Explanation of EM-fields using SR

[A.T.]

What do you think about this explanation of EM-fields using SR?

https://www.youtube.com/watch?v=1TKSfAkWWN0

 

[Nugatory]

Narrator slips up a few times and says "moving" instead of "moving relative to", but other than that it's pretty good.

 

[Noyhcat]

I don't immediately get why the separation of the negatively charged particles doesn't contract from the man's reference frame, as they are moving relative to him, and therefore there would be a negative overall charge.

 

[pervect]

It seems like an attempt to popularze Purcell.

I have a suspicion that it will confuse it's target's audience , but I'm not sure if that can be helped, considering that it's an attempt to reach as many people as possible (and hence put the "target" as low as possible).

Personally, I'd save this sort of explanation for someone who is sophisticated enough to work out what charge densities are required (as measured in the lab and comoving frames, the comoving frames of the electrons being different from the lab frame) to ensure electrical neutrality in the lab frame. Perhaps I'm being pessemistic, perhaps not.

 

[WannabeNewton]

Purcell should be popularized in my opinion. It's the best EM text I know of.

As for Noyhcat, take a look here: http://physics.weber.edu/schroeder/mrr/MRRtalk.html as it explains things in a much more lucid manner than does the video commenter in my opinion.

 

[Dale]

I don't immediately get why the separation of the negatively charged particles doesn't contract from the man's reference frame, as they are moving relative to him, and therefore there would be a negative overall charge.

That tends to be the point that confuses most of the people that actually understand the argument being presented, so kudos on understanding the argument.

The spacing of the electrons in the wire frame is determined by the observed fact that the wire is uncharged in the wire frame. This is a "boundary condition" that can be experimentally controlled.

For example, instead of having an uncharged wire you could give the wire an excess positive charge by putting it at a very high voltage. If you did that then the spacing between electrons in the wire frame would be greater than the spacing between protons.

Once the spacing is determined in the wire frame, then it is determined in all frames.

 

[A.T.]

As for Noyhcat, take a look here: http://physics.weber.edu/schroeder/mrr/MRRtalk.html as it explains things in a much more lucid manner than does the video commenter in my opinion.

Thanks for the link, but I don't think it adresses Noyhcat's point, as it also starts out with the current already flowing and the wire being neutral in the lab frame.

Let's start with a wire without a current. It is neutral too, so the distances between pos. and neg. charges are equal here. Now what happens when a current starts flowing? Judging by the video and your link only one type of charge starts moving and gets contracted in the lab frame. So the wire should become charged in the lab frame, which is not the case.

Shouldn't both charge types move in opposite directions in the lab frame? Then they are contracted by the same amount, and the wire remains neutral in the lab frame. Only when you move relative to the wire the contractions become different, and the wire becomes charged. The problem is of course that they identify positive charges with protons which cannot move in the lab frame, instead with the electron holes that are moving opposite to the electrons in the lab frame.

 

[pervect]

I don't immediately get why the separation of the negatively charged particles doesn't contract from the man's reference frame, as they are moving relative to him, and therefore there would be a negative overall charge.

That tends to be the point that confuses most of the people that actually understand the argument being presented, so kudos on understanding the argument.

The spacing of the electrons in the wire frame is determined by the observed fact that the wire is uncharged in the wire frame. This is a "boundary condition" that can be experimentally controlled.

This is related to what I was trying to say earlier.

There are three possibilities:

1) Ignore the issue, which is what the video has done. Then you'll get questions like Noyhcat's.
2) Try to explain this in the video - which will raise the bar on the target audience
3) Raise the bar on the target as far as the "target audience" is concerned.

Overall, I favor 3, because ignoring the issue doesn't really work, and I'm afraid I don't know how to do 2) (explain the issue) without violating 3) (raising the height of the target audience).

 

[WannabeNewton]

You're right A.T. in that the link doesn't answer that particular question in a straightforward manner. Do you have access to Griffiths book? Perhaps his explanation would be to your liking. Check out section 12.3.1.

 

[A.T.]

I'm afraid I don't know how to do 2) (explain the issue) without violating 3) (raising the height of the target audience).

