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Speed of Light and the Causality Problem

  1. Oct 23, 2007 #1

    Zan

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    Supposedly nothing can travel faster than the speed of light because this would violate causality and produce paradoxes. Someone on Planet Alpha will travel to Planet Beta at ten times the speed of light, so that to someone watching from planet Gamma it will LOOK like the person was on Beta before Alpha. But isn't this just a problem with APPEARANCES -- and no more strange than when a stick in water LOOKS like it is where it isn't? How is causality violated or a paradox produced -- just because some observer misinterprets the data before him, and confuses appearance with reality?
     
  2. jcsd
  3. Oct 23, 2007 #2
  4. Oct 23, 2007 #3

    OOO

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    Well, in a sense, it is just appearances. In one frame you may arrange a line of instantaneously flashing light bulbs. When your mate travels at some relativistic speed, he will report to you, that he has seen the flashes in succession and that the movement from one flash to the next was faster than the speed of light.

    But, as an informed physicist, you know that there is no action travelling faster than light, and so you tell your mate that the causality between the flashes had only been apparent to him, and not real.

    Thus, what is appearances, is not just the "velocity" of spacelike separated events (light flashes) but rather the causality between them, which your mate did insinuate implicitely and which can not exist.

    That's right. No paradoxes arise if special relativity is interpreted correctly (no causal relationship along spacelike separation).
     
    Last edited: Oct 23, 2007
  5. Oct 23, 2007 #4
    OOO, that's not right. The OP is asking about actual superluminal travel, not just shadows.
     
  6. Oct 23, 2007 #5

    JesseM

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    Do you understand about the relativity of simultaneity? If two events are simultaneous in one frame, then in another frame one happens before the other. In particular, if I send a signal "instantaneously" in my frame so that the event of my sending it and the event of your receiving it happen simultaneously in my frame, then if you are moving away from me, in your frame the event of your receiving the signal happened before the event of my sending it. And if you then send a reply which is "instantaneous" in your frame, so the event of your sending the reply happens simultaneously with the event of my receiving it, then in my frame I receive the reply before you sent it. The end result is that it's possible for me to receive your reply before I sent the first message, so if my first message was about the winning lottery numbers and I received your reply before that, I could know the numbers in advance. If you're familiar with how Minkowski diagrams work, there are some good ones illustrating this type of situation here.

    Of course this only works if we assume that every frame is able to send signals instantaneously--if there's only one frame that can do this, you can have FTL without causality problems. But in that case you'd have a preferred reference frame, violating the first postulate of relativity which says the laws of physics work the same way in every inertial frame. So, of relativity, causality, and FTL, you can pick two, but all three can't be valid at once.
     
  7. Oct 23, 2007 #6

    OOO

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    Sorry for my misunderstanding. I got the impression that he asked about the relativity of simultaneity and I thought superluminal travel just served as a tool for asking his question.

    (but hopefully you didn't want to dispute the actual content of my post, did you.)
     
    Last edited: Oct 23, 2007
  8. Oct 23, 2007 #7
    It would be more precise to say that "Lorentz transformations + FTL" means violation of causality. The principle of relativity (even if you add the invariance of the speed of light to it) does not immediately imply Lorentz transformations.

    Eugene.
     
  9. Oct 23, 2007 #8

    JesseM

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    How do you mean? Normally the Lorentz transformations are derived from the two postulates--what are the loopholes that would allow both postulates to be true but not give you a Lorentz-invariant theory?
     
  10. Oct 23, 2007 #9

    Ich

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    Well said!
    Why not?
     
  11. Oct 27, 2007 #10
    All proofs that I am aware of use some additional assumptions. For example, they often assume that events (whose times and positions in different frames are considered) are associated with non-interacting systems (such as colliding free particles or intersecting light rays). The insufficiency of the two postulates is clear already from the fact that the second postulate (the invariance of the speed of light) has relevance only to specific kinds of systems - light pulses and has nothing to say about behavior of other systems, such as interacting systems of massive particles.

    If you like, we can take apart your favorite "proof" of Lorentz transformations together.

    Eugene.
     
  12. Oct 27, 2007 #11

    JesseM

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    Well, of course it's true that the second postulate deals only with light, but wouldn't the dynamical laws governing any other system (such as the equations that would allow us to predict the dynamics of any system of interacting massive particles, given their initial conditions) be covered by the first postulate?

    I suppose since the postulates deal with inertial systems, you do have to assume that inertial coordinate systems are possible, which wouldn't be true globally in curved spacetime, for example. I'm not sure how to best define an inertial coordinate system, perhaps just a coordinate system where anything with a constant coordinate position will experience no G-forces, and where as Einstein says in his 1905 paper "the equations of Newtonian mechanics hold good" (in the appropriate limits of low relative velocities and scales where quantum effects are unimportant). One could also define them in terms of measurements on a physical grid of rigid rulers that were not experiencing G-forces, with clocks attached to each point on the rulers and synchronized using the Einstein synchronization convention. However we define them, would you say that if we assume inertial coordinate systems are possible, then that assumption plus the two postulates is enough to derive the Lorentz transformations, or would you say this is still not enough?
     
