History of Topics in Special Relativity/Four-potential

Overview
The Electromagnetic four-potential $$A^\alpha$$ combines both an electric scalar potential and a magnetic vector potential into a single four-vector satisfying the Lorenz gauge condition:


 * $$\begin{matrix}A^{\alpha}=\left(\phi/c,\mathbf{A}\right), & \ \left[\partial_{\mu}A^{\mu}=0\right]\\

(a) & (L) \end{matrix}$$

Its product with the D'Alembert operator can be related to the four-current $$J^\beta$$ or the four-divergence of the electromagnetic tensor $$F^{\alpha\beta}$$ representing Maxwell equations:


 * $$\begin{matrix}\square A^{\beta} & =\mu_{0}J^{\beta} & =\partial_{\alpha}F^{\alpha\beta}\\

& (b) & (c) \end{matrix}$$

and by forming the exterior derivative (four-curl) it produces the electromagnetic tensor:


 * $$(d)\quad F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$

The four-potential has the solution:


 * $$(e)\quad A^{\alpha}=-\frac{1}{4\pi^{2}}\int\frac{J^{\alpha}}{x^{2}+y^{2}+z^{2}-t^{2}}dx\,dy\,dz\,dt$$

The electric part of (b) was given by Riemann in 1858, while the complete equation (b) together with the Lorenz gauge condition of the four-potential was given by L. Lorenz in 1867, all of which was popularized by H. A. Lorentz in 1892. Solution (e) of the four-potential was given by, while the Lorentz transformation of all components of (a) was given by and. The modern treatment of the four-potential was given by and was elaborated by, , , , , , and embedded in a generally covariant treatment of electromagnetism by  and.

Kirchhoff, Riemann, Lorenz, Lorentz (1857-1892)
Gustav Kirchhoff (1857) defined the continuity equation $$\frac{du}{dx}+\frac{dv}{dy}+\frac{dw}{dz}=-\frac{1}{2}\frac{d\epsilon}{dt}$$ for electric density and the "Kirchhoff gauge condition” $$\frac{dU}{dx}+\frac{dV}{dy}+\frac{dW}{dz}=\frac{1}{2}\frac{d\Omega}{dt}$$ for potentials in which ”u,v,w” depend on U,V,W. This is similar to the Lorentz gauge condition (L) with a sign change, yet Kirchhoff was still thinking in terms of Weber's electrodynamics involving actions at a distance.

In a lecture given in 1858, published 1867, Bernhard Riemann defined the retarded electric potential ”U” satisfying


 * $$\frac{d^{2}U}{dt^{2}}-\alpha\alpha\left(\frac{d^{2}U}{dx^{2}}+\frac{d^{2}U}{dy^{2}}+\frac{d^{2}U}{dz^{2}}\right)+\alpha\alpha4\pi\varrho=0$$

This corresponds to the electrical part of the inhomogeneous electromagnetic wave equation (b1), but Riemann didn't discuss the corresponding magnetic part (b2) in his model.

Ludvig Lorenz (1867) gave the first gave a complete formulation of the electromagnetic potential. Elaborating on Kirchhoff's work, Lorenz redefined u, v, w by replacing U, V, W with the retarded magnetic potentials $$\alpha,\beta,\gamma$$ and $$\Omega$$ by the retarded electric potential $$\overline{\Omega}$$, independently derived Maxwell's equations, and formulated the following conditions


 * $$\begin{matrix}\frac{du}{dx}+\frac{dv}{dy}+\frac{dw}{dz}=-\frac{1}{2}\frac{d\epsilon}{dt} & (1)\\

\frac{d\overline{\Omega}}{dt}=-2\left(\frac{d\alpha}{dx}+\frac{d\beta}{dy}+\frac{d\gamma}{dz}\right)\text{ or }\frac{d\alpha}{dx}+\frac{d\beta}{dy}+\frac{d\gamma}{dz}=-\frac{1}{2}\frac{d\overline{\Omega}}{dt} & (2)\\ \left(\Delta_{2}-\frac{d^{2}}{a^{2}dt^{2}}\right)\int\int\int\frac{dx'dy'dz}{r}\phi\left(t-\frac{r}{a},x',y',z'\right)=-4\pi\phi(t,x,y,z) & (3)\\ \left(\Delta_{2}=\frac{d^{2}}{dx^{2}}+\frac{d^{2}}{dy^{2}}+\frac{d^{2}}{dz^{2}}\right) \end{matrix}$$

Equations (1) is the continuity condition for the electromagnetic four-current, (2) the Lorenz gauge condition of the four-potential (L), while (3) is used to define the inhomogeneous electromagnetic wave equation.

