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 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.

Transformation by reciprocal radii

Development in the 19th century

preserving angles between circles were first discussed by Durrande, with Quetelet and Plücker 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 applied them on spheres with coordinates in the course of developing the method of inversion in electrostatics. Joseph Liouville demonstrated its mathematical meaning by showing that it belongs to the conformal transformations producing the following quadratic form:
Liouville himself and more extensively Sophus Lie showed that the related conformal group can be differentiated : For instance, includes the Euclidean group of ordinary motions; scale or similarity transformations in which the coordinates of the previous transformations are multiplied by ; and gives Thomson's transformation by reciprocal radii :
Subsequently, Liouville's theorem was extended to dimensions by Lie and others such as Darboux :
This group of conformal transformations by reciprocal radii preserves angles and transforms spheres into spheres or hyperspheres. It is a 6-parameter group in the plane R² which corresponds to the Möbius group of the extended complex plane, a 10-parameter group in space R³, and a 15-parameter group in R⁴. In R² it represents only a small subset of all conformal transformations therein, whereas in R²⁺ⁿ it is identical to the group of all conformal transformations therein, in accordance with Liouville's theorem. Conformal transformations in R³ were often applied to what Darboux called "pentaspherical coordinates" by relating the points to homogeneous coordinates based on five spheres.

Oriented spheres

Another method for solving such sphere problems was to write down the coordinates together with the sphere's radius. This was employed by Lie in the context of Lie sphere geometry which represents a general framework of sphere-transformations conserving lines of curvature and transforming spheres into spheres. The previously mentioned 10-parameter group in R³ related to pentaspherical coordinates is extended to the 15-parameter group of Lie sphere transformations related to "hexaspherical coordinates" by adding a sixth homogeneous coordinate related to the radius. Since the radius of a sphere can have a positive or negative sign, one sphere always corresponds to two transformed spheres. It is advantageous to remove this ambiguity by attributing a definite sign to the radius, consequently giving the spheres a definite orientation too, so that one oriented sphere corresponds to one transformed oriented sphere. This method was occasionally and implicitly employed by Lie himself and explicitly introduced by Laguerre. In addition, Darboux brought the transformations by reciprocal radii into a form by which the radius r of a sphere can be determined if the radius of the other one is known:
Using coordinates together with the radius was often connected to a method called "minimal projection" by Klein, which was later called "isotropy projection" by Blaschke emphasizing the relation to oriented circles and spheres. For instance, a circle with rectangular coordinates and radius in R² corresponds to a point in R³ with coordinates. This method was known for some time in circle geometry and can be further differentiated depending on whether the additional coordinate is treated as imaginary or real: was used by Chasles, Möbius, Cayley, and Darboux ; was used by Cousinery, Druckenmüller, and in the "cyclography" of Fiedler, therefore the latter method was also called "cyclographic projection" – see E. Müller for a summary. This method was also applied to spheres by Darboux, Lie, or Klein. Let and be the center coordinates and radii of two spheres in three-dimensional space R³. If the spheres are touching each other with same orientation, their equation is given
Setting, these coordinates correspond to rectangular coordinates in four-dimensional space R⁴:
In general, Lie showed that the conformal point transformations in Rⁿ correspond in R^n-1 to those sphere transformations which are contact transformations. Klein pointed out that by using minimal projection on hexaspherical coordinates, the 15-parameter Lie sphere transformations in R³ are simply the projections of the 15-parameter conformal point transformations in R⁴, whereas the points in R⁴ can be seen as the stereographic projection of the points of a sphere in R⁵.

Relation to electrodynamics

and Ebenezer Cunningham showed that the electromagnetic equations are not only Lorentz invariant, but also scale and conformal invariant. They are invariant under the 15-parameter group of conformal transformations in R⁴ producing the relation
where includes as time component and as the speed of light. Bateman also noticed the equivalence to the previously mentioned Lie sphere transformations in R³, because the radius used in them can be interpreted as the radius of a spherical wave contracting or expanding with, therefore he called them "spherical wave transformations". He wrote:
Depending on they can be differentiated into subgroups:
correspond to mappings which transform not only spheres into spheres but also planes into planes. These are called [|Laguerre transformations/inversions] forming the Laguerre group, which in physics correspond to the Lorentz transformations forming the 6-parameter Lorentz group or 10-parameter Poincaré group with translations.
represents scale or similarity transformations by multiplication of the space-time variables of the Lorentz transformations by a constant factor depending on. For instance, if is used, then the transformation given by Poincaré in 1905 follows:
However, it was shown by Poincaré and Einstein that only produces a group that is a symmetry of all laws of nature as required by the principle of relativity, while the group of scale transformations is only a symmetry of optics and electrodynamics.
Setting particularly relates to the wide conformal group of transformations by reciprocal radii. It consists of elementary transformations that represent a generalized inversion into a four-dimensional hypersphere:
which become real spherical wave transformations in terms of Lie sphere geometry if the real radius is used instead of, thus is given in the denominator.
Felix Klein pointed out the similarity of these relations to Lie's and his own researches of 1871, adding that the conformal group doesn't have the same meaning as the Lorentz group, because the former applies to electrodynamics whereas the latter is a symmetry of all laws of nature including mechanics. The possibility was discussed for some time, whether conformal transformations allow for the transformation into uniformly accelerated frames. Later, conformal invariance became important again in certain areas such as conformal field theory.

Lorentz group isomorphic to Möbius group

It turns out that also the 6-parameter conformal group of R², which in turn is isomorphic to the 6-parameter group of hyperbolic motions in R³, can be physically interpreted: It is isomorphic to the Lorentz group.
For instance, Fricke and Klein started by defining an "absolute" Cayley metric in terms of a one-part curvilinear surface of second degree, which can be represented by a sphere whose interior represents hyperbolic space with the equation
where are homogeneous coordinates. They pointed out that motions of hyperbolic space into itself also transform this sphere into itself. They developed the corresponding transformation by defining a complex parameter of the sphere
which is connected to another parameter by the substitution
where are complex coefficients. They furthermore showed that by setting, the above relations assume the form in terms of the unit sphere in R³:
which is identical to the stereographic projection of the -plane on a spherical surface already given by Klein in 1884. Since the substitutions are Möbius transformations in the -plane or upon the -sphere, they concluded that by carrying out an arbitrary motion of hyperbolic space in itself, the -sphere undergoes a Möbius transformation, that the entire group of hyperbolic motions gives all direct Möbius transformations, and finally that any direct Möbius transformation corresponds to a motion of hyperbolic space.
Based on the work of Fricke & Klein, the isomorphism of that group of hyperbolic motions to the Lorentz group was demonstrated by Gustav Herglotz. Namely, the Minkowski metric corresponds to the above Cayley metric, if the spacetime coordinates are identified with the above homogeneous coordinates
by which the above parameter become
Herglotz concluded, that any such substitution corresponds to a Lorentz transformation, establishing a one-to-one correspondence to hyperbolic motions in R³. The relation between the Lorentz group and the Cayley metric in hyperbolic space was also pointed out by Klein as well as Pauli. The corresponding isomorphism of the Möbius group to the Lorentz group was employed, among others, by Roger Penrose.