What do you think about my idea in post #7, with both charge types moving in opposite directions in the wire's frame? Does it work out quantitatively? I don't think it would make the video more difficult to understand.

 

[WannabeNewton]

That's what Griffiths does in the aforementioned section.

 

[A.T.]

That's what Griffiths does in the aforementioned section.

What is the title of the book?

 

[WannabeNewton]

Introduction to Electrodynamics by David Griffiths: https://www.amazon.com/dp/013805326X/?tag=pfamazon01-20

 

[A.T.]

Introduction to Electrodynamics by David Griffiths: https://www.amazon.com/dp/013805326X/?tag=pfamazon01-20

Thanks, I will try to check it out. But the idea with both charge types moving is basically correct?

 

[atyy]

I like the video and DaleSpam's answer. One has to specify what is happening in one frame of reference, eg. there is a current in the wire, and the wire containing the current is uncharged in the lab frame. Relativity is a relationship between frames of reference, so if you specify what happens in one frame, it tells you what happens in another frame.

 

[A.T.]

I like the video and DaleSpam's answer. One has to specify what is happening in one frame of reference, eg. there is a current in the wire, and the wire containing the current is uncharged in the lab frame. Relativity is a relationship between frames of reference, so if you specify what happens in one frame, it tells you what happens in another frame.

Yes, with the clarifications mentioned by DaleSpam it makes sense. But otherwise many will try to extrapolate the presented mechanism, to see what happens when a current starts/stops flowing.

 

[atyy]

Maybe something like this:

We know that in the lab frame we can set up a wire with a current in it. We know we can do this in a way such that the wire is electrically neutral, because the wire neither attracts nor repels a charged particle that is stationary in the lab frame. Since we have set up the wire to be electrically neutral in the lab frame, the distance between positive charges is the same as the distance between negative charges in the lab frame. Now, will the wire attract or repel a charged particle that is moving in the lab frame? If the only force that affects charged particles is the electric force, since the wire is electrically neutral in the lab frame, it will neither attract nor repel a moving charged particle.

 

[WannabeNewton]

Thanks, I will try to check it out. But the idea with both charge types moving is basically correct?

Yeah.

 

[universal_101]

Thanks, I will try to check it out. But the idea with both charge types moving is basically correct?

Yeah.

And how do you explain, if the current carrying wire is made of highly doped n-type semiconductor, when you don't have any holes to go by.

 

[universal_101]

The spacing of the electrons in the wire frame is determined by the observed fact that the wire is uncharged in the wire frame. This is a "boundary condition" that can be experimentally controlled.

Once the spacing is determined in the wire frame, then it is determined in all frames.

I think role of physics is in finding relations(mostly logical) between observations, for example what should we observe if we stop the current, and we all know it is an observed fact that the wire still remains electrically neutral. The point is, according to the SR length contraction explanation, it should not be neutral when we stop the current if it were to be neutral when there was a current.

 

[Dale]

I think role of physics is in finding relations(mostly logical) between observations, for example what should we observe if we stop the current, and we all know it is an observed fact that the wire still remains electrically neutral. The point is, according to the SR length contraction explanation, it should not be neutral when we stop the current if it were to be neutral when there was a current.

That would only be true if the electrons were rigidly attached to each other, which they are not. Charge carriers in a conductor are, by definition, very mobile and able to change their position and spacing in response to any fields.

In the wire frame the wire is electrically neutral. The spacing of the charges must reflect that boundary condition.

 

[Dale]

What do you think about my idea in post #7, with both charge types moving in opposite directions in the wire's frame? Does it work out quantitatively?

Yes, it works out quantitatively. Using units where c=1 the current four-vector or four-current is $J=(\rho,\mathbf{j})$ where ρ is the charge density and j is the current density.

Suppose that we have a flow of unit charges (no opposite charges) each separated by a distance of 0.5 and traveling at a speed of 0.5 c. That gives a four-current of J=(2,1,0,0). Now, if we boost that to .5 c we find J'=(1.732,0,0,0), which corresponds to unit charges at rest with a spacing of 0.577. The γ factor at .5 c is 1.155, and so the spacing of 0.577 at rest transforms to a spacing of .5 at .5 c. You can play around with other speeds, charges, and distances to convince yourself that the four-current is a proper four-vector. Regardless of the combination of charge, separation, and velocity, the four-current always transforms correctly.