These methods were popularized by Hendrik Lorentz (1892) who incorporated them into his theory of electrons and immobile aether, in which fields and electrons were strictly separated


 * $$\begin{matrix}\chi=-\frac{1}{4\pi V^{2}}\int\frac{1}{r}F\left(t-\frac{r}{V},x',y',z'\right)d\tau'\\

V^{2}\Delta\chi-\frac{\partial^{2}\chi}{\partial t^{2}}=F(t,x,y,z)\\ \left(\Delta=\frac{\partial^{2}}{dx^{2}}+\frac{\partial^{2}}{dy^{2}}+\frac{\partial^{2}}{dz^{2}}\right)\\ \hline \begin{align}V^{2}\Delta\omega-\frac{\partial^{2}\omega}{\partial t^{2}} & =\rho_{0}\\ V^{2}\Delta\chi_{1}-\frac{\partial^{2}\chi_{1}}{\partial t^{2}} & =\rho_{0}\mathrm{x}\\ V^{2}\Delta\chi_{2}-\frac{\partial^{2}\chi_{2}}{\partial t^{2}} & =\rho_{0}\mathrm{y}\\ V^{2}\Delta\chi_{3}-\frac{\partial^{2}\chi_{3}}{\partial t^{2}} & =\rho_{0}\mathrm{z} \end{align} \end{matrix}$$

Herglotz (1904)
Gustav Herglotz – similar to Arthur W. Conway in 1903 – showed that the wave equation in terms of potential $$\varphi$$


 * $$\frac{\partial^{2}\varphi}{\partial t^{2}}-c^{2}\left(\frac{\partial^{2}\varphi}{\partial x^{2}}+\frac{\partial^{2}\varphi}{\partial y^{2}}+\frac{\partial^{2}\varphi}{\partial z^{2}}\right)=0$$

has the solution in terms of the complex variable $$\tau$$


 * $$\begin{matrix}\varphi=\frac{c}{4\pi^{2}i}\int\frac{s(t-\tau)d\tau}{R_{t-\tau}^{2}-c^{2}\tau^{2}}\\

\left[R^{2}=(x-x')^{2}+(y-y')^{2}+(z-z')^{2}\right] \end{matrix}$$

from which he obtained the retarded potential $$\Phi$$ in terms of charge density $$\varrho$$ and four coordinates $$(x',y',z',t-\tau)$$ as follows


 * $$\Phi=\frac{c}{4\pi^{2}i}\int d\tau\int\frac{\varrho\left(x',y',z',t-\tau\right)}{R^{2}-c^{2}\tau^{2}}dv$$

Using these results, Herglotz went on to determine the force exerted by an electron on another one by defining the potential $$P(T,\alpha)$$, which he interpreted as the "ordinary four-dimensional mutual potential of two three-dimensional spherical disks located in four-dimensional space", in which $$\alpha$$ is the radius and T the distance of their centers.

called Herglotz's equations "the most natural representation of electrodynamic potential in the sense of relativity theory", which is notable because this solution was given by Herglotz already in 1904 before the spacetime representations of Poincaré and Minkowski. Sommerfeld also remarked that Minkowski privately told him that the four-dimensional symmetry of electrodynamics is latently contained and mathematically applied in Herglotz's paper.

