The reality is that Sir Knight put his sword through your head and you have been completely and thoroughly discredited:
http://www.youtube.com/watch?v=dhRUe-gz690
You're becoming like Terry, linking to YouTube videos rather than making a relevant point.
Wrong, Maxwell's equations are covariant INDEPENDENT of notation. Heck, all the macroscopic laws of physics are covariant.
Yes, Nature is independent of how we formalise our description, just like Nature doesn't care whether you and I communicate in English or French or Sindar.
But that also means that if you have an expression which is not invariant under coordinate transformations then its not formalised in the proper way or its incomplete. Picking out a single component of Maxwell's equations or from Maxwell's tensor will be such an example. Under Lorentz transforms you can convert electric fields into magnetic and vice versa, which means that if you didn't know about magnetic fields and you only considered electric you'd have an incomplete view of the physics you're attempting to describe, ie the choice of coordinate ends up defining a particular form of the electric field component of the underlying things you're examining.
You provided the wave equation applied to an electric field. Under a Lorentz transformation that electric field can be made into a linear combination of electric and magnetic fields, so clearly something is amiss and that something is what I previously said, by picking out a single component in a particular coordinate system you've extracted
part of a system and your choice of coordinates has defined that 'part'. Unless you include the other parts you can't expect to have a covariant formalism because your choice of coordinates determines the part you're considering so you must have
all parts.
Hence why tensor expressions for Maxwell's work (and the non-abelian extensions into Yang Mills theory) involve all components. $$nabla^{1}F_{12}$$ is not covariant, but $$\nabla^{a}F_{ab}$$ as a whole is.
Your comment about Nature being covariant doesn't contradict anything I've said, it actually gels perfectly with my point that
your choice of coordinates when you give that expression for the electric field wave equation is
violating the "Coordinate choices don't matter" principle.
The point that you and Guest have missed (since you both appear not to understand the notion of covariance correctly)
I can't speak for Guest but it happens that general relativity was my best course in my 4th year and my PhD was in extra dimensional formalisms in string theory. Coordinates and the structures they have was my area of research. Clearly people whose job it is to know whether I know my stuff disagree with you.
For example, the Lorentz transforms preserve covariance , the Galilei, Tangherlini, Shubert crackpot transforms do not.
Galilean transformations include as a subgroup rotations, which do preserve the Lorentz structure.
There's a difference between those transformations which leave the metric
invariant and those which change its form. This is a point you previously demonstrated you didn't get when you said there's no metric involved. If Eugene is talking about Euclidean space then the metric is Euclidean. If he's talking about Minkowski space in the SR formalism then its the Minkowski metric. Either way if his space has a notion of distance then unless its pathological there'll be a metric.
A rotation in Euclidean space or a Lorentz transform in Minowski space-time leave the metric unchanged, ie $$g \to M \cdot g \cdot M^{\top} = g$$. If g is Euclidean metric then $$M \in SO(N)$$ and if $$g = \eta$$ then $$M \in SO(N-1,1)$$. Notice how both of these are linear operations represented by a matrix? But what about something else like going from Cartesians to polars? That
isn't a linear operation, you can't write it as a matrix and the metric
changes. In the 2d Cartesian to polars you have $$ds^{2} = dx^{2}+dy^{2} = dr^{2} + r^{2}d\theta^{2}$$. This is a transformation on the coordinates which doesn't alter anything, it is just a different way of describing things. From this point of view the change to polars is
valid, despite it not being a transformation which leaves the metric's form invariant. Instead all covariant and contravariant indices transform with powers of the Jacobian.
Changing from Cartesians to polars in Maxwell's equations in this context is entirely valid, despite the metric also changing. From this point of view, which is what Guest and I have been talking about, Maxwell's equations are covariant. From this point of view you can change coordinates with a Galilean transform and Maxwell's equations keep their form, because the Jacobian is valid.
That is what covariance means. It is distinct from the space of transformations which are expressible as linear operators and which leave some particular choice of inner product, typically the metric form $$g(v_{1},v_{2})$$, invariant.
For instance if you do a Lorentz transform from $$(t,x) \to (t',x')$$ then it preserves the space-time interval, $$ds^{2} = -dt^{2} + dx^{2} = -(dt')^{2} + d(x')^{2}$$, you just as ' to all your labels. This is a very special kind of coordinate transformations, precisely for that reason. do a change of coordinates of the Euclidean 2d space interval to polars, $$(x,y) \to (r,\theta)$$ and you don't get an invariant expression, $$ds^{2} = dx^{2}+dy^{2} = dr^{2} + r^{2}d\theta^{2}$$. Still utterly valid in terms of covariance. You can write Maxwell's equations in polar coords and you'll find you can still express them as $$\nabla^{a}F_{ab} = 0$$ for a,b being $$r,\theta$$ etc, rather than $$x,y$$ etc.
