Ball wave transformation

Spherical wave transformations (English: spherical wave transformations ) can take the form of spherical waves and the laws of optics and electrodynamics in all inertial invariant. They correspond to the conformal group of "transformations through reciprocal radii" in connection with the spherical geometry of Sophus Lie, which was already known in the 19th century . They were first used in 1909 by Harry Bateman and Ebenezer Cunningham and were named by Bateman. Since time is used as the fourth dimension in spherical wave transformations in the sense of the Minkowski space , they have a certain analogy to the Lorentz transformations of the special theory of relativity . It turns out that the conformal group contains the Lorentz group and Poincaré group as subgroups, the latter representing a symmetry of all natural laws including mechanics , while the conformal group is only valid for certain areas such as electrodynamics.

A special case of Lie's spherical geometry is the “transformation through reciprocal directions” or Laguerre inversion, which was also known in the 19th century and is the generating operator of the group of Laguerre transformations. It not only depicts spheres in spheres, but also levels in levels. If time is used here as the fourth dimension, there is a close analogy to the Lorentz transformation and the Lorentz group, as Bateman, Cartan or Poincaré showed.

Transformation through reciprocal radii

Development in the 19th century

Inversions that get the angles of circles were first discussed by Durrande (1820), whereby Quetelet (1827) and Plücker (1828) wrote down the corresponding transformation formulas, with the inversion radius: ${\displaystyle k}$

${\displaystyle x^{\prime }={\frac {k^{2}x}{x^{2}+y^{2}}},\quad y^{\prime }={\frac {k^{2}y}{x^{2}+y^{2}}},}$

Such inversions were later referred to as reciprocal radius transformations. They became better known when Thomson (1845, 1847) applied them to spheres with coordinates , and thus was able to solve problems of electrostatics . Joseph Liouville (1847, 1850) clarified its mathematical meaning by showing that it belongs to the conformal transformations . It creates the following relationship between quadratic forms of the differentials : ${\displaystyle x,y,z}$

${\displaystyle \delta x^{\prime 2}+\delta y^{\prime 2}+\delta z^{\prime 2}=\lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}\right)}$.

He himself and, in a much more general way, Sophus Lie (1871) found that the associated transformation group can be divided into different types depending on the choice of : the Euclidean group of ordinary movements; Correspondence maps ; and at the transformations result from reciprocal radii: ${\displaystyle \lambda }$ ${\displaystyle \lambda =1}$ ${\displaystyle \lambda \neq 1}$ ${\displaystyle \lambda =k^{4}/\left(x^{2}+y^{2}+z^{2}\right)^{2}}$

${\displaystyle x^{\prime }={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}}},\quad y^{\prime }={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}}},\quad z^{\prime }={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}}}}$

Lie (1871) and others such as Gaston Darboux (1878) also expanded the group to include dimensions so that: ${\displaystyle n}$

${\displaystyle \delta x_{1}^{\prime 2}+\dots +\delta x_{n}^{\prime 2}=\lambda \left(\delta x_{1}^{2}+\dots +\delta x_{n}^{2}\right)}$.

An essential property of the conformal transformations through reciprocal radii is that they preserve angles and transform spheres into spheres (see conformal group , Möbius transformation ). It is a 6-parameter group in level R ² , a 10-parameter group in room R ³ , and a 15-parameter group in R ⁴ . In R ^{2 it} represents only a small part of all conforming transformations, but in R ^{2 + n it} is identical to all conforming transformations according to a theorem of Liouville. The conformal transformations in R ³ were often related to “pentaspheric coordinates” according to Darboux (1873). These are homogeneous coordinates based on five spheres that are assigned to the points.

Oriented spheres

Another way to calculate circle and sphere problems was to use rectangular coordinates along with the radius. This was used by Lie (1871) in the context of "Lieschen Kugelgeometrie", which contains contact transformations with which lines of curvature are conserved and spheres are transformed into spheres. The previously mentioned conformal 10-parameter group in R ³ with pentaspheric coordinates is extended to the 15-parameter group of Lies spherical transformations, whereby according to Klein (1893) "hexaspheric coordinates" are to be used from now on, as a sixth homogeneous coordinate is added which refers to the radius. However, since the radius can be positive or negative depending on the sign, there are always two transformed spheres for one sphere. In order to remove this ambiguity, it is advantageous to use only a certain sign for the radius, whereby the spheres receive a certain orientation, and consequently one oriented sphere is transformed into another oriented sphere. This method was occasionally and implicitly used by Lie (1871), and specifically introduced by Laguerre (1880). Darboux (1887) also wrote the transformation by reciprocal radii in a form where the radius of the other sphere could be determined from the radius of one sphere with a certain sign:

