# Invariance of the Lorentz transform

tina21
Homework Statement:
Prove the invariance of the electromagnetic wave equation by showing that the corresponding differential operator is an invariant.
Relevant Equations:
d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2 = d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2
of course y and z terms are invariant but for the x and t terms I am getting an additional factor of 1/1-v^2/c^2

## Answers and Replies

Homework Helper
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Homework Statement: Prove the invariance of the electromagnetic wave equation by showing that the corresponding differential operator is an invariant.
Homework Equations: d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2 = d^2/dx^2+d^2/dy^2+d^2/dz^2-1/c^2d^2/dt^2

of course y and z terms are invariant but for the x and t terms I am getting an additional factor of 1/1-v^2/c^2
Can you use Latex to show what you got?

https://www.physicsforums.com/help/latexhelp/

tina21
I haven't used latex before. I hope these images are okay

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Homework Helper
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That doesn't look right. I prefer to differentiate a trial function. Imagine we have a function ##f(t', x')##, where ##t' = \gamma(t - vx/c^2), \ x' = \gamma(x - vt)##.

Now, we differentiate ##f## with respect to ##x## using the chain rule:

##\frac{\partial f}{\partial x} = \frac{\partial f}{\partial t'}\frac{\partial t'}{\partial x} + \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial x} = \frac{\partial f}{\partial t'}(-\gamma v/c^2) + \frac{\partial f}{\partial x'}(\gamma)##

Now you need to take the second derivative of ##f## by repeating this process - and remember that you have to differentiate both terms using the chain rule, so you will get cross terms in ##\frac{\partial^2 f}{\partial t' \partial x'}##.

At the end, you can remove the trial function to leave, for example:

##\frac{\partial}{\partial x} = (-\gamma v/c^2)\frac{\partial }{\partial t'} + (\gamma)\frac{\partial}{\partial x'}##

That's what you get if you only wanted the first derivative.

Note that you can do the chain rule on differentials, but I prefer to have a function to differentiate, and then take the function away when I'm finished.

Last edited:
tina21
tina21
yes, I tried doing it too. Made it a whole lot easier and I got the answer. Thanks so much !