Stupid question about invariance of maxwell equations

In summary: Thanks for all your help !I'm not sure if the second equality just holds for LT. Isn't this just a general propertyof tensor-transformation ?No, what I used there is the definition of a Lorentz transformation:\Lambda_a{}^c\Lambda^a{}_d=\delta^c_dThis might look more familiar to you if you multiply by \eta_{bc}\Lambda_a{}^c\eta_{bc}\Lambda^a{}_d=\delta^c_d\eta_{bc}=\eta_{bd}and
  • #1
kron
8
0
Hi,

as you all know one can write the Maxwell-equations in covariant form, namely:

[tex]\partial_a F^{ab} = \frac{4\pi }{c} j^{b} [/tex]

and

[tex]\partial_a G^{ab}=0[/tex]

where [tex]\textbf{G}[/tex] is the dual Tensor to [tex]\textbf{F}[/tex].

Now the two simple questions.
I can see that they are invariant, because I have a 4-Vector on both sides, and so the rhs and lhs
will transform in the same way, right ?
So the equation will have in another frame exactly the same form.

But on the other hand this equations would be invariant under all such transformations, not only
Lorentztransformations ?

I don't get it..

Thanks
 
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  • #2
All "such" transformations? Do you mean all linear transformations?

[tex]\partial'_a F'^{ab}=\Lambda_a{}^c\Lambda^a{}_d\Lambda^b{}_e \partial_c F^{de}=\delta^c_d\Lambda^b{}_e \partial_c F^{de}=\Lambda^b{}_e \partial_c F^{ce}=\Lambda^b{}_e j^e=j'^b[/tex]

The second equality only holds for Lorentz transformations.
 
  • #3
What do you mean by "all such" transformations?

What is meant by "such"? What do you refer to? example?

Lorentz transformations is boosts in space-time, rotations, parity and time-reversal.

Secondly, the quantities you wrote are not invariant under LT, they transforms as Lorentz 4vectors. The invariants under Lorentz transformation are Lorentz scalars, such as:
[tex]F^{\mu\nu}F_{\mu \nu}[/tex] and
[tex]x^{\mu}x_{\mu}[/tex] .

Those are invariants under Lorentz transformations. A 4vector such as the 4current
[tex]j^\alpha[/tex] transforms:
[tex]j'^\alpha = \Lambda ^\alpha_\beta j^\beta \neq j^\alpha [/tex]
So the 4-current is not a Lorentz invariant.

But you are correct that the "equations are invariant".
 
  • #4
kron said:
But on the other hand this equations would be invariant under all such transformations, not only
Lorentztransformations ?

I don't get it..

Thanks

I've been trying to sort this issue out as well.

Some facts that play a role in this puzzle.
- The above equations (or something close to it) are associated with terms like "Generally Covariant Form" or "pre-metric".
- The metric plays a role in defining the "duality" relationship between F and G, which is associated with the constitutive relations. (Do you use E,B or E,B,D,H?)
 
  • #5
Fredrik said:
All "such" transformations? Do you mean all linear transformations?

[tex]\partial'_a F'^{ab}=\Lambda_a{}^c\Lambda^a{}_d\Lambda^b{}_e \partial_c F^{de}=\delta^c_d\Lambda^b{}_e \partial_c F^{de}=\Lambda^b{}_e \partial_c F^{ce}=\Lambda^b{}_e j^e=j'^b[/tex]

The second equality only holds for Lorentz transformations.

Yes I mean linear transformations, sorry.
My problem is, what can I say about the [tex]\Lambda^b{}_e[/tex]-matrices ?
I can put anything I want in the lambda's, because the equations (not the 4-vectors) will
remain invariant.

I'm not sure if the second equality just holds for LT. Isn't this just a general property
of tensor-transformation ?

malawi_glenn said:
What do you mean by "all such" transformations?

What is meant by "such"? What do you refer to? example?

Lorentz transformations is boosts in space-time, rotations, parity and time-reversal.

Secondly, the quantities you wrote are not invariant under LT, they transforms as Lorentz 4vectors. The invariants under Lorentz transformation are Lorentz scalars, such as:
[tex]F^{\mu\nu}F_{\mu \nu}[/tex] and
[tex]x^{\mu}x_{\mu}[/tex] .

Those are invariants under Lorentz transformations. A 4vector such as the 4current
[tex]j^\alpha[/tex] transforms:
[tex]j'^\alpha = \Lambda ^\alpha_\beta j^\beta \neq j^\alpha [/tex]
So the 4-current is not a Lorentz invariant.