Once you have convinced yourself of that then it follows from linear algebra that if A and B are vectors (e.g. the four-current) and L is a linear transform (e.g. the Lorentz transform) then A+B=C implies that $L \cdot A+L \cdot B=L \cdot (A+B)=L \cdot C$ so it doesn't matter how you split up your current four-vector into positive charges moving one way and negative charges moving another way, as long as the sum is correct, the Lorentz transform will give you the correct four-current in any other frame.

 

[universal_101]

That would only be true if the electrons were rigidly attached to each other, which they are not. Charge carriers in a conductor are, by definition, very mobile and able to change their position and spacing in response to any fields.

In the wire frame the wire is electrically neutral. The spacing of the charges must reflect that boundary condition.

So if it is the spacing of the electrons that keeps changing while switching on and off the current, what do you suggest happens for a closed loop of current wire! ,where does the extra electrons go or come from ?

 

[Dale]

So if it is the spacing of the electrons that keeps changing while switching on and off the current, what do you suggest happens for a closed loop of current wire! ,where does the extra electrons go or come from ?

If any additional electrons are required then they would come from the battery or other power source. However, usually no additional electrons are required. Usually, charge is just redistributed around the loop.

 

[universal_101]

If any additional electrons are required then they would come from the battery or other power source. However, usually no additional electrons are required. Usually, charge is just redistributed around the loop.

How does a particular redistribution of charges(electrons), as a result of the presence or absence of net electric field in the wire in a single direction, cancels out the effect of length contraction due to motion of electrons everywhere. A particular redistribution can only cancel out the effects at a particular point and not everywhere.

And you already know we don't always need the batteries to produce the current !

 

[Dale]

How does a particular redistribution of charges(electrons), as a result of the presence or absence of net electric field in the wire in a single direction, cancels out the effect of length contraction due to motion of electrons everywhere. A particular redistribution can only cancel out the effects at a particular point and not everywhere.

Consider an uncharged square loop of unit length (units where c=1) and width carrying a unit current clockwise. In the top wire the four-current is Jt=(0,1,0,0). In the right wire the four-current is Jr=(0,0,-1,0). In the bottom wire the four-current is Jb=(0,-1,0,0). And in the left wire the four current is Jl=(0,0,1,0).

Now, if we go to a frame moving at .6 c wrt the wire then the four-currents become Jt'=(-.75,1.25,0,0), Jr'=(0,0,-1,0), Jb'=(.75,-1.25,0,0), Jl'=(0,0,1,0). So the excess negative charge density on the top is balanced by an excess positive charge density on the bottom, with no net charge for the whole loop.

And you already know we don't always need the batteries to produce the current !

Clearly not. I just mentioned it because there are times when there is a net charge and in such cases the charge comes from the power source. I never said that batteries are always needed, and in fact, I specifically said that usually the power source is not needed for the explanation.

 

[DrGreg]

So if it is the spacing of the electrons that keeps changing while switching on and off the current, what do you suggest happens for a closed loop of current wire! ,where does the extra electrons go or come from ?

In the rest frame of the wire, the spacing of electrons does not change, so no extra electrons are needed.

In an inertial frame in which the wire is moving and the electrons along one straight part of the wire are at rest, electrons in a different part of the closed loop (the return wire, if you like) are not at rest. So, in this frame, the electron density isn't constant: it's lower where the electrons are at rest and higher where the electrons are moving. Averaged out over the whole loop, the total number of electrons is unchanged.

I posted the diagram below over 18 months ago in a thread you took part in:

Click here for explanation in old thread

 

[universal_101]

Consider an uncharged square loop of unit length (units where c=1) and width carrying a unit current clockwise. In the top wire the four-current is Jt=(0,1,0,0). In the right wire the four-current is Jr=(0,0,-1,0). In the bottom wire the four-current is Jb=(0,-1,0,0). And in the left wire the four current is Jl=(0,0,1,0).

