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

[GregAshmore]

This paper by Cacioppo and Gangopadhyaya deals with two variations on the Pole and Barn paradox. Both paradoxes are successfully resolved. However, with regard to the first paradox, an electromagnetic wave comes into being in one frame, but not in the other. I consider this to be a paradox in its own right.

The scenario is a battery which slides on a rail. At one end of the battery is the positive terminal; the negative terminal is at the other end. There is a small gap between each terminal and the rail.

At a certain spot on the rail is a copper plate. The rest length of the plate is the same as the rest length of the battery. The thickness of the plate is the same as the gap between the terminals and the rail.

The battery moves along the rail at some fraction of light speed. An observer at rest on the rail will find that the battery is shorter than the plate, so that for some elapsed time both terminals are in contact with the plate. An observer at rest on the battery will find that the plate is shorter than the battery, so that there is never a time at which both terminals are in contact with the plate.

The paradox is presented this way: In the rail frame, the battery is shorted, but in the battery frame it is not.

The paradox is resolved by this note: "[In the rail frame a] circuit is not made until the negative terminal reaches the copper section, at which time the information that the negative terminal is now on the copper section must reach the forward (positive) terminal in order for the circuit to be completed." It is then shown that the information, which travels at the speed of light, cannot reach the positive terminal before it loses contact with the plate. Hence, the battery is not shorted.

While it is true to say that the battery is not shorted, that is not the whole story. The information that the second terminal has come into contact with the plate travels from terminal to terminal in the form of an electromagnetic wave. That wave comes into existence at the instant the rear terminal makes contact with the plate, and ceases to exist when the front terminal loses contact with the plate.

In the battery frame, there is never a time when both terminals are in contact with the plate.

The electromagnetic wave is a physical entity that exists in the rail frame, but not in the battery frame.

Have I got this right?

 

[pervect]

We had a roughly similar paradox in the past. My observation then was that lumped circuit theory didn't really work relativistically.

Specifically, Kirkhoff's current law takes a hit (though the relativistic continuity equations work just fine, it's no longer true that wires are neutral in all frames). More relevantly, Kirchoff's voltage law also takes a hit the intergal of the voltage around the loop depends on the observer.

I'm not sure where to go for further information - I do recall that my old E&M textbook derived lumped circuit theory from Maxwell's equations as an approximation valid under the correct circumstances, but I don't recall how they did it.

I advocated using non-lumped circut theory - if you have a nearby groundplane, you can then treat the copper strip in your example as a transmission line.

It may or may not be possible to "patch up" lumped circuit theory somehow (perhaps using retarded potentials) so that it makes sensible relativistic predictions. However, I'm not sure how to do this. The paper's treatment is a bit superficial.

Going back to the transmission line approach - what will happen when the positive terminal battery reaches the transmision ine is that some current will flow. If the internal inductance of the battery is low enough, the current flow would be equal to V/Z, V being the voltage of the battery and Z being the characteristic impedance of the transmision line it's charging.

The details of what happens are rather complex and I'd need to calcluate them, but basically the copper sheet woul wind up mostly being charged up first to the potential of th positive end of the battery +V, then discharged to the potential of the negative end of the battery. I'm not sure if the charging would be complete or not without more analysis, but some charging would occur.

So basically current would be flowing, but it would be mostly controlled by capacitance and /or inductance.

 

[Mentz114]

The electromagnetic wave is a physical entity that exists in the rail frame, but not in the battery frame.

In this case we could have a bomb paradox. If a detector is wired to a bomb in each frame, then they must both detect or not detect the wave.

The waves begin when the first electrode touches the copper - one backwards through the battery, and one forward through the copper. This happens in both frames.

 

[Dale]

That wave ... ceases to exist when the front terminal loses contact with the plate.

I don't think this is correct. Can you justify it?

 

[GregAshmore]

I've read all three replies. I'll respond in this one reply.

