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Simultaneity, contraction and interval.

  1. Mar 14, 2007 #1
    Hi...

    I know that this question will be found little bit stupid, but unfortunatly I can't see where I made a mistake.

    Let's have stick with light sources on both ends. In stick's reference frame the flashes from both sources are simultaneous. In laboratory ref. frame they are not. However in laboratory ref. frame the stick is shorter. From these facts we can write that x < x', t' = 0 and t is not equal 0. (add deltas)
    The invariance of the interval says that t^2 - x^2 = t'^2 - x'^2. From that t^2 - x^2 = 0 - x'^2 but since t^2 is positive and x^2 is smaller than x'^2 we can not find any noncomplex equality.

    Please explain me where I made the mistake or what is wrong with this thinking. I will be really glad,
     
  2. jcsd
  3. Mar 14, 2007 #2
    x should be the distance between the places where the flashes occur in the lab frame, which will not be the same as the length of the stick, since the stick moves between the flashes.
     
  4. Mar 15, 2007 #3
    I assume that your primed variables refer to the stick's frame.

    Although the length of the stick does appear to be shortened to the lab observer, with respect to whom the stick is in relative motion, the flashes occur at different points that are separated by a distance greater than the length of the stick. Remember, the flashes are NOT simultaneous to this observer. So, if one flash occurs at t = t1, then the second flash occurs at t =t2, i.e., after a time interval t2-t1, by which time the stick would have moved a distance v(t2-t1).

    [tex](\Delta x')^2 = (\Delta x)^2 - (\Delta t)^2, (\Delta x) > (\Delta x')[/tex]
     
  5. Mar 15, 2007 #4

    daniel_i_l

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    The mistake is that you're using the equation for a time-like interval when the two events are a space-like interval. A TMI (time-like interval) is defined by two events that can be connected by a ray of light. In this can the two events make a SLI because a ray of light can never go through both of them. The equation for a SLI is:
    s^2 = dx^2 - dt^2
     
  6. Mar 15, 2007 #5
    It doesn't matter in what order you write the coordinates. (as long you use the same order at all times).

    [itex]Space-like: \left(\Delta x\right)^2 > \left(\Delta t\right)^2[/itex]
    [itex]Time-like: \left(\Delta x\right)^2 < \left(\Delta t\right)^2[/itex]
     
  7. Mar 15, 2007 #6

    jtbell

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    To see this, apply the Lorentz transformation separately to both ends of the stick. Let S be the lab frame, S' be the stick frame, and let [itex]L_0[/itex] be the proper length of the stick. Further, let the stick be traveling to the right (+x direction) with speed v in the lab frame, and let the left end of the stick be located at the origin of both frames at t = t' = 0.

    In the stick frame:

    [tex]x^\prime_L = 0; t^\prime_L = 0; x^\prime_R = L_0; t^\prime_R = 0[/tex]

    In the lab frame:

    [tex]x_L = 0; t_L = 0[/tex]

    [tex]x_R = \gamma \left( x^\prime_R + v t^\prime_R \right)[/tex]

    [tex]t_R = \gamma \left( t^\prime_R + \frac {v x^\prime_R}{c^2} \right)[/tex]

    Calculate [itex]x_R[/itex] and [itex]t_R[/itex], and then calculate

    [tex]c^2 (t_R - t_L)^2 - (x_R - x_L)^2 = c^2 t_R^2 - x_R^2[/tex]

    and

    [tex]c^2 (t^\prime_R - t^\prime_L)^2 - (x^\prime_R - x^\prime_L)^2 = c^2 (t^\prime_R)^2 - (x^\prime_R)^2[/tex]

    which should be equal. You should also find that [itex]x_R - x_L = \gamma L_0[/itex] but this is not the length of the stick in S (which is [itex]L_0 / \gamma[/itex]) because the two x's are measured at different times in S. You can show that the two distances are consistent by calculating how far the stick moves during the time interval between the flashes.
     
    Last edited: Mar 15, 2007
  8. Mar 15, 2007 #7
    Thank you very much for your time. Now I see it and I feel little bit ashamed. Next time I should think little bit more...:)
     
  9. Mar 18, 2007 #8

    ZapperZ

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    Please note that you should NOT respond to crackpottery. Use the REPORT button. If not, the Mentors will have to not only clean up the crackpot post, but also YOUR posts that would have been left hanging.

    Zz.
     
  10. Mar 18, 2007 #9
    If you want people to use the report button in response to crackpots you might consider modifying the instructions on the report page. In particular,

    was what stopped me from actually reporting the post. Apologies for the extra work, but I was simply following your instructions.
     
  11. Mar 19, 2007 #10

    pervect

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    I reported the post as being suspect, but I wasn't sure from the initial post how bad the potential problem was, so I also posted a reply. So I'm probably the person ZapperZ is refering to.

    Is this really such a bad idea (report and respond), as long as I don't care overmuch what happens to the reply? I didn't want to spend a lot of time on a reply, but I thought a short response might draw out the OP a bit, to show if he was a hardcore crackpot type or someone whom we might give some lattitude to.

    I agree that the text on the "report bad post" button should be clarified - I've been assuming that significant violations of PF guidelines should be reported, so that a moderator can take whatever action seems best to them.
     
  12. Mar 19, 2007 #11

    ZapperZ

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    It isn't MY instructions. It has been suggested by several of us that the instruction be modified. It just hasn't been implimented as it isn't on the top priority of the list of things that need to be done on PF.

    I hope that THIS has clarified what the REPORT button should be used for.

    Zz.
     
  13. Mar 19, 2007 #12

    ZapperZ

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    PF has a very strict guidelines against crackpottery. In several instances before, people were responding to the crackpottery while other legitimate posts were also imbeded among such discussion. When the crackpot post was removed, you got all of these posts that are actually left hanging that also had to be deleted, without removing the in-between discussion. This makes it time consuming to figure out which is which.

    So before one responds to a crackpot post, keep in mind that there's a very good chance that that post will be deleted. Do you want to waste time and effort addressing such a thing?

    Use the REPORT button whenever you see something that violates the PF Guidelines. That is what the button is for, regardless of what it says it is for right now. Is everyone clear on that now?

    Zz.
     
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