A SR experiment in which an entity exists in frame A but not frame B

[pervect]

The authors were not terribly clear about what the "wave" was, which is why I don't care for the paper much.

My interpretation of what they probably meant is that the wave is a physical wave of charge which flows through the metal strip, which as I mentioned before acts approximately like a transmission line.If that's not what they meant, I'm at a bit of a loss as to what they were trying to say.

[add]
It may be the case that what they are referring to as a wave is the so-called retareded potential, http://en.wikipedia.org/wiki/Retarded_potential
[end add]

Without a ground return, the problem is tricky to analyze and rather suspectible to the details of the environment. But if you envision it occurring out in an empty vacuum with nothing nearby, and you idealize the battery, basically you expect that when the positive terminal of the battery touches the metal, there will be a physical wave of + charge transmitted to the plate from the battery, which will spread at the speed of light (assuming a vacuum dielectric) through the metal. The plate will act like a transmission line with no ground plane (or a ground plane at infinity).

The battery will acquire a net negative charge in the process, which will spread through the casing of the battery at the speed of light as well.

The capacitance of the battery to infinity multipled by its charge will be equal to the capacitance of the charged section of plate to infinity multiplied by its charge.

With a non-ideal battery, you'll need to model its internal structure as well. Which would add a lot of complexity to the problem - but you might need to, since the dimensions of the battery has a longer proper length than the wire.

Basically,whichever approach you use, you'll need to carry out your analysis using Maxwell's equations, and not circuit theory.

That's the other reason I don't like the paper - they don't make this (rather obvious) fact clear, they don't even mention Maxwell's equations.
[add]But, if they are using the retarded potential formalism and just not making it clear, their results will automatically satisfy Maxwell's equations
[end add]Actually, as I think about it, the transmission line approach is still an approximation to what Maxwell's equations will predict. The transmission line approach assumes, for instance, that no energy is radiated away. The actual metal strip will radiate, so it will be a combination of transmission line and antenna..

The overall behavior should be very similar, but I suspect it would require a computer analysis to really solve for exactly what happens via Maxwell's equations if you want to model it to a high level of detial to include antenna losses and what sort of electromagnetic signal gets radiated exactly.

Also, if you include realistic boundary conditions, it will be rather sensitive to the environment. Transmission lines with ground planes at infinity don't have any huge theoretical obstacles (they still obey Maxwsell's equations) but when you try to actually build them their behavior will change at the wave of a hand (literally). Having a ground plane helps stabilize the behavior relative to the enviornment a lot.

[add]
So, in conclusion, there should be a physical wave of charge traveling at "c", and also a mathematical "retarded potential" which might also be said to move at "c".

 

[GregAshmore]

But, the conditions I set are sufficient to resolve the paradox. They focus on relativity of simultaneity. Your additional level of preciseness (and that of the author of the OP reference) turns the thread into a discussion of circuit theory. Is it really worth it?

Yes, it certainly is, in my view. SR is presented as able to correctly represent and predict the dynamic behavior of electrical components at high relative velocities. Seeing it do so, even if only in calculations of hypothetical arrangements of components, would be pretty cool, for lack of a better term.

The details intrigue me. For example, it is not enough to calculate that the net charge transferred from terminal to terminal is the same in both frames. The progression of the electromagnetic field must be one and the same in both frames. By that I mean that there can be only one "actual" or "real" history for the em field. (Of course, the observed values of the field will be different from frame to frame at any chosen world point, according to the transformation equations. There must be one history, transformed into any number of frames.)

Taking this idea one step further, the history of any individual electron must be the same in both frames. We cannot have an electron in motion relative to its neighbors in one frame, and stationary relative to its neighbors in the other. The reason for this particular stipulation will be clear in a moment.

It is not clear to me that my qualitative analysis would satisfy the requirement of one reality. It, too, is likely to be overly simplistic. (I take pervect's most recent post as indicating the same: the situation is highly complex.)

If we consider the em field, it might be possible to construct a common history from the two "frame narratives" in my qualitative analysis. The history of the em would have a common beginning and end in the two frames. In the beginning, the charge is flowing out of the positive terminal; in the end charge is flowing into the negative terminal.

