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Dipartimento Energia
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UNIFICATION OF TWO DIMENSIONAL SPECIAL AND GENERAL RELATIVITY
BY MEANS OF HYPERCOMPLEX NUMBERS
FRANCESCO CATONI
Centro Ricerche Casaccia, Roma
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RT/ERG/95/11
ENTE PER LE NUOVE TECNOLOGIE, L'ENERGIA E L’AMBIENTE
Dipartimento Energia
UNIFICATION OF TWO DIMENSIONAL SPECIAL AND GENERAL RELATIVITY
BY MEANS OF HYPERCOMPLEX NUMBERS
FRANCESCO CATONI
Centro Ricerche Casaccia, Roma
RT/ERG/95/11
Testo pervenuto nell’ottobre 1995
I contenuti tecnico-scientifici dei rapporti tecnici dell'ENEA rispecchiano I'opinione degli autori e non necessariamente quella dell'Ente.
DISCLAIMER
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ABSTRACT
An extension of complex numbers arid functions of complex variable is proposed through the properties of the related finite and infinite Lie groups. This is accomplished by hypercomplex number systems following the elementary algebra rules. More precisely, the functions of such systems define an infinite Lie group.The functional transformations of a particular two dimensional hypercomplex number system, holding the wave equation invariant (and, then, the speed of light constant) are considered as a generalization of Lorentz group describing accelerated frames.According to General Relativity, such frames can represent physical fields. A physical interpretation of a theorem due to Bianchi, by which hypercomplex number systems are related to flat Riemann spaces, is shown to connect these systems to General Relativity.The investigations by generalized Special Relativity and by General Relativity give the same expected results allowing to speak of unification.
[Special Relativity, General Relativity, Graded Lie Groups, Hypercomplex Functions PACS numbers 03.30, 04.20, 02.20.T]
RIASSUNTOUnificazione in due dimension! della Relativita Ristretta con la
Generate
Le proprieta di gruppo delle variaoili complesse e delle loro funzioni sono estese ai sistemi di numeri ipercomplessi che seguono le regole dell’algebra elementare.Le trasformazioni funzionali coilegate ad un particolare sistema a due unita lasciano invariante l’equazione delle onde e possono, quindi, essere considerate come una estensione delle trasformazioni di Lorentz in grade di descrivere i campi.Un’interpretazione fisica di un teorema dovuto a Bianchi, che collega i sistemi di numeri ipercomplessi agli spazi con curvatura nulla, permette di collegare questi sistemi alia Relativita Generate.I risultati ottenuti mediante la Relativita Ristretta e la Generale coinci- dono e questo permette di parlare di Unificazione.
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Contents1 Introduction 7
2 Hypercomplex variables and Lie groups 7
3 Unification of Special and General Relativity 11
A Lie groups and hypercomplex number systems 14
B The functions of hypercomplex variables 16
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1 IntroductionThe wide use of complex numbers, far behind their algebraic introduction as square root of negative numbers, always stimulated their extensions.The most important ones are probably the N dimensional space complexification and Hamilton’s quaternions. The former plays an important role in mathematical physics [1, 2]; the latters, though introduced [3, 4] from an algebraic point of view, have been used in a wider meaning [5]. Unfortunately, quaternions do not satisfy the product commutative property that is fundamental for the kind of extension of complex numbers we will treat ahead. In particular, complex numbers and functions of a complex variable can be related to finite and infinite Lie group and an extension preserving the group properties will be argued. This will be shown for the so called hyper complex numbers that follow the elementary algebra rules (associative, distributive and commutative) for the sum as well as for the product. A geometrical and physical interpretation of this extension will be given and a unification of two dimensional Special and General Relativity will be proposed as a first application.