First, to DaleSpam: No, I cannot justify my statement that the wave ceases to exist when the front terminal loses contact with the plate. As I wrote the post, I figured that the wave would not degrade instantaneously. I started to say something about the wave degrading over time, but I don't know enough about the dynamics of an open circuit to say how that degradation would proceed. I decided to waffle a bit and use the potentially ambiguous word "when". I did this on the grounds that, in the context of this discussion, it doesn't matter too much how the wave degrades; the main thing is that it exists in one frame and not the other.

To pervect and Mentz144: As I thought about the problem, I wondered whether the leading terminal would begin to charge the plate as it made contact, even before the trailing terminal made contact. I decided to follow the lead of the paper, which does not begin to consider transfer of "information" until both terminals have made contact.

Presumably, the decision to ignore (or assume no) movement of charge until both terminals are in contact with the plate was a conscious simplification of the scenario. This simplification allowed the authors to make a point about the transmission speed of electrical signals, but at the cost of introducing another [seeming] paradox.

I found the paper while I was looking for a solution to a problem in Taylor and Wheeler. In T&W, there is an open circuit with a battery, lamp, and two rails. One rail is tied to the negative terminal of the battery. The other rail is tied to one end of the lamp filament (yes, it's shown as a wound filament). There is a gap in one rail. The slider is exactly as wide as the gap when both are at rest. So, when both are at rest, the lamp glows steadily. The question is whether the lamp will flicker as the slider whizzes by.

I (finally) understand length contraction and relativity of simultaneity enough to be sure that the slider will be out of contact with the rail for some period of time. Then it became a question of how the electrical circuit would behave. As I thought about the inductance in the filament, capacitance in the battery, etc, etc, all in a relativistic system, I got a headache. So I did what you can't do in class--I asked Mr. Google for help. I found the paper referenced above, which uses a simplified version of the apparatus in T&W. I also found the paper by G. P. Sastry on which the T&W problem is based. The two papers use the same method to resolve the paradox.

Now, it's obvious that there is no simple solution to the problem. Yet, a simple solution was offered. This reminds me of why I was so skeptical of what I read about relativity as I started: Most of the experiments in the books have never been run in a lab. It's easy to say things when you don't have to run the experiment to prove that you are right.

I'm mostly past that skepticism, having worked enough problems, and plotted the results on a spacetime diagram, to see that the theory is self-consistent, and to understand why the behavior of high speed particles (for example) does support the theory.

Still, it would be nice if people didn't pretend that everything is cut and dried.

 

[Dale]

the main thing is that it exists in one frame and not the other.
...
Still, it would be nice if people didn't pretend that everything is cut and dried.

I agree with your point about the oversimplification, but I am highly skeptical of the claim that a wave exists in one frame and not the other. I think that their analysis resolving the paradox was over simplified, but I think your analysis introducing a new paradox is also over simplified.

 

[PeterDonis]

The electromagnetic wave is a physical entity that exists in the rail frame, but not in the battery frame.

No, this is not correct. The EM wave exists in both frames, and in both frames, it fails to reach the positive terminal in time to complete a circuit.

Call the event when the negative terminal of the battery makes contact with the copper plate event N. Call the event when the positive terminal of the battery *loses* contact with the copper plate event P. By hypothesis, events N and P are spacelike separated.

At event N, an EM wave is created and propagates towards the positive terminal of the battery at the speed of light. However, since event P is spacelike separated from event N, the EM wave can't possibly reach the positive terminal of the battery before event P (where "before" here refers to the time ordering of events on the positive battery terminal's worldline, so the statement I just made is frame invariant). So the circuit can never be completed; but the EM wave emitted from event N exists in both frames.

 

[PeterDonis]

That wave comes into existence at the instant the rear terminal makes contact with the plate, and ceases to exist when the front terminal loses contact with the plate.

This is probably where the confusion lies. The wave does not cease to exist when the positive terminal loses contact with the plate; it ceases to exist when it reaches the positive terminal and discovers that the positive terminal has already lost contact with the plate. In other words, there some event on the positive terminal's worldline, call it event W, where the EM wave emitted at event N arrives; and this event is *after* event P (where again, this is a frame invariant statement because both events lie on the same timelike worldline). Event W is where the EM wave ceases to exist, and this is true in both frames.