The history in the middle of the episode is a bit trickier. By middle, I mean the period in which: a) the charged plate is in contact with neither terminal in the battery frame; and b) a constant (?) current is flowing in the plate frame. The magnitude of the em field might be the same in both frames--in one frame due to static charge, and in the other frame due to constant current.

However, even if the magnitude of the field is the same in the two frames, can the particle history be the same, with current in one frame but not the other?

 

[Dale]

The progression of the electromagnetic field must be one and the same in both frames. By that I mean that there can be only one "actual" or "real" history for the em field. (Of course, the observed values of the field will be different from frame to frame at any chosen world point, according to the transformation equations. There must be one history, transformed into any number of frames.)

That is guaranteed by the fact that Maxwells equations are invariant under the Lorentz transform.

 

[Mentz114]

But, the conditions I set are sufficient to resolve the paradox. They focus on relativity of simultaneity. Your additional level of preciseness (and that of the author of the OP reference) turns the thread into a discussion of circuit theory. Is it really worth it?

If you're referring to the link you gave showing some times on the clocks - I have to say I'm not convinced by that argument. There's no doubt that in the track frame both electrodes are in contact for a time, and in the battery frame they are not. There's no simultaneity problem with that. The resolution comes about by considering the flows of charge, and the fact that the time for the circuit to close is greater in the track frame than the battery frame.

 

[pervect]

If you're referring to the link you gave showing some times on the clocks - I have to say I'm not convinced by that argument. There's no doubt that in the track frame both electrodes are in contact for a time, and in the battery frame they are not. There's no simultaneity problem with that. The resolution comes about by considering the flows of charge, and the fact that the time for the circuit to close is greater in the track frame than the battery frame.

The notion of "a circuit" to my mind, isn't Lorentz invariant because it depends on the notion of simultaneity.

Which is why I say that circuit theory isn't quite the same thing as Maxwell's equations.

The paper seems to have some other defintion of a "circuit" in mind, perhaps using retarded potentials. But it's difficult to see exactly what they had in mind, at least from my first reading it wasn't terribly clear.

In any case, I think it's clear that charge will flow. It might help to replace the battery with a pair of charged spheres, one with a net + charge and the other with a net - charge, and analyze the resulting problem. The battery will be sort of like that, except there will be some mechanism that transports charge at some rate to attempt to keep the charge on the spheres constant. Said mechanism will not be and cannot be perfect.

 

[Mentz114]

The notion of "a circuit" to my mind, isn't Lorentz invariant because it depends on the notion of simultaneity.

Which is why I say that circuit theory isn't quite the same thing as Maxwell's equations.

The paper seems to have some other defintion of a "circuit" in mind, perhaps using retarded potentials. But it's difficult to see exactly what they had in mind, at least from my first reading it wasn't terribly clear.

In any case, I think it's clear that charge will flow. It might help to replace the battery with a pair of charged spheres, one with a net + charge and the other with a net - charge, and analyze the resulting problem. The battery will be sort of like that, except there will be some mechanism that transports charge at some rate to attempt to keep the charge on the spheres constant. Said mechanism will not be and cannot be perfect.

Well, you don't seem to be disagreeing with anything in my post, so it seems a bit ungracious to disagree with your assertion circuit theory is not Lorentz invariant.

If the battery potential is 10v and it has 10 ohm internal resistance, when the terminals are short circuited we get a steady current of 1 amp on our am(p)meter. Every observer will agree on that reading. If we had instruments that could detect potentials in the battery rest frame, the readings on those instruments would be invariant. So in what way could we detect this LT failure ?

 

[pervect]

Well, you don't seem to be disagreeing with anything in my post, so it seems a bit ungracious to disagree with your assertion circuit theory is not Lorentz invariant.

If the battery potential is 10v and it has 10 ohm internal resistance, when the terminals are short circuited we get a steady current of 1 amp on our am(p)meter. Every observer will agree on that reading. If we had instruments that could detect potentials in the battery rest frame, the readings on those instruments would be invariant. So in what way could we detect this LT failure ?

Every observer will agree on the reading, but not all observers will agree on I = dQ/dt, t being coordinate time. This is I believe how 3-current is defined (in terms of coordinate time). Would you disagree?

3-Current through a wire isn't any sort of Lorentz invariant, it's just the component of a 4-vector. The 4-vector as a whole is of course Lorentz invariant. But circuit theory doesn't use the 4-vector approach, it uses the non-covariant 3-vector.

