What Actually is Length Contraction?

[Sturk200]

When I first learned special relativity it was on an elementary level. I was told that "space itself" contracts in a moving reference frame. Now I am studying electrodynamics out of Griffiths. I just read the derivation of the Lienard-Wiechert potential and the fields for a moving charge. Griffiths on the field of a moving charge: "In the forward and backward directions E is reduced by a factor of (1-v^2/c^2)..." So now putting special relativity aside, doesn't this reduction in the field strength imply that the intermolecular forces would be weakened and thus that the intermolecular separations would be contracted along the direction of motion? Does special relativity account for this within the framework of the Lorentz transformation, or is this "Lienard-Wiechert contraction" a separate effect that has to be calculated independently to find an accurate result?

 

[strangerep]

You can't "put special relativity aside" in this case. The Lienard-Wiechert potential is derived within the framework of SR, i.e., as a relativistically correct version of the EM potential produced by a moving point source.

 

[Sturk200]

"

Except it was derived in 1898 and SR in 1905...? Griffiths doesn't invoke SR at all in deriving the Lienard-Wiechert potential.

 

[strangerep]

Except it was derived in 1898 and SR in 1905...?

The Lorentz transformations, and various puzzles surrounding it, were being investigated earlier than 1898. See, e.g., the history section in Wikipedia's article on LTs.

Griffiths doesn't invoke SR at all in deriving the Lienard-Wiechert potential.

I don't have a copy of Griffiths, but if it's similar to the derivation in the Wikipedia article on Lienard-Wiechert I'm guessing it relies on Maxwell's equations, which are SR-compatible. (I.e., SR-like features are already built-in to Maxwell's equations.)

 

[Dale]

So now putting special relativity aside, doesn't this reduction in the field strength imply that the intermolecular forces would be weakened and thus that the intermolecular separations would be contracted along the direction of motion?

Intermolecular separations (at equilibrium) are definitely contracted along the direction.

The question of forces is a little more complicated.

Does special relativity account for this within the framework of the Lorentz transformation, or is this "Lienard-Wiechert contraction" a separate effect that has to be calculated independently to find an accurate result?

The LW potentials are fully relativitistic. You don't need to calculate anything else independently.

 

[Sturk200]

I'm guessing it relies on Maxwell's equations, which are SR-compatible. (I.e., SR-like features are already built-in to Maxwell's equations.)

Of course it relies on Maxwell's equations. I don't think that means it has SR "built in."

Intermolecular separations (at equilibrium) are definitely contracted along the direction.

The question of forces is a little more complicated.

The LW potentials are fully relativitistic. You don't need to calculate anything else independently.

Thanks for your reply. I guess my question is the inverse of the one you answered. Let there be a measuring rod traveling in some direction at constant velocity. SR tells you how much the length of the rod contracts (measured in stationary frame) due to the changing geometry of spacetime, but as I learned it SR doesn't mention anything about contraction due to intermolecular spacing. Since there definitely is contraction due to changes in intermolecular spacing, why doesn't SR mention that? Is it just sort of built into the math and nobody talks about it? You say that LW potentials are fully relativistic, so I'm assuming that means you don't have to add any calculations to SR to account for intermolecular effects. But I still don't understand why SR doesn't mention intermolecular contraction explicitly. Maybe I just learned a watered down version...

 

[strangerep]

Of course it relies on Maxwell's equations. I don't think that means it has SR "built in."

Then you need to learn more about SR, and the subject of symmetry groups of dynamical equations.

Anyway, it sounds like I'm not the right person to help you. Bye.

 

[PeterDonis]

SR tells you how much the length of the rod contracts (measured in stationary frame) due to the changing geometry of spacetime

No, it doesn't. The geometry of spacetime in SR is the same everywhere--flat Minkowski spacetime. SR tells you how much the length of the rod contracts due to the different "angle in spacetime" at which the length is being measured (the "angle" is due to relative velocity and is a hyperbolic angle).

as I learned it SR doesn't mention anything about contraction due to intermolecular spacing

It may not have been mentioned, but SR length contraction is contraction due to a change in intermolecular spacing. When measured at a different angle in spacetime, the spaces between the molecules along the direction of motion of the object are shorter, just like the spaces between any other objects.

