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For starters let's talk about Maxwell's Equations; we know that Maxwell's Equations are invariant under Lorentz's Transformation, after all this is why all the relativity business got started. To state that a law is invariant under Lorentz's transformation is equivalent to state that the same law is true for any observer in any reference frame (that is if I understand the meaning of the word invariant correctly in this context).
Problem is that this isn't manifestly true for Maxwell's Equation; to solve this problem we write Maxwell's equations using tensors, this kind of evidently invariant formulation of Maxwell's Equation is so nice that it deserves its own name: Covariant formulation of classical electromagnetism. (To be precise this term refers to the expression of all laws of electromagnetism in an invariant way, but still Maxwell's Equations are the main ones.)
But why writing a law using tensors should imply that that law is surely invariant under Lorentz's transformation? This is because: tensors are invariant under any coordinate transformation1, so any law wrote in the form: tensor equals tensor is sure to be invariant under any transformation, Lorentz's one included.(Lorentz's transformation operates on four vectors ecc. So on tensor with indices scaling from $0$ to $3$, so of course the tensors in the manifestly invariant laws should scale between the same range.)

This is what I currently understand about this topic, is this correct?

Also, and maybe mainly, I don't get why the invariant formulation of electromagnetism is called covariant; seems to me that the term covariant is out of place here.

And at last: if my reasoning is correct laws written in the form: tensor equals tensor should be simultaneously invariant under any kind of coordinate transformation, not just the Lorentz's one; so even if Lorentz's transformation were to have another completely different form still laws written with tensors, such as Maxwell's equation, would be invariant anyway. This seems really strange to me, is this true? I mean: this would mean that for example Maxwell's equation would be invariant no matter the kind of transformation; so there would be no special relation with Lorentz's transformation whatsoever.


[1]: Of course the components of the tensor are not constant under an arbitrary transformation, but the components change in such a way to compensate the change of the basis vectors, so the tensor overall remains the same.

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Suppose you have two systems $S$ and $S'$ in each system we have coordinates and vector potentials $x^\mu$, $x'^\mu$, $A^\mu$ and $A'^\mu$. Since both $x$ and $A$ are vectors we know they transform like:

$x^\mu\rightarrow\left(\Lambda^\mu{}_\nu\right)x^\nu$,

this also gives us the transformation law for the derivatives:

$\frac{\partial}{\partial x^\mu}\rightarrow \left(\Lambda^{-1}\right)_\mu{}^\nu\frac{\partial}{\partial x^\nu}$

Which is the same as the transformation of $x_\mu$. Now if you consider a law like Maxwell's Equations:

$\partial_\mu F^{\mu\nu}-\mu_0J^\nu=0$

This will transform like:

$\partial'_\mu F'^{\mu\nu}-\mu_0J'^\nu=\left(\Lambda^{-1}\right)^\rho{}_\nu\left(\partial_\mu F^{\mu\nu}-\mu_0J^\nu\right)=0$

Which means that both equations written in terms of the coordinates and vectors of each system predict the same behavior, because the equations in the primed system can be shown to reduce to those in the unprimed system.

The discussion before was mainly for Special Relativity, or linear coordinate transformations. The fully covariant equations for a generic coordinate transformation are slightly more complex Maxwell's Equations in Curved Spacetime, but the underlying idea is the same, the root of the issue lays on the fact that derivatives do not transform like $x_\mu$ in curvilinear coordinates, they require an extra piece to transform like a properly Covariant Derivative.

But why writing a law using tensors should imply that that law is surely invariant under Lorentz's transformation?

The law itself is not invariant, it will transform accordingly, if it has one index it will transform like a vector, and so on. But the key point is that the law in one system predicts he same as the law in any other.

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