Now, if we go to a frame moving at .6 c wrt the wire then the four-currents become Jt'=(-.75,1.25,0,0), Jr'=(0,0,-1,0), Jb'=(.75,-1.25,0,0), Jl'=(0,0,1,0). So the excess negative charge density on the top is balanced by an excess positive charge density on the bottom, with no net charge for the whole loop.

Seriously! I thought we were discussing switching the current on and off, what you are describing is Lorentz invariant nature of current in two different frames.

Clearly not. I just mentioned it because there are times when there is a net charge and in such cases the charge comes from the power source. I never said that batteries are always needed, and in fact, I specifically said that usually the power source is not needed for the explanation.

I think i was rather vague in my last post, what I meant was where do you get the extra electrons when the current is produced by the changing magnetic field, or let's say a current carrying superconducting wire, where current is due to changing magnetic field.

 

[Dale]

Seriously! I thought we were discussing switching the current on and off, what you are describing is Lorentz invariant nature of current in two different frames.

Your posts are rather vague. You should carefully specify what you are interested in including the reference frame. I believe that I answered the question you asked.

Regarding switching a current on or off, there is no net charge without current. There is no net charge with current. So what is the confusion?

what I meant was where do you get the extra electrons when the current is produced by the changing magnetic field, or let's say a current carrying superconducting wire, where current is due to changing magnetic field.

What extra electrons? The point of the exercise is to show that there aren't any extra electrons.

 

[universal_101]

In the rest frame of the wire, the spacing of electrons does not change, so no extra electrons are needed.

In an inertial frame in which the wire is moving and the electrons along one straight part of the wire are at rest, electrons in a different part of the closed loop (the return wire, if you like) are not at rest. So, in this frame, the electron density isn't constant: it's lower where the electrons are at rest and higher where the electrons are moving. Averaged out over the whole loop, the total number of electrons is unchanged.

I posted the diagram below over 18 months ago in a thread you took part in:

Click here for explanation in old thread

Well, there is NO need to move the wire, or consider a frame in which the wire is moving. The problem is not with the change in reference frames where there is a constant current carrying loop. The problem is with the one frame of reference(i.e. wire's) and once current being on and the other time being off. And the difference between the two scenarios is that you can't observe the current to be zero NO matter which frame of reference you choose. I'm questioning the later scenario where it can be done.

 

[Dale]

I'm still not sure what you are asking then. In the wire's frame with no current the wire is uncharged. In the wire's frame with current the wire is uncharged. There is no difference in charge density in either case, so no excess or deficit of charges to be accounted for. If you aren't asking about any other frame then the scenario seems completely and obviously unobjectionable.

Can you clearly and explicitly state what your objection is?

 

[universal_101]

I don't immediately get why the separation of the negatively charged particles doesn't contract from the man's reference frame, as they are moving relative to him, and therefore there would be a negative overall charge.

I'm asking same as the above concerned, except i don't get it ever.

That tends to be the point that confuses most of the people that actually understand the argument being presented, so kudos on understanding the argument.

The spacing of the electrons in the wire frame is determined by the observed fact that the wire is uncharged in the wire frame. This is a "boundary condition" that can be experimentally controlled.

For example, instead of having an uncharged wire you could give the wire an excess positive charge by putting it at a very high voltage. If you did that then the spacing between electrons in the wire frame would be greater than the spacing between protons.

Once the spacing is determined in the wire frame, then it is determined in all frames.

Your response, ignoring the issue and instead you take your starting point with the current already flowing, ignoring how did you get there(which is incompatible with the current model and SR, and this is what i thought we were discussing).

This is related to what I was trying to say earlier.

There are three possibilities:

1) Ignore the issue, which is what the video has done. Then you'll get questions like Noyhcat's.
2) Try to explain this in the video - which will raise the bar on the target audience
3) Raise the bar on the target as far as the "target audience" is concerned.

Overall, I favor 3, because ignoring the issue doesn't really work, and I'm afraid I don't know how to do 2) (explain the issue) without violating 3) (raising the height of the target audience).

Pervect addressing the issue, but lost me on the difference between 2 and 3.

