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Partial Differential Equation in Special Relativity

  1. Feb 25, 2017 #1
    1. The problem statement, all variables and given/known data
    (a) Light waves satisfy the wave equation ##u_{tt}-c^2u_{xx}## where ##c## is the speed of light.
    Consider change of coordinates $$x'=x-Vt$$ $$t'=t$$
    where V is a constant. Use the chain rule to show that ##u_x=u_{x'}## and ##u_{tt}=-Vu_{x'}+u_{t'}##
    Find ##u_{xx},u_{tt},## and hence ##u_{tt}-c^2u_{xx}##, in terms of derivatives with respect to ##x'## and ##t'##.
    Deduce that if ##u## satisfies the wave equation in ##x,t## coordinates, then it does not satisfy the same equation in the ##x',t'## coordinates.
    3. The attempt at a solution
    So I've worked out that $$u_{xx}=u_{x'x'}$$ and $$u_{tt}=u_{t't'}+v^2(u_{x'x'})-2v(u_{x'xt})$$ so technically ##u_{tt}-c^2u_{xx}## expressed in terms of derivatives with respect to ##x'## and ##t'## would just be $$u_{t't'}+v^2(u_{x'x'})-2v(u_{x'xt})-c^2(u_{x'x'})=0$$ right?
    But how do I do the bit where question says "Deduce that if ##u## satisfies the wave equation in ##x,t## coordinates, then it does not satisfy the same equation in the ##x',t'## coordinates." I don't know where to start with this?Would this be done conceptually or mathematically?
     
  2. jcsd
  3. Feb 25, 2017 #2

    kuruman

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    Start with the wave equation.
     
  4. Feb 25, 2017 #3
    Can you give a little bit more hints than that please? Thanks
     
  5. Feb 25, 2017 #4

    kuruman

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  6. Feb 27, 2017 #5
    I give up. Can you guide me through this. Please. Thanks
     
    Last edited: Feb 27, 2017
  7. Feb 27, 2017 #6
    Is it simply due to the fact that ##u_{xx}-c^2u_{tt} \neq u_{x'x'}-c^2u_{t't'} ##, so that if ##u## satisfies the wave equation in ##x##,##t## coordinates, then it does not satisfy the same equation in the ##x'##,##t'## coordinates? Or do I deduce it mathematically?
     
  8. Feb 27, 2017 #7

    kuruman

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    If you calculate ##u_{xx}-c^2u_{tt}## and ##u_{x'x'}-c^2u_{t't'}## and it turns out that the two expressions are not equal, then you have "deduced it mathematically."
     
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