Relative Simultaneity and Electrodynamics: Lorentz Maths

[Austin0]

I am interested in how the Lorentz maths were derived from the Maxwell electrodynamic and field equations. But not in a struct mathemetical sense as the math is outside my range but on a simpler conceptual level. For eg. contraction seems to have relevance wrt electron electrostatic fields and their interactions.
Is there any relevance of relative simultaneity in the calculations of electrodynamics?
Was the mathematical expression of relative simultaneity in any way derived from the Maxwell maths or could it be??
Or is it directly a consequence of the clock synchronization convention that was added later with no correlation to electrodynamics?
Thanks

 

[granpa]

http://en.wikipedia.org/wiki/History_of_special_relativity

 

[Mentz114]

I am interested in how the Lorentz maths were derived from the Maxwell electrodynamic and field equations. But not in a struct mathemetical sense as the math is outside my range but on a simpler conceptual level. For eg. contraction seems to have relevance wrt electron electrostatic fields and their interactions.
Is there any relevance of relative simultaneity in the calculations of electrodynamics?
Was the mathematical expression of relative simultaneity in any way derived from the Maxwell maths or could it be??
Or is it directly a consequence of the clock synchronization convention that was added later with no correlation to electrodynamics?
Thanks

I don't think you need Maxwell's equations to show that the Lorentz transformation is the proper length conserving transformation of Minkowski spacetime.

The interesting thing about relativistic electrodynamics is that electric and magnetic fields 'mix' when viewed from a moving frame. When a frame is boosted, space and time 'mix'

t' = Yt + vYx
x' = Yx + vYt

and a similar thing happens to E and B. If we have 3 electric fields E_x, E_y and E_z and we get a velocity v in the z-direction, then the new fields are (Y is the gamma from the z boost)

E'_x = YE_x
E'_y = YE_y
E_z = E_z

and now there are magnetic fields where there were none,

B_y = vYE_x
B_x = vYE_y

I've omitted the constants that convert B -> E for clarity.

 

[Austin0]

I don't think you need Maxwell's equations to show that the Lorentz transformation is the proper length conserving transformation of Minkowski spacetime.

The interesting thing about relativistic electrodynamics is that electric and magnetic fields 'mix' when viewed from a moving frame. When a frame is boosted, space and time 'mix'

t' = Yt + vYx
x' = Yx + vYt

and a similar thing happens to E and B. If we have 3 electric fields E_x, E_y and E_z and we get a velocity v in the z-direction, then the new fields are (Y is the gamma from the z boost)

E'_x = YE_x
E'_y = YE_y
E_z = E_z

and now there are magnetic fields where there were none,

B_y = vYE_x
B_x = vYE_y

I've omitted the constants that convert B -> E for clarity.

From what I have read so far in the link grandpa provided it seems that the basis for
t' = Yt + vYx
did appear much earlier in the form of what they called local time. So it appears the real change is the addition of the gamma transformation factor but the essence of relative simultaneity appeared even before Lorentz. I would not bet large sums on the correctness of my understanding here but so far it seems like this is the basis for simultaniety and the clock convention came later as a rational implementation of this.
I still want to learn more regarding the meaning of "local time" in electrdynamics and more of how relative simultaneity would practically relate to particles and fields etc. in the same way as contraction.
Thanks for your explication ,,,food for thought

 

[Austin0]

http://en.wikipedia.org/wiki/History_of_special_relativity

Thanks for the helpful link. ANy additional insights or comments about it would be great.

 

[Meir Achuz]

"I am interested in how the Lorentz maths were derived from the Maxwell electrodynamic and field equations."
They haven't been and don't follow from Maxwell. Lorentz hoped that would happen, but no one has done it.

"Is there any relevance of relative simultaneity in the calculations of electrodynamics?"
Not until relativity is added.

"Was the mathematical expression of relative simultaneity in any way derived from the Maxwell maths or could it be??"
Only in the sense that in order for Maxwell to be relativistic, simultaneity has to be relative."clock synchronization convention"
What does this mean?

 

[atyy]

At a conceptual level, you can try "How to teach special relativity'" on p67 of http://books.google.com/books?id=FGnnHxh2YtQC&dq=bell+unspeakable&source=gbs_navlinks_s The handwavy argument has to do with the contraction of the electric field of a moving point charge. The argument has a hole because it needs a system of charges to have a unique equilibium configuration, which isn't true in classical electrostatics. I've heard an argument that tries to fix this by using quantum mechanics, saying that many systems have unique ground states. However, the quantum theory of Maxwell's equations does not hold to arbitrarily high energies, so one might object to using a mathematically unsound theory to derive the Lorentz transformations. I wonder if this can be done by saying that QED is a sound and unique theory (has an infrared fixed point) at low energies, so we can use that to derive the Lorentz transformations (ie. use the canonical form where Lorentz covariance is not manifest, and show that it is equivalent to the covariant form)?

Maxwell's equations are invariant under the Poincare group, and a larger group called the conformal group. The restriction to the Poincare group for special relativity comes when massive fields are considered, in addition to the massless field of Maxwell.

 

[Austin0]

"I am interested in how the Lorentz maths were derived from the Maxwell electrodynamic and field equations."
They haven't been and don't follow from Maxwell. Lorentz hoped that would happen, but no one has done it.

"Is there any relevance of relative simultaneity in the calculations of electrodynamics?"

Not until relativity is added.

Then at relativistic velocities does it directly relate to the moving fields and particles and if so how?

"clock synchronization convention"

What does this mean?

The Einstein convention of synchronization through calculations based on reflected light transmission.