${\displaystyle {\begin{aligned}x^{\prime }&={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}-r^{2}}},\quad &z^{\prime }&={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}-r^{2}}},\\y'&={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}-r^{2}}},&r^{\prime }&={\frac {\pm k^{2}r}{x^{2}+y^{2}+z^{2}-r^{2}}}.\end{aligned}}}$

The specification of the coordinates together with the radius was often linked to a method that Klein (1893) called "minimal projection" , whereby Blaschke (1926) later used the term "isotropic projection" to clarify the context for orientation. If a circle is given with coordinates and the radius in R ² , then according to the minimal projection they correspond to a point with the coordinates in R ³ . This method has been known for a long time in the context of circular geometry (but without a clear orientation) and can be further subdivided depending on whether the additional coordinate is imaginary or real: can be found in Chasles (1852), Möbius (1857), Cayley (1867), Darboux (1872); can be found in Cousinery (1826), Druckermüller (1842) and in Fiedler's "Zyklographie" (1882), which is why it is also called "cyclographic projection" - see also E. Müller (1910). This method has now been applied to spheres by Darboux (1872), Lie (1871), and Klein (1893). Let the center coordinates as well as the radii of two spheres and in the three-dimensional space R ^{3 be} given. If the balls touch in the same direction, their equation is given by ${\displaystyle x,y}$ ${\displaystyle r}$ ${\displaystyle x,y,z}$ ${\displaystyle z=ir}$ ${\displaystyle z=r}$ ${\displaystyle x,y,z,r}$ ${\displaystyle x',y',z',r'}$

${\displaystyle (x-x')^{2}+(y-y')^{2}+(z-z')^{2}-(r-r')^{2}=0}$.

If set, they correspond to the following rectangular coordinates in a four-dimensional space R ⁴ : ${\displaystyle t=ir}$

${\displaystyle (x-x')^{2}+(y-y')^{2}+(z-z')^{2}+(t-t')^{2}=0}$.

In general, Lie (1871) was able to show that the conformal point transformations in R ⁿ (composed of movements, similarities, and transformations through reciprocal radii) in R ^n-1 correspond to those spherical transformations which are touch transformations . Klein (1893) also showed, using the minimal projection on hexaspheric coordinates, that the 15-parameter transformations of Lies spherical geometry in R ^{3 are} a simple image of the conforming 15-parameter transformations in R ⁴ , while the points of R ^{4 are} again can be viewed as the stereographic projection of points on a sphere in R ⁵ .

Relation to electrodynamics

Harry Bateman and Ebenezer Cunningham (1909) showed that the electromagnetic equations are not only Lorentz invariant , but also scale-invariant or conformally invariant in the above sense . That is, they are invariant under the 15-parameter group of conformal transformations by reciprocal radii in R ⁴ , which produces the following equation:

${\displaystyle \delta x^{\prime 2}+\delta y^{\prime 2}+\delta z^{\prime 2}+\delta u^{\prime 2}=\lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}+\delta u^{2}\right),}$

where contains the time and the speed of light according to the Minkowski space. Bateman also recognized the analogy to the transformations of Lies spherical geometry in R ³ , since the radius of this spherical geometry can be interpreted as the radius of an expanding or contracting spherical wave, which is why he called it a "spherical wave transformation". Bateman was referring to Darboux's variant of minimal projection and wrote: ${\displaystyle u=ict}$ ${\displaystyle t}$ ${\displaystyle c}$ ${\displaystyle r}$ ${\displaystyle ct}$

"When we use Darboux's representation of a point in by a spherical wave in , the group 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. " ${\displaystyle S_{4}}$ ${\displaystyle S_{3}}$ ${\displaystyle G_{15}}$

Analogous to the approach of Liouville and Lie, Cunningham was able to show that this group can be further subdivided depending on the choice of : ${\displaystyle \lambda }$

(a) not only transforms spheres into spheres, but also planes into planes. It contains the Lorentz transformation, because it is the extension of the Euclidean group of classical mechanics to the 6-parameter Lorentz group or 10-parameter Poincaré group with translations. ${\displaystyle \lambda =1}$ ${\displaystyle G_{10}}$

(b) are scale or similarity transformations . They correspond to the multiplication of the Lorentz transformation with a scale factor dependent on. With , for example, the form given by Poincaré (1905) results: ${\displaystyle \lambda \neq 1}$ ${\displaystyle \lambda }$ ${\displaystyle l={\sqrt {\lambda }}}$

${\displaystyle x^{\prime }=\gamma l\left(x-vt\right),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=\gamma l\left(t-x{\frac {v}{c^{2}}}\right)}$.