If you boost a current loop you get the situation in https://www.physicsforums.com/showthread.php?t=631446, in which the 3-current varies from I*gamma to I/gamma.

The easy situation to analyze is a transverse boost, where because of time dilation, the current drops by a factor of gamma.

The case of a parallel boost gets a bit trickier - rather than get off topic I suggest referring to the original "boosting a current loop" post. BUt the point is that no, not everyone agrees that the current is equal to I in this situation. At least by my reckoning.

Ohms law is another example of something used in circuit theory that is not Lorentz invariant. You can write it in tensor form, but the standard circuit theory version isn't written that way.

Kirchoff's voltage law has some issues, too, I believe. The definition of "voltage around the loop" depends on then notion of simultaneity used. I believe that these isssues are highly relevant to the "paradox" in question.

So there are a lot of elements of circuit theory that are not Lorentz invariant -though Maxwell's equations certinaly are.

Let me point out on the experimental level that it would be a rather unusual battery that had a 10 ohm internal impedance at all. frequences from 0-100 Ghz. (It might be possible to design one with careful enough construction techniques - it doesn't seem fundamentally impossible like designing a battery with no internal impedance).

If one is trying to actually understand the actual physics of a real battery in the situation, the DC impedance of the battery would be mostly irrelevant. The lowest frequency of interest would be 1 / (2 pi t), t being the time of contact. So if t is on the order of 1 ns (a rather long wire or battery of 1 foot), you'd be in the Ghz region already as the lowest frequency of interest.

Actual circuits usually have capacitors across the battery placed at strategic locations to keep the impedance low across a range of frequencies, they don't rely on a battery having a low impedance at "high" frequencies. Usually you use a lot of such "bypass" capacitors, you scatter them around the circuit board strategically, near the source of things you need to bypass.

What's really important at high frequencies is capacitance - and lead inductance.

This suggests some obvious (to my mind) simplifications of the original thought experiment to avoid batteries (just use a capacitor, that's what you'd use realistcally anyway) but perhaps this is drifting away from the point (though if you wanted to actually carry out an experiment, it would be essential).

I still believe that the simple resolution of the issue is to say that Maxwell's equations are Lorentz invariant, but lumped circuit theory has various and sundry issues. There might be a way to cast circuit theory in a provably Lorentz invariant form, but I'm not aware of anyone who has actually done this.

 

[Mentz114]

Pervect, thanks for the detailed breakdown of the issues, some of which you've aired in previous posts here and elsewhere. The point I'm trying to make is that it is not relevant that no-one has modeled all these phenomena in a relativistic way because relativity already contains the resolution.

What I mean is this - the physical events that take place can be thought of as a skein of time-like and null worldlines representing electrons and other involved matter, and em radiation. The configuration of these 4D curves is the physics. And it is fundamental to SR that no relativistic effect can alter this.

For instance in the simple case of the electron moving relative to an electric field. Some observers see a magnetic field why ? Because otherwise the worldline of the electron would look different in that frame if it were not there. So relativistic effects are there to preserve the integrity of the fundamental configuration.

So we know there can be no paradox in the battery-track scenario, and efforts to explain this are purely academic exercises - and not being able to explain it actually means nothing at all.

Although it can be fun and instructive.

 

[pervect]

Pervect, thanks for the detailed breakdown of the issues, some of which you've aired in previous posts here and elsewhere. The point I'm trying to make is that it is not relevant that no-one has modeled all these phenomena in a relativistic way because relativity already contains the resolution.

What I mean is this - the physical events that take place can be thought of as a skein of time-like and null worldlines representing electrons and other involved matter, and em radiation. The configuration of these 4D curves is the physics. And it is fundamental to SR that no relativistic effect can alter this.

For instance in the simple case of the electron moving relative to an electric field. Some observers see a magnetic field why ? Because otherwise the worldline of the electron would look different in that frame if it were not there. So relativistic effects are there to preserve the integrity of the fundamental configuration.

So we know there can be no paradox in the battery-track scenario, and efforts to explain this are purely academic exercises - and not being able to explain it actually means nothing at all.

Although it can be fun and instructive.

I agree there isn't any paradox. I'm hoping that pointing out the correct way to get results will be helpful to some people.

