Maxwell eqs invariant under other transforms

• zwoodrow
In summary, the proof is that the Lorentz transformation is the only transformation that preserves the speed of light in a coordinate system moving at a uniform velocity with respect to another reference frame.
zwoodrow
Has anyone ever seen a proof that lorentz transforms are the only transforms for maxwells equations to remain invariant between two reference frames moving at a uniform velocity with respect to each other?

The proof is simple.
Max's equations contain the constant c.
This leads to the speed of light being c in any coordinate system.
The LT is the only transformation between two frames moving at a uniform velocity with respect to each other that preserves the speed c.

ive done that derivation before, i was wondering if anyone has proved that the lorentz transform is the unique solution.

zwoodrow said:
Has anyone ever seen a proof that lorentz transforms are the only transforms for maxwells equations to remain invariant between two reference frames moving at a uniform velocity with respect to each other?

This statement isn't technically true as written. The full set of transformations that leave Maxwell's equations form-invariant is the Poincare group, which is larger than the group of Lorentz boosts.

If you want to change the statement to a statement that the Poincare group is the only possible group, then I think you might still need to clarify your assumptions in order to rule out doubly special relativity: http://en.wikipedia.org/wiki/Doubly_special_relativity

Meir Achuz said:
The proof is simple.
Max's equations contain the constant c.
This leads to the speed of light being c in any coordinate system.
The LT is the only transformation between two frames moving at a uniform velocity with respect to each other that preserves the speed c.

That's an interesting proof, but it only applies to a pure vacuum. The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)

Last edited:
Meir Achuz said:
The proof is simple.

I think there's a big gap in your argument between this step:
Meir Achuz said:
Max's equations contain the constant c.
This leads to the speed of light being c in any coordinate system.

and this one:
Meir Achuz said:
The LT is the only transformation between two frames moving at a uniform velocity with respect to each other that preserves the speed c.

The final statement asserts uniqueness, but you haven't proved uniqueness.

The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)

Is this generally accepted?? Never knew about it if accurate...

The full set of transformations that leave Maxwell's equations form-invariant is the Poincare group, which is larger than the group of Lorentz boosts.

Exactly as discussed in Wikipedia, post #4...

I am reliably informed ...

... that the form of Maxwell's equations are invariant under not only the ten dimensional Poincare group (generated by boosts, rotatations, and translations) but also under the E^{1,3} conformal group, which happens to be SO(2,4), a fifteen dimensional Lie group. The extra generators of the Lie algebra can be given as a scaling transformation and certain "toral rolls" (higher dimensional generalizations of certain Moebius transformations); the corresponding transformations are conformal but do not preserve the Minkowski interval.

The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)

Naty1 said:
Is this generally accepted?? Never knew about it if accurate...
Yes, look at the equation for Gauss' law. It is almost always given in 2 variations, one for use in a vacuum using the symbol E (electric displacement) and the second with D (electric displacement field) for general use.

If the medium, ie. vacuum, contains no dielectric materials and is isotropic, non-dispersive and uniform then

$$D = \varepsilon_0E$$

where $$\varepsilon_0$$ is the vacuum permittivity.

Since the presence of any charged particle effectively makes the medium dispersive, one must then determine the actual value of the permittivity which will depend on many factors such as the positions of the charges and presence of EM fields. Macroscopically, within a uniform material, one can experimentally determine an approximate value.

Last edited:
PhilDSP said:
That's an interesting proof, but it only applies to a pure vacuum. The addition of any charged object changes the permittivity (from a constant that produces the speed of light to a tensor producing dispersion and a different group velocity)
The presence of a medium provides a preferred system, the rest system of the medium, so the Lorentz transformation does not have the same meaning as it does in vacuum.

clem said:
The presence of a medium provides a preferred system, the rest system of the medium, so the Lorentz transformation does not have the same meaning as it does in vacuum.

We may not be entirely synchronized on the meaning of "medium". I'm thinking of it just as Maxwell, Heavyside, Thomson and others did a century ago as the *thing* that fluccuates when an EM wave is in motion. We know from Poincare's analysis that such a medium must be stationary in all frames with regard to the object that either emits or receives radiation.

In that regard there is no difference between the medium in a vacuum or within a steel bar for example. Even the steel bar consists volume-wise almost entirely of vacuum.

1. How are Maxwell equations invariant under other transforms?

The Maxwell equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields. They are invariant under other transforms, meaning that they have the same form regardless of the coordinate system used to describe them. This is because the equations are based on fundamental physical principles that are independent of the specific coordinates chosen.

2. What are the other transforms that Maxwell equations are invariant under?

The Maxwell equations are invariant under a variety of transforms, including Galilean transformations, Lorentz transformations, and general coordinate transformations. These transforms are used to describe different reference frames, such as moving or accelerating frames, and help to ensure that the equations hold true in all situations.

3. Why is it important for Maxwell equations to be invariant under other transforms?

The invariance of the Maxwell equations under other transforms is a crucial aspect of their usefulness in describing the behavior of electromagnetic fields. It allows the equations to be applied in a wide range of situations and reference frames, making them a powerful tool for understanding and predicting the behavior of electromagnetic phenomena.

4. Can the invariance of Maxwell equations be demonstrated mathematically?

Yes, the invariance of Maxwell equations under other transforms can be mathematically demonstrated using tensor analysis. This involves showing that the equations have the same form and properties in different coordinate systems, which is a key characteristic of invariance.

5. How does the invariance of Maxwell equations relate to the principle of relativity?

The principle of relativity states that the laws of physics should be the same in all inertial reference frames. The invariance of Maxwell equations under other transforms is consistent with this principle, as it ensures that the equations hold true in all reference frames and do not depend on a specific frame of reference.

• Special and General Relativity
Replies
101
Views
3K
• Special and General Relativity
Replies
32
Views
2K
• Special and General Relativity
Replies
144
Views
6K
• Special and General Relativity
Replies
30
Views
2K
• Special and General Relativity
Replies
1
Views
1K
• Special and General Relativity
Replies
32
Views
3K
• Special and General Relativity
Replies
5
Views
2K
• Special and General Relativity
Replies
47
Views
3K
• Special and General Relativity
Replies
4
Views
797
• Special and General Relativity
Replies
8
Views
5K