If, however, a definite setting is made, the group property is only given for (the Lorentz group), as was shown by Poincaré and Einstein. Only this is compatible with the relativity principle for all laws of nature, while the group of conformal and similarity transformations with different only symmetrically depicts individual areas such as optics and electrodynamics. ${\displaystyle l}$ ${\displaystyle l=1}$ ${\displaystyle l}$

(c) Finally, there is the most general variant, namely the conformal group of transformations through reciprocal radii, which represent inversions into a hypersphere: ${\displaystyle \lambda =k^{4}/\left(x^{2}+y^{2}+z^{2}+u^{2}\right)^{2}}$

${\displaystyle {\begin{aligned}x^{\prime }&={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}+u^{2}}},\quad &z^{\prime }&={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}+u^{2}}},\\y'&={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}+u^{2}}},&u^{\prime }&={\frac {k^{2}u}{x^{2}+y^{2}+z^{2}+u^{2}}}.\end{aligned}}}$

They become real spherical wave transformations within the framework of Lies spherical geometry, if the real radius is used instead of what is given in the denominator. Bateman and Cunningham also discussed the possibility that conformal transformations enable the transition to constantly accelerated frames of reference, which they and later authors questioned. Felix Klein (1921) referred to the close connection between the methods of Batemans and Cunningham and the methods of projective geometry , but like Einstein he noted that the conformal group is only valid to a limited extent in comparison to the Lorentz group: ${\displaystyle ct}$ ${\displaystyle u=ict}$ ${\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$

“For physics, however , this does not have the same meaning as its subgroup, that of the Lorentz group. The reason for this is that only the latter is a generalization of classical mechanics (to which it passes if the speed of light is set to infinity), but a general physics must include both mechanics and electrodynamics. Einstein occasionally expressed this factual relationship to me as follows: The transformation through reciprocal radii preserves the form of Maxwell's equations, but not the relationship between coordinates and the measurement results of scales and clocks. " ${\displaystyle G_{15}}$ ${\displaystyle G_{10}}$ ${\displaystyle G_{10}}$

The concept of conformal mapping has regained importance in specialized areas of modern physics, especially in conformal field theories such as some quantum field theories .

Transformation through reciprocal directions

Development in the 19th century

As described, coordinates have already been used in connection with conformal transformations together with radii of certain signs, which gave circles and spheres a certain orientation. There was now a special transformation or geometry within the Lies spherical geometry, which was mainly formulated by Edmond Laguerre (1880) and referred to by him as “transformation through reciprocal directions”. Then he laid the foundations of a geometry-oriented spheres and surfaces by 1885 . According to Darboux and Bateman, similar relationships were previously discussed by Albert Ribaucour (1870) and Lie (1871). Stephanos (1881) showed that Laguerre's geometry is a special case of Lies spherical geometry. He also used quaternions (1883) to depict Laguerre's oriented spheres .

Lines, circles, surfaces or spheres that have to be traversed in a certain sense are called half-straight lines (direction), half-circles (cycle), half-surfaces, hemispheres, etc. The tangent is the half-line that intersects a cycle at a point, provided that both elements have the same direction at this point of contact. The transformation through reciprocal directions now maps oriented spheres below and oriented planes below and leaves the "tangential distance" of two cycles (the distance between the contact points of one of their common tangents) invariant, and also conserves the lines of curvature. Laguerre (1882) transformed two cycles under the following conditions: their power line is the transformation axis, and their common tangents are parallel to two fixed directions of the mutually transformed half lines (he called this special method "transformation through reciprocal half lines"). If and are the radii of the cycles, and the distances of their centers from the axis, we get: ${\displaystyle R}$ ${\displaystyle R'}$ ${\displaystyle D}$ ${\displaystyle D'}$