 

[Nugatory]

I decided to waffle a bit and use the potentially ambiguous word "when"

In a paradox that involves relativity of simultaneity, the word "when" is not 'potentially' ambiguous - it is just plain ambiguous, and the paradox is created by exploiting that ambiguity. The paradox comes from the ambiguity, so when you decide to waffle a bit you aren't just taking a convenient verbal short-cut to describe a problem - you are creating the problem.
[EDIT]or at least making it possible to create the problem

 

[GregAshmore]

In a paradox that involves relativity of simultaneity, the word "when" is not 'potentially' ambiguous - it is just plain ambiguous, and the paradox is created by exploiting that ambiguity. The paradox comes from the ambiguity, so when you decide to waffle a bit you aren't just taking a convenient verbal short-cut to describe a problem - you are creating the problem.
[EDIT]or at least making it possible to create the problem

No, the paradox does not come from the ambiguity about the demise of the wave. The paradox arises at the creation of the wave, because the wave is created in one frame and not the other. The manner of its demise is not relevant to the paradox.

The authors treat the wave as coming into being at the instant that both terminals come into contact with the plate. This only happens in the rest frame of the plate. In the rest frame of the battery, there is never a time when both terminals are in contact with the plate; therefore, the wave never exists.

Granted, the paradox of the wave only comes about as the result of the simplification that the authors have made. If that simplification is unjustified, then the paradox of the wave goes away. Then, too, if that simplification is unjustified the authors have not resolved the paradox of the shorting battery.

 

[Nugatory]

The authors treat the wave as coming into being at the instant that both terminals come into contact with the plate.

"at the instant" sure sounds to me like an improperly specified "when"

 

[Dale]

The paradox arises at the creation of the wave, because the wave is created in one frame and not the other.

This is what I doubt. I think that any wave must be created in both frames.

 

[GregAshmore]

"at the instant" sure sounds to me like an improperly specified "when"

The discussion is of the incident as viewed from the rest frame of the plate. So the authors are talking about "when" in that frame.

 

[GregAshmore]

This is what I doubt. I think that any wave must be created in both frames.

The wave, as defined by the authors, comes into being at the instant in the rest frame of the plate at which both terminals come into contact with the plate. There is no instant in the rest frame of the battery at which both terminals are in contact with the plate, so the wave as defined by the authors cannot be created in that frame.

[Edit] It is perhaps important to note that the authors do not speak of an electromagnetic wave. They speak of information being transmitted from terminal to terminal at the speed of light. That transmission begins at the instant in the rest frame of the plate at which both terminals come into contact with the plate.

 

[Dale]

It is perhaps important to note that the authors do not speak of an electromagnetic wave.

I think that is important. You are the only one talking about a wave which exists in one frame and not in another. Whether or not there is a wave is irrelevant in one frame and important in the other frame, but that doesn't mean that the wave doesn't exist in both frames.

 

[GregAshmore]

Whether or not there is a wave is irrelevant in one frame and important in the other frame

Understood.

but that doesn't mean that the wave doesn't exist in both frames.

If the wave exists in a frame, it is due to some cause.
For each frame, let us inquire as to the cause of the wave.

What is the cause of the wave in the rest frame of the plate?
Here is the description in the paper:

the positive terminal contacts the copper section first, followed by the negative terminal. A circuit is not made until the negative terminal reaches the copper section, at which time the information that the negative terminal is now on the copper section must reach the forward (positive) terminal in order for the circuit to be completed.

The cause of the wave in the rest frame of the plate is the presence of both terminals on the plate. Presumably this is due to the establishment of a potential [edit] in the plate [end edit] from terminal to terminal.

What is the cause of the wave in the rest frame of the battery?
If the wave exists in the rest frame of the battery, it cannot be due to the same cause as in the rest frame of the plate, because in the rest frame of the battery there is never a time when both terminals are in contact with the plate.