That is what covariance means, its a much much larger set of transformations than those which leave the inner product associated to the vector space your space-time's tangent spaces define.
The Schrodinger equation is simple to understand in Cartesians but to do the Hydrogen atom its better to change into spherical polars. The result is a very different $$\Delta$$ form but its perfectly valid because the end results are the same.
The precise equation I gave you guys to work on has been used by Lorentz to prove that Galilei transforms do not preserve covariance while developing the transforms named after him (that do preserve covariance). You learned this early on in your SR class but you have obviously forgotten.
You need to realise there's a difference between inner product invariance and covariance. Inner product invariance involves acting on elements of the (typically) Hilbert space with linear operators and then deducing which space of linear operators leaves the inner product invariant (this is how you get orthogonal, unitary and symplectic matrices). Covariance involves a non-singular Jacobian. For spaces with 'nice' inner products (like Euclidean or Lorentzian ones) you automatically get a valid Jacobian associated with the linear transform because you have $$|J| = det(M)$$ where J is the Jacobian and M the linear transform. I did this for the Lorentz group in a previous post but it obviously passed you by
.
You two guys are so brainwashed by the tensor notation that you missed the basics.
And somehow it didn't come up when I submitted papers for publication in reputable journals or in my PhD viva. Weird that, its almost like people who know what they're talking about think I know what I'm talking about.
Sure, PROVIDED that the theory that you are examining has a valid metric allowing you to calculate the Christoffel symbols. Shubert's doesn't.
So you're saying Eugene isn't working in either Euclidean or Lorentzian space? A change of coordinates has nothing to do with a fundamental change in the metric. Defining coordinates doesn't have anything to do with defining a metric, only how you define the metric's components
in those coordinates.
A good course in GR will cover things like how $$g : TM \times TM \to \mathbb{R}$$, ie $$g(X,Y) \in \mathbb{R}$$ for $$X,Y \in TM$$ is a coordinate free definition. Only when you pick coordinates do you get components $$g_{ab} = g(e_{a},e_{b})$$ for $$X = X^{a}e_{a}$$ and likewise for Y. Doing a change in coordinates $$e_{a} \to f_{a}$$ induces a change in the components but the overall expression $$g(X,Y)$$ is unchanged. Eugene is saying he wants to use different coordinates to the usual ones, which has no impact on whether or not the metric exists or whether $$g(X,Y)$$ exists.
But you CANNOT construct the tensor formulation in the Shubert formalism because it has no valid metric. You need to go back to your books and re-read the chapter on general covariance.
Firstly a tensor can be defined without reference to a metric, which is why covariance is a larger notion than transformations which leave the metric unchanged. A tensor is a multilinear operator, it doesn't mention 'metric' in the definition. The whole covariant and contravariant concepts are definable without a metric being known or even existing. When you go eventually get to metric spaces and the like then you can define a
smaller class of
linear operators which satisfy $$g_{ab} = M^{c}_{a}g_{cd}M^{d}_{b}$$, which is much more restrictive than the general way a (2,0) tensor would transform (look up tensor densities).
You've gotten these two concepts jumbled together.
No, what I am criticizing you is for being shallow. You look at things for a few minutes, you cu and paste something that you remember from your school days and, when challenged to get deeper, you cut and paste some more of the same tripe.
So I'm shallow for not giving a crank much attention, a crank you also agree is a crank? I guess the hypocrisy you're displaying is lost on you given you'd concluded within a dozen or two posts from joining this forum that I'm a hack and Guest is a crank?
You're welcome to search for threads I've started, you'll notice they tend to be a touch higher level than the average one and I display working understanding of the relevant concepts. I don't post my work specifically, I only ask questions related to it when I think someone here will have more first hand experience with things which I might find of interest but if you want a discussion on the specifics of say the last paper I got published in a reputable journal (of which I was the only author) we could start another thread.
He's exposed your shallowness.
Irony meter just went off the scale....
But when you and Guest254 show up with red herring claims of his theory being general covariant, it doesn't help the thread.
I've been saying that
assuming his transformation has a non-singular Jacobian then things like Maxwell's equations in their well known covariant tensor formalism would be unchanged in their tensor formulation when you go from your favourite coordinate system to his. Whether or not they can be written as a linear operator and that linear operator leaves whatever metric might be relevant invariant is entirely another
irrelevant matter.
You can't use generalized coordinates in the Shubert framework, you dork. His theory doesn't have a metric.
Here's a question to think about which should help you realise your mistake : Where in the definition of the coordinate transformation in 2D of Cartesians to polars is a metric required or even mentioned?