In fact, as well known, the Lorentz transformation of Special Relativity is linear and represents a change of inertial frames. It will be considered a generalization by functional transformations that keep the speed of light constant.If we refer space-time to cartesian coordinates the inertial motion is represented by straight lines and, by the linear Lorentz transformation, these lines remain straight. From a geometrical point of view a functional transformation represents a transformation of curvilinear coordinates by which the straight lines in one reference frame become curved lines in the other. From a physical point of view, these lines represent a non inertial motion, i.e. a motion in a field. The question arising is whether such fields have a physical meaning: this will be proved in the two-dimensional case (one spatial coordinate and time). This can lead to a generalized Special Relativity that will be extended to General Relativity.
As most reference books on traditional Lie groups and hypercomplex number systems have been neither reprinted nor translated from the original editions, some basic concepts on these subjects are reported in appendix A, that it is advisable to read before sec. 2.In Sec. 2 a geometrical and physical interpretation of the relation between finite Lie groups and hypercomplex systems is proposed and applied to the two dimensional numbers; then, the infinite group properties of functions of complex and hypercomplex variables (and their physical interpretation) are shown.In the last section (3) a first application of these ideas is described. The two dimensional Special Relativity is generalized by an infinite group of functions of hypercomplex variable and the General Relativity is related to hypercomplex systems by a physical interpretation of a theorem due to Bianchi. Both approches lead to the same result.
2 Hypercomplex variables and Lie groupsThe definitions of traditional Lie groups [6, 7] and hypercomplex systems [8, 9] (and their relations) are recalled in App. A. Let us see now their geometrical and physical meaning.
Complex numbers are known to be used for the representation of plane vectors and the related linear algebra [10]: z =■ x + iy is interpreted as a vector of components x and y,
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and versors 1 and i. In vector form, it can be written z — la; + iy, where x and y are the current coordinates of the plane. In order to show how this is related to finite group properties of complex numbers, consider the multiplicative group (18) that can be expressed as z\ = az = (or + + iy) and, in the familiar linear algebra, as:
( X\ \ _ ( Or ~di \ ( X \\y\ ) V ar j \y j
The constant a play the role of an operator representing an orthogonal axis rotation with a homogeneous dilatation (homothety). These are the permissible vector transformations in a Euclidean plane.Then we can use complex numbers to describe plane vector algebra because the vectors are, usually, represented in an orthogonal coordinate system and the complex number multiplicative group is, with unitary dilatation, the Euclidean one. It is known that the multiplicative group is usually studied in complex analysis by a polar coordinate system. The reason is that polar coordinates are equivalent to complex exponential transformations and so on a logarithmic plane the multiplication is transformed into a sum. The exponential transformation will be the best way to study the multiplicative group for all hypercomplex systems.
In order to study the multiplicative groups (appendix A) of other hypercomplex numbers, Klein’s relation (“Erlangen Program”) [11] between “geometries” and continuous groups needs now to be recalled. In fact all the geometries (elementary, projective, etc.) are based on an equivalence principle. All transformations producing equivalent forms can be regarded as a group. Klein’s idea was to consider geometries as theories of related group invariants. This idea can be inverted and applied to hypercomplex systems because (18) is a linear transformation and most geometries are related to linear transformations [11].The general linear transformation y1 = cjgX13, with the only condition |c$| / 0, defines the N2 parameter affine geometry whose invariant is the quadratic form of two point distance. The allowed transformations are equivalent to arbitrary changes of coordinate axes.To assume the Pythagorean form of the distance as invariant, the allowed transformations must be the orthogonal ones, the parameters become N(N—1)/2 and the_Euclidean geometry is obtained. Note that adding the N parameters of the translational group, the N(N + l)/2 resulting number of parameters can be related to all of the allowed movements in a homogeneous space.
It is now interesting to compare, for various dimensions (N) the parameter number of Euclidean geometry (PEG) with the one related to the geometry associable to multiplicative groups of hypercomplex systems (PHG):
PHG[N} 2 3 4 5 6PEG[N(N - l)/2] 1 3 6 10 15
The parameter number is the same only for N = 3 but, as will be shown ahead, there is no equivalence between the two “geometries”. For N > 3 the PHG shows less permissible transformations, thus representing the allowed movements in an unomogeneous space.
