Euclidean invariance of the wave equation

This post follows James Rohlf’s Modern Physics from α to Z0 (p.104-105). See also the slides here.

The Galilean transformation without the time component is the Euclidean transformation for three-dimensional geometry (see here). Euclidean transformations are applied here to 3D space and 3D time.

Start with the standard configuration for relativity. The space dependent Euclidean transformation is

$x'=x-vt,\;&space;y'=y,\;&space;z'=z$

The dual time dependent Euclidean transformation is

$t'=t-wx,\;&space;s'=s,\;&space;r'=r$

where r, s, and t are time axes and w = 1/v. The y, z, s, and r axes are trivially invariant. Consider the one-dimensional wave equation for an electric field in a vacuum:

$\frac{\partial^{2}E_{i}}{\partial&space;x^{2}}=&space;\frac{1}{c^{2}}\frac{\partial^{2}E_{i}}{\partial&space;t^{2}}$

The first derivative of the space dependent part of the Euclidean transformation is as follows:

$\frac{\partial&space;x'}{\partial&space;x}=1,\;&space;\frac{\partial&space;t'}{\partial&space;x}=0$

$\frac{\partial&space;E_{i}}{\partial&space;x}=&space;\frac{\partial&space;E_{i}}{\partial&space;x'}\frac{\partial&space;x'}{\partial&space;x}+&space;\frac{\partial&space;E_{i}}{\partial&space;t'}\frac{\partial&space;t'}{\partial&space;x}=&space;\frac{\partial&space;E_{i}}{\partial&space;x'}(1)+\frac{\partial&space;E_{i}}{\partial&space;t'}(0)$

$\frac{\partial&space;E_{i}}{\partial&space;x}=\frac{\partial&space;E_{i}}{\partial&space;x'}$

The second derivative of the space dependent part of the Euclidean transformation is as follows:

$\frac{\partial}{\partial&space;x}\frac{\partial&space;E_{i}}{\partial&space;x}=&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;x'^{2}}\frac{\partial&space;x'}{\partial&space;x}+&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;x'\partial&space;t'}\frac{\partial^{2}&space;t'}{\partial&space;x}$

$\frac{\partial^{2}&space;E_{i}}{\partial&space;x^{2}}=&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;x'^{2}}(1)+&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;t'\partial&space;x'}(0)=&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;x'^{2}}$

The first derivative of the time dependent part of the Euclidean transformation is as follows:

$\frac{\partial&space;x'}{\partial&space;t}=0,\;&space;\frac{\partial&space;t'}{\partial&space;t}=1$

$\frac{\partial&space;E_{i}}{\partial&space;t}=&space;\frac{\partial&space;E_{i}}{\partial&space;x'}\frac{\partial&space;x'}{\partial&space;t}+&space;\frac{\partial&space;E_{i}}{\partial&space;t'}\frac{\partial&space;t'}{\partial&space;t}=&space;\frac{\partial&space;E_{i}}{\partial&space;x'}(0)+&space;\frac{\partial&space;E_{i}}{\partial&space;t'}(1)$

$\frac{\partial&space;E_{i}}{\partial&space;t}=\frac{\partial&space;E_{i}}{\partial&space;t'}$

The second derivative of the time dependent part of the Euclidean transformation is as follows:

$\frac{\partial}{\partial&space;t}\frac{\partial&space;E_{i}}{\partial&space;t}=&space;\frac{\partial}{\partial&space;t'}\frac{\partial&space;E_{i}}{\partial&space;x'}&space;\frac{\partial&space;x'}{\partial&space;t}+&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;t'^{2}}\frac{\partial&space;t'}{\partial&space;t}$

$\frac{\partial^{2}&space;E_{i}}{\partial&space;t^{2}}=&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;x'\partial&space;t'}(0)+&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;t'^{2}}(1)=&space;\frac{\partial^{2}&space;E_{i}}{\partial&space;t'^{2}}$

The full result is as follows:

$\frac{\partial^{2}&space;E_{i}}{\partial&space;x'^{2}}=&space;\frac{1}{c^{2}}\frac{\partial^{2}&space;E_{i}}{\partial&space;t'^{2}}$

The conclusion is that the one-dimensional wave equation for an electric field in a vacuum is form invariant under the Euclidean (or Galilean) transformations with space and time duality.