As seen in the rest frame of the battery, the wave comes into being when the leading edge of the plate reaches the negative terminal. At that time, the trailing edge of the plate has already passed the positive terminal; only the negative terminal is in contact with the plate. What would cause an electromagnetic wave to arise at that time?

 

[Nugatory]

As seen in the rest frame of the battery, the wave comes into being when the leading edge of the plate reaches the negative terminal. At that time, the trailing edge of the plate has already passed the positive terminal; only the negative terminal is in contact with the plate. What would cause an electromagnetic wave to arise at that time?

That's what always happens when you touch a conductor to one terminal of a battery. As long as there is any potential gradient in a conductor, charge flows in whatever way levels out the gradient; that's why in a static situation all parts of a conductor are at the same potential.

Usually we assume that the leveling happens instantaneously (all parts of the copper plate reach the same potential at the same time - there are those evil and treacherous words again!) but in this problem the speeds involved are high enough and the times short enough that we have to allow for the propagation time.

 

[Dale]

As seen in the rest frame of the battery, the wave comes into being when the leading edge of the plate reaches the negative terminal.

As seen in both frames, that is when a wave would start. Why would it only be the rest frame of the battery?

The positive terminal has already charged the plate, regardless of whether it is still in contact or not. Capacitors retain their charge even when removed from the source.

 

[PeterDonis]

The wave, as defined by the authors, comes into being at the instant in the rest frame of the plate at which both terminals come into contact with the plate.

I don't think that's quite what they meant to say. Their description is vague, since they don't actually talk about what a "circuit" is, and as you note, they don't actually talk about an EM wave, but just about the "information" that the negative terminal is touching the copper propagating back to the positive terminal. But here's what I think they were trying to describe:

(1) When the positive terminal contacts the copper plate, an EM wave starts propagating from the positive terminal, through the plate, back towards the negative terminal. (As DaleSpam and Nugatory have noted, this EM wave charges the plate.)

(2) When the negative terminal contacts the plate, a "partial circuit" is formed, and an EM wave starts propagating back *through the battery* towards the positive terminal. (As DaleSpam noted, this EM wave is caused by the charge on the plate.)

(3) When the positive terminal loses contact with the copper plate, the wave that was propagating from it starts to die out; one could think of this as the end of an EM wave train propagating back through the plate towards the negative terminal.

(4) When the EM wave from the negative terminal reaches the positive terminal, it finds that it can't close the circuit because the positive terminal has lost contact with the plate.

(If the positive terminal were still in contact with the plate when the EM wave from the negative terminal reached it, the circuit would be closed, and what would close it would be the EM wave from negative to positive through the battery connecting up with the EM wave from positive to negative through the copper plate.)

 

[MikeLizzi]

I appear to have a different take on this problem. As far as I am concerned, the circuit never closes either from the reference frame of the battery or the plate. It's an issue of relativity of simultaneity.

Here is a web page with pictures and real calculated numbers

http://www.relativitysimulation.com/Documents/BatteryParadox2.htm

 

[PeterDonis]

the circuit never closes either from the reference frame of the battery or the plate.

I think everyone is in agreement on this. What's described on the page you linked to is basically what I described in posts #7 and #19, except that they are using a weird kind of clock synchronization. The issue appears to be more in the details of what would constitute the circuit closing, and what happens when a circuit is "partially closed" but not yet fully closed.

 

[Mentz114]

MikeLizzi, please point out my error, because these diagrams indicate ( to me ) that in the track frame, the -ve electrode touches the copper while the +ve electrode is in contact. The rest length of the battery is the same as the rest length of the copper. The relative velocity is 0.80c.

There seems little doubt that the resolution is by treating the copper as a capacitor and looking at the charge/discharge cycle in each frame. The transient current will produce an EM pulse in both frames - and I'll bet the pulses are related by an LT boost. The state of the system will return to its initial state when the battery has passed, and there's no reason to expect any paradox.

