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, k {\displaystyle k} being the radius of inversion:
x ′ = k 2 x x 2 + y 2 , y ′ = k 2 y x 2 + y 2 {\displaystyle x^{\prime }={\frac {k^{2}x}{x^{2}+y^{2}}},\quad y^{\prime }={\frac {k^{2}y}{x^{2}+y^{2}}}} . These inversions were later called "transformations by reciprocal radii", and became better known when Thomson (1845, 1847) applied them on spheres with coordinates x , y , z {\displaystyle x,y,z} 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:
δ x ′ 2 + δ y ′ 2 + δ z ′ 2 = λ ( δ x 2 + δ y 2 + δ z 2 ) {\displaystyle \delta x^{\prime 2}+\delta y^{\prime 2}+\delta z^{\prime 2}=\lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}\right)} . Liouville himself and more extensively Sophus Lie (1871) showed that the related conformal group can be differentiated (Liouville's theorem): For instance, λ = 1 {\displaystyle \lambda =1} includes the Euclidean group of ordinary motions; λ ≠ 1 {\displaystyle \lambda \neq 1} scale or similarity transformations in which the coordinates of the previous transformations are multiplied by λ {\displaystyle {\sqrt {\lambda }}} ; and λ = k 4 / ( x 2 + y 2 + z 2 ) 2 {\displaystyle \lambda =k^{4}/\left(x^{2}+y^{2}+z^{2}\right)^{2}} gives Thomson's transformation by reciprocal radii (inversions):
x ′ = k 2 x x 2 + y 2 + z 2 , y ′ = k 2 y x 2 + y 2 + z 2 , z ′ = k 2 z x 2 + y 2 + z 2 {\displaystyle x^{\prime }={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}}},\quad y^{\prime }={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}}},\quad z^{\prime }={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}}}} . Subsequently, Liouville's theorem was extended to n {\displaystyle n} dimensions by Lie (1871) and others such as Darboux (1878):
δ x 1 ′ 2 + ⋯ + δ x n ′ 2 = λ ( δ x 1 2 + ⋯ + δ x n 2 ) {\displaystyle \delta x_{1}^{\prime 2}+\dots +\delta x_{n}^{\prime 2}=\lambda \left(\delta x_{1}^{2}+\dots +\delta x_{n}^{2}\right)} . 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 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 (corresponding to the Möbius transformations in higher dimensions) therein, in accordance with Liouville's theorem. Conformal transformations in R3 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 R3 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:
x ′ = k 2 x x 2 + y 2 + z 2 − r 2 , z ′ = k 2 z x 2 + y 2 + z 2 − r 2 , y ′ = k 2 y x 2 + y 2 + z 2 − r 2 , r ′ = ± k 2 r x 2 + y 2 + z 2 − r 2 . {\displaystyle {\begin{aligned}x^{\prime }&={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}-r^{2}}},\quad &z^{\prime }&={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}-r^{2}}},\\y'&={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}-r^{2}}},&r^{\prime }&={\frac {\pm k^{2}r}{x^{2}+y^{2}+z^{2}-r^{2}}}.\end{aligned}}}
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 x , y {\displaystyle x,y} and radius r {\displaystyle r} in R2 corresponds to a point in R3 with coordinates x , y , z {\displaystyle x,y,z} . 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: z = i r {\displaystyle z=ir} was used by Chasles (1852), Möbius (1857), Cayley (1867), and Darboux (1872); z = r {\displaystyle z=r} 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 x , y , z , r {\displaystyle x,y,z,r} and x ′ , y ′ , z ′ , r ′ {\displaystyle x',y',z',r'} 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
( x − x ′ ) 2 + ( y − y ′ ) 2 + ( z − z ′ ) 2 − ( r − r ′ ) 2 = 0 {\displaystyle (x-x')^{2}+(y-y')^{2}+(z-z')^{2}-(r-r')^{2}=0} . Setting t = i r {\displaystyle t=ir} , these coordinates correspond to rectangular coordinates in four-dimensional space R4:
( x − x ′ ) 2 + ( y − y ′ ) 2 + ( z − z ′ ) 2 + ( t − t ′ ) 2 = 0 {\displaystyle (x-x')^{2}+(y-y')^{2}+(z-z')^{2}+(t-t')^{2}=0} . In general, Lie (1871) showed that the conformal point transformations in Rn (composed of motions, similarities, and transformations by reciprocal radii) correspond in Rn-1 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 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 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 G 15 {\displaystyle G_{15}} (transformations by reciprocal radii) in R4 producing the relation
