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. Transformation by reciprocal radii. Development in the 19th century. Inversions preserving angles between circles were first discussed by Durrande (1820), with Quetelet (1827) and Plücker (1828) writing down the corresponding transformation formula, formula_1 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 formula_3 in the course of developing the method of inversion in electrostatics. Joseph Liouville (1847) 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 (1871) showed that the related conformal group can be differentiated (Liouville's theorem): For instance, formula_5 includes the Euclidean group of ordinary motions; formula_6 scale or similarity transformations in which the coordinates of the previous transformations are multiplied by formula_7; and formula_8 gives Thomson's transformation by reciprocal radii (inversions): Subsequently, Liouville's theorem was extended to  formula_10 dimensions by Lie (1871) and others such as Darboux (1878): This group of conformal transformations by reciprocal radii preserves angles and transforms spheres into spheres or hyperspheres (see Möbius transformation, conformal symmetry, special conformal transformation). It is a 6-parameter group in the plane R which corresponds to the Möbius group of the extended complex plane, a 10-parameter group in space R, and a 15-parameter group in R. In R it represents only a small subset of all conformal transformations therein, whereas in R it is identical to the group of all conformal transformations (corresponding to the  Möbius transformations in higher dimensions) therein, in accordance with Liouville's theorem. Conformal transformations in R were often applied to what Darboux (1873) 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 (1871) in the context of Lie sphere geometry which represents a general framework of sphere-transformations (being a special case of contact transformations) conserving lines of curvature and transforming spheres into spheres. The previously mentioned 10-parameter  group in R related to pentaspherical coordinates is extended to the 15-parameter group of Lie sphere transformations related to "hexaspherical coordinates" (named by Klein in 1893) 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 (1871) himself and explicitly introduced by Laguerre (1880). In addition, Darboux (1887) 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 (1893), which was later called "isotropy projection" by Blaschke (1926) emphasizing the relation to oriented circles and spheres. For instance, a circle with rectangular coordinates formula_13 and radius formula_14 in R corresponds to a point in R with coordinates formula_3. This  method was known for some time in circle geometry (though without using the concept of orientation) and can be further differentiated depending on whether the additional coordinate is treated as imaginary or real: formula_16 was used by Chasles (1852), Möbius (1857), Cayley (1867), and Darboux (1872); formula_17 was used by Cousinery (1826), Druckenmüller (1842), and in the "cyclography" of Fiedler (1882), therefore the latter method was also called "cyclographic projection" – see E. Müller (1910) for a summary. This method was also applied to spheres by Darboux (1872), Lie (1871), or Klein (1893). Let formula_18 and formula_19 be the center coordinates  and radii of two spheres in three-dimensional space R. If the spheres are touching each other with same orientation, their equation is given Setting formula_21, these coordinates correspond to rectangular coordinates in four-dimensional space R: In general, Lie (1871) showed that the conformal point transformations in R (composed of motions, similarities, and transformations by reciprocal radii) correspond in R to those sphere transformations which are contact transformations. Klein (1893) pointed out that by using minimal projection on hexaspherical coordinates, the 15-parameter Lie sphere transformations in R are simply the projections of the 15-parameter conformal point transformations  in R, whereas the points in R can be seen as the stereographic projection of the points of a sphere in R. Relation to electrodynamics. Harry Bateman and Ebenezer Cunningham (1909) 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 formula_23 (transformations by reciprocal radii) in R producing the relation where formula_25 includes formula_26 as time component and formula_27 as the speed of light. Bateman (1909) also noticed the equivalence to the previously mentioned Lie sphere transformations in R, because the radius  formula_14 used in them can be interpreted as the radius formula_29 of a spherical wave contracting or expanding with formula_27, therefore he called them "spherical wave transformations". He wrote: Depending on formula_31 they can be differentiated into subgroups: (a) formula_5 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. (b) formula_33 represents scale or similarity transformations by multiplication of the space-time variables  of the Lorentz transformations by a constant factor depending on formula_31. For instance, if formula_35 is used, then the transformation given by Poincaré in 1905 follows: However, it was shown by Poincaré and Einstein that only formula_37 produces a group that is a symmetry of all laws of nature as required by the principle of relativity (the Lorentz group), while the group of scale transformations is only a symmetry of optics and electrodynamics. (c) Setting formula_38 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 formula_29 is used instead of formula_25, thus formula_42 is given in the denominator. Felix Klein (1921) 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 R (i.e. the Möbius group composed of automorphisms of the Riemann sphere), which in turn is isomorphic to the 6-parameter