But you are correct that the "equations are invariant".

Sorry for my sloppiness, yes I mean linear transformations and of course the invariance of
the equations and not 4-vectors or the field-strength tensor or something like that.
robphy said:
I've been trying to sort this issue out as well.

Some facts that play a role in this puzzle.
- The above equations (or something close to it) are associated with terms like "Generally Covariant Form" or "pre-metric".
- The metric plays a role in defining the "duality" relationship between F and G, which is associated with the constitutive relations. (Do you use E,B or E,B,D,H?)

I use E and B like in Jackson. It's just all a little bit too short in Jackson (for my purpose).
So would it be right to say the above two equations are invariant under all linear transformations which
include the Lorentz transformations ?

Thanks for all your help !
 
Last edited:
  • #6
kron said:
I'm not sure if the second equality just holds for LT. Isn't this just a general property
of tensor-transformation ?
No, what I used there is the definition of a Lorentz transformation:

[tex]\Lambda_a{}^c\Lambda^a{}_d=\delta^c_d[/tex]

This might look more familiar to you if you multiply by [itex]\eta_{bc}[/itex]

[tex]\Lambda_a{}^c\eta_{bc}\Lambda^a{}_d=\delta^c_d\eta_{bc}=\eta_{bd}[/tex]

and then rewrite the left-hand side as

[tex]\Lambda_a{}^c\eta_{bc}\Lambda^a{}_d=\Lambda_a{}_b\Lambda^a{}_d=\Lambda^c{}_b\eta_{ca}\Lambda^a{}_d[/tex]

So now we have

[tex]\Lambda^c{}_b\eta_{ca}\Lambda^a{}_d=\eta_{bd}[/tex]

which clearly is just the bd component of the matrix equation

[tex]\Lambda^T\eta\Lambda=\eta[/tex]
 
Last edited:
  • #7
Ok, thanks Fredrik.

I still could say one sums over the index 'a' so I don't have to transform that as a
Tensor with three indices but as a "normal" 4-vector.

[tex]\partial'_a F'^{ab}=j'^b[/tex]

But if I would take an arbitray linear transformation for that I wouldn't get the desired result,
because the primed quantities on both sides wouldn't have terms that cancel (if I would expand them into
their relations with the unprimed ones), so that one would
get the unprimed quantites on both sides as expected.
That was pretty much where I got stuck. And I think that this is
what is really meant by an equation is lorentz invariant.

But how is it possible to see, that this transformation will turn out, that the
extra terms on both sides cancel ? I mean on the right I have a 4-vector, on the left I have the
divergence of a second rank tensor, which is also (a different) 4-vector.

Edit:

I mean I can not see that the divergence of the field strength tensor on the left
and the 4-vector on the right have a priori this transformation relation.

Thanks
 
Last edited:

Related to Stupid question about invariance of maxwell equations

1. What is the invariance of Maxwell's equations?

The invariance of Maxwell's equations refers to the fact that these fundamental equations of electromagnetism do not change under certain transformations, such as changing the reference frame or the units of measurement. This property allows these equations to accurately describe the behavior of electromagnetic phenomena in any situation.

2. Why is the invariance of Maxwell's equations important?

The invariance of Maxwell's equations is important because it ensures that the laws of electromagnetism are universal and consistent, regardless of the specific conditions or circumstances. This allows scientists to confidently use these equations to make predictions and develop new technologies.

3. Are there any exceptions to the invariance of Maxwell's equations?

There are some situations where the invariance of Maxwell's equations may not hold, such as in extreme conditions like near a black hole or at very small scales. However, these exceptions are typically accounted for by incorporating additional factors into the equations or by using more advanced theories, such as quantum electrodynamics.

4. How was the invariance of Maxwell's equations discovered?

The invariance of Maxwell's equations was first recognized by James Clerk Maxwell in the 1860s when he developed his theory of electromagnetism. It was later confirmed through experiments and observations, and has since been extensively studied and tested by scientists.

5. Can the invariance of Maxwell's equations be violated?

While the invariance of Maxwell's equations is a fundamental property of electromagnetism, there are some theories that suggest it may be violated in certain situations, such as in high energy interactions or in alternative theories of gravity. However, these ideas are still speculative and have not been conclusively proven or accepted by the scientific community.

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