${\displaystyle D^{2}-D^{\prime 2}=R^{2}-R^{\prime 2},\quad D-D'=\alpha (R-R'),\quad D+D'={\frac {1}{\alpha }}(R+R'),}$

with the transformation:

${\displaystyle {\begin{aligned}D'&={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}},\\R'&={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}.\end{aligned}}}$

Darboux (1887) also derived the same formulas from the transformation through reciprocal directions in a slightly different notation (with and ), and also used the and coordinates: ${\displaystyle z=D}$ ${\displaystyle k=\alpha }$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle {\begin{aligned}x'&=x,\quad &z'&={\frac {1+k^{2}}{1-k^{2}}}z-{\frac {2kR}{1-k^{2}}},\\y'&=y,&R'&={\frac {2kz}{1-k^{2}}}-{\frac {1+k^{2}}{1-k^{2}}}R,\end{aligned}}}$

With

${\displaystyle {\begin{aligned}z'+R'&={\frac {1+k}{1-k}}(z-R),\\z'-R'&={\frac {1-k}{1+k}}(z+R),\end{aligned}}}$

whereby he got the relationship:

${\displaystyle x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}}$.

As mentioned, oriented spheres in R ^{3 can be represented} by points in a four-dimensional space R ⁴ using isotropic (minimal) projection, which is particularly important for Laguerre's geometry. E. Müller (1898) based his presentation on the fact that “the oriented spheres can be clearly mapped onto the points of a flat manifold of four dimensions” (comparing this with Fiedler's “cyclograpy”). He also systematically investigated the relationship between the transformations through reciprocal radii ("inversion on a sphere") and the transformations through reciprocal directions ("inversion on a plane spherical complex"). Based on Müller's work, Smith (1900) examined the same transformations and the related "group of geometry of reciprocal directions". With reference to Klein's (1893) treatment of minimal projection, he pointed out that this group is isomorphic to the group of all displacements and symmetry transformations in the space of four dimensions. Smith received the same transformation as Laguerre and Darboux in slightly different notation:

${\displaystyle p'={\frac {\kappa ^{2}+1}{\kappa ^{2}-1}}p-{\frac {2\kappa }{\kappa ^{2}-1}}R,\quad R'={\frac {2\kappa }{\kappa ^{2}-1}}p-{\frac {\kappa ^{2}+1}{\kappa ^{2}-1}}R}$

with the relationships:

${\displaystyle \kappa ={\frac {R'-R}{p'-p}},\quad p^{\prime 2}-p^{2}=R^{\prime 2}-R^{2}.}$

Laguerre inversion and Lorentz transformation

In 1905 Henri Poincaré and Albert Einstein showed that the Lorentz transformation of the special theory of relativity (with ) ${\displaystyle c=1}$

${\displaystyle x'={\frac {x-vt}{\sqrt {1-v^{2}}}},\quad y'=y,\quad z'=z,\quad t'={\frac {t-vx}{\sqrt {1-v^{2}}}}}$

leaves the expression invariant. Einstein pointed out that a spherical wave in one inertial system remains a spherical wave in all other inertial systems. Poincaré also showed that the Lorentz transformation can be understood as a rotation in a four-dimensional space, and Hermann Minkowski was able to deepen this insight considerably (see history of special relativity ). ${\displaystyle x^{2}+y^{2}+z^{2}-t^{2}}$

As shown above, the relationship is also invariant under the transformation through reciprocal directions or half-lines, which was later often referred to as the Laguerre inversion. The relationship to the Lorentz transformation has been noted by various authors. Bateman (1910) argued that this transformation (which he attributed to Ribaucour) is "identical" to the Lorentz transformation. In 1912 he wrote that, especially in the form given by Darboux (1887), it corresponds formally to the Lorentz transformation in -direction, provided that ,, and the -terms are replaced by velocities. He also designed geometric representations of the relativistic spheres of light using spherical systems. However, Kubota (1925) replied to Bateman that the Laguerre inversion is involutive in contrast to the Lorentz transformation. To make both equivalent it is necessary to combine the Laguerre inversion with a reversal of the direction of the cycles. ${\displaystyle x^{2}+y^{2}+z^{2}-R^{2}}$ ${\displaystyle z}$ ${\displaystyle R=ct}$ ${\displaystyle R'=ct'}$ ${\displaystyle k}$