Let us now study what “geometry” can be related to the “hyperbolic number” system (see App. A). From now on, the second unit of these numbers will be called j and the second
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variable t; so it is posedz = x+jt (1)
By the formal method, leading to Euler’s formula for the complex variable, the exponential function of the hyperbolic system can be introduced [2], assuming as valid the same properties of real and complex exponential:
exp (x + jt) = exp x exp jt = exp iEf-= exp x f E ^7 + E $i+i)i) =
( ,21 OO 21 1 j \exp x ( E (5ij? + 3 E (2IT1)! ) = exp x (cosh t + j sinh t). (2)
The exponential transformation will be
X + jT => exp x (cosh t + j sinh t) (3)
As in complex analysis, this transformation can be inverted and the definition of logarithmicfunction is obtained: ______
X + jT => In y/x2 — t2 + j ta,nh~1(t/x) (4)
The multiplicative group can now be studied.Writing the hyperbolic constant a = ar + jaj as:aT + jaj = ja| (cosh 6 + j sinh 6) where |a| = ^Ja2 — aj ; 0 = tanh_1(aj/ar) the multiplicative group becomes:
z'= x' + jt' — |a| exp jO(x + jt) = |a|(cosh B + j sinh 0)z (5)
As usual in complex analysis, equating the “real” and “imaginary” parts in this expression, the Lorentz transformation of Special Relativity is found [10]. Note that relation (5) is usually obtained by the following steps:. introduction of imaginary time t1 = it. equivalence, in this form, of the Lorentz invariance with the rotation invariance . introduction of hyperbolic functions by their equivalence with imaginary circular functions. All these steps are only formal. The direct description by hyperbolic numbers has the meaning of symmetry (or invariants) preservation. Indeed, also the Lorentz invariant (space-time distance) is the same for hyperbolic numbers (17); then the “geometry” associated to the multiplicative group of hyperbolic numbers is the Space-Time geometry. Within the limits of my knowledge, the first algebraic description of Special Relativity, directly by these numbers (called “Perplex numbers”), was introduced in [12], but without evidence of group or simmetry preservation properties.
Although every hypercomplex system is related to a different multiplicative group, considering systems containing the unity (App. A) all multiplicative groups will have a common (homothetic) subgroup. For this reason, the three parameter group of the three dimensional hypercomplex numbers can not be equivalent to the Euclidean group. Adding the homoth- eties to the Euclidean group, the only hypercomplex number system related to the Euclidean geometry is seen to be the one of ordinary complex numbers.Consider, now, an important relation between functions of complex and hypercomplex variables and infinite Lie groups.
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According to Riemann [2]: a function w(z) = u(x,y) + iv(x, y) is considered a function of the complex variable z if its derivative is independent of direction. From the derivative condition dw = w'(z)dz a relation follows between the partial derivatives of w, v, known as Cauchy-Riemann’s (C-R) equations. As any composition of system solutions is still a solution, this system defines (App. A) an infinite continuous group. In Lie’s language the C-R equations can be interpreted as infinitesimal transformations of the group [6]. It is also well known that u and v, thanks to C-R conditions, can be considered the components of a plane harmonic vector field. The symmetries of a two dimensional electric field are the ones of an infinite group.The C-R conditions show that u, v depend on the only variable z rather than on the two variables x, y. Furthermore, as one component u or v determines the other, a physical interpretation arises. In fact, considering a three dimensional conservative field represented either by a function (the potential), or by three functions (the field), it is evident the difference between the “informative content” of these two representations: if the “information” needed to describe the field were contained in one function only, the components of the field would be obtained from one of them. This occurs for fields described by functions of complex variable and, as will be seen, for functions of hypercomplex variables.Even the latter can be defined following Riemann, assuming their derivability. This cames from a theorem, due to Scheffers, [13] which states that for functions of hypercomplex variable associated to a distributive number system (with unity), the necessary and sufficient conditions for the existence of differential and integral calculus are that the number system be commutative and associative.Therefore the structure constants must satisfy (15) and (16). For hypercomplex numbers following the algebra rules, functions exist and (App. B) satisfy the GCR conditions (19). The system solutions are functions whose composition satisfies the same differential system, so that functions of hypercomplex variables can be associated to an infinite Lie group. Then the hypercomplex number systems following the algebra rules can be considered an extension of complex numbers also due to their infinite group properties.
