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Why must the Lorentz Transformations Be Linear?

  1. Sep 7, 2006 #1
    All derivations of the Lorentz Transformations I've seen assume a linear transformation between coordinates. Why must this be the case? Thanks.
     
  2. jcsd
  3. Sep 7, 2006 #2

    Hurkyl

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    A transformation is linear if and only if it keeps the origin fixed, and it maps lines to lines.
     
  4. Sep 7, 2006 #3

    robphy

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    In particular, a Lorentz transformation must preserve the nature of timelike lines, which represent inertial particles.
     
  5. Sep 7, 2006 #4
    So if the transformation were not linear, then an object would appear to be accelerating in one inertial frame but moving at a constant velocity in another. This would mean F=ma doesn't hold in both frames, a violation of a relativity assumption. Does that argument sound good to you guys?
     
  6. Sep 7, 2006 #5
    linearity of the LET

    I used to say to my students that the LET should be linear because to a pair of space-time coordinates in one inertial reference frame should correspond a single pair of space-time coordinates in an other one.
     
  7. Sep 7, 2006 #6

    Hurkyl

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    LET? Oh, you mean Lorentz-Einstein transforms, not Lorentz Ether Theory.

    That happens with any 1-1 transformation...
     
  8. Sep 8, 2006 #7
    linearity

    of couse! in any consistent theory.
    sine ira et studio:smile: :
     
  9. Sep 8, 2006 #8

    Meir Achuz

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    Yes, the LT is linear so that if there is no accelartion in one LF, there will no acceleration in any other LF. But, don't talk about F=ma in SR.
     
  10. Sep 8, 2006 #9
    why no F=ma in SR?
     
  11. Sep 8, 2006 #10
    I guess it's more appropriate to say F=dp/dt.
     
  12. Sep 8, 2006 #11

    Meir Achuz

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    a and dp/dt are related by:
    [itex]\frac{d\bf p}{dt}=\frac{d}{dt}(m{\bf v}\gamma)
    = m\frac{d}{dt}\left[\frac{\bf v}
    {\sqrt{1-{\bf v}^2}}\right]
    =m\gamma^3[{\bf a}+{\bf v\times(v\times a)}].[/itex]
     
  13. Sep 10, 2006 #12
    What? How do you figure that a frame can be accelerated to one inertial frame but not accelerated to another inertial frame. This makes no sense. F=ma holds in all inertial frames. However, as mentioned above, mass can vary. If mass did not vary then, due to the speed limit of c, a given force in an inertial frame @ .9999c would obviously yield a much smaller acceleration than that same force at .00001c, and if only acceleration changed due to a given force then the postulate of special relativity would be violated. But, at very high speeds the lack of acceleration is made up for in a large increase in mass, so regardless of speed, a given force yields the same momentum.
     
    Last edited: Sep 10, 2006
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