Spherical wave transformation

Spherical wave transformations leave the form of spherical waves as well as the laws of optics and electrodynamics invariant in all inertial frames. They were defined between 1908 and 1909 by Harry Bateman and Ebenezer Cunningham, with Bateman giving the transformation its name. They correspond to the conformal group of "transformations by reciprocal radii" in relation to the framework of Lie sphere geometry, which were already known in the 19th century. Time is used as fourth dimension as in Minkowski space, so spherical wave transformations are connected to the Lorentz transformation of special relativity, and it turns out that the conformal group of spacetime includes the Lorentz group and the Poincaré group as subgroups. However, only the Lorentz/Poincaré groups represent symmetries of all laws of nature including mechanics, whereas the conformal group is related to certain areas such as electrodynamics. In addition, it can be shown that the conformal group of the plane (corresponding to the Möbius group of the extended complex plane) is isomorphic to the Lorentz group. A special case of Lie sphere geometry is the transformation by reciprocal directions or Laguerre inversion, being a generator of the Laguerre group. It transforms not only spheres into spheres but also planes into planes. If time is used as fourth dimension, a close analogy to the Lorentz transformation as well as isomorphism to the Lorentz group was pointed out by several authors such as Bateman, Cartan or Poincaré. Inversions preserving angles between circles were first discussed by Durrande (1820), with Quetelet (1827) and Plücker (1828) writing down the corresponding transformation formula, being the radius of inversion: These inversions were later called "transformations by reciprocal radii", and became better known when Thomson (1845, 1847) applied them on spheres with coordinates in the course of developing the method of inversion in electrostatics.