Poincaré (1905/6)
Henri Poincaré in July 1905, published 1906, showed that the four quantities related to the electromagnetic potential (defined in relation to the components of the four-current using the D'Alembert operator) in different frames are related to each other by Lorentz transformations


 * $$\begin{matrix}\square^{\prime}\psi^{\prime}=-\rho^{\prime},\quad\square^{\prime}F^{\prime}=-\rho^{\prime}\xi^{\prime},\dots\\

\hline \psi^{\prime}=\frac{k}{l}(\psi+\epsilon F),\ F^{\prime}=\frac{k}{l}(F+\epsilon\psi),\ G^{\prime}=\frac{1}{l}G,\ H^{\prime}=\frac{1}{l}H\\ \left[k=\frac{1}{\sqrt{1-\epsilon^{2}}},\ l=1,\ \square^{\prime}=\sum\frac{d^{2}}{dx^{\prime2}}-\frac{d^{2}}{dt^{\prime2}}\right] \end{matrix}$$

satisfying the Lorenz gauge condition


 * $$\frac{d\psi^{\prime}}{dt^{\prime}}+\sum\frac{dF^{\prime}}{dx^{\prime}}=0$$

Even though Poincaré didn't directly use four-vector notation in this case, his quantities are the components of the four-potential in arbitrary inertial frames.

Marcolongo (1906)
Roberto Marcolongo, citing Poincaré, defined the general Lorentz transformation $$\alpha,\beta,\gamma,\delta$$ of the components of the four-potential $$\mathbf{J},\varphi$$:


 * $$\begin{matrix}(J_{x},J_{y},J_{z})=\mathbf{J},\ (J_{x}^{\prime},J_{y}^{\prime},J_{z}^{\prime})=\mathbf{J}'\\

\hline J_{x'}^{\prime}=\alpha_{1}\mathbf{J}_{x}+\beta_{1}\mathbf{J}_{y}+\gamma_{1}\mathbf{J}_{z}-i\delta_{1}\varphi\\ \dots\\ \varphi'=i\alpha_{4}\mathbf{J}_{x}+i\beta_{4}\mathbf{J}_{y}+i\gamma_{4}\mathbf{J}_{z}+\delta_{4}\varphi \end{matrix}$$

equivalent to the components of (a) and the relation to the components of the four-current


 * $$\begin{matrix}(J_{x},J_{y},J_{z})=\mathbf{J},\ (J_{x}^{\prime},J_{y}^{\prime},J_{z}^{\prime})=\mathbf{J}'\\

\hline \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'\\ \Box\mathbf{J}'_{x}=\Box\left(\alpha_{1}\mathbf{J}_{x}+\beta_{1}\mathbf{J}_{y}+\gamma_{1}\mathbf{J}_{z}-i\delta_{1}\varphi\right),\dots \end{matrix}$$

equivalent to the components of (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-potential with $$\psi_{1},\ \psi_{2},\ \psi_{3}$$ as the components of the vector potential, and $$\Phi$$ as scalar potential:


 * $$\psi\Rightarrow\left(\psi_{1},\ \psi_{2},\ \psi_{3},\psi_{4}\right)\ \psi_{4}=i\Phi$$

equivalent to (a), pointed out the relation to the four-current using the D'Alembert operator


 * $$\square\psi_{j}=-\varrho_{j}\ (j=1,2,3,4),$$

equivalent to (b), as well as the relation to the electromagnetic tensor (which he called "Traktor”) by setting the exterior derivative (four-curl):


 * $$\psi_{jk}=\frac{\partial\psi_{k}}{\partial x_{j}}-\frac{\partial\psi_{j}}{\partial x_{k}}$$

equivalent to (d).