δ x ′ 2 + δ y ′ 2 + δ z ′ 2 + δ u ′ 2 = λ ( δ x 2 + δ y 2 + δ z 2 + δ u 2 ) {\displaystyle \delta x^{\prime 2}+\delta y^{\prime 2}+\delta z^{\prime 2}+\delta u^{\prime 2}=\lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}+\delta u^{2}\right)} , where u = i c t {\displaystyle u=ict} includes t {\displaystyle t} as time component and c {\displaystyle c} as the speed of light. Bateman (1909) also noticed the equivalence to the previously mentioned Lie sphere transformations in R3, because the radius r {\displaystyle r} used in them can be interpreted as the radius c t {\displaystyle ct} of a spherical wave contracting or expanding with c {\displaystyle c} , therefore he called them "spherical wave transformations". He wrote:
When we use Darboux's representation of a point in S 4 {\displaystyle S_{4}} by a spherical wave in S 3 {\displaystyle S_{3}} , the group G 15 {\displaystyle G_{15}} becomes the group of spherical wave transformations which transform a spherical wave into a spherical wave. This group of transformations has been discussed by S. Lie; it is the group of transformations which transform lines of curvature on a surface enveloped by spherical waves into lines of curvature on the surface enveloped by the corresponding spherical waves. Depending on λ {\displaystyle \lambda } they can be differentiated into subgroups: (a) λ = 1 {\displaystyle \lambda =1} 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) λ ≠ 1 {\displaystyle \lambda \neq 1} represents scale or similarity transformations by multiplication of the space-time variables of the Lorentz transformations by a constant factor depending on λ {\displaystyle \lambda } . For instance, if l = λ {\displaystyle l={\sqrt {\lambda }}} is used, then the transformation given by Poincaré in 1905 follows:
x ′ = γ l ( x − v t ) , y ′ = l y , z ′ = l z , t ′ = γ l ( t − x v c 2 ) {\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)} . However, it was shown by Poincaré and Einstein that only l = 1 {\displaystyle l=1} 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 λ = r 4 / ( x 2 + y 2 + z 2 + u 2 ) 2 {\displaystyle \lambda =r^{4}/\left(x^{2}+y^{2}+z^{2}+u^{2}\right)^{2}} 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:
x ′ = k 2 x x 2 + y 2 + z 2 + u 2 , z ′ = k 2 z x 2 + y 2 + z 2 + u 2 , y ′ = k 2 y x 2 + y 2 + z 2 + u 2 , u ′ = k 2 u x 2 + y 2 + z 2 + u 2 , {\displaystyle {\begin{aligned}x'&={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}+u^{2}}},\quad &z'&={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}+u^{2}}},\\y'&={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}+u^{2}}},&u'&={\frac {k^{2}u}{x^{2}+y^{2}+z^{2}+u^{2}}},\end{aligned}}}
which become real spherical wave transformations in terms of Lie sphere geometry if the real radius c t {\displaystyle ct} is used instead of u = i c t {\displaystyle u=ict} , thus x 2 + y 2 + z 2 − c 2 t 2 {\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}} 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 R2 (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 R3, 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
z 1 2 + z 2 2 + z 3 2 − z 4 2 = 0 {\displaystyle z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0} , where z 1 , z 2 , z 3 , z 4 {\displaystyle z_{1},\ z_{2},\ z_{3},\ z_{4}} 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 ξ {\displaystyle \xi } of the sphere
ξ = z 1 + i z 2 z 4 − z 3 {\displaystyle \xi ={\frac {z_{1}+iz_{2}}{z_{4}-z_{3}}}}
which is connected to another parameter ξ ′ {\displaystyle \xi '} by the substitution
ξ ′ = α ξ + β γ ξ + δ {\displaystyle \xi '={\frac {\alpha \xi +\beta }{\gamma \xi +\delta }}}
where α , β , γ , δ {\displaystyle \alpha ,\beta ,\gamma ,\delta } are complex coefficients. They furthermore showed that by setting z 1 : z 2 : z 3 : z 4 = X : Y : Z : 1 {\displaystyle z_{1}:z_{2}:z_{3}:z_{4}=X:Y:Z:1} , the above relations assume the form in terms of the unit sphere in R3:
X 2 + Y 2 + Z 2 = 1 , ξ = X + i Y 1 − Z {\displaystyle X^{2}+Y^{2}+Z^{2}=1,\quad \xi ={\frac {X+iY}{1-Z}}} . which is identical to the stereographic projection of the ξ {\displaystyle \xi } -plane on a spherical surface already given by Klein in 1884. Since the substitutions ξ , ξ ′ {\displaystyle \xi ,\xi '} are Möbius transformations (German: Kreisverwandtschaften) in the ξ {\displaystyle \xi } -plane or upon the ξ {\displaystyle \xi } -sphere, they concluded that by carrying out an arbitrary motion of hyperbolic space in itself, the ξ {\displaystyle \xi } -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
z 1 = x , z 2 = y , z 3 = z , z 4 = t