 

[Sturk200]

No, it doesn't. The geometry of spacetime in SR is the same everywhere--flat Minkowski spacetime. SR tells you how much the length of the rod contracts due to the different "angle in spacetime" at which the length is being measured (the "angle" is due to relative velocity and is a hyperbolic angle).

Sorry, I always get my words mixed up on this.

It may not have been mentioned, but SR length contraction is contraction due to a change in intermolecular spacing. When measured at a different angle in spacetime, the spaces between the molecules along the direction of motion of the object are shorter, just like the spaces between any other objects.

But why don't you have to calculate the contraction due to changes in the strengths of the fields of individual molecules (a la LW potential) and sum them over the entire object? Is this effect accounted for implicitly by the spacetime rotation?

 

[Dale]

SR tells you how much the length of the rod contracts (measured in stationary frame) due to the changing geometry of spacetime, but as I learned it SR doesn't mention anything about contraction due to intermolecular spacing.

It is the same thing.

It doesn't matter if the length is the distance between molecules, the distance between ends of a rod, or the distance between stars. All lengths of any type exhibit length contraction.

 

[Sturk200]

It is the same thing.

So the coordinate transformation of SR is actually calculating all the LW contracted fields between the molecules and summing them up and telling us the net effect this has on the length of the object? Could you get the same answer by explicitly calculating the length contraction using the retarded potential between molecules, assuming this were computationally feasible?

 

[Dale]

Of course it relies on Maxwell's equations. I don't think that means it has SR "built in."

Maxwell's equations have SR built in, so anything based on Maxwell's equations also has SR built in.

 

[Dale]

So the coordinate transformation of SR is actually calculating all the LW contracted fields between the molecules and summing them up and telling us the net effect this has on the length of the object? Could you get the same answer by explicitly calculating the length contraction using the retarded potential between molecules, assuming this were computationally feasible?

Assuming that there are no other forces involved besides EM forces on classical point charges, yes.

Maxwell's equations are more general than the LW potentials since they apply to other charge configurations. And QED is more general than Maxwell's equations since it applies to quantum scales. And SR is more general than QED since it applies to the weak and strong forces too. But wherever LW applies, yes, you could get the same answer either way.

 

[jartsa]

Electrodynamics says that a moving system of a positive and a negative charge is time dilated and length-contracted? That's quite interesting.

Let's consider a relativistic water droplet. If the longitudinal surface tension is weaker than the transverse surface tension, then the droplet tends to become longer. That does not sound right, so we conclude that it is not so that longitudinal intermolecular forces inside a moving droplet become weaker.

Now we just have to find out what "In the forward and backward directions E is reduced" really means.

Special relativity says that a moving field is contracted, so if a field changes by 10 % per one meter at rest, then at speed 0.87 c the "same" field changes 10 % per half meter, and maybe about 20% per one meter.

SR says that in different frames the "same" point of an E-field is at different distance from the charge, while electrodynamics says that in different frames the "same"point of an E-field is not at different distance from the charge, but the EM-field strength is different in different frames?

 

[PeterDonis]

Special relativity says that a moving field is contracted

No, it doesn't. The way EM fields transform under a Lorentz transformation is not a simple "length contraction" of a static field.

 

[Dale]

Let's consider a relativistic water droplet. If the longitudinal surface tension is weaker than the transverse surface tension, then the droplet tends to become longer. That does not sound right, so we conclude that it is not so that longitudinal intermolecular forces inside a moving droplet become weaker.

The surface tension forces and the forces inside the droplet all transform the same way. It isn't about "sounding right", it is about working through the math.

SR says that in different frames the "same" point of an E-field is at different distance from the charge, while electrodynamics says that in different frames the "same"point of an E-field is not at different distance from the charge, but the EM-field strength is different in different frames?

SR and electrodynamics say the same thing. SR is built into electrodynamics. The difference is only that SR is more general and applies to situations where electrodynamics does not.