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The covariant form of Maxwell's equations is Lorentz invariant.

$$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$$

$$\partial_{\alpha}F_{\beta\gamma} + \partial_{\beta}F_{\gamma \alpha} + \partial_{\gamma} F_{\alpha \beta}=0.$$

However, in Griffiths book on electrodynamics, he says that Maxwell's equations in vector calculus form will allow us to predict the motions of stuff in a way that's the same in all reference frames, although what one considers a magnetic field in one frame may be considered a electric field in another.

Can someone show how to prove this?

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    $\begingroup$ Easy way: Show equivalence to the relativistic notation. $\endgroup$ – Danu Jan 4 '16 at 17:56
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The comments is completely wrong, and that is why questions should not be answered in comments, wrong answer in comments can not be downvoted.

The covariant form of Maxwell's equations:

$$\partial_{\alpha}F^{\alpha\beta} = \mu_{0} J^{\beta}$$

$$\partial_{\alpha}F_{\beta\gamma} + \partial_{\beta}F_{\gamma \alpha} + \partial_{\gamma} F_{\alpha \beta}=0$$

are indeed Lorentz invariant, in particular you wrote them in a way where you know how everything transforms.

But Maxwell tells you how the fields evolve, given the charge and current. It doesn't tell you the motion of charges. For that you'd need something completely different, like the Lorentz Force Law and some relativistic version of Newton's Laws.

Griffiths is saying that if you write down Maxwell in any frame then you can use that (to find the field and then e.g. use the Lorentz Force Law and $\vec F=d\vec p/dt$) to find the kinematics of objects. And these predictions for different frames will be the kinematics two different frames would describe for the same events.

But Maxwell in vector form didn't tell us how the electric and magnetic fields transform between frames. We know the equations should be the same, but we don't know how the fields, the solutions, should change. So you should take it backwards and say that the fields should transform in a way that gives us the same dynamics. And that happens because we define the fields in terms of forces. So we need the Lorentz Force law, for example, in order to know what the fields should be in the different frames in the vector form.

So we have to assert how they transform to make it so they give the same dynamics for motions of stuff when we combine with Lorentz Force and some laws of motion.

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The "motion of stuff" is not the same in all reference frames. Each frame has it's own coordinate system, and so the physical process you are looking at will be described by different variables. (This is true even when you think of Galilean coordinate systems).

What Griffiths means is that you can use the vector form of Maxwell's equations along with the relativistic equations of motion for the particles involved in any frame you like, and they will describe the same physical process (just from the point of view of two different frames).

A couple of points to aid understanding of this:

  • If two reference frames are used to describe the same physical process then you should be able to transform the solution in one frame into the other frame and get the same answer.
  • Maxwell's equations are Lorentz invariant (they "look" exactly the same when written down on paper!) even when they are written in vector form. To prove this you have to transform all variables in those equations (fields, current-density and charge-density) using the appropriate Lorentz transformations for those variables. The reason the tensor notation is used is because you never have to worry about proving this if the law can be written in tensor form you already know it's Lorentz invariant.
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