Some Remarks on “A Contribution to Electrodynamics” by Bernhard Riemann

Abstract

Around 1850, the idea originated that electromagnetic forces between moving charges in circuits are propagated with the velocity of light. After such a speculation by C. F. Gauss in 1845, B. Riemann, in 1858, suggested the inhomogeneous wave equation in 3-dimensional space for the modeling of this propagation. He found a particular solution replacing Coulomb’s potential, now called the retarded potential. His attempt failed to derive from this solution Weber’s action-at-a-distance potential. Riemann withdrew his pertinent paper before it became printed. After a description of some aspects of research by Gauss, Weber and Riemann, a likely reason for Riemann’s withdrawal is specified differing from recent suggestions by historians of mathematics.

Appendix 1

Electric field $\overset{\rightarrow }{E}$ and magnetic field $\overset{\rightarrow }{B}$ are combined in the the field tensor of the electromagnetic field $F = F_{ik} dx^i\wedge dx^k~ (i, k = 0, 1,2,3)~ ,$ which can be expressed by the 4-potential $A = A_i dx^i$ as $F= dA~$ with (in components) $A_{i}\simeq (\phi ,-\overset{\rightarrow }{A})$ where $\phi$ is the scalar, $\overset{\rightarrow }{A}$ the vector potential. Thus, $F_{ik}= \partial _{i} A_k -\partial _{k} A_i~ ,$ with $F_{0k}\simeq \overset{\rightarrow }{E}= -\overset{\rightarrow }{\nabla }{\phi }-\frac{1}{c}\frac{\partial \overset{\rightarrow }{A}}{\partial t}~, F_{\mu \nu }~ (\mu , \nu = 1, 2, 3) \rightarrow \overset{\rightarrow }{B} = \overset{\rightarrow }{\nabla }\times \overset{\rightarrow }{A}~$. From the first of Maxwell’s equations :

$$\begin{aligned} \partial _{l}F^{il}= \frac{4\pi }{c}j^{i}~~, ~~\partial _{l} F^{* il}= 0 \end{aligned}$$

(20)

with $F^{ik} = \eta ^{ir} \eta ^{ks} F_{rs} ~, ~F^{* ik}= \frac{1}{2}\epsilon ^{iklm} F_{lm}~,$ the Minkowski metric $\eta _{ik}$, and the 4-current $J^i \simeq ( c\rho , \overset{\rightarrow }{j})$, we obtain:

$$\begin{aligned} \partial ^{i} \partial _{l}A^{i} - \partial _{l} \partial ^{l}A^{i} = \frac{4\pi }{c}j^{i} ~. \end{aligned}$$

(21)

As the vector potential is determined only up to gauge transformations $A \rightarrow A' = A + d\lambda$ with a scalar function $\lambda$, a so-called gauge condition may be added. Taking the Lorenz gauge $\partial _{l}A^{l}=0$, from (2) the inhomogeneous wave equation follows:

$$\begin{aligned} \square A^{i} = -\frac{4\pi }{c}j^{i} \end{aligned}$$

(22)

with $\square = \partial _s \partial ^s= \eta ^{rs}\partial _r \partial _s$. The Lorentz gauge condition then leads to $\partial _sj^{s}= 0$, i.e., to the equation for the conservation of electrical charge. For the scalar potential, then

$$\begin{aligned} \square \phi =\frac{1}{c^2}\frac{\partial ^2 \phi }{\partial t ^2} - \overset{\rightarrow }{\nabla }\cdot \overset{\rightarrow }{\nabla } \phi = -4\pi \rho ~. \end{aligned}$$

(23)

For a static electric field, Poisson’s equation follows with the Coulomb potential

$$\begin{aligned} \phi (x) = \frac{1}{4\pi } \int d^3x' \frac{\rho (x')}{|\overset{\rightarrow }{x}-{\overset{\rightarrow }{x'}}|}~. \end{aligned}$$

(24)

The retarded potential is a particular solution of (23):

$$\begin{aligned} \phi (x) = \frac{1}{4\pi } \int d^3x' \frac{\rho (x', t- \frac{|\overset{\rightarrow }{x}-\overset{\rightarrow }{x'}|}{c})}{|\overset{\rightarrow }{x}-{\overset{\rightarrow }{x'}|}}~ \end{aligned}$$

(25)

vanishing at spacelike infinity. It replaces Coulomb’s potential for an arbitrarily time-dependent charge distribution.

Retarded and advanced solutions are combined in:

$$\begin{aligned} A^{i} =\frac{2}{c} \int d^{4}x' \theta (x'-x)\delta [(x^{s}-x'^{s})(x_{s}-x'_{s})] j^{i} \end{aligned}$$

(26)

with Dirac’s $\delta$-distribution and the characteristic function $\theta (x'-x)= 0, +1$ or $0, -1$ selecting directions into the future and past lightcone [10]. With the expression for the electrical current

$$\begin{aligned} j^{i}= c e \int _{-\infty }^{+\infty } ds~ u^{i} \delta ^4 (x-x')~, \end{aligned}$$

(27)

where $u^{i}\simeq \gamma (c, \overset{\rightarrow }{v})$, $\gamma = (1-\frac{v^2}{c^2})^{-\frac{1}{2}}$, and e the electrical charge of a point particle, then the so-called Liénard-Wiechert potential results:

$$\begin{aligned} \phi = \frac{e}{|\overset{\rightarrow }{x}-{\overset{\rightarrow }{x'}}|} (1- \frac{1}{c}\overset{\rightarrow }{n}\cdot \overset{\rightarrow }{v})^{-1}~, ~\overset{\rightarrow }{A}= \frac{e \overset{\rightarrow }{v}}{|\overset{\rightarrow }{x}-{\overset{\rightarrow }{x'}}|} (1- \frac{1}{c}\overset{\rightarrow }{n}\cdot \overset{\rightarrow }{v} )^{-1}~, \end{aligned}$$

(28)

with $\overset{\rightarrow }{v}, {\overset{\rightarrow }{x'}}$ taken at the retarded time; $\overset{\rightarrow }{n}= ({\overset{\rightarrow }{x}-\overset{\rightarrow }{x'}}) (|{\overset{\rightarrow }{x}-\overset{\rightarrow }{x'}}|)^{-1}$.

(28) is different from Riemann’s Ansatz (16) criticized by Wiechert; cf. (17) in Sect. 4.