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,\; y'=y,\; z'=z

The dual time dependent Euclidean transformation is

t'=t-wx,\; s'=s,\; 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 x^{2}}= \frac{1}{c^{2}}\frac{\partial^{2}E_{i}}{\partial t^{2}}

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

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

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

\frac{\partial E_{i}}{\partial x}=\frac{\partial E_{i}}{\partial x'}

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

\frac{\partial}{\partial x}\frac{\partial E_{i}}{\partial x}= \frac{\partial^{2} E_{i}}{\partial x'^{2}}\frac{\partial x'}{\partial x}+ \frac{\partial^{2} E_{i}}{\partial x'\partial t'}\frac{\partial^{2} t'}{\partial x}

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

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

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

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

\frac{\partial E_{i}}{\partial t}=\frac{\partial E_{i}}{\partial t'}

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

\frac{\partial}{\partial t}\frac{\partial E_{i}}{\partial t}= \frac{\partial}{\partial t'}\frac{\partial E_{i}}{\partial x'} \frac{\partial x'}{\partial t}+ \frac{\partial^{2} E_{i}}{\partial t'^{2}}\frac{\partial t'}{\partial t}

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

The full result is as follows:

\frac{\partial^{2} E_{i}}{\partial x'^{2}}= \frac{1}{c^{2}}\frac{\partial^{2} E_{i}}{\partial 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.