Abstract
After the introduction of the electric four-current vector field as the basic tool for describing a set of moving charges, Maxwell equations are formulated, first in terms of the electromagnetic field tensor, before being expressed in terms of the electric and magnetic fields with respect to some observer. It is shown that, as a consequence of Maxwell equations, the electric charge is conserved. The solution to Maxwell equations is searched by means of a four-potential 1-form. The associated gauge choice is discussed, and it is shown that within Lorenz gauge, Maxwell equations reduce to a d’Alembert equation for the four-potential. Its general solution is presented in terms of Green functions and is applied to the case of a single particle, leading to the Liénard–Wiechert solution. The Coulombian part and the radiative part of this solution are presented. Finally, it is shown that Maxwell equations can be derived from a principle of least action.
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- 1.
From this point of view, denoting the action of δ A on f by an integral as in (18.3) is an abuse of notation often used in physics.
- 2.
It is denoted by t ′ to keep t for the coordinate orthogonal to \(\mathcal{S}\) in \(\mathcal{W}\).
- 3.
Let us recall that the exterior derivative has been introduced in Sect. 15.5, the Hodge dual in Sect. 14.5 and specifically for \(\boldsymbol{F}\) in Sect. 17.2.5.
- 4.
Let us recall that the space of 3-forms is four-dimensional; cf. (14.40).
- 5.
James Clerk Maxwell (1831–1879): Scottish theoretical physicist, famous for having unified electricity, magnetism and optics.
- 6.
The magnetic potential will be introduced in Sect. 18.5.2.
- 7.
Oliver Heaviside (1850–1925): English physicist and mathematician; self-taught, he contributed to many domains of electromagnetism and mathematics (vector analysis, differential equations).
- 8.
Heinrich Hertz (1857–1894): German physicist, author of many works in electromagnetism and famous for having experimentally shown the existence of electromagnetic waves.
- 9.
Let us recall that for a hypersurface, closed means compact and without boundary.
- 10.
Compact and without boundary, as, for instance, a sphere.
- 11.
This is true at least in classical electrodynamics; things are different in the quantum regime, where \(\boldsymbol{A}\), or more precisely its integral, is directly involved in the measure of a phenomenon called the Aharonov–Bohm effect (cf., e.g. Sect. 12.3.3 of Le Bellac (2006)).
- 12.
Cf. (18.48) and Remark 15.4 p. 504.
- 13.
Note that we have taken the metric dual to get the 1-form \(\boldsymbol{A}\) instead of the vector \(\overrightarrow{\boldsymbol{A}}\).
- 14.
Alfred-Marie Liénard (1869–1958): French physicist and engineer; director of École des Mines in Paris from 1929 to 1936.
- 15.
Emil Wiechert (1861–1928): German geophysicist; he knew quite well Hilbert (p. 361), Minkowski (p. 26) and Sommerfeld (p. 27), because the four of them made their studies at the University of Königsberg and later on were professors at the University of Göttingen.
- 16.
Notably via the identity \(\overrightarrow{\boldsymbol{n}}\boldsymbol{ \times }_{\boldsymbol{u_{0}}}[(\overrightarrow{\boldsymbol{n}} -\overrightarrow{\boldsymbol{V }}/c)\boldsymbol{ \times }_{\boldsymbol{u_{0}}}\overrightarrow{\boldsymbol{\gamma }}] = (\overrightarrow{\boldsymbol{n}} \cdot \overrightarrow{\boldsymbol{\gamma }})(\overrightarrow{\boldsymbol{n}} -\overrightarrow{\boldsymbol{V }}/c) - (1 -\overrightarrow{\boldsymbol{n}} \cdot \overrightarrow{\boldsymbol{V }}/c)\overrightarrow{\boldsymbol{\gamma }}\).
- 17.
One often says simply Lagrangian, instead of Lagrangian density.
- 18.
Klein stands for the Swedish physicist Oskar Klein (1894–1977) and not for the German mathematician Felix Klein mentioned in Chap. 7 (cf. p. 255).
- 19.
Let us recall that in the present case, \(\det g = -1\) since we are using inertial coordinates.
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Gourgoulhon, É. (2013). Maxwell Equations. In: Special Relativity in General Frames. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37276-6_18
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