 

[MikeLizzi]

I think everyone is in agreement on this. What's described on the page you linked to is basically what I described in posts #7 and #19, except that they are using a weird kind of clock synchronization. The issue appears to be more in the details of what would constitute the circuit closing, and what happens when a circuit is "partially closed" but not yet fully closed.

Oh, good. Sometimes I have trouble following (understanding) other peoples posts.

With regards to the graphics I referenced and clock synchronization;

In the scene, a reference frame is defined with origin at the center of the scene (call it S) at time t=0. Its velocity with respect to us, the observers is zero.

The plate is at rest with respect to S and at position +x. So the plate has no length contraction and all clocks attached to it display the same value.

The battery is moving to the right at .866c with respect to S at some distance –x. The battery has length contraction and clocks attached to the battery at different locations display different values (proper time). The clocks on the battery are running at half the rate of the clocks on the plate.

With regards to the meaning of “closed circuit” I would offer the definition that the circuit is closed when the following condition applies:
1. The plate is in contact with terminal t1 when the proper time for the clock on t1 displays tc
2. The plate is in contact with terminal t2 when the proper time for the clock on t2 displays tc
And that never happens.

 

[MikeLizzi]

MikeLizzi, please point out my error, because these diagrams indicate ( to me ) that in the track frame, the -ve electrode touches the copper while the +ve electrode is in contact. The rest length of the battery is the same as the rest length of the copper. The relative velocity is 0.80c.

There seems little doubt that the resolution is by treating the copper as a capacitor and looking at the charge/discharge cycle in each frame. The transient current will produce an EM pulse in both frames - and I'll bet the pulses are related by an LT boost. The state of the system will return to its initial state when the battery has passed, and there's no reason to expect any paradox.

There is nothing wrong with your diagrams as far as I can see. They say that, in the reference frame of the battery, the time intervals when the terminals are touching the plate don’t overlap. In the reference frame of the plate, the time intervals do overlap. The issue is really the definition of a closed circuit. I offered a definition in my response to PeterDonis.

 

[PeterDonis]

With regards to the meaning of “closed circuit” I would offer the definition that the circuit is closed when the following condition applies:
1. The plate is in contact with terminal t1 when the proper time for the clock on t1 displays tc
2. The plate is in contact with terminal t2 when the proper time for the clock on t2 displays tc
And that never happens.

It's true that this never happens in the scenario as given, because the plate is shorter than the battery in the battery's rest frame. However, this is not the strictly correct meaning of "closed circuit", because it takes time for current to travel from one end of the battery or the plate to the other.

The strictly correct meaning of "closed circuit" is that the events I labeled N and P in previous posts--the event of the negative battery terminal making contact with the plate, and the event of the positive battery terminal losing contact with the plate--are timelike separated. (If they are null separated, we have an "edge case" where there is only a closed circuit for an instant.) In the scenario as it was given, neither of our conditions for a closed circuit are satisfied; but one could construct scenarios where your condition was satisfied but mine was not.

 

[MikeLizzi]

It's true that this never happens in the scenario as given, because the plate is shorter than the battery in the battery's rest frame. However, this is not the strictly correct meaning of "closed circuit", because it takes time for current to travel from one end of the battery or the plate to the other.

The strictly correct meaning of "closed circuit" is that the events I labeled N and P in previous posts--the event of the negative battery terminal making contact with the plate, and the event of the positive battery terminal losing contact with the plate--are timelike separated. (If they are null separated, we have an "edge case" where there is only a closed circuit for an instant.) In the scenario as it was given, neither of our conditions for a closed circuit are satisfied; but one could construct scenarios where your condition was satisfied but mine was not.

I think our conditions are identical. Yours is just phrased in more sophisticated language.
I can try any version of the OP's problem that anyone thinks will violate my conditions using my simulation program. This is, after all, just the "Pole-in-the-Barn" paradox with some diversionary electrical circuitry thrown in. I've run hundreds of versions of the "Pole-in-the-Barn" paradox during program testing.

