History of Topics in Special Relativity/Four-current

Overview
The Four-current is the four-dimensional analogue of the 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 speed of light, $$U^{\alpha}$$ the four-velocity, ρ is the charge density, $$\rho_{0}$$ the rest charge density, and j the conventional 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 and. It was explicitly formulated in modern form by and reformulated in different notations by, , , , , ,. The generally covariant form was first given by and.

Poincaré (1905/6)
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, 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)
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, 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 spherical wave transformations), was given by 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)
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}^{\prime2}}=\varrho_{0}\right] \end{matrix}$$

equivalent to four-current (a).

Sommerfeld (1910)
In influential papers on 4D vector calculus in relativity, 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{align}P & =\mathfrak{Div}\mathrm{Rot}\ \Phi=\mathfrak{Div}\ f\\

-P & =\square\Phi,\ (\mathrm{Div}\ \Phi=0)\\ \mathrm{Div}\ P & =0 \end{align} \\ \left[\begin{align}\mathrm{Rot} & =\text{exterior product}\\ \mathrm{Div} & =\text{divergence four-vector}\\ \mathfrak{Div} & =\text{divergence six-vector}\\ \square & =\text{D'Alembert operator} \end{align} \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)
Gilbert Newton Lewis (1910) devised an alternative 4D vector calculus based on Dyadics which, however, never gained widespread support. The four-current is a “1-vector”:


 * $$\begin{align}\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{align} $$

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


 * $$\begin{matrix}\begin{align}\lozenge\lozenge\times\mathbf{m} & =\mathbf{q}\\

\lozenge\mathbf{M} & =\mathbf{q}\\ \lozenge^{2}\mathbf{m} & =-\mathbf{q} \end{align} \\ \begin{align}\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{align} \\ \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 Edwin Bidwell Wilson used only real coordinates, writing the above expressions as


 * $$\begin{matrix}\begin{align}\lozenge\cdot\mathbf{M} & =4\pi\mathbf{q}\\

\lozenge^{2}\mathbf{m} & =-4\pi\mathbf{q} \end{align} \\ \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, 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{align}P & =\varDelta iv\ (\mathfrak{M})\\ -P & =\square\Phi\ (Div\ \Phi=0)\\ Div\ (P) & =0 \end{align} \\ \left[\begin{align}\mathfrak{Rot} & =\text{exterior product}\\ Div & =\text{divergence four-vector}\\ \varDelta iv & =\text{divergence six-vector}\\ \square & =\text{D'Alembert operator} \end{align} \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)
Ludwik Silberstein devised an alternative 4D calculus based on 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{align}\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{align} \\ \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)
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), 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{align}\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{align} \\ \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.