Maybe something like this:

We know that in the lab frame we can set up a wire with a current in it. We know we can do this in a way such that the wire is electrically neutral, because the wire neither attracts nor repels a charged particle that is stationary in the lab frame. Since we have set up the wire to be electrically neutral in the lab frame, the distance between positive charges is the same as the distance between negative charges in the lab frame. Now, will the wire attract or repel a charged particle that is moving in the lab frame? If the only force that affects charged particles is the electric force, since the wire is electrically neutral in the lab frame, it will neither attract nor repel a moving charged particle.

Again taking the first option.

I'm still not sure what you are asking then. In the wire's frame with no current the wire is uncharged. In the wire's frame with current the wire is uncharged. There is no difference in charge density in either case, so no excess or deficit of charges to be accounted for. If you aren't asking about any other frame then the scenario seems completely and obviously unobjectionable.

Can you clearly and explicitly state what your objection is?

I think this summary shows what is or was my concern.

 

[Drakkith]

That would only be true if the electrons were rigidly attached to each other, which they are not. Charge carriers in a conductor are, by definition, very mobile and able to change their position and spacing in response to any fields.

In the wire frame the wire is electrically neutral. The spacing of the charges must reflect that boundary condition.

Does this mean that since electrons aren't rigid bodies, the distance between them doesn't change when they move?

Imagine we have a bunch of space probes (or whatever object you like) moving through space in a very long line, all at the same velocity. Observer A is moving parallel to the line of probes and at the same velocity while observer B is our stationary rest frame. Is the distance between the probes different for observers A and B?

 

[universal_101]

Imagine we have a bunch of space probes (or whatever object you like) moving through space in a very long line, all at the same velocity. Observer A is moving parallel to the line of probes and at the same velocity while observer B is our stationary rest frame. Is the distance between the probes different for observers A and B?

Yes of-course, atleast in the present understanding of the situation in mainstream and that is because of the length contraction and supposedly it(LC) can also increase the density of matter by making it contract in the direction of relative motion !

Edit-Oh! And I forgot to mention that you cannot observe this increase in density, but it is there!

 

[TrickyDicky]

Universal, your question was basically addressed by A.T. in #7.

However I'm also curious about the situation with no holes you mentioned and nobody replied to:

And how do you explain, if the current carrying wire is made of highly doped n-type semiconductor, when you don't have any holes to go by.

Would someone care to address it?

 

[harrylin]

What do you think about this explanation of EM-fields using SR?

https://www.youtube.com/watch?v=1TKSfAkWWN0

He pretends that his explanation applies to the electromagnet in his hand. But as far as I can tell, that electromagnet creates a magnetic field in any inertial frame; it cannot be transformed away. Therefore, that explanation of magnetism looks simply misleading to me.

 

[universal_101]

He pretends that his explanation applies to the electromagnet in his hand. But as far as I can tell, that electromagnet creates a magnetic field in any inertial frame; it cannot be transformed away. Therefore, that explanation of magnetism looks simply misleading to me.

His explanation is about force(electromagnetic) and not about magnetic field, i.e. magnetic force can be zero even if the magnetic field is present.

 

[universal_101]

However I'm also curious about the situation with no holes you mentioned and nobody replied to:

Would someone care to address it?

There is NO need to address it, because hall effect clearly shows that for most of the metals the current carrying charges are electrons(mostly) and not holes. Except for beryllium(p-type semiconductors) etc. where holes dominate as charge carriers. But the point is you won't see any cross voltage in hall effect if the negative and positive charge carriers are supposed to be exactly equal.

 

[harrylin]

His explanation is about force(electromagnetic) and not about magnetic field, i.e. magnetic force can be zero even if the magnetic field is present.

No, that video pretends to give an explanation about magnets and magnetism. But in fact, it doesn't, not really.
To make it clearer, we can put a positively charged dog in rest with the wire, next to it. Now the cat, using the frame that is co-moving with the electrons as rest frame, has to explain the lack of net force on the dog despite the electric field. The cat can only explain this by the magnetic field of the moving ions and which must exactly compensate the electric field force.

 

[A.T.]

No, that video pretends to give an explanation about magnets and magnetism.

It is about the relationship between Coulomb force and Lorentz force across different frames.