 

[PeterDonis]

I think our conditions are identical.

No, they're not. Your condition says that event P must be later than event N in the battery's rest frame. It is perfectly possible for that to be the case but still have events N and P spacelike separated; just have the time interval between N and P, in the battery's rest frame, be less than the light-travel time between the terminals in the same frame (i.e., the distance in that frame divided by c). If this is the case, your condition is met but mine is not.

 

[MikeLizzi]

No, they're not. Your condition says that event P must be later than event N in the battery's rest frame. It is perfectly possible for that to be the case but still have events N and P spacelike separated; just have the time interval between N and P, in the battery's rest frame, be less than the light-travel time between the terminals in the same frame (i.e., the distance in that frame divided by c). If this is the case, your condition is met but mine is not.

Wow. Maybe I stated my conditions poorly because the requirement you stated isn't required.

Can you give me some numbers for starting conditions? Then I can see for myself.

 

[GregAshmore]

I think we would all agree that the solution presented in the paper is oversimplified, in that it deals with flow of charge only while both terminals are in contact with the plate.

That simplification is the root of my argument that there exists an em wave in one frame but not the other, for under the simplified conditions, charge flows in one frame but not the other. However, my objection is to an overly simplistic approach to a problem, not to SR itself.

To resolve the paradox of the shorting battery, I will consider the flow of charge at each of the four events in the sequence. (I do not try to define "closed circuit".)

In the following I summarize the events qualitatively. My conclusion is that in both frames some quantity of charge is transferred from the positive terminal to the negative terminal of the battery. Whether that quantity of transferred charge would be calculated to be the same in both frames I'm not prepared to say. I will note, however, that the total elapsed time during which charge flows is greater in the rest frame of the plate than in the rest frame of the battery.

Assume that at the start of the experiment the plate is at the same potential as the negative terminal of the battery.

The four events are these:
EvA: Positive terminal of battery meets leading edge of plate.
EvB: Positive terminal of battery meets trailing edge of plate.
EvC: Negative terminal of battery meets leading edge of plate.
EvD: Negative terminal of battery meets trailing edge of plate.

The events are plotted on a spacetime diagram, below. The diagram is scaled with c = 1.
In the statement of the problem, the rest lengths of the plate and battery are equal. For this discussion, the lengths are set to 2. Thus, it takes 2 units of time for light to travel the rest length of the plate or battery.

To keep the graph readable, the relative speed is set to 0.6.

In the rest frame of the plate:

EvA: Positive terminal of battery meets leading edge of plate.
Position = 0
Time = 0
Charge begins to flow from the positive terminal of battery into the plate.



EvC: Negative terminal of battery meets leading edge of plate.
Position = 0
Time = 2.667

In the 2.667 units of elapsed time since its inception at event EvA, the wave of charge has had more than enough time to travel the 2.0 units of plate length. There is therefore no place on the plate that has not been affected to some degree by the charge from the battery. This means that at every point on the plate the potential is now higher than the potential at the negative terminal of the battery.

Because the positive terminal is still in contact with the plate, charge continues to flow from the positive terminal of the battery into the plate.

Because every place on the plate is at a higher potential than the negative terminal of the battery, charge begins to flow from the plate into the negative terminal of the battery.



EvB: Positive terminal of battery meets trailing edge of plate.
Position = 2.000
Time = 3.333

Charge stops flowing from the positive terminal of the battery into the plate.
Total time that charge flowed into the plate = 3.333 units.

Charge continues to flow from the plate into the negative terminal of the battery.



EvD: Negative terminal of battery meets trailing edge of plate.
Position = 2.000
Time = 6.000

Charge stops flowing from the plate into the negative terminal of the battery.
Total time that charge flowed out of the plate = 3.333 units.



In the rest frame of the battery:
EvA: Positive terminal of battery meets leading edge of plate.
Position = 0
Time = 0

Charge begins to flow from the positive terminal of battery into the plate.