Unfortunately, the dedicated "paradox hunter" seems to be mostly beyond reach (perhaps someday one will surprise me though by listening and understanding.)

The "paradox hunter", by focusing on ever-more complex problems, manages to avoid confronting the underlying issues (mostly related to the relativity of simultaneity) that cause their problems in understanding the theory.

Basically the way to actually learn a theory is to study simple examples, not complex ones.

Complex examples are good for creating fear, uncertainty, and doubt, debating, and generally "blowing smoke", but aren't usually very good for learning.

 

[Mentz114]

I agree there isn't any paradox. I'm hoping that pointing out the correct way to get results will be helpful to some people.

Of course.

My post is a bit whimsical if not tautological and off-topic.

However, I don't rate simultaneity as a major cause of confusion. For example, in the classic barn-pole scenario, the paradox is said to arise from a simultaneity issue. But actually there is no frame in which pressing the button and the door closing can be simultaneous. The resolution is that the time between these events ( which happen in the barn frame) is hugely dilated in the pole frame. So the resolution is by looking at clock rates - not simultaneity. I'm occupied for the next 12 hours, so I won't be able to reply to a storm of protest, should one arise.

 

[ghwellsjr]

I haven't read every detail of this thread but I don't think any of you have considered the fact that there is an EM wave and currents flowing in the copper plate even before the plate reaches a battery terminal which makes the analysis even more complicated.

 

[GregAshmore]

That is guaranteed by the fact that Maxwells equations are invariant under the Lorentz transform.

Not if, at the event in question, the conditions stated for one frame are fundamentally different than the conditions stated for the other frame. In the present discussion, the paper referenced in the original post stated (in effect) that, at one and the same event: a) charge is transferred from terminal to plate in the rest frame of the plate; b) no charge is transferred from terminal to plate in the rest frame of the battery.

Unfortunately, the dedicated "paradox hunter" seems to be mostly beyond reach (perhaps someday one will surprise me though by listening and understanding.)

The "paradox hunter", by focusing on ever-more complex problems, manages to avoid confronting the underlying issues (mostly related to the relativity of simultaneity) that cause their problems in understanding the theory.

Basically the way to actually learn a theory is to study simple examples, not complex ones.

Complex examples are good for creating fear, uncertainty, and doubt, debating, and generally "blowing smoke", but aren't usually very good for learning.

The problem in this case is that the simple approach leads to a new paradox. I didn't bring up transmission lines or antennas or capacitors; you folks introduced complexity in order to solve the problem. (I don't fault you for doing so.) I was quite happy with the solution in the referenced paper for a couple of days. It was only as I tried to picture the solution on a spacetime diagram that I realized that the em wave in the one frame does not exist in the other.

I haven't read every detail of this thread but I don't think any of you have considered the fact that there is an EM wave and currents flowing in the copper plate even before the plate reaches a battery terminal which makes the analysis even more complicated.

I suppose so, due to the field surrounding the moving terminal (as seen from the rest frame of the plate).

I doubt that there is a simple solution to the question posed in the referenced paper, or to the question posed in the similar problem in Taylor and Wheeler. Indeed, the qualitative analysis which I presented a few posts back has the same problem as the solution offered by the authors of the referenced paper.

The problem is this:
In a simple solution, we assume that when (in a frame) a terminal is not in contact with the plate, no charge is transferred from that terminal to the plate. If we choose any event from the middle part of the sequence, we will find that neither terminal is touching the plate in the rest frame of the battery, and both terminals are touching the plate in the rest frame of the plate. Consequently, in any simple solution, we will find charge passing from terminal to plate in the one frame but not the other. We will thus have an em field in the neighborhood of the terminal that exists in one frame but not the other.

 

[PeterDonis]

If we choose any event from the middle part of the sequence, we will find that neither terminal is touching the plate in the rest frame of the battery, and both terminals are touching the plate in the rest frame of the plate.

No, this is *not* what we will find. What we will find is that the events "from the middle part of the sequence" in the battery rest frame, where neither terminal is touching the plate, are *different* from the events where the terminals *are* touching the plate, in either frame. You mentioned spacetime diagrams; drawing one will make it obvious that whether or not a given terminal is touching the plate at a given event is frame invariant.

