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How to determine which frame measures "larger" dx?

  1. Feb 21, 2015 #1
    First time posting in this section. I understand that this question could possibly be an old and common question about Lorentz Transformation, however I failed to find useful discussions or instructions online.

    Assuming that there're 2 frames ##S, S'## where ##S'## moves along the ##x_{+}## axis of ##S## at constant speed ##v##. The frames coincide at ##<0,0,0,0>## (as well as their rectilinear coordinate axes) for a starting event ##P## and then measure ##<x, y, z, t>## and ##<x', y', z', t'>## respectively for event ##Q##.

    According to Lorentz Transform I shall have:

    ##x'=\gamma \cdot (x-vt)##
    ##t'=\gamma \cdot (t-\frac{vx}{c^2})##

    where ##\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}##

    Now that ##\frac{dx'}{dx} = \gamma## which indicates that ##\frac{dx'}{dx}## is independent of the direction of ##v## along the ##x## axis. However it doesn't make sense to me here. If I introduce a 3rd frame ##S''## which moves along the ##x_{-}## direction of ##S## at a constant speed ##v##, then should I get ##\frac{dx''}{dx}=\gamma## as well and further ##dx'' = dx'## (which should NOT hold bcz ##S'## and ##S''## are dynamic to each other)? Did I make a mistake in the calculation?

    I'm quite confused for how "some degree of symmetry" (for relation of ##dx, dx', dx''## above) could be achieved if Lorentz Transformation is true. Any help will be appreciated :)
     
  2. jcsd
  3. Feb 21, 2015 #2

    DrGreg

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    When there is more than one variable involved, it's not true that[tex]
    dx' = \frac{dx'}{dx}\,dx
    [/tex]The correct equation is [tex]
    dx' = \frac{\partial x'}{\partial x}\,dx + \frac{\partial x'}{\partial y}\,dy + \frac{\partial x'}{\partial z}\,dz + \frac{\partial x'}{\partial t}\,dt
    [/tex]
     
  4. Feb 22, 2015 #3
    Hi @DrGreg, I'm not sure how I should interpret your answer. Of course you're right about that ##dx' = \frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy + \frac{\partial x'}{\partial z} dz + \frac{\partial x'}{\partial t} dt##, however if ##dx, dy, dz, dt## are mutually independent then ##\frac{dx'}{dx} = \gamma## should still hold. Do you imply that ##dx, dy, dz, dt## can NEVER be mutually independent for any events ##P## and ##Q##?

    I tried to put the classical thought experiment where ##P## is "emission of light along x+ axis of ##S##(as well as ##S'##)" and ##Q## is "detection of light somewhere in space-time of ##S##(as well as ##S'##)" into calculation. Now I have ##x=ct## of ##S## as variable dependency but I'm still lost in the maths :(

    Maybe I shall re-describe the question this way: 2 observers Alice and Bob who remain still in frame ##S## and ##S'## respectively where ##S## and ##S'## are the same as stated in my original question. Alice learns Special Relativity and he figures out that currently for 2 specific events ##P, Q## Bob measures larger(or smaller maybe, haven't figured this out) time-elapsed ##dt'## than his own measurement ##dt##. Can Alice just reverse the direction of his ##x##-axis(i.e. rotate around the ##z##-axis) and say that "from now on Bob measures smaller ##dt'## than my ##dt##"? In short is measurement dependent upon "choice of coordinate axes" or "alignment of coordinate axes"?
     
  5. Feb 22, 2015 #4

    DrGreg

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    To put it more precisely, ##dx' = \gamma dx## is true provided ##dt = 0##. (In your question ##dy = dz = 0## always, so we can ignore those.)

    Similarly, ##dx'' = \gamma dx## is true provided ##dt = 0##. When you come to compare S' with S'', however, you can only conclude that ##dx'' = dx'## whenever ##dt = 0##, when really what you are interested is when ##dt' = 0## (or ##dt'' = 0## if you are comparing S' with S'').
     
  6. Feb 22, 2015 #5
    @DrGreg, that makes sense. Do you minding taking a look at this as well(quoted from my previous reply)?

    2 observers Alice and Bob who remain still in frame ##S## and ##S′## respectively where ##S## and ##S′## are the same as stated in my original question. Alice learns Special Relativity and he figures out that currently for 2 specific events ##P,Q## Bob measures larger(or smaller maybe, haven't figured this out) time-elapsed ##dt′## than his own measurement ##dt##. Can Alice just reverse the direction of his ##x##-axis(i.e. rotate around the ##z##-axis) and say that "from now on Bob measures smaller ##dt′## than my ##dt##"? In short is measurement dependent upon "choice of coordinate axes" or "alignment of coordinate axes"?
     
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