i) Show that the wave equation:
[( -1/c^2) d^2/dt^2 + d^2/dx^2 + d^2/dy^2 + d^2/dz^2 ]u(t,x,y,z) = 0
is invariant under a Lorentz boost along the x-direction, i.e. it takes the same form as a partial differential equation in the new coordinates. [Use the chain rule in two variables.]
ii) Show that the wave equation is not invariant under a Galilei transformation.
second order partial derivative equation. I can't get Latex to work right now, or I'd write it out, here's a link to it on another thread https://www.physicsforums.com/showthread.php?t=216869
g=gamma, the lorentz factor
The Attempt at a Solution
So I just haven't ever done this, and I am not super sure where to start.
I tried rearranging the x' and t' eqs to read
x=x' / g + vt
then using the chain rule which gave me
dt/dt' = 1/g
dx/dx' = 1/g
with both the second derivatives being 0
Plugging that into the equations gave me an extra factor of 1/g^2 on the second derivatives with respect to x and t. I am fairly certain this is the wrong approach anyway. I just don't know where to plug what.
I know this is 100% a math exercise, but it's my HW, and I figure this is the best place to put it.
edit: I realized that this is a multivariable problem, and I have been treating it as a single variable problem i.e. u=u(x'(x,t),y',z',t'(x,t))
du/dx = (du/dx')(dx'/dx)+(du/dt')(dt'/dx)
but now if I want to take the second derivative of that i.e.
d^2/dx^2 u = d/dx [(du/dx')(dx'/dx)+(du/dt')(dt'/dx)]
how do I evaluate d/dx(du/dx') and d/dx(du/dt') I know to product rule the rest, but i don't quite remember if there's anything fancy about taking the derivative of a function with respect to a variable whose already had it's derivative taken by function depending on that variable.