 

[TrickyDicky]

There is NO need to address it

Why did you ask in the first place then?

because hall effect clearly shows that for most of the metals the current carrying charges are electrons(mostly) and not holes. Except for beryllium(p-type semiconductors) etc. where holes dominate as charge carriers. But the point is you won't see any cross voltage in hall effect if the negative and positive charge carriers are supposed to be exactly equal.

So I guess the explanation A.T and WN commented is an oversimplification that doesn't really answer Noyhcat's problem.
My own take on this is that you guys are too hung up on the spacing between charges and its putative length contraction issue, just using the customary assumption that charges are so close together that they can be considered a continuous line of charge takes away the problem, I mean this assumption can be taken also in the electrostatic-current off set up.

 

[Dale]

Your response, ignoring the issue and instead you take your starting point with the current already flowing, ignoring how did you get there(which is incompatible with the current model and SR, and this is what i thought we were discussing).

I didn't ignore the issue. I directly addressed it. The charge on the wire is under experimental control. In particular, I said:

For example, instead of having an uncharged wire you could give the wire an excess positive charge by putting it at a very high voltage. If you did that then the spacing between electrons in the wire frame would be greater than the spacing between protons.

To be more explicit, consider a wire with self-capacitance of 1 pF. If I raise it to 1 MV then it will have an excess charge of 1 μC. This is an experimentally observed fact in the lab frame, and the spacing in the lab frame must conform to that fact (further than the spacing of the protons). Once you have determined the spacing in the lab frame, then you can use the Lorentz transform to determine the spacing in any other frame.

I hope you see now what I mean by the fact that the spacing in the lab frame is a boundary condition which is used to determine the spacing in other frames. Many other people understood the explanation, so I am not sure what is not "clicking" for you. It would help if you would be more descriptive of your particular concern.

 

[TrickyDicky]

I didn't ignore the issue. I directly addressed it. The charge on the wire is under experimental control. In particular, I said:

For example, instead of having an uncharged wire you could give the wire an excess positive charge by putting it at a very high voltage. If you did that then the spacing between electrons in the wire frame would be greater than the spacing between protons.

To be more explicit, consider a wire with self-capacitance of 1 pF. If I raise it to 1 MV then it will have an excess charge of 1 μC. This is an experimentally observed fact in the lab frame, and the spacing in the lab frame must conform to that fact (further than the spacing of the protons). Once you have determined the spacing in the lab frame, then you can use the Lorentz transform to determine the spacing in any other frame.

I hope you see now what I mean by the fact that the spacing in the lab frame is a boundary condition which is used to determine the spacing in other frames. Many other people understood the explanation, so I am not sure what is not "clicking" for you. It would help if you would be more descriptive of your particular concern.

This seems a kind of awkward way to say that we impose the condition that the charge density, in this case charge per length unit must be the same both with the apparatus on and off, IOW charge must be conserved as we all know.
Universal seems to think this is incompatible with SR's length contraction (he is of course wrong) and how exactly this is made compatible(what mechanism makes sure that the spacing between charges is both compatible with length contraction and charge conservation) is what he says you are ignoring.
Something similar motivated Maxwell's introduction of the displacement current.

 

[WannabeNewton]

He pretends that his explanation applies to the electromagnet in his hand. But as far as I can tell, that electromagnet creates a magnetic field in any inertial frame; it cannot be transformed away. Therefore, that explanation of magnetism looks simply misleading to me.

I really suggest you purchase this book: https://www.amazon.com/dp/1107014026/?tag=pfamazon01-20

 

[Dale]

Does this mean that since electrons aren't rigid bodies, the distance between them doesn't change when they move?

Yes. Consider Bell's spaceship scenario. The spaceships are not rigidly connected, so it is possible to set up an acceleration profile such that the distance between them doesn't change in the launch frame. Once you have established (as an imposed boundary condition) the distance in one frame, then you can transform to any other frame (e.g. the momentarily co-moving inertial frame) to find the distance in that other frame.

Imagine we have a bunch of space probes (or whatever object you like) moving through space in a very long line, all at the same velocity. Observer A is moving parallel to the line of probes and at the same velocity while observer B is our stationary rest frame. Is the distance between the probes different for observers A and B?

