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Show that a nonlinear transformation preseves velocity

  1. Nov 17, 2015 #1
    1. The problem statement, all variables and given/known data
    I have a particle moving with uniform velocity in a frame ##S##, with coordinates $$ x^\mu , \mu=0,1,2,3. $$
    I need to show that the particle also has uniform velocity in a frame ## S' ##, given by
    $$x'^\mu=\dfrac{A_\nu^\mu x^\nu + b^\mu}{c_\nu x^\nu + d}, $$
    with ## A_\nu^\mu,b^\mu,c_\nu x^\nu,d ## constant.


    2. Relevant equations
    I don't think these are very relevant because they're not the transformations for the question but
    $$\Delta x = \gamma(\Delta x' + v\Delta t')$$
    $$\Delta t = \gamma(\Delta t' + v\Delta x'/c^2)$$
    $$\Delta x' = \gamma(\Delta x - v\Delta t)$$
    $$\Delta t' = \gamma(\Delta t - v\Delta x/c^2)$$

    3. The attempt at a solution
    I wrote the ## S' ## coordinates out and using ## x'^0=t'##,##x'^1=x'##,##x'^2=y'##,##x'^3=z' ##, try to calculate the velocities but I don't think it's right. I'm not sure how to show a transformation preserves the particle velocity. Could anyone point me how to show this for the Lorentz transformations, and then I could try to do it for my transformations?
     
    Last edited: Nov 17, 2015
  2. jcsd
  3. Nov 17, 2015 #2

    BvU

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    Oh, and for inline math we use ## \#\# ## to start and end, not ##$##
     
  4. Nov 17, 2015 #3
    I've fixed that now, thanks. Did you have something typed before "Oh,"?
     
  5. Nov 17, 2015 #4

    BvU

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    Ummm, no ... :frown:
     
  6. Nov 17, 2015 #5

    strangerep

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    "Uniform velocity" means zero acceleration. So you must show that if $$\frac{dv^i}{dt} \equiv \frac{d^2 x^i}{dt^2} = 0$$then
    $$\frac{dv'^i}{dt'} \equiv\frac{d^2 x'^i}{dt'^2} = 0$$ (where ##i=1,2,3##).

    BTW, what is the context of this problem? It's actually a classic -- the fractional linear transformations are known to be the most general transformations which preserve inertial motion. :-)

    Not sure how much of a hint I should give you, so I'll start with this:

    Work with the differentials, i.e., find ##dx^i## and ##dt## separately, then take their quotient to find an expression for ##v##. Take differential ##dv## similarly, and take its quotient with ##dt## to find the acceleration.

    Further hints: use the chain rule to compute the various differentials.
     
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