The relationship between the Lorentz transformation and the Laguerre inversion can be demonstrated as follows (see HR Müller (1948) for an analogous formulation in a slightly different notation). Laguerre's 1882 inversion formulas (equivalent to Darboux's 1887 formulas) are

${\displaystyle D'={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}},\quad R'={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}.}$

is now set

${\displaystyle {\frac {2\alpha }{1+\alpha ^{2}}}=w}$

so follows

${\displaystyle {\frac {1-\alpha ^{2}}{1+\alpha ^{2}}}={\sqrt {1-w^{2}}},\quad {\frac {2\alpha }{1-\alpha ^{2}}}={\frac {w}{\sqrt {1-w^{2}}}},}$

Because of this and by setting of , the Laguerre inversion takes the form of a Lorentz transformation, with the difference that the expression of the ordinary Lorentz transformation is reversed to : ${\displaystyle D=x,D'=x',R=t,R'=t'}$ ${\displaystyle t-vx}$ ${\displaystyle wx-t}$

${\displaystyle x'={\frac {x-wt}{\sqrt {1-w^{2}}}},\quad t'={\frac {wx-t}{\sqrt {1-w^{2}}}}}$.

According to Müller, the usual Lorentz transformation results from an even number of such Laguerre inversions which change the sign. In this way, first an inversion into the plane can be carried out which is inclined at a certain angle with respect to the plane, and then another inversion into the plane . See section # Laguerre group isomorphic to Lorentz group for more details on the relationship of the Laguerre inversion to other variants of Laguerre transformations. ${\displaystyle \pi _{1}}$ ${\displaystyle \pi }$ ${\displaystyle \pi }$

Lorentz transformation in Laguerre geometry

Timerding (1911) used Laguerre's concept of oriented spheres to derive and represent the Lorentz transformation. Using a sphere with a radius and the distance of its center from the central plane, he arrived at the following relations between this and a corresponding second sphere: ${\displaystyle r}$ ${\displaystyle x}$

${\displaystyle x'+r'={\sqrt {\frac {1+\lambda ^{2}}{1-\lambda ^{2}}}}(x+r),\quad {\frac {x'-r'}{x'+r'}}={\frac {1-\lambda }{1+\lambda }}\cdot {\frac {x-r}{x+r}}}$

from which the transformation follows

${\displaystyle {\sqrt {1-\lambda ^{2}}}\cdot x'=x-\lambda r,\quad {\sqrt {1-\lambda ^{2}}}\cdot r'=r-\lambda x.}$

By setting and it becomes the Lorentz transformation. ${\displaystyle \lambda =v/c}$ ${\displaystyle r=ct}$

Following Timerding and Bateman, Ogura (1913) analyzed a Laguerre transformation of the form:

${\displaystyle \alpha '=\alpha {\frac {1}{\sqrt {1-\lambda ^{2}}}}-R{\frac {\lambda }{\sqrt {1-\lambda ^{2}}}},\quad \beta '=\beta ,\quad \gamma '=\gamma ,\quad R'=\alpha {\frac {-\lambda }{\sqrt {1-\lambda ^{2}}}}+R{\frac {1}{\sqrt {1-\lambda ^{2}}}}}$,

which becomes the Lorentz transformation by

${\displaystyle {\begin{aligned}x&=\alpha ,&y&=\beta ,&z&=\gamma ,&R&=ct,\\x'&=\alpha ',&y'&=\beta ',&z'&=\gamma ',&R'&=ct',\end{aligned}}}$     ${\displaystyle \lambda ={\frac {v}{c}}}$.

He explained that the Laguerre transform in the sphere manifold is equivalent to the Lorentz transform in the space-time manifold.