group of hyperbolic motions (i.e. isometric automorphisms of a hyperbolic space) in R, can be physically interpreted: It is isomorphic to the Lorentz group. For instance, Fricke and Klein (1897) 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 formula_44 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 formula_45 of the sphere which is connected to another parameter formula_47 by the substitution where formula_49 are complex coefficients. They furthermore showed that by setting formula_50, the above relations assume the form in terms of the unit sphere in R: which is identical to the stereographic projection of the formula_45-plane  on a spherical surface already given by Klein in 1884. Since the substitutions formula_53 are Möbius transformations () in the formula_45-plane or upon the formula_45-sphere, they concluded that by carrying out an arbitrary motion of hyperbolic space in itself, the formula_45-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 (and consequently of the Möbius group) to the Lorentz group was demonstrated  by Gustav Herglotz (1909). Namely, the Minkowski metric corresponds to the above Cayley metric (based on a real conic section), 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 R. The relation between the Lorentz group and the Cayley metric in hyperbolic space was also pointed out by Klein (1910) as well as Pauli (1921). The corresponding isomorphism of the Möbius group to the Lorentz group was employed, among others, by  Roger Penrose. Transformation by reciprocal directions. Development in the 19th century. Above, the connection of conformal transformations with coordinates including the radius of spheres within Lie sphere geometry was mentioned. The special case formula_5 corresponds to a sphere transformation given by Edmond Laguerre (1880-1885), who called it the "transformation by reciprocal directions" and who laid down the foundation of a geometry of oriented spheres "and planes". According to Darboux and Bateman, similar relations were discussed before by Albert Ribaucour (1870) and by Lie himself (1871). Stephanos (1881) pointed out that Laguerre's geometry is indeed a special case of  Lie's sphere geometry. He also represented Laguerre's oriented spheres by quaternions (1883). Lines, circles, planes, or spheres with radii of certain orientation are called by Laguerre half-lines, half-circles (cycles), half-planes, half-spheres, etc. A tangent is a half-line cutting a cycle at a point where both have the same direction. The transformation by reciprocal directions transforms oriented spheres into oriented spheres and oriented planes into oriented planes, leaving invariant the "tangential distance" of two cycles (the distance between the points of each one of their common tangents), and also conserves the lines of curvature . Laguerre (1882) applied the transformation to two cycles under the following conditions: Their radical axis is the axis of transformation, and their common tangents are parallel to two fixed directions of the half-lines that are transformed into themselves (Laguerre called this specific method the "transformation by reciprocal half-lines", which was later called "Laguerre inversion"). Setting formula_61 and formula_62 as the radii of the cycles, and formula_63 and formula_64 as the distances of their centers to the axis, he obtained: with the transformation: Darboux (1887) obtained the same formulas in different notation (with formula_67 and formula_68) in his treatment  of the "transformation by reciprocal directions", though he included the formula_69 and formula_70 coordinates as well: with consequently he obtained the relation As mentioned above, oriented spheres in R can be represented by points of four-dimensional space R using minimal (isotropy) projection, which became particularly important in Laguerre's geometry. For instance, E. Müller (1898) based his discussion of oriented spheres on the fact that they can be mapped upon the points of a plane manifold of four dimensions (which he likened to Fiedler's "cyclography" from 1882). He systematically compared the transformations by reciprocal radii (calling it "inversion  at a sphere") with the transformations by reciprocal directions (calling it "inversion at a plane sphere complex"). Following Müller's paper, Smith (1900) discussed Laguerre's transformation and the related "group of the geometry of reciprocal directions". Alluding to Klein's (1893) treatment of minimal projection, he pointed out that this group "is simply isomorphic with the group of all displacements and symmetry transformations in space of four dimensions". Smith obtained the same transformation as Laguerre and Darboux in different notation, calling it "inversion into a spherical complex": with the relations Laguerre inversion and Lorentz transformation. In 1905 both Poincaré  and Einstein pointed out that the Lorentz transformation of special relativity (setting formula_76) leaves the relation formula_78 invariant. Einstein stressed the point that by this transformation a spherical light wave in one frame is transformed into a spherical light wave in another one. Poincaré showed that the Lorentz transformation can be seen as a rotation in four-dimensional space with time as fourth coordinate, with Minkowski deepening this insight much further (see History of special relativity). As shown above, also Laguerre's transformation by reciprocal directions or half-lines – later called Laguerre inversion – in the form given by Darboux (1887 ) leaves the expression formula_79 invariant. Subsequently, the relation to the Lorentz transformation was noted by several authors. For instance, Bateman (1910) argued that this transformation (which he attributed to Ribaucour) is "identical" to the Lorentz transformation. In particular, he argued (1912) that the variant given by Darboux (1887) corresponds to the Lorentz transformation in formula_80 direction, if formula_81, formula_82, and the formula_1 terms are replaced by velocities. Bateman (1910) also sketched geometric representations of relativistic light spheres using such spherical systems. However, Kubota (1925) responded to Bateman by arguing that the Laguerre inversion is involutory whereas the Lorentz transformation is  not. He concluded that in order to make them equivalent, the Laguerre inversion has to be combined with a reversal of direction of the cycles. The specific relation between the Lorentz transformation and the Laguerre inversion can also be demonstrated as follows (see H.R. Müller (1948) for analogous formulas in different notation). Laguerre's inversion formulas from 1882 (equivalent to those of Darboux in 1887) read: by setting it follows finally by setting formula_87 the Laguerre inversion becomes very similar to the Lorentz transformation except that the expression formula_88 is reversed into formula_89: According to Müller, the Lorentz transformation can  be seen as the product of an even number of such Laguerre inversions that change the sign. First an inversion is conducted into plane formula_91 which is inclined with respect to plane formula_92 under a certain angle, followed by another inversion back to formula_92. See section #Laguerre group isomorphic to Lorentz group for more details of the connection between the Laguerre inversion to other variants of Laguerre transformations. Lorentz transformation within Laguerre geometry. Timerding (1911) used Laguerre's concept of oriented spheres in order to represent and derive the Lorentz transformation. Given a sphere of radius formula_14, with formula_69 as  the distance between its center and the central plane, he obtained the relations to a corresponding sphere resulting in the transformation By setting formula_98 and formula_99, it becomes the Lorentz transformation. Following Timerding and Bateman, Ogura (1913) analyzed a Laguerre transformation of the form which become the Lorentz transformation with He stated that "the Laguerre transformation in sphere manifoldness is equivalent to the Lorentz transformation in spacetime manifoldness". Laguerre group isomorphic to Lorentz group. As shown above, the group of conformal point transformations in R (composed of motions, similarities, and inversions) can be related by minimal projection to the group  of contact transformations in R transforming circles or spheres into other circles or spheres. In addition, Lie (1871, 1896) pointed out that in R there is a 7-parameter subgroup of point transformations composed of motions and similarities, which by using minimal projection corresponds to a 7-parameter subgroup of contact transformations in R transforming circles into circles. These relations were further studied by Smith (1900), Blaschke (1910), Coolidge (1916) and others, who pointed out the connection to Laguerre's geometry of reciprocal directions related to oriented lines, circles, planes and spheres. Therefore, Smith (1900) called it the "group of  the geometry of reciprocal directions", and Blaschke (1910) used the expression "Laguerre group". The "extended Laguerre group" consists of motions and similarities, having 7 parameters in R transforming oriented lines and circles, or 11 parameters in R transforming oriented planes and spheres. If similarities are excluded, it becomes the "restricted Laguerre group" having 6 parameters in R and 10 parameters in R, consisting of orientation-preserving or orientation-reversing motions, and preserving the tangential distance between oriented circles or spheres. Subsequently, it became common that the term Laguerre group only refers to the restricted Laguerre group. It was also noted  that the Laguerre group is part of a wider group conserving tangential distances, called the "equilong group" by Scheffers (1905). In R the Laguerre group leaves invariant the relation formula_103, which can be extended to arbitrary R as well. For instance, in R it leaves invariant the relation formula_104. This is equivalent to relation formula_105 in R by using minimal (isotropy) projection with imaginary radius coordinate, or cyclographic projection (in descriptive geometry) with real radius coordinate. The transformations forming the Laguerre group can be further differentiated into "direct Laguerre transformations" which are related to motions preserving both the tangential distance  as well as the sign; or "indirect Laguerre transformations" which are related to orientation-reversing motions, preserving the tangential distance with the sign reversed. The Laguerre inversion first given by Laguerre in 1882 is involutory, thus it belongs to the indirect Laguerre transformations. Laguerre himself did not discuss the group related to his inversion, but it turned out that every Laguerre transformation can be generated by at most four Laguerre inversions and every direct Laguerre transformation is the product of two involutory transformations, thus Laguerre inversions are of special importance because they are generating operators of the entire Laguerre group . It was noted that the Laguerre group is indeed isomorphic to the Lorentz group (or the Poincaré group if translations are included), as both groups leave invariant the form formula_106. After the first comparison of the Lorentz transformation and the Laguerre inversion by Bateman (1910) as mentioned above, the equivalence of both groups was pointed out by Cartan in 1912 and 1914, and he expanded upon it in 1915 (published 1955) in the French version of Klein's encyclopedia. Also Poincaré (1912, published 1921) wrote: Others who noticed this connection include Coolidge (1916), Klein & Blaschke (1926), Blaschke (1929), H.R. Müller , Kunle & Fladt (1970), Benz (1992). It was recently pointed out: ---- ----