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

  1. Jan 25, 2013 #1
    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?
     
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  3. Jan 25, 2013 #2

    pervect

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    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.
     
  4. Jan 25, 2013 #3

    Mentz114

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    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.
     
    Last edited: Jan 25, 2013
  5. Jan 25, 2013 #4

    Dale

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    I don't think this is correct. Can you justify it?
     
  6. Jan 26, 2013 #5
    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.
     
  7. Jan 26, 2013 #6

    Dale

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    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.
     
  8. Jan 26, 2013 #7

    PeterDonis

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    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.
     
  9. Jan 26, 2013 #8

    PeterDonis

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    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.
     
  10. Jan 26, 2013 #9

    Nugatory

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    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
     
  11. Jan 26, 2013 #10
    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.
     
  12. Jan 26, 2013 #11

    Nugatory

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    "at the instant" sure sounds to me like an improperly specified "when"
     
  13. Jan 26, 2013 #12

    Dale

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    This is what I doubt. I think that any wave must be created in both frames.
     
  14. Jan 26, 2013 #13
    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.
     
  15. Jan 26, 2013 #14
    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.
     
  16. Jan 26, 2013 #15

    Dale

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    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.
     
  17. Jan 26, 2013 #16
    Understood.

    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 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?
     
    Last edited: Jan 26, 2013
  18. Jan 26, 2013 #17

    Nugatory

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    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.
     
  19. Jan 26, 2013 #18

    Dale

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    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.
     
  20. Jan 26, 2013 #19

    PeterDonis

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    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.)
     
    Last edited: Jan 27, 2013
  21. Jan 26, 2013 #20
    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
     
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