Laguerre group isomorphic to the Lorentz group

As shown above, the group of conformal transformations (composed of motions, similarities , and inversions) in R ⁿ can be related by minimal projection to the group of touch transformations in R ^n-1 , which transform circles and spheres into other circles and spheres. Furthermore, Lie (1871, 1896) showed that there is a 7-parameter subgroup of infinitesimal similarity transformations (composed of movements and similarities) which, through minimal projection in R ^2, corresponds to a 7-parameter group of infinitesimal touch transformations that transform circles into circles . It corresponds to Laguerre's geometry of reciprocal directions, which is why Smith (1900) called it the “group of the geometry of reciprocal directions”, or “Laguerre group” according to Blaschke (1910), which together with Coolidge (1916) and others their properties investigated in the context of the Laguerre geometry of oriented spheres and planes. The (extended) Laguerre group consists of movements and similarities, and has 7 parameters in R ² (oriented lines and circles are transformed) or 11 parameters in R ³ (oriented planes and spheres are transformed). If similarities are excluded, because the result is the (narrower) Laguerre group with 6 parameters in R ² or 10 parameters in R ³ , which leaves the tangential distances invariant, consists of movements and assignments, and transforms oriented lines, circles, planes and spheres . When the Laguerre group is referred to in the following, then the closer Laguerre group is meant. The Laguerre group is not the only group that leaves tangential distances invariant, but is part of the more extensive “equilongous group” according to Scheffers (1905).

In R ² the Laguerre group leaves the relationship invariant, whereby this relationship can be extended to any R ⁿ . In R ³ , for example, the relationship is invariant. If minimal (isotropic) projection with imaginary radius coordinate or cyclographic projection with real radius coordinate is used in the context of the representing geometry , this expression is equivalent to in R ⁴ with the radius as coordinate. The transformations that make up the Laguerre group can be further subdivided into “actual Laguerre transformations” that relate to movements and contain both the tangential distance and the sign; or “improper Laguerre transformations” that refer to movements that reverse orientation (“assignments”) and that retain the tangential distance but reverse the sign. The Laguerre inversion (i.e. the transformation which was first given by Laguerre in 1882) is involutorial and thus belongs to the improper Laguerre transformations. Laguerre himself does not use the group term in his investigations, but the inversion he gives is of fundamental importance, since every Laguerre transformation can be generated from a maximum of four Laguerre inversions and every direct Laguerre transformation is the product of two involutorial transformations, which is why the entire Laguerre group can be generated from Laguerre inversions. ${\displaystyle dx^{2}+dy^{2}-dr^{2}}$ ${\displaystyle dx^{2}+dy^{2}+dz^{2}-dr^{2}}$ ${\displaystyle dx^{2}+dy^{2}+dz^{2}+dr^{2}}$ ${\displaystyle r}$

It was also found that the Laguerre group is actually isomorphic to the Lorentz group (or the Poincaré group if translations are included), since both expressions leave as invariant. After Bateman had referred to the connection with the Lorentz transformation as early as 1910 ( see above ), Cartan showed the correspondence of the two groups in a brief paper from 1912, again in 1914, and in a general way in 1915 (published in 1955) in the French Version of the Encyclopedia of Mathematical Sciences . Poincaré (1912, published 1921) also wrote: ${\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-dx_{4}^{2}}$

"Mr. Cartan recently gave a curious example. We know the importance that the so-called Lorentz group has in mathematical physics; it is this group on which the new ideas about the principle of relativity and the dynamics of the electron are based. On the other hand, Laguerre once introduced a group of transformations into geometry that turns spheres into spheres. These two groups are isomorphic, so that, from a mathematical point of view, these two theories, one physical and the other geometrical, have no essential difference. "

- Henri Poincaré, 1912

Other authors who pointed this out are, for example, Coolidge (1916), Klein & Blaschke (1926), Blaschke (1929), HR Müller , Kunle and Fladt (1970), Benz (1992), Pottmann, Grohs, Mitra (2009).

See also

• History of the Lorentz Transformation

Original sources

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Supporting documents:

1. ↑ ^{a b c} Bateman (1908); Bateman (1909); Cunningham (1909)
2. ↑ ^{a b c} Bateman (1910b), p. 624
3. ↑ ^{a b} Poincaré (1912), p. 145
4. ↑ Liouville (1847); Liouville (1850a); Liouville (1850b)
5. ↑ ^{a b} Liouville (1850b)
6. ↑ ^{a b c d e} Lie (1871); Lie (1872)
7. ↑ Darboux (1872), p. 282
8. ^ Lie (1872), p. 183
9. ↑ ^{a b} Klein (1893), p. 474
10. ↑ ^{a b} Laguerre (1881); Laguerre (1905), pp. 592-684 (works from 1880 to 1885).
11. ↑ Darboux (1887), p. 225
12. ↑ ^{a b c} Klein (1893), p. 473
13. ↑ Darboux (1872), pp. 343-349, 369-383
14. ↑ Bateman (1912), pp. 328 and 336
15. ↑ ^{a b} Darboux (1872), p. 366
16. ↑ Lie (1871), pp. 201ff; Lie (1872), p. 186; Lie & Scheffers (1896), pp. 433-444
17. ↑ Bateman (1909), pp. 225, 240; (1910b), p. 623
18. ↑ Bateman (1912), p. 358
19. ↑ Cunningham (1914), pp. 87-89
20. ↑ Poincaré (1906), p. 132.
21. ↑ Klein (1910/21)
22. ↑ Darboux (1887), p. 259
23. ^ Ribaucour (1870)
24. ↑ Stephanos (1881)
25. ↑ Stephanos (1883)
26. ↑ Laguerre (1882), p. 550.
27. ↑ Laguerre (1882), p. 551.
28. ↑ Darboux (1887), p. 254
29. ↑ E. Müller (1898), see footnote p. 274.
30. ↑ ^{a b} Smith (1900), p. 172
31. ^ Smith (1900), p. 159
32. ↑ Bateman (1912), p. 358
33. ↑ Bateman (1910a), see footnote pp. 5-7
34. ↑ Kubota (1925), see footnote p. 162.
35. ↑ ^{a b} H.R. Müller (1948), p. 349
36. ↑ Timerding (1911), p. 285
37. ↑ Ogura (1913), p. 107
38. ↑ Lie (1871), pp. 201ff; Lie (1872), pp. 180-186; Lie & Scheffers (1896), p. 443
39. ↑ ^{a b} Blaschke (1910)
40. ↑ Blaschke (1910), pp. 11-13
41. ↑ Blaschke (1910), p. 13
42. ↑ Cartan (1912), p. 23
43. ↑ Cartan (1914), pp. 452-457
44. ↑ Poincare (1912), p. 37: M. Cartan en a donné récemment un exemple curieux. On connaît l'importance en Physique Mathématique de ce qu'on a appelé le groupe de Lorentz; c'est sur ce groupe que reposent nos idées nouvelles sur le principe de relativité et sur la Dynamique de l'Electron. D'un autre côté, Laguerre a autrefois introduit en géométrie un groupe de transformations qui changent les sphères en sphères. Ces deux groupes sont isomorphes, de sorte que mathématiquement ces deux théories, l'une physique, l'autre géométrique, ne présentent pas de différence essentielle .
45. ^ HR Müller (1948), p. 337

literature

Textbooks, encyclopedic entries, historical introductions:

• Bateman, Harry : The mathematical analysis of electrical and optical wave motion on the basis of Maxwell's equations . University Press, Cambridge 1915.
• Benz, Walter : Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces Third Edition . Springer, 1992/2005, ISBN 3-0348-0420-2 , pp. 133-175.
• Blaschke, Wilhelm : Lectures on Differential Geometry and Geometric Basics of Einstein's Theory of Relativity, Vol. 3 . Springer, Berlin 1929, doi : 10.1007 / 978-3-642-50823-3 .
• Élie Cartan , Gino Fano: La théorie des groupes continus et la géométrie . In: Encyclopédie des sciences mathématiques pures et appliquées . 3.1, 1915, pp. 39-43. (Only pages 1-21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)
• Thomas E. Cecil: Laguerre geometry . In: Lie sphere geometry . Springer, 1992/2008, ISBN 0-387-74655-2 , pp. 37-46.
• Coolidge, Julian : A treatise on the circle and the sphere . Clarendon Press, Oxford 1916.
• Cunningham, Ebenezer : The principle of relativity . University Press, Cambridge 1914.
• Fano, Gino : Continuous geometric groups. The group theory as a geometrical classification principle . In: Encyclopedia of Mathematical Sciences . 3.1.1, 1907, pp. 289-388.
• Kastrup, HA: On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics . In: Annals of Physics . 520, No. 9-10, 2008, pp. 631-690. arxiv : 0808.2730 . bibcode : 2008AnP ... 520..631K . doi : 10.1002 / andp.200810324 .
• Klein, Felix : Introduction to Higher Geometry I 1893.
• Felix Klein , Wilhelm Blaschke: Lectures on higher geometry . Springer, Berlin 1926. (Klein's lectures from 1893 updated and edited by Blaschke in 1926.)
• Kunle H .; Fladt K .: Erlangen program and higher geometry - Laguerre geometry . In: Heinrich Behnke (Ed.): Fundamentals of Mathematics: Geometry . MIT Press, 1926, pp. 460-516.
• Müller, Emil : The different coordinate systems . In: Encyclopedia of Mathematical Sciences . 3.1.1, 1910, pp. 596-770.
• Pedoe, Daniel : A forgotten geometrical transformation . In: L'Enseignement Mathématique . 18, 1972, pp. 255-267. doi : 10.5169 / seals-45376 .
• Rougé, André: Relativité restreinte: la contribution d'Henri Poincaré . Editions Ecole Polytechnique, 2008, ISBN 2-7302-1525-5 .
• Walter, Scott: Figures of light in the early history of relativity . In: To appear in Einstein Studies, D. Rowe, ed., Basel: Birkhäuser 2012.
• Warwick, Andrew: Cambridge mathematics and Cavendish physics: Cunningham, Campbell and Einstein's relativity 1905-1911 Part I: The uses of theory . In: Studies in History and Philosophy of Science Part A . 23, No. 4, 1992, pp. 625-656. doi : 10.1016 / 0039-3681 (92) 90015-X .
• Warwick, Andrew: Masters of Theory: Cambridge and the Rise of Mathematical Physics . University of Chicago Press, Chicago 2003, ISBN 0-226-87375-7 .

