Tach, you appear to have either completely ignored almost all of my post or you simply didn't understand it. I'm leaning towards the former because even you should have managed to understand it when I
repeatedly said I wasn't defending Eugene's work but making a broader point, yet you continue in your reply to say "Show me how Eugene's equations are right!!!", both failing to grasp the points I'm making about covariance and the very clear statement I made about how Eugene's work is
irrelevant to the point I'm making.
This is in the category "not even wrong" since it was the lack of covariance of the Maxwell laws under Galilei transforms that sparked the quest for the discovery of the Lorentz transforms. You have all your basics thoroughly screwed up. So, no, the Maxwell equations are not covariant in the Galilei formalism. They are in the Einstein-Lorentz formalism.
I explained the difference between a general coordinate transformation, which need not be a linear operator and whose properties are examined in a particular way, and a linear operator which leaves an inner product invariant. You failed to respond to any of it and you appear to have failed in understanding it.
I'm well aware of how Maxwell's equations are Lorentz invariant. This is
not the same as saying "The only coordinate transformations valid are Lorentz ones". You can write Maxwell's equations in polar coordinates (and many physicists do for things like magnetohydrodynamics), which aren't a Lorentz transformation away from Cartesians, and still have valid expressions.
Maxwell's equations are Lorentz invariant in the sense of the inner product invariance. It's a little over the top but the most general way of writing down a Yang Mills gauge theory Lagrangian is along the lines of $$\mathcal{L} = \textrm{Tr}(F_{\mu\nu}F^{\mu\nu}) = F^{a}_{\mu\nu}F^{a}^{\mu\nu}$$ where 'a' is the Lie algebra generator index. The Lie algebra structure has a particular kind of symmetry, to do with the Killing form, and the space-time indices have a different kind. I am willing to go into the Killing form kind if you wish but lets stick to the space-time indices for the time being. The Lagrangian density can be rewritten as $$F_{\rho\xi}\eta^{\rho \lambda} \eta^{\xi\phi} F_{\lambda \phi}$$. Suppose now we act on the space-time indices with a linear operator in the manner of $$v_{\mu} \to M_{\mu}^{\nu}v_{\nu}$$ and $$w^{\nu} \to M^{\nu}_{\mu}w^{\mu}$$. Doing this for the F indices and factorising in a particular way it follows (the algebra is too lengthy to type here, I dislike typing excessive LaTeX on forums) that the Lagrangian density is invariant if $$M\cdot \eta \cdot M^{\top} = \eta$$ (if you don't see why, ask).
Thus the form of Maxwell's equations picks out a set of linear operators as 'special', which due to the form of the equations can be reexpressed as those linear operators which leave the metric (ie the associated inner product) invariant. Galilean transforms, in general (though the rotational subgroup does), don't do this specific thing.
However, if you wish to do a coordinate transformation via a Galilean transform, you're welcome to do so on Maxwell's equations. The specific form of the individual equations will change, this is not a surprise. For instance, the form of the Laplacian $$\Delta = \sum_{j} \frac{\partial^{2}}{\partial x_{j}^{2}}$$ changes completely when you go into polars (too long to type out, see
here). The fact the form has changed isn't a problem, it doesn't mean writing the parameters of space in terms of spherical polars is wrong, just as using a Galilean transform to construct new space parameters will lead to a new expression which isn't 'wrong' either.
In the case of the inner product preserving Lorentz transformation if you have a set of equations involving say the electric and magnetic fields E and B in terms of parameters t,x,y,z then afterwards you'll have a set of equations which are
exactly the same but with E changed to E', B to B' and (t,x,y,z) to (t',x',y',z'). That's 'special' because if you changed to polar coordinates you obviously don't just change x to r, y to $$\theta$$ etc but never the less the coordinate transformation is a valid one and one which doesn't change the
tensor structure of Maxwell's equations.
Despite repeated explanation from myself, Guest and przyk you haven't demonstrated you even see the distinction, never mind understand it.
Let's cut the crap and prove that the Jacobian for the Shubert formalism is valid . Roll up your sleeves for once and do some calculations. (hint: You won't be able to prove the Jacobian is valid. It is obvious why).
Where did I say I thought it was? I said
assuming the Jacobian is valid then Eugene's transformation doesn't alter the tensor structure of Maxwell's equations. I repeatedly explained that the point about Jacobians is not dependent on the specific form of Eugene's expressions.
Seriously, how many times do I have to say something as simple as "Eugene's work is irrelevant to the point I'm making" before you get that its
irrelevant to the point I'm making?
Why don't you cut the crap and calculate the operator $$\nabla^{a}$$? You have the Shubert formalism at your disposal. Start with the Christoffel symbols. Enough blathering , let's see some calculations. You have two ways to show that you really know what you are talking about.
Actually I have three ways, one of which is explaining the mistake you've made and then elaborating on the specifics of that mistake and how you could correct your understanding if you bothered. And that's the route I've just gone down.
When you come back from your ban please don't come out with this "Show that Eugene's equations work!!" strawman
again. Repeatedly I've said your mistake is nothing to do with Eugene's equations and I've explained it in the much wider context of tensor calculus and inner product spaces. This isn't a matter of "Opps, I forget to carry the two, turns out that transformation doesn't leave the metric invariant", its a matter of "The validity of a coordinate transformation is
independent of the metric".
Eugene said:
Journals don't publish obvious results that high school students can understand even though professional physicists cannot.
The cry of the internet crank.