Railguns and Gauge Invariance Paul J. Cote and Mark A. Johnson of Benet Laboratories, Army Research, Engineering and Development Command wrote a series of short papers on the vector potential arising from their attempts to solve engineering problems with railguns.  One of those papers titled "Groupthink and the blunder of the gauges" has, as one of its conclusions, the provocative claim:

a. The dynamic form of Gauss' law contains the hidden law of physics
  $$\nabla\cdot E_I = (\partial^2\varphi_c/\partial t^2)/c^2$$ This is
  one of the universal laws of electromagnetism and is needed to
  supplement Faraday's law of induction in order to permit a full
  definition of $$E_I$$ It can be viewed as a missing Maxwell equation.

Where
$$E_I$$ is the induced electric field.
I've found only one critique -- an anonymous critique at that -- of this and it seems to be a critique of the entire series of papers by Cote and Johnson on the grounds that they misunderstood the subtleties of gauge invariance.
Is this critique valid?
Does the third version of "Groupthink and the blunder of the gauges" render the critique invalid?
 A: I ended up spending an hour or so wading through the papers, so here's my main conculsion from that:


*

*No, I don't think the critique of the papers is wrong;

*Nor do I think that the basic algebraic arguments in the three papers addressed by the critique are wrong either. The authors describe a particular "decomposition" of the electromagnetic fields into a static part and dynamic part. To make this particular decomposition work, they find that a particular gauge choice is helpful and in the process they find they need "missing Maxwell equations". This "missing" equation is nothing more than their gauge condition - exactly analogously with the Lorenz gauge condition $\nabla . A + c^{-2}\partial_t\,\phi =0$;

*Where the papers go monumentally wrong is here: they claim that they have found new physics, as evidenced by their "newly found" so called "missing Maxwell equations". In effect, they are claiming that a particular gauge choice is more valid than others and that their gauge condition defines a physical phenomenon;

*They go on to say that the rest of the physicist community has committed a fundamental "blunder" and only their perspective is the right perspective to look at electrodynamics from.


So 3. and 4. are where the papers go haywire. A scientifically valid version of claim 3 would read something like the following:

"We find a new equation $X$ to extend standard Maxwellian electrodynamics. This equation foretells result $Y$ to our proposed experiment $Z$, whereas standard Maxwellian electrodynamics foretells the different experimental result $Y^\prime$"