EvB: Positive terminal of battery meets trailing edge of plate.
Position = 0
Time = 2.667

Charge stops flowing from the positive terminal of the battery into the plate.
Total time that charge flowed into the plate = 2.667 units.


EvC: Negative terminal of battery meets leading edge of plate.
Position = -2.000
Time = 3.333

Because the charging of the plate began at its leading edge, the leading edge of the plate is now at a higher potential than the negative terminal of the battery. Therefore, charge begins to flow from the plate into the negative terminal of the battery.



EvD: Negative terminal of battery meets trailing edge of plate.
Position = -2.000
Time = 6.000

Charge stops flowing from the plate into the negative terminal of the battery.
Total time that charge flowed out of the plate = 2.6667 units.



This image shows the four events as the battery slides by the plate.
Pole leading = positive terminal of battery
Pole trailing = negative terminal of battery
Barn = plate

 

[PeterDonis]

Wow. Maybe I stated my conditions poorly because the requirement you stated isn't required.

Here's how you stated the condition originally:

With regards to the meaning of “closed circuit” I would offer the definition that the circuit is closed when the following condition applies:
1. The plate is in contact with terminal t1 when the proper time for the clock on t1 displays tc
2. The plate is in contact with terminal t2 when the proper time for the clock on t2 displays tc

In the battery's rest frame, coordinate time is equal to proper time for both terminals (because their clocks are synchronized in that frame). Terminal t1 is negative, and terminal t2 is positive (at least, I'm assuming that was your intended labeling--but actually the condition as you state it is symmetric between t1 and t2 so it doesn't really matter anyway). So #1 says that event N (negative terminal making contact with plate) has to happen before time tc, and #2 says that event P (positive terminal losing contact with plate) has to happen after time tc. Thus, event N must occur before event P in the battery's rest frame.

Can you give me some numbers for starting conditions? Then I can see for myself.

Here are two sets of numbers; the first gives results consistent with the scenario as stated in the OP--neither of our closed circuit conditions are satisfied--while the second satisfies your condition but not mine. All numbers are given in the battery rest frame.

Common Definitions

The spatial origin of the frame is equidistant between the positive and negative terminals of the battery. The time origin is at the instant when the spatial origin passes the geometric center of the copper plate (i.e., battery and plate are mutually "centered" on each other). The battery has rest length 2, so the negative terminal is at x = -1, and the positive terminal is at x = 1. The plate is moving in the negative x direction in the battery rest frame;. We use units where c = 1.

Scenario A

In this scenario, the plate has rest length 2, the same as the battery, and the relative velocity of the plate and the battery is 0.866, so the length contraction factor is 2, and the plate appears to be of length 1 in the battery rest frame. This leads to the following coordinates for events N and P:

Event N: (t, x) = (0.5, -1)

Event P: (t, x) = (-0.5, 1)

Note that event N happens after event P in this frame, and the events are spacelike separated, so neither of our "closed circuit" conditions is met.

Scenario B

In this scenario, the plate has rest length 4, and the relative velocity of the plate and the battery is 0.6, so the length contraction factor is 1.25, and the plate appears to be of length 3.2 in the battery rest frame. The event coordinates now are:

Event N: (t, x) = (-0.6, -1)

Event P: (t, x) = (0.6, 1)

Note that event N happens *before* event P in this frame, so your "closed circuit" condition is met. But the events are still spacelike separated: a light signal emitted from event N will reach the positive terminal at x = 1 *after* event P. So my closed circuit condition is *not* met; and the circuit will not actually close, because by the time the current within the battery that is initiated at event N reaches the positive terminal, that terminal is no longer in contact with the plate, so the circuit can't be completed.

 

[PeterDonis]

my objection is to an overly simplistic approach to a problem, not to SR itself.

I agree with this objection.

(I do not try to define "closed circuit".)

Actually, your analysis makes the good point that the term "closed circuit" is ambiguous; a better approach is to ask whether any net charge is transferred between the battery terminals.

My conclusion is that in both frames some quantity of charge is transferred from the positive terminal to the negative terminal of the battery.

