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Prove that s^2=(s')^2 using the Lorentz Transformation

  1. Aug 28, 2014 #1
    1. The problem statement, all variables and given/known data I am learning special relativity and we came across the invariant quantity s = x2 - (ct)2. Our professor wants us to prove it. I admit that this is a proof and belongs in the mathematics section but I didn't see an Algebra section and this is most easily identified by those learning special relativity.

    The assignment simply states

    "Prove s2 = s'2"


    2. Relevant equations
    s2= x2-(ct)2

    [itex]\gamma[/itex]=[1-([itex]\frac{v}{c}[/itex])2]-1/2

    x' = [itex]\gamma[/itex](x-vt)

    t' = [itex]\gamma[/itex](t-(vx/c2)

    3. The attempt at a solution

    My textbook is telling me in one sentence that if we apply the lorentz transformation to x and t then s2 = s'2.....so I did that...

    I choose to start with s'2 = x'2-(ct')2

    Applying the lorentz transformation to x' and t' our equation becomes...

    s'2 = ([itex]\gamma[/itex](x-vt))2-(c([itex]\gamma[/itex](t-(vx/c2))2

    Expanding what we have takes us to...

    s'2 = ([itex]\gamma[/itex]2(x2-2vt+(vt)2)-(c2[itex]\gamma[/itex]2(t2)-2(v/c2)x+(v2/c4)x2))

    If I combine some terms...

    s'2 = [itex]\gamma[/itex]2[x2(1-(v2/c4)+t2(v-1)+2v((x/c2)-t)]

    From here I tried a couple of different things on scratch paper but I couldn't see particular direction that would simplify it all down. Am I just not being patient enough and not seeing that it gets worse before it gets better?

    Thanks in advance.
     
  2. jcsd
  3. Aug 28, 2014 #2
    You made a bunch of algebra errors. The last equation you had that was correct was: s'2 = x'2-(ct')2

    Chet
     
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