There is a condition, written down in 1867, that every student of electromagnetism meets sooner or later. It relates the two potentials of the electromagnetic field, and it is the reason those potentials can be made to propagate causally, at the speed of light, rather than adjusting themselves instantaneously across all of space.
It is named after the man who published it. Almost everyone attributes it to somebody else.
What a gauge is
Readers who already work with gauge freedom can skip to the next section. For everyone else, the idea is worth setting out plainly, because the whole story turns on it.
Electromagnetism can be described in two ways. One uses the fields directly — the electric field E and the magnetic field B, the things that push on a charge and that an instrument can measure. The other uses potentials: a scalar φ and a vector A, from which the fields are obtained by differentiation.
The awkward fact about the second description is that it is not unique. Given any charge and current distribution, there is no single correct pair (φ, A). There are infinitely many, and they all describe exactly the same physical situation. Take any sufficiently smooth function χ of position and time, and transform:
The potentials change. The fields do not. Every measurable consequence is identical.
The closest everyday parallel is the choice of a zero for height. To work out the potential energy of a book on a shelf, you have to decide where you are measuring from — the floor, the ground outside, sea level, the centre of the Earth. Each choice gives a different number, and none of them is the right one. But the difference in energy between the shelf and the floor comes out the same no matter which you pick, and that difference is the thing you can actually measure when the book falls. The zero is a convention. The drop is physics.
Gauge freedom is the same situation, with a much larger space of conventions available. Choosing a gauge means picking one member of the family and agreeing to work with it. A quantity is called observable when it comes out the same regardless of that choice — and only observables can correspond to something an experiment could detect. A quantity that changes when you change gauge is an artefact of the bookkeeping, however useful it may be along the way.
The condition at the top of this page is one such choice. It does not add physics. It fixes a convention — and, as it turns out, a particularly good one.
Copenhagen, 1867
Ludvig Valentin Lorenz was born in Helsingør in 1829 and became the first Danish theoretical physicist to gain an international reputation. In 1867 he published, in the Philosophical Magazine, a paper with the title On the Identity of the Vibrations of Light with Electrical Currents.
He was working from Maxwell’s theory, which he had read closely. What he obtained were general integral solutions of the field equations in which the finite speed of light is built in explicitly: the potential at a point now depends on what the sources were doing earlier, at a time separated by exactly the light-travel delay. These are the retarded potentials.
The gauge condition falls out of this construction rather than being imposed on it. That is the point that gets lost when it is described as one convenient choice among several: it is the choice under which the potentials themselves respect causality, propagating outward at c like everything else in the theory.
Maxwell, for his part, did not receive the work warmly.
Göttingen, 1861
There is an earlier thread, reported by the historian Edmund Whittaker in his History of the Theories of Aether and Electricity. Bernhard Riemann had discussed substantially the same condition in lectures in 1861 — six years before Lorenz, and never in print.
This is a familiar shape in the history of science, and it cuts in an unexpected direction here. It does not diminish Lorenz’s claim; it strengthens it. Priority in science attaches to publication, not to private insight, precisely because publication is what makes an idea available to everyone else. Riemann thought of it. Lorenz put it where it could be used.
Which makes what happened next harder to excuse.
The other Lorentz
Hendrik Antoon Lorentz was Dutch, born in 1853, and a physicist of the first rank in his own right: the transformations that bear his name, the electron theory of matter, the Nobel Prize in 1902 shared with Zeeman. He is one of the figures without whom twentieth-century physics does not happen.
He also had nothing to do with the gauge condition. When Lorenz published it in 1867, Lorentz was fourteen years old.
The attribution nonetheless drifted to him, and stuck, for two reasons that reinforce each other. The first is simply that the names sound alike and differ by one letter. The second is more interesting: the Lorenz condition happens to be Lorentz-invariant. It is compatible with special relativity, unchanged in form under a change of inertial frame — which is exactly what one would expect of something Lorentz had produced, and which is in fact a coincidence of a Danish physicist writing four decades before relativity existed.
An error that plausible is very hard to dislodge. It appears in serious scientific literature and in reference texts, Feynman’s among them, and it has needed a small specialist literature of its own to document: van Bladel in 1991, Nevels and Shin in 2001, and the historical review by Jackson and Okun in the same year.
It is a clean case study in how attribution actually works. A name does not win in collective memory by being correct. It wins by being familiar, and by fitting the story the reader already expects.
Why the choice matters
Gauge freedom means no choice is more true than another. It does not mean every choice is equally convenient, and the differences are substantial.
The Coulomb gauge sets the divergence of A to zero. It reduces the equation for the scalar potential to an ordinary Poisson equation, which makes electrostatic problems straightforward — at the cost that φ then appears to respond instantaneously to a change in the charge distribution anywhere in the universe. Nothing observable propagates faster than light; the apparent instantaneity is confined to the potential and cancels in the fields. But you have to know that, and keep track of it.
The Lorenz gauge does not have that feature. Both potentials satisfy wave equations, both propagate at c, and causality is manifest in the formalism rather than something one has to argue for afterwards.
If gauges are lenses, this is the sense in which they differ: not that one shows more of the object, but that each is ground for a different kind of work. The Coulomb gauge is the right lens for a static problem. The Lorenz gauge is the right lens for anything that has to travel.
It is worth calling it by the name of the man who published it.
Sources
- L. V. Lorenz, “On the Identity of the Vibrations of Light with Electrical Currents,” Phil. Mag., series 4, vol. 34, pp. 287–301, 1867.
- E. T. Whittaker, A History of the Theories of Aether and Electricity, vol. 1. London: Nelson, 1910; rev. 1951.
- J. Van Bladel, “Lorenz or Lorentz?,” IEEE Antennas and Propagation Magazine, vol. 33, no. 2, p. 69, 1991.
- R. Nevels and Chang-Seok Shin, “Lorenz, Lorentz, and the Gauge,” IEEE Antennas and Propagation Magazine, vol. 43, no. 3, pp. 70–71, 2001.
- J. D. Jackson and L. B. Okun, “Historical Roots of Gauge Invariance,” Rev. Mod. Phys., vol. 73, pp. 663–680, 2001.
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