On your assumptions (which seem to me to be reasonable), there would be some net charge transferred for any relative velocity of plate and battery (less than the speed of light).

Whether that quantity of transferred charge would be calculated to be the same in both frames I'm not prepared to say.

Since the transferred charge is in principle directly observable, it should be the same in both frames by the principle of relativity. I haven't worked out in detail how this comes out, but my guess is that it has to do with the voltage between the terminals being different in the different frames; in the battery frame, when the total time of charge flow is smaller, the voltage is higher, hence the current flow is higher, so the total charge transferred is the same.

 

[MikeLizzi]

Here's how you stated the condition originally:



In the battery's rest frame, coordinate time is equal to proper time for both terminals (because their clocks are synchronized in that frame). Terminal t1 is negative, and terminal t2 is positive (at least, I'm assuming that was your intended labeling--but actually the condition as you state it is symmetric between t1 and t2 so it doesn't really matter anyway). So #1 says that event N (negative terminal making contact with plate) has to happen before time tc, and #2 says that event P (positive terminal losing contact with plate) has to happen after time tc. Thus, event N must occur before event P in the battery's rest frame.

Here are two sets of numbers; the first gives results consistent with the scenario as stated in the OP--neither of our closed circuit conditions are satisfied--while the second satisfies your condition but not mine. All numbers are given in the battery rest frame.

Common Definitions

The spatial origin of the frame is equidistant between the positive and negative terminals of the battery. The time origin is at the instant when the spatial origin passes the geometric center of the copper plate (i.e., battery and plate are mutually "centered" on each other). The battery has rest length 2, so the negative terminal is at x = -1, and the positive terminal is at x = 1. The plate is moving in the negative x direction in the battery rest frame;. We use units where c = 1.

Scenario A

In this scenario, the plate has rest length 2, the same as the battery, and the relative velocity of the plate and the battery is 0.866, so the length contraction factor is 2, and the plate appears to be of length 1 in the battery rest frame. This leads to the following coordinates for events N and P:

Event N: (t, x) = (0.5, -1)

Event P: (t, x) = (-0.5, 1)

Note that event N happens after event P in this frame, and the events are spacelike separated, so neither of our "closed circuit" conditions is met.

Scenario B

In this scenario, the plate has rest length 4, and the relative velocity of the plate and the battery is 0.6, so the length contraction factor is 1.25, and the plate appears to be of length 3.2 in the battery rest frame. The event coordinates now are:

Event N: (t, x) = (-0.6, -1)

Event P: (t, x) = (0.6, 1)

Note that event N happens *before* event P in this frame, so your "closed circuit" condition is met. But the events are still spacelike separated: a light signal emitted from event N will reach the positive terminal at x = 1 *after* event P. So my closed circuit condition is *not* met; and the circuit will not actually close, because by the time the current within the battery that is initiated at event N reaches the positive terminal, that terminal is no longer in contact with the plate, so the circuit can't be completed.

Oh, now I think I get what you mean.
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?

 

[PeterDonis]

But, the conditions I set are sufficient to resolve the paradox.

Only for an oversimplified view of what the "paradox" is. On a more comprehensive view, I'm not sure either of our conditions are the right ones. See below.

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?

As GregAshmore's follow-up post shows, the real question is whether there is any net charge transferred, and if his analysis is correct, neither of our conditions tells whether that happens, and how much.

 

[Dale]

After some thought, I don't think that it makes sense to try to formulate a relativistic definition of what constitutes a closed circuit. A closed circuit is part of circuit theory, and circuit theory rests on three assumptions, two of which seem to be violated for a relativistic circuit. Specifically, that none of the elements have a net charge and more importantly that electrical effects happen instantaneously (so called small circuit assumption).

 

[PeterDonis]

After some thought, I don't think that it makes sense to try to formulate a relativistic definition of what constitutes a closed circuit.

I agree; GregAshmore's latest post shows that a better question to ask is whether any net charge is transferred between the battery terminals.