 

[GregAshmore]

No, this is *not* what we will find. What we will find is that the events "from the middle part of the sequence" in the battery rest frame, where neither terminal is touching the plate, are *different* from the events where the terminals *are* touching the plate, in either frame. You mentioned spacetime diagrams; drawing one will make it obvious that whether or not a given terminal is touching the plate at a given event is frame invariant.

My statement makes no sense: "If we choose any event from the middle part of the sequence, we will find that neither terminal is touching the plate in the rest frame of the battery, and both terminals are touching the plate in the rest frame of the plate."

It makes no sense because two spatially separated terminals cannot be at one event.

The mental image that I had when I made the statement was based on the qualitative explanation several posts back. The times and positions in that explanation are from the spacetime diagram shown in that post. Even so, as I started to say what I meant in more precise terms, I found that the mental image I had formed of the process is not in agreement with the spacetime diagram. I'll need some time to think this through more carefully.

 

[pervect]

The problem in this case is that the simple approach leads to a new paradox. I didn't bring up transmission lines or antennas or capacitors; you folks introduced complexity in order to solve the problem. (I don't fault you for doing so.) I was quite happy with the solution in the referenced paper for a couple of days. It was only as I tried to picture the solution on a spacetime diagram that I realized that the em wave in the one frame does not exist in the other.

Which "em" wave are you referring to?

The "wave" of moving charges exists in all frames. There's also the "retarded potential".

I would guess that you're talking about the "em wave" in the paper, and my best guess is that they're talking about the retarded potential, so I'll answer it assuming that's what you meant, and assuming that's what the paper meant. (If you're complaining about the paper, I have to agree it's not terribly clear what it meant).

If you meant something else, we can have another go-around.

The retarded potential is a bit like a voltage. It's not really directly measurable (at least not classically). It's not uniquely defined because of the gauge condition, so you might regard it as a mathematical abstraction.

If you regard tensors as "existing" then it exists. If you require that it be able to be measured with some instrument, then it's just a mathematical abstraction (in all frames) because no instrument can measure the gauge part of it, the gauge part can be set arbitrarily (within certain rules.

There's no interpretation of the retarded potential that meets your criterion of "existing in one frame and not another" that I can see. you can regarded as "existing" or "being a mathematical abstraction" sensibly, depending on the details of what you mean by "exist", but the question of its "existence" doesn't depend on the frame in any way.

I doubt that there is a simple solution to the question posed in the referenced paper, or to the question posed in the similar problem in Taylor and Wheeler. Indeed, the qualitative analysis which I presented a few posts back has the same problem as the solution offered by the authors of the referenced paper.

The problem is this:
In a simple solution, we assume that when (in a frame) a terminal is not in contact with the plate, no charge is transferred from that terminal to the plate. If we choose any event from the middle part of the sequence, we will find that neither terminal is touching the plate in the rest frame of the battery, and both terminals are touching the plate in the rest frame of the plate. Consequently, in any simple solution, we will find charge passing from terminal to plate in the one frame but not the other. We will thus have an em field in the neighborhood of the terminal that exists in one frame but not the other.

The way I see it is this. This is a variant of the barn and the pole paradox. In the barn and the pole paradox, we learn that rigid bodies are idealizations that don't exist.

In this refinement of the barn-and-pole paradox, we learn that lumped circuit elements don't exist. They are rather similar to rigid bodies, in that they are overly simple.

A rigid body is defined by a small set of numbers - it's position, and rotation. The equations that model it are simple differential equations.

A nonrigid body is defined by a "fluid". The equations that model it are partial differential equations.

The lumped circuit elements are also defined by a small set of numbers - charge for a lumped capacitor, current for a lumped inductor.

Their equations of lumped circuit elements, in ordinary circuit theory, are described by ordinary differntial equations.

This is only an approximation. Real, physical circuit elements need to be described by fields. The equations that describe these fields are partial differential equations, Maxwell's equations.

Lumped circuit elements, are, like rigid bodies, only approximations. The actual description of a bodies state requires more than a few numbers.

If you start to draw off charge from a capacitor, let's say you put a discharging wire on the left side of the plate, the voltage on the right side of the plate does not jump instantaneously, faster than light. The charge has to flow across the plate, through the wire.

You can try to make a "paradox" out of this. Nothing can move faster than light but in your lumped circuit model, the right side of the plate discharges instantaneously when you connect the wire to the left side of the plate.

