# Partial Differential Equation in Special Relativity

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1. Feb 25, 2017

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. Feb 25, 2017

### kuruman

3. Feb 25, 2017

Can you give a little bit more hints than that please? Thanks

4. Feb 25, 2017

### kuruman

5. Feb 27, 2017

I give up. Can you guide me through this. Please. Thanks

Last edited: Feb 27, 2017
6. Feb 27, 2017

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?
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."