Born (1909)
Max Born (1909) defined the four-potential and the Lorenz gauge condition


 * $$\begin{matrix}\left(\Phi_{x},\ \Phi_{y},\ \Phi_{z},\ i\Phi\right)\Rightarrow\left(\Phi_{1},\ \Phi_{2},\ \Phi_{3},\ \Phi_{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 that its product with the D'Alembert operator corresponds to the four-current $$\varrho_{\alpha}$$


 * $$\frac{\partial}{\partial x_{\alpha}}\sum_{\beta=1}^{4}\frac{\partial\Phi_{\beta}}{\partial x_{\beta}}-\sum_{\beta=1}^{4}\frac{\partial^{2}\Phi_{\alpha}}{\partial x_{\beta}^{2}}=\varrho_{\alpha}\quad\left(\sum_{\beta=1}^{4}\frac{\partial\Phi_{\beta}}{\partial x_{\beta}}=0\right)$$

equivalent to (b), and its exterior derivative (four-curl) forming the electromagnetic tensor


 * $$f_{\alpha\beta}=\frac{\partial\Phi_{\beta}}{\partial x_{\alpha}}-\frac{\partial\Phi_{\alpha}}{\partial x_{\beta}}$$

equivalent to (d).

Bateman (1909/10)
A discussion of four-potential 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:


 * $$A_{x}=\beta(A'_{x}-v\Phi'),\ -\Phi=\beta(vA'_{x}-\Phi'),\ A_{y}=A'_{y},\ A_{z}=A'_{z},\ \left[\beta=\frac{1}{\sqrt{1-v^{2}}}\right]$$

forming the following invarant relations using differential four-position and four-current:


 * $$\begin{matrix}A_{x}dx+A_{y}dy+A_{z}dz-\Phi dt\\

\rho\left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi\right]dx\ dy\ dz\ dt \end{matrix}$$

Sommerfeld (1910)
In influential papers on 4D vector calculus in relativity, Arnold Sommerfeld simplified Minkowski's spacetime formalism and defined the four-potential $$\Phi$$ in relation to the four-current P and the electromagnetic tensor (six-vector) f together with the Lorenz gauge condition:


 * $$\begin{matrix}\Phi\Rightarrow\left(\Phi_{x}=\mathfrak{A}_{x},\ \Phi_{y}=\mathfrak{A}_{y},\ \Phi_{z}=\mathfrak{A}_{z},\ \Phi_{l}=i\varphi\right)\\

\hline \begin{align}\mathrm{Rot}\ \Phi & =f\\ \mathfrak{Div}\mathrm{Rot}\ \Phi & =\mathfrak{Div}\ f=P\\ \square\Phi & =-P\\ \mathrm{Div}\ \Phi & =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}\\ & l=ict \end{align} \right] \end{matrix}$$

equivalent to (a,b,c,d), with the Herglotz solution


 * $$4\pi^{2}\Phi=\int\frac{P}{R^{2}}d\Sigma,\ \left[R^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}+\left(l-l_{0}\right)^{2}\right]$$

equivalent to (e). He also formulated the "electro-kinetic potential” as the scalar product with the four-current


 * $$(P\Phi)$$

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-potential is a “1-vector”


 * $$\begin{align}\mathbf{m} & =\mathbf{a}+i\phi\mathbf{k}_{4}\\

& =a_{1}\mathbf{k}_{1}+a_{2}\mathbf{k}_{2}+a_{3}\mathbf{k}_{3}+i\phi\mathbf{k}_{4} \end{align} $$

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


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

\lozenge\times\lozenge\times\mathbf{m} & =0\\ \lozenge\times\mathbf{m} & =\mathbf{M}=\mathbf{E}+\mathbf{H}\\ \lozenge\mathbf{m} & =0\\ \lozenge^{2}\mathbf{m} & =-\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 (b,c,d).

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


 * $$\begin{matrix}\begin{align}\lozenge^{2}\mathbf{m} & =-4\pi\mathbf{q}\\

\mathbf{M} & =\lozenge\times\mathbf{m}\\ \lozenge\times\mathbf{M} & =\lozenge\times\lozenge\times\mathbf{m}=0\\ \lozenge\cdot\mathbf{M} & =\lozenge\cdot\mathbf{m}(\lozenge\times\mathbf{m})=\lozenge(\lozenge\cdot\mathbf{m})-(\lozenge\cdot\lozenge)\mathbf{m}\\ \lozenge\mathbf{m} & =0 \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 (b,c,d).