That is, they would make a falsifiable prediction of an experimental result.
It is not wrong to propose an unconventional way of looking at known physics - often such new perspectives are helpful, and indeed the first of the papers critiqued does just this, although I don't think that they are the first to look deeply into a decomposition of the EM field in this way: I recall thinking along these lines and I'm doubly sure I'm not particularly original in this regard. The authors do, however, write their way of thinking up in quite a lucid way, so that their first paper, "New Perspectives on Classical Electromagnetism" is, albeit haughtily so, a helpful contribution to physics. As you say, Feynman did exactly the same as the authors' paper 1: for him (and for most of us), there was a most natural gauge choice: the Lorenz invariant one. 
Any gauge can be awkward and mind bending, and I, for example, find it mind bending to think of some EM phenomena in terms of the Lorenz gauge. But Nature does not care about my intellectual shortcomings. We all have our pet ways of thinking about things - ways of thinking that are the most helpful to the particular thinker in question, and we all think differently, so naturally there are many different pet ways of thinking. But there is only one physics: that of Nature, so let us look at what experiment tells us about the electromagnetic interaction.
So far, almost everything that is known about EM interaction is through experiments observing the effect of the EM field on the state of motion of electrically charged bodies. We have an equation for the Lorentz force:
$$\mathrm{d}_\tau P= q \,\mathcal{F}\, V\tag{1}$$
where $P$ is the four-momentum of the body whose state of motion $V$ (the velocity four-vector) is being acted on by the field (the Faraday tensor $\mathcal{F}$). This is the simplest Lorentz-invariant equation we can write for a changing state of motion of a charge $q$ and so far all experiments have confirmed its correctness (note the $t$ in Lorentz - this is Hendrik Lorentz as opposed to Ludvig Lorenz whom the Lorenz gauge is named after). It follows that the physically meaningful description of the electromagnetic field for these experiments is the Faraday tensor  $\mathcal{F}$ and nothing else. At a quantum level, elementary particles like the electron have an intrinsic magnetic dipole moment, so that their spin state precesses in a magnetic field. Thus one has to add this precession to (1). But still, it is experimentally found that only the components of  $\mathcal{F}$ are involved here.
Maxwell's equations describe $\mathcal{F}$ and nothing else. It so happens that the Faraday tensor is a so called exact differential form and can be expressed as a (exterior) derivative of another vector field $A$. Thus, as far as experimental results - i.e. REAL physics goes - any transformation wrought on $A$ is utterly immaterial if the transformation does not change $\mathcal{F} = \mathrm{d} A$, for such a transformation has no bearing on the experimentally witnessed effect of the EM field on the state of motion of charge. Maxwells equations describing the total $\mathcal{F}$ are indisputably gauge invariant. Particular decompositions of $\mathcal{F}$ are not gauge invariant - but a particular decomposition is a mathematical tool, not a description of nature.
There is one other experimental result we must address, and that is the Aharonov Bohm effect: the peculiar interference pattern where electrons are diffracted in a plane past tiny tubes of magnetic flux and the experiment seems to say that the electron's phase depends on whether it passed above or below the flux tubes. That is, the electron's phase is anholonomic, or depends on the path taken. The phase change of an electron as it circles a flux tube is given by $\oint \vec{A}\cdot\mathrm{d}\vec{r}$. This is what the critique means by the statement only the "... electomagnetic field and holonomy integrals ... have physical meaning". Although this curious effect seems to bestow some kind of physical reality on $\vec{A}$ (and thus, through the Lorentz transformation on $\phi$ as well), the phase change attributable to the flux tube $\oint \vec{A}\cdot\mathrm{d}\vec{r}$ is also gauge invariant: $\vec{A}$ gets an arbitrary gradient $\nabla \chi$ added to it in a gauge transformation, which gradient clearly does not contribute to the phase difference around the flux tube. So Maxwell's equations + $\oint \vec{A}\cdot\mathrm{d}\vec{r}$ are still gauge invariant. Moreover, although people seem to prefer to talk about $\vec{A}$ in the AB effect, one can equally well write the holonomy intergral - through Stokes's theorem - as a flux of $\vec{B}$ through the loop, so one could argue we still have only  $\mathcal{F}$ as physically meaningful. This latter standpoint, although logically defensible as not begetting contradiction, is a bit weird because there is no $\vec{B}$ where the electrons go, only a flux of $\vec{B}$ through the closed path. Talking about $\vec{A}$ here seems a little less weird - and more local - than talking about $\vec{B}$. But Nature is weird, and ultimately, experiment so far is gauge invariant.
The authors of the papers make a great deal about the causalilty "contradictions" in the Coulomb gauge. As the critique says, this is not a problem if the deviation from causality is annulled on passing from $A$ to the physical $\mathcal{F}$. Moreover, in the Brill paper referenced by the critique, there is a very fine discussion of how this acausality resolves itself. But talking about causality when talking about even Maxwell's equations alone is futile, for Maxwells equations are not in themselves causal. You cannot derive causality from Maxwell's equations, for any solution of Maxwell's equations can be run backwards in time and still be a solution. That is, Maxwell's equations are invariant with respect to $t\mapsto-t$. For every retarded potential solution of the ME, there is an equally valid advanced potential solution, and indeed Feynman formulated his (ultimately causal) Feynman-Wheeler Absorber Theory by looking at solutions that were $\frac{1}{2}(R+A)$ - $R$ is the retarded potential and $A$ the advanced. Causality comes from the boundary conditions we impose on Maxwell's equations.
Lastly, it might be worth calculating the Baez Crank Index for these papers: the first is quite reasonable, the "...Peculiarity of the the Coulomb Gauge" and "Groupthink and the blunder of the gauges" score, IMO, at least on items 17, 19 and 37 of the Baez index. There are many sloppy statements such as (page 2, "New Perspectives on Classical Electromagnetism"):

"It is important to stress here that the standard approach treats the two terms in parentheses on the left hand side of Eq. (7) as independent and unrelated"

Not so: they are related by the gauge transformation; they together have a continuous, $U(1)$ degree of freedom - quite different from being "unrelated"), but not many of us are brilliant technical communicators, and all (or at least I certainly do) say silly things like this that we don't quite mean before rereading one's work or having it checked by a colleague. Without the claims that everyone else is wrong, one would overlook such slipups as a reviewer and simply suggest a better phrasing. But it is a bit rich to be claiming no-one else appreciates subtlety when you make such slip ups yourself. In my own early career, I had the misfortune to deal with the military as a contract scientist / engineer. The kind of haughty disdain for all unwashed nonmilitary people ("the literature", "conventional scientists" and all the rest) was a daily experience for me. Unfortunately the kind of sentiments bespoken by "Groupthink and the blunder of the gauges" do not surprise me in the least. 