But there isn't any "paradox". There is only a model that's insufficiently advanced - a model that's trying to describe what needs to be describe by fields and partial differential equatons by "avereages" of the fields and ordinary differential equations.

 

[GregAshmore]

Which "em" wave are you referring to?

The wave that is implied by what the paper calls "information" transmitted from negative to positive pole at the speed of light. This must mean some transfer of charge, I think.

The "wave" of moving charges exists in all frames.

I need to understand how this is so. You all have told me that it is a matter of understanding the relativity of simultaneity. I thought I did understand the relativity of simultaneity. As has happened a hundred times in this relativity business, I did not understand. I will take some comfort in the fact that I have not completely misunderstood. What was entirely opaque to me is now fairly luminous, though still out of focus. I'll work through the spacetime diagram some more. I see what I did wrong; now I'll see if I can do it right.

(If you're complaining about the paper, I have to agree it's not terribly clear what it meant).

chuckle.

You can try to make a "paradox" out of this.

I was not trying to make a paradox--just not able to see how it made sense in both frames. As I say, I'll chew on the spacetime diagram some more. I'm finding it difficult to think about this at all because I can't get away from saying "when", "before", "after". Even on the spacetime diagram, you can't watch the process from beginning to end. I get twisted up, and say things that contradict what I already know, such as talking about one event when at least two must be involved.

 

[PeterDonis]

I'll work through the spacetime diagram some more. I see what I did wrong; now I'll see if I can do it right.

I think the diagrams that Mentz114 gave in post #22 are a good start:

https://www.physicsforums.com/showpost.php?p=4244336&postcount=22

 

[pervect]

The wave that is implied by what the paper calls "information" transmitted from negative to positive pole at the speed of light. This must mean some transfer of charge, I think.

My take on what this means is that it means the Lienard Wiechart potential
http://en.wikipedia.org/w/index.php?title=Liénard–Wiechert_potential&oldid=518131167

Also known as the "retarded potential"

http://en.wikipedia.org/w/index.php?title=Retarded_potential&oldid=530495062

To get the potential, you need to add up the contributing potentials for all charges. Initially, all the charges should be in the battery, so at the start the LW potetial is determined by the battery.

The electric part of it varioulsy called E or $\varphi$ is basically a voltage. You can measure it with a voltmeter except for an additive constant.

If you'll look at the definition, you'll see that because of the retarded time, the L.W. potential of a charge incorporates lightspeed propagation delays. So the position of a charge now doesn't add to the potential until "later", later being determined by the lightspeed delay in the frame you choose to do the analysis.

The magnetic part of it is usually called A. I'm not sure how much to say about A,perhaps it's best to read the wiki article and see if you have any questions about it. Failing that (i.e. if the wiki is so much goobley gook as far as you're concerned) you can tell us if you know what div, grad, and curl are. If you do, we might be able to say a bit more about A.

(E,A) forms a perfectly valid 4-vector (i.e. a tensor). If you regard tensors as "existing", then it "exists". But I'm not sure of your philosophy here.

 

[Dale]

Not if, at the event in question, the conditions stated for one frame are fundamentally different than the conditions stated for the other frame. In the present discussion, the paper referenced in the original post stated (in effect) that, at one and the same event: a) charge is transferred from terminal to plate in the rest frame of the plate; b) no charge is transferred from terminal to plate in the rest frame of the battery.

I certainly didn't read it that way, perhaps you can point out exactly where you drew that idea from.

However, if someone did propose such a scenario then the charges for a) and b) are not related by a boost so we wouldn't expect the resulting EM fields or any other aspect of the scenarios to be related by a boost. That doesn't contradict my assertion above, it is just not relevant to such a scenario.

 

[GregAshmore]

I certainly didn't read it that way, perhaps you can point out exactly where you drew that idea from.

I can't talk right now, I'm chewing...on the spacetime diagram. I expect that your original suggestion that the wave/signal is present in both frames will turn out to be correct.

 

[GregAshmore]

My take on what this means is that it means the Lienard Wiechart potential. Also known as the "retarded potential"

If you'll look at the definition, you'll see that because of the retarded time, the L.W. potential of a charge incorporates lightspeed propagation delays. So the position of a charge now doesn't add to the potential until "later", later being determined by the lightspeed delay in the frame you choose to do the analysis.

