A non-zero divergence for the E field means that there are charge sources and/or sinks, i.e. it is no longer empty space.

yep, but my sentence is about the wave equation: is it possible to demonstrate that adding an irrotational field, we have still a solution of the wave equation? this is also related to the fact that wave ewuation and maxwell equation are not equivalent, there are more solutions of the wave equations which do not satsfy the maxwellequation

@AdrianoDelVincio 'the fact that wave ewuation and maxwell equation are not equivalent,' They are equivalent.

I am really confused, because my book states the opposite: starting from the maxwell equation in empty space, we can obtain the wave equation, however adding an irrotational field to a solution is still a solution of the wave equation, is this correct?

You can add any field that obeys the wave equation, and the sum will still obey it. This addition can be a field with a non-zero curl, or a field with a zero curl; this does not matter. The important thing is that it changes in time in a proper way, to obey the wave equation.

@my2cts this is incorrect, in addition to the wave equation we require seperate conditions. Such as obeying faradays law and gauss law. This fixes specific orientations of E and B, and their related magnitudes.

@jensenpaull All you need is the wave equation for the potential $A^\mu$.

@my2cts Sorry, this is not correct, one must have an additional condition like the Lorentz or Coulomb gauge. Wave phenomena are far more general than those admitted by the Maxwell equations. See the relevant section in J.D. Jackson's venerable text.

@AlbertusMagnus All you need is current conservation. The wave equation then implies the Lorenz condition.

@my2cts Well, then it is true that the wave equation alone is not sufficient. Current conservation is the Lorenz condition, i.e. $\partial_\mu A^\mu=0$. Thus, requiring only that $A^\mu$ satisfy the wave equation is not enough.

@AlbertusMagnus Charge-current conservation is not a condition on the field. It is a property of the matter that sources the field.

@my2cts There is literally no difference; saying "condition on the field" or saying "property of matter" are two different ways of saying the same thing. At any rate, each to his own!

You are mixing up the properties of photons with those of charged matter. I know this is the mainstream way of thinking, but I am surprised how physicists repeat this. Just consider this: charge conservation is an experimental fact, while the Lorenz condition is a convention in the mainstream line of thinking. How can these two be the same?

@my2cts the lorenz gauge condition still specifies additional properties that the inhomogenous wave equation does not. This is very different than saying that the inhomogenous wave equations for the vector and scalar potential satisfy maxwells equations and is equivalent. The lorenz gauge condition implies the inhomogenous wave equation, the converse is not true

@my2cts Physicists keep repeating this because it is useful, in the framework of classical EM, photons do not exist! The two are the same in that the convention is a representation of the experimental fact. No matter how you slice it, your initial comment was at the very least incomplete. Thats my 99 cents! LOL!

@AlbertusMagnus 'Usefulness' is opinion based. 'the convention is a representation of the experimental fact'. Nice joke. Conventions cannot represent experimental facts.

@my2cts Well you got me on that one, but that was not the issue of this argument anyways. It is indeed better to say: we have the convention because of the experimental fact.

@my2cts Will you at least agree to say that: "the wave equation along with the experimental fact of charge conservation are equivalent to the Maxwell equations", rather than: "they are equivalent"?

@AlbertusMagnus How is that different? It sounds all the same gobbledegook to me. Note that I respect your right to base your opinion on a convention.

You have got to be kidding me!

With that last statement I completely agree! I mean "the wave equation along with the experimental fact of charge conservation are equivalent to the Maxwell equations". In vacuum that is. In media you have to add linear response theory.

So I think the deal with you is that from your point of view, if one has an $A^\mu$, then it is an actual physical entity (since by definition it is a vector potential) and thus is always linked to the physical facts so that bringing them up is sort of redundant. I think some of us were trying to say that you can't just take a vector called $A^\mu$ that satisfies the wave equation and promote it to the status of vector potential. D'accord?

@AlbertusMagnus It is possible that at some point in the future charge conservation is compromised. Then Maxwell theory would no longer hold, as it requires charge conservation. Speculation, sure.

I am gonna take that as a yes!

"so that bringing them up is sort of redundant." I don't get that part. In my view $A^\mu$ is one-to-one connected to $j^\mu$. The restriction $\partial_\mu j^\mu=0$ then implies the Lorenz condition.

Yeah that is exactly what I am getting at, if you always see that $A\mu$ is physical then it automatically satisfies the wave Maxwell equations.

@AlbertusMagnus It solves Maxwell's equation if charge is conserved, which so far is always the case. In the hypothetical case that charge were _not_conserved $\partial_\mu A^\mu = \chi$, some scalar. So there is a one to one correspondence of charge _non_conservation with a specific gauge. $F^{\mu\nu}$ would be unaffected by charge non-conservation. Shall we move to chat?