Yes. They are related by the Lorentz transform. Once you specify the distance for either observer A or for observer B then the distance for the other one is uniquely determined.

 

[Dale]

This seems a kind of awkward way to say that we impose the condition that the charge density, in this case charge per length unit must be the same both with the apparatus on and off

Yes. Hopefully between my awkward way and your simple way it gets through to him.

 

[Drakkith]

Yes. Consider Bell's spaceship scenario. The spaceships are not rigidly connected, so it is possible to set up an acceleration profile such that the distance between them doesn't change in the launch frame. Once you have established (as an imposed boundary condition) the distance in one frame, then you can transform to any other frame (e.g. the momentarily co-moving inertial frame) to find the distance in that other frame.

I don't know what an imposed boundary condition is. Could you elaborate on that?

Yes. They are related by the Lorentz transform. Once you specify the distance for either observer A or for observer B then the distance for the other one is uniquely determined.

Okay. Now, if I take two probes, one right in front of the other, and accelerate them at exactly the same rate until they reach some arbitrary velocity, will the distance between them, according to themselves, be different after the acceleration, or will it remain the same as before? (Trying to understand Bell's spaceship scenario a bit better)

 

[Dale]

I don't know what an imposed boundary condition is. Could you elaborate on that?

The laws of physics are differential equations. Differential equations don't have a unique solution. To get a unique solution you have to impose additional constraints which are known as boundary conditions. These additional constraints contain the description of the particular physical scenario to which you want to apply the physical laws and are generally considered to be "given" in the problem scenario.

For example, in projectile problems you use the law of physics $-mg=m d^2x/dt^2$ which does not give a unique solution until you supply the initial position and velocity of the projectile. The initial position and velocity are the boundary conditions which describe the particular problem and allow you to obtain a unique solution.

Similarly, the Lorentz transform doesn't give us a unique solution until we completely specify the problem. We cannot know the distance in one frame until it is specified in sufficient detail in another frame. I call that specification a "boundary condition" (taking a bit of license), in keeping with the term's use elsewhere for describing the specific problem, even though the Lorentz transform is not usually specified as a differential equation.

Okay. Now, if I take two probes, one right in front of the other, and accelerate them at exactly the same rate until they reach some arbitrary velocity, will the distance between them, according to themselves, be different after the acceleration, or will it remain the same as before? (Trying to understand Bell's spaceship scenario a bit better)

If the distance is the same in the launch frame then the distance will be different in the momentarily co-moving inertial frame (greater).

 

[universal_101]

I didn't ignore the issue. I directly addressed it. The charge on the wire is under experimental control. In particular, I said:

For example, instead of having an uncharged wire you could give the wire an excess positive charge by putting it at a very high voltage. If you did that then the spacing between electrons in the wire frame would be greater than the spacing between protons.

I think what you are suggesting is unbalance of the charge, but ofcourse if a part of conductor is positively charged then spacing between electron and proton is different, because there are more protons and less electrons and they all have to share the same volume, so their respective spacing changes accordingly.

Once you have determined the spacing in the lab frame, then you can use the Lorentz transform to determine the spacing in any other frame.

Why would I Lorentz transform anything, Lorentz transform is for analyzing a particular situation from different reference frames. As I mentioned earlier you can't have any Lorentz transform for the situation when switching the current on and off.

I hope you see now what I mean by the fact that the spacing in the lab frame is a boundary condition which is used to determine the spacing in other frames. Many other people understood the explanation, so I am not sure what is not "clicking" for you. It would help if you would be more descriptive of your particular concern.

you want me to understand something without even talking about it. Why don't you just accept that switching the current on and off makes two different situations, which can be Lorentz transformed separately but cannot be transformed into each other.

 

[harrylin]

It is about the relationship between Coulomb force and Lorentz force across different frames.

If the movie said that, then I would have no objection - it does neatly, although too simplistic, illustrate how EM fields appear differently in different frames. I also like the way of presentation, it's cool. If he had added my charged dog to his charged cat, that would have been really cool. This topic has a lot of similarity to elements of Bell's spaceship example (and a little also with Ehrenfest's rotating disc, in view of the coil).