I get this part, conceptually. As to the rest, I'm afraid it is out of range for me mathematically at this point. How far I will pursue the math will be decided later. (If I keep making progress at the historical rate, not far.)

Honestly, I've never had an experience like this in my life. I've been working at understanding relativity for over five years--not full time, and not to the exclusion of other spare time interests, but five years just the same. I've been confused by other subjects, certainly. But in those cases, I never got it, I knew hadn't gotten it, and I moved on. With relativity, I read, I think I get it, I open my mouth, and I find that I am just plain wrong. That scenario has played out more times than I care to recall.

I have made genuine progress while actively working problems. I left off working problems for a while due to certain circumstances. I'm back at it, and, in spite of this most recent stumble, I believe that I will get this figured out.

I have not been able to form a mental picture of what happens in a relativistic episode that involves anything more than isolated particles. I can see the progression in either frame separately, but I cannot fuse the two into one process. It's the mix-up of time that gets me. I accept that the two seemingly contradictory narratives do not actually contradict each other, but until I can get a sense of the one reality that the two narratives express, I will continue to make dumb mistakes.

...(E,A) forms a perfectly valid 4-vector (i.e. a tensor). If you regard tensors as "existing", then it "exists". But I'm not sure of your philosophy here.

This matter of the philosophy of reality is why I keep chipping away at the rock pile of relativity. I have a particular question I need to answer, a question at the junction of physics and metaphysics. I won't get into it here; I understand and agree to the rule that says we stick to physics on this forum.

 

[GregAshmore]

I spent some time studying the spacetime diagram for this plate and battery setup. To help me see it better, I doubled the rest lengths, shifted the start position and time to put the "middle" of the process at the origin, and added some time projection lines. The improved diagram is below. (I also found that saving the screenshot as a png avoids the smearing effect that I had with jpg.)

I'll make two observations now.

1. By what I consider the most natural reading of the referenced paper, the authors are saying that, in a given frame, an electrical signal will come into existence only when both terminals are in contact with the plate. In this scenario, that happens only in the rest frame of the plate. Therefore, an electrical signal exists in one frame but not the other. That cannot happen. In the OP, I called this a paradox. Now, I'll just call it incorrect. Perhaps the authors meant something else; then my complaint does not apply. As to whether they were thinking of retarded potential, perhaps they were. I don't think that changes the outcome of an em wave in one frame but not the other, based on what they actually said in the paper.

2. I was clearly wrong when I said that a terminal could be in contact with the plate in one frame but not the other. I saw that two days ago as I prepared my response to PeterDonis. What I was not able to do at that time was visualize the "combined" process, as opposed to two separate and independent processes. That's not a very good way to say what I have in mind, but that will have to do for now.

Looking at a specific event on the spacetime diagram, I can give you an idea of what I meant about one reality for the two frames, down to the particle level. (I don't refer to QED; I have a big enough headache already--and no desire to put my foot deeper into my mouth.)

At the origin of the spacetime diagram:
a) In the rest frame of the plate, both terminals have been in contact with the plate for some time.

b) In the rest frame of the battery, neither terminal has been in contact with the plate for some time.

Given the complexity of the problem, I am not going to say that current is flowing in the rest frame of the plate, but no current is flowing in the rest frame of the battery. Indeed, I am not going to make any prediction at all about the state of the plate.

What I will do is point out that in both frames the positive terminal (only) was in contact with the plate for some time, and thus some amount of charge passed into the plate. In the rest frame of the plate, when the second terminal comes into contact, the plate is already "primed" with charge, and thus the flow of charge would tend to increase. On the other hand, in the rest frame of the battery, when the terminal loses contact with the plate the flow of charge would tend to decrease.

I understand enough about inductance to know that sudden changes in current are resisted; not to mention the other complexities that have been talked about. So I am not making any disparaging remarks about relativity. What I am pointing out is that it is not at all obvious to someone who is not on the "inside" that everything will work out in practice so that when the state of the field in the plate is calculated from the perspective of the rest frame of the plate, it will match (after transformation) the state calculated from the perspective of the rest frame of the battery.

On the other hand, there is not much in this hypothetical case that is practical. So there is no point in getting too worked up about it. The only comment I can make of practical value is to suggest that physics textbooks for beginners would do better not to gloss over obvious difficulties. At the least, simplifications should be stated as assumptions.