Von Laue (1911/13)
In the first textbook on relativity in 1911, Max von Laue elaborated on Sommerfeld's methods and explicitly introduced the term four-potential (Viererpotential) $$\Phi$$ in terms of vector potential $$\mathfrak{A}$$ and scalar potential $$\varphi$$, showing its showed its relation to the four-current P and the electromagnetic tensor (six-vector) $$\mathfrak{M}$$ together with the Lorenz gauge condition


 * $$\begin{matrix}\left(\mathfrak{A},\varphi\right)\Rightarrow\Phi\\

\hline \begin{align}\mathfrak{Rot}\Phi & =\mathfrak{M}\\ \mathfrak{Div}\mathrm{Rot}\ \Phi & =\varDelta iv\ (\mathfrak{M})=P\\ \square\Phi & =-P\\ Div\ \Phi & =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,b,c,d).

In the second edition (preface dated 1912, published 1913), von Laue also formulated the Herglotz solution:


 * $$\Phi=\frac{1}{4\pi^{2}}\int\frac{P}{R^{2}}d\Sigma$$

equivalent to (e).

Silberstein (1911)
Ludwik Silberstein devised an alternative 4D calculus based on Biquaternions which, however, never gained widespread support. He defined the “potential-quaternion” (i.e. four-potential) $$\Phi$$ in relation to the “current-quaternion” (i.e. four-current) C and the “electromagnetic bivector” (i.e. field tensor) $$\mathbf{F}$$


 * $$\begin{matrix}\Phi=i\phi+\mathbf{A}\\

\hline \begin{align}-\mathrm{D_{c}}\Phi & =\mathbf{F}\\ \Box\Phi & =-\mathrm{C},\ \left(\mathrm{S}\mathrm{D}_{c}\Phi=0\right) \end{align} \\ \left[\begin{matrix}\mathrm{D}=\frac{\partial}{\partial l}-\nabla,\ \mathrm{D}_{c}=\text{conjugate}\mathrm{D}\\ \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}} \end{matrix}\right] \end{matrix}$$

equivalent to (a,b,c,d).

Kottler (1912)
Friedrich Kottler defined the four potential $$\Phi_{\alpha}$$ and its relation to four-current $$\mathbf{P}^{(\beta)}$$, electromagnetic field-tensor $$F_{\alpha\beta}$$, and the Herglotz solution


 * $$\begin{matrix}\Phi_{1}=\mathfrak{A}_{x},\quad\Phi_{2}=\mathfrak{A}_{y},\quad\Phi_{3}=\mathfrak{A}_{z},\quad\Phi_{4}=i\varphi\\

\hline F_{\alpha\beta}=\frac{\partial}{\partial x^{(\alpha)}}\Phi_{\beta}-\frac{\partial}{\partial x^{(\beta)}}\Phi_{\alpha}\\ \Box\Phi_{\alpha}=-\mathbf{P}^{(\alpha)}\\ 4\pi^{2}\Phi_{\alpha}(y)=\int dx^{(1)}dx{}^{(2)}dx^{(3)}dx^{(4)}\frac{\mathbf{P}^{(\alpha)}(x)}{R^{2}}\\ \left[R^{(\alpha)}=x^{(\alpha)}-y^{(\alpha)}\right] \end{matrix}$$

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


 * $$\begin{matrix}F_{\alpha\beta}=\frac{\partial\Phi_{\beta}}{\partial x^{(\alpha)}}-\frac{\partial\Phi_{\alpha}}{\partial x^{(\beta)}}=\Phi_{\beta/\alpha}-\Phi_{\alpha/\beta}.\\

\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}$$

Einstein (1916)
In 1916, after finishing his general relativity, Albert Einstein also used the electromagnetic potential $$\varphi_{\nu}$$ as a covariant four-vector, relating it to the covariant six-vector of the electromagnetic field (i.e. electromagnetic field tensor):

$$\varphi_{\nu}\Rightarrow F_{\varrho\sigma}=\frac{\partial\varphi_{\varrho}}{\partial x_{\sigma}}-\frac{\partial\varphi_{\sigma}}{\partial x_{\varrho}}$$

equivalent to (a,d).