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

The dual time dependent Euclidean transformation is

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:

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

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

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

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

The full result is as follows:

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.