(YP)Y\end{matrix}}} ,Silberstein (1911) edit 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 {\displaystyle \mathbf {F} } and “potential-quaternion” (i.e. four-potential) Φ {\displaystyle \Phi } [R 21]
C = ρ ( ι + 1 c p ) = ι ρ d q d l C = D F = − ◻ Φ S D c C = 0 [ D = ∂ ∂ l − ∇ , D D c = ◻ = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 + ∂ 2 ∂ l 2 ] {\displaystyle {\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) edit w:Friedrich Kottler defined the four-current P ( α ) {\displaystyle \mathbf {P} ^{(\alpha )}} and its relation to four-velocity V ( α ) {\displaystyle V^{(\alpha )}} , four-potential Φ α {\displaystyle \Phi _{\alpha }} , four-force F α {\displaystyle F_{\alpha }} , electromagnetic field-tensor F α β {\displaystyle F_{\alpha \beta }} , stress-energy tensor S α β {\displaystyle S_{\alpha \beta }} :[R 22]
P ( 1 ) = ρ v x c = i ρ 0 V ( 1 ) , P ( 2 ) = ρ v y c = i ρ 0 V ( 2 ) , P ( 3 ) = ρ v z c = i ρ 0 V ( 3 ) , P ( 4 ) = i ρ = i ρ 0 V ( 4 ) ∑ h = 1 4 ∂ F g h ∂ x ( h ) = P ( g ) , ◻ Φ α = − P ( α ) F α ( y ) = ∑ β F α β ( y ) P ( β ) ( y ) 1 − w 2 / c 2 [ ∑ β F α β ( y ) P ( β ) ( y ) = ∑ β F α β ( y ) ∑ γ ∂ ∂ y ( γ ) F β γ ( y ) = ∑ β ∂ ∂ y ( β ) S α β , ρ 0 = ρ 1 − v 2 / c 2 ] {\displaystyle {\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 α β {\displaystyle c_{\alpha \beta }} [R 23]
∑ c ( 1 α ) ∑ β , γ c ( β γ ) Φ α / β γ = − P ( α ) etc . [ ∑ β , γ c ( β γ ) Φ β / γ = 0 ] {\displaystyle {\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) edit 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):[R 24]
ϱ 0 d x ν d s = 1 − g ϱ 0 d x ν d t {\displaystyle \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
∑ ν ∂ ∂ x ν ( − g ⋅ φ μ ν ) = ϱ 0 d x μ d t ∂ H x ∂ y − ∂ H y ∂ z − ∂ E x ∂ t = u x … … ∂ E x ∂ x + ∂ E y ∂ z + ∂ E x ∂ z = ϱ [ ϱ 0 d x μ d t = u μ ] {\displaystyle {\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 μ ν {\displaystyle g_{\mu \nu }} being the Minkowski tensor.