  13. Oct 27, 2007 #12
    Of course, inertial frames of reference are possible, and I don't have any objections against the first and second Einstein's postulate. However, this is not sufficient to prove Lorentz transformations. If you have a proof that does not involve any additional assumptions (most importantly, it shouldn't tacitly assume that coordinates (x,t) refer to events in non-interacting systems) I would be very interested to see it.


    Eugene.
     
  14. Oct 27, 2007 #13

    JesseM

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    Well, can we imagine as a thought-experiment that we physically instantiate the coordinate system using rigid rulers that have clocks set at each point along the rulers, where the clocks are synchronized using light as in the Einstein clock synchronization convention? In this case, to say an event happened at (x,t) would simply mean that it happened in the immediate locality of the mark "x" on a ruler that lies on or parallel to the x-axis, and that the clock at that marking read "t" at the moment the event occurred...it wouldn't matter what the event actually was (whether it referred to an event in an interacting system or a non-interacting one, for example), just what ruler-marking and clock-tick it happened next to (and for the purposes of a thought experiment 'next to' can mean something like 'infinitesimally close to'). Is this acceptable as the basis for a derivation or do you think it involves additional assumptions beyond "inertial frames of reference are possible"?
     
    Last edited: Oct 27, 2007
  15. Oct 27, 2007 #14
    That's the unjustified assumption I was talking about.

    Let us consider, for definiteness, two kinds of events. Events of the first kind are collisions of billiard balls. Each collision is characterized by coordinates (x,t) in the reference frame O and (x',t') in the moving reference frame O'. (let us also assume that the balls have very small radius and that measurement uncertainties are irrelevant). The balls move freely before and after the collisions. In this case, I can agree that (x,t) and (x't') are related to each other by usual Lorentz transformation formulas.

    For events of the second kind let us put some (positive) electric charge on each of the balls. The charges are not too high, so the balls can still collide with each other, but now there is some repulsion between them that makes their trajectories curvilinear. My claim is that in this case coordinates (x,t) and (x',t') of balls' collisions cannot be related by Lorentz formulas. There should be some corrections that take into account the interaction between balls.

    Special relativity makes an assumption that Lorentz formulas are exactly valid in both cases. Where this assumption comes from? How is it justified?

    Eugene.
     
  16. Oct 27, 2007 #15

    Integral

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    So you reject Einstein's 1905 paper?
     
  17. Oct 27, 2007 #16
    "Improve" is a better word.

    Eugene.
     
  18. Oct 27, 2007 #17

    JesseM

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    I don't get it. Were you willing to grant my suggestion that we could assign coordinates based on local readings on an actual ruler/clock system like the one I described? Do you agree that when events happen "next to" each other (in the idealized sense of being infinitesimally close) this property is transitive, so if A happens next to B and B happens next to C, then A happens next to C? If so, just consider some event E that happens at coordinates (x,t). What this means is that event E happens next to the event of the clock at position x on the ruler showing a time t. Likewise, if we say that (x,t) transforms to (x',t') in another coordinate system, this means that the event (clock which is at position x on the ruler shows a time t) on the first ruler/clock system happens next to the event (clock which is at position x' on the ruler shows a time t') on the second ruler/clock system which is in motion relative to the first one. So, if events being next to one another is transitive, then this automatically means E happens next to the event (clock which is at position x' on the ruler shows a time t') on the second ruler/clock system, so it must have coordinates (x',t'). All you really have to worry about is which clock-readings and ruler-markings on one system happen next to which clock-readings and ruler-markings on the other, the event E is actually irrelevant.
     
  19. Oct 27, 2007 #18
    Let me formulate your statement in a slightly different way. Suppose that N and I are two events that occur next to each other (they have the same coordinates (x,t)) in the reference frame O. N is an event (collision) with non-interacting balls, and I is an event (collision) with interacting balls. Then your assumption is that these two events will be seen next to each other in all other reference frames O', i.e. [itex] (x', t')_N = (x', t')_I [/itex]. Is it what you are saying?

    I agree that if this assumption is made, then we can forget about the nature of events N and I and interactions in corresponding physical systems. Then Lorentz transformations acquire universal interaction-independent status, and they can be regarded as "geometrical" transformations of global space-time coordinates. Then entire formalism of special relativity follows.

    But is this assumption obvious? Not for me. Could you please explain why do you believe that this assumption is true?

    Eugene.
     
  20. Oct 27, 2007 #19

    JesseM

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    Well, we're idealizing "next to" to mean infinitesimally close...are you saying two events can be infinitesimally close in one coordinate system but not another? This would seem tantamount to saying that physical facts about things like collisions can be different in different coordinate systems--for example, suppose we have two very small physical clocks and N represents the event of one clock showing some time t and I represents the event of the other clock showing some time t', and each clock will stop when an object collides with it, mustn't there be single frame independent truth about what time each shows when they collide and therefore what times they'll show after stopping?

    Also, isn't rejecting the notion that "infinitesimally close" is frame-independent tantamount to rejecting that spacetime can be treated as a manifold with a metric on it? Metrics are supposed to provide some frame-independent notion of "distance" between points on a manifold (or paths between points), no?
     
    Last edited: Oct 27, 2007
  21. Oct 27, 2007 #20

    Dale

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    Causality is frame invariant, so there is no problem.
     
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