I'm not quite sure from where I first heard this: that the proper spelling is Lorenz gauge and Lorentz transformation. The former is named after a Dane by the name of Ludwig Valentin Lorenz, the latter is the Dutch Hendrik Antoon Lorentz.

The experience was remarkable purely because it was one of those cases that the author made his point, in a footnote, by pointing the citation to another footnote in a different book authored by someone else.

And I remember that second someone else was somebody really famous. But that's about all I remember.

The gauge condition without the "t" is a condition in electric-magnetism and Yang-Mills theory that requires the gauge fields to be completely divergence free. Since any element of the Lie algebra commutes with itself, the gauge condition is usually written as

D^μA_μ = ∂^μA_μ = 0

where the Einstein summation convention is assumed and the sum is over all space-time indices. This is to be compared against the Coulomb gauge condition

∂^jA_j = 0

where the summation is over only space indices.

The gauge condition is used in analysis of the equations arising from those theories because us analysts have a hard time dealing with equivalence classes. Basically, the idea is that nature has some inherent symmetry so that two things that are written mathematically different can be physically the same. (Think about different points of view. The same one-way street can be described as allowing cars to go from right to left or left to right depending on which side of the street you are standing on.) To make the discussion of mathematical ideas precise, we want to fix a point of view (say, looking at the traffic from the north or east side of the street). In that manner, we can pin down mathematically the "right" way of describing a physical manifestation. The gauge condition does exactly that.

The transformation with a "t", on the other hand, describes the space-time symmetry of Minkowski space/special relativity. Confusingly enough, it also first arose in the context of electromagnetism (which is one of the contributing factors why many people gets them confused). Whereas the Lorenz gauge is a statement about the internel symmetry of photons, the Lorentz transformation makes precise what sort of external symmetries of spacetime are allowed in relativity theory.

Yesterday, during his seminar on local well-posedness for the Maxwell-Dirac system, Sigmund Selberg made a comment about the proper spelling of Lorenz, which drew a chuckle from several members of the audience.

Today, as a sort of weird confluence of ideas, a paper with the title "Ludwig Valentin Lorenz is the discoverer of the 'Lorenz gauge'" was posted to the arXiv.

And lastly, neither of them has anything to do with Edward Lorenz, of the Lorenz attractor fame (chaos theory, dynamical systems, fractals, weather prediction).