Individual evidence

1. ↑ Kastrup (2008)
2. ↑ ^{a b} Walter (2012)
3. ^ Warwick (1992), (2012)
4. ↑ ^{a b c d} Fano: Continuous geometric groups. The group theory as a geometrical classification principle. 1907, pp. 318-320.
5. ↑ ^{a b} Müller (1910), Chapter 25
6. ↑ Pedoe (1972)
7. ↑ ^{a b} Cartan (1915), pp. 39-43
8. ↑ ^{a b c} Coolidge (1916), p. 422, is the invariant distance between two points in R ⁴ . ${\displaystyle \textstyle {\sqrt {(x-x')^{2}+(y-y')^{2}+(z-z')^{2}+(r-r')^{2}}}}$
9. ↑ ^{a b} Klein & Blaschke (1926), pp. 253-262
10. ↑ ^{a b} Blaschke (1929), Chapter 4
11. ↑ ^{a b} Kunle and Fladt (1970), p. 481
12. ↑ ^{a b} Benz (1992), Chapter 3.17
13. ↑ Kastrup (2008), Section 2.2
14. ↑ Kastrup (2008), Section 2.3
15. ↑ Fano (1907), pp. 312-315
16. ↑ E. Müller (1910), pp. 706-712
17. ↑ Kastrup (2008), Section 2.4
18. ↑ E. Müller (1910), p. 706
19. ^ Fano (1907), p. 316
20. ↑ E. Müller (1910), pp. 706-712
21. ↑ Müller (1910), p. 717
22. ↑ Klein & Blaschke (1926), pp. 246–248
23. ↑ E. Müller (1910), pp. 706-707, especially note 424.
24. ↑ Klein & Blaschke (1926), p. 258
25. ↑ Klein & Blaschke (1926), p. 253
26. ↑ Kastrup (2008), Section 1.1
27. ↑ Cunningham (1914), pp. 87-88
28. ^ Cunningham (1914), p. 88
29. ↑ Cunningham (1914), pp. 88-89
30. ↑ Kastrup (2008), Section 5.2
31. ↑ Kastrup (2008), section 6
32. ^ Walter (2012), Section 1
33. ^ Coolidge (1916), p. 355
34. ↑ Pedoe (1972), p. 256
35. ↑ Coolidge (1916), Chapters 10 & 11
36. ↑ ^{a b} Cecil (1992)
37. ^ ^{A b} Coolidge (1916), pp. 369 & 415
38. ^ Coolidge (1916), pp. 370-372
39. ↑ Cartan (1915), p. 40
40. ↑ Cartan (1915), p. 42, is the invariant square of the tangential distance between two oriented spheres. ${\displaystyle (x-x')^{2}+(y-y')^{2}+(z-z')^{2}-(t-t')^{2}}$
41. ↑ Blaschke (1910), p. 13
42. ^ Coolidge (1916), p. 372
43. ^ Coolidge (1916), p. 378, p. 382
44. ↑ Blaschke (1910), p. 15
45. ↑ Rougé (2008), pp. 127–128
46. ^ Pottmann, Grohs, Mitra (2009)