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 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 similar to the Lorentz transformation of special relativity. It turns out that the conformal group includes the Lorentz group and the Poincaré group as subgroups, but only the latter represent symmetries of all laws of nature including mechanics, whereas the conformal group is only related to certain areas such as electrodynamics.
Contents
- Development in the 19th century
- Oriented spheres
- Relation to electrodynamics
- Laguerre inversion and Lorentz transformation
- Lorentz transformation within Laguerre geometry
- Laguerre group isomorphic to Lorentz group
- References
A special case of Lie sphere geometry is the "transformation by reciprocal directions" or Laguerre inversion, being a generator of the group of Laguerre transformations (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 and the Lorentz group was pointed out by several authors such as Bateman, Cartan or Poincaré.
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,
These inversions were later called "transformations by reciprocal radii", and became better known when Thomson (1845, 1847) applied them on spheres with coordinates
Liouville himself and more extensively Sophus Lie (1871) showed that the related conformal group can be differentiated (Liouville's theorem): For instance,
Subsequently, Liouville's theorem was extended to
This group of conformal transformations by reciprocal radii preserves angles and transforms spheres into spheres or hyperspheres (see also conformal symmetry, Möbius transformation, and generally the Lie group). It is a 6-parameter group in the plane R2, 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 (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 conformal group with respect to pentaspherical coordinates in R3, is extended to the 15-parameter group of Lie sphere transformations with respect 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
Setting
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
where
When we use Darboux's representation of a point in
Depending on
(a)
(b)
However, it was shown by Poincaré and Einstein that only
(c) Setting
which become real spherical wave transformations in terms of Lie sphere geometry if the real radius
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.
Development in the 19th century
Above, the connection of conformal transformations with coordinates including the radius of spheres within Lie sphere geometry was mentioned. In relation to this, a special sphere transformation was 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"). Setting
with the transformation:
Darboux (1887) obtained the same formulas in different notation (with
with
consequently he obtained the relation
As mentioned above, oriented spheres in R3 can be represented by points of four-dimensional space R4 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
leaves the relation
As shown above, also the Transformation by reciprocal directions or half-lines – later called Laguerre inversion – leaves the expression
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
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
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
resulting in the transformation
By setting
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 Rn (composed of motions, similarities, and inversions) can be related by minimal projection to the group of contact transformations in Rn-1 transforming circles or spheres into other circles or spheres. In addition, Lie (1871, 1896) pointed out that in R3 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 R2 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 R2 transforming oriented lines and circles, or 11 parameters in R3 transforming oriented planes and spheres. If similarities are excluded, it becomes the "restricted Laguerre group" having 6 parameters in R2 and 10 parameters in R3, 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 R2 the Laguerre group leaves invariant the relation
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
Mr. Cartan has recently given a curious example. We know the importance in mathematical physics of what has been called the Lorentz group; it is this group upon which our new ideas on the principle of relativity and the dynamics of the electron are based. On the other hand, Laguerre once introduced into geometry a group of transformations that change the spheres into spheres. These two groups are isomorphic, so that mathematically these two theories, one physical, the other one geometric, show no essential difference.
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:
A Laguerre transformation (L-transform) is a mapping which is bijective on the sets of oriented planes and oriented spheres, respectively, and preserves tangency between plane and sphere. L-transforms are more easily understood if we use the so-called cyclographic model of Laguerre geometry. There, an oriented sphere