These functions can describe fields with “symmetries” defined by the structure constants. In fact, fields are defined by the potential of a point source, i.e. the Green function of the related partial differential equation. Just like Laplace equation derives from CR conditions, so GCR conditions define a set of second order partial differential equations depending on structure constants (the same occurs for the related Green functions).
The functions of hypercomplex variables can be considered an extension of functions of complex variable, in a simply connected domain, also due to the following properties:
1. The derivative of a function of hypercomplex variable is a function of hypercomplex variable.
2. From any component wa of a hypercomplex function can be found all the other ones, which are defined to within an arbitrary constant term.
3. A primitive, such that all the wa can be determined as a linear combination of the partial derivative (generalized gradient), can be obtained from any component wa of a hypercomplex function.
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4. The functions of a hypercomplex variable x can be obtained by powers (or power series) of x.
5. The dependence on N variables is apparent because it can be described by only one variable with the components linked to each other by the symmetries defined by the structure constants.
Based on the previous considerations, the structure of space-time, made unhomogeneous by sources, could possibly be described by the “geometry” of a hyper complex system. A first encouraging result will be discussed in the following section.
3 Unification of Special and General RelativityConsider now the canonic hyperbolic numbers (1) that, from now on, will be called Lorentz numbers.Their functions w(z) = u(x,t) + jv(x,t) satisfy the following system of partial differential equations [14], where the comma will stand for derivation with respect to the following variables:
= =which, in turn, can be named CRL (Cauchy-Riemann conditions for Lorentz functions). All these functions satisfy the wave equation that is invariant for the transformations (x, t) => (u, v), with u, v satisfying the CRL conditions. The relation between the Lorentz functions and wave equation is the same as that between functions of complex variable and the Laplace equation [14, 15].From a physical point of view, this means that in any of the curvilinear coordinate system, introduced by these functional transformations, the speed of light is constant.Therefore, t can be referred to a physical normalized time variable. It is found again that, as the group of complex functions describes the symmetries of two spatial variables in harmonic fields, so the symmetries between one spatial variable and time are described by the infinite group of these transformations. It is natural to call this infinite group: “ Generalized Lorentz- Poincare two dimensional group”.
It will be now shown that the motion in a central field is described by functions of these Lorentz variables.
As an introductive example, the central field of a point source in a bidimensional space is found by means of ordinary complex variables. We shall deal with the problem of finding a solution, that must be constant on a circle, of the Laplace equation UtXX + f/yy = 0. This problem is usually solved [15] by a polar coordinate transformation, but the use of a complex exponential transformation, that has the same symmetries of the polar one, leaves the Laplace equation invariant; then we can solve the equation:
U,xx + Uyy =0 (6)
where ______X + iY = ln(x + iy) = In \jx2 + y2 + itsxT1(y/x).
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The U constancy on a circle means independence of the rotation angle Y, then U,yy = 0, equation (6) becomes Utxx = 0, and the solution is U = aX + b = a In y/x2 + y2 + b i.e. the potential of a point charge (the Green function for the Laplace equation).
For the x,t variables, the experimental evidence of symmetry is the invariance under the Lorentz transformation, given by the (5) with |a| — 1. Looking for a solution of the wave equation U<xx — U$t — 0 independent of the hyperbolic angle, we proceed as before. By means of the hyperbolic logarithmic transformation (4), and the symmetry relation, we must solve U,xx = 0 obtaining:
U = aX + b = a In yfx2 — t2 + b (7)In the x, t plane the equipotential U = const, expressed by a parameter r, will be:
x = g cosh r ; t — g sinh t (8)i.e. the hyperbolic motion [16].This is the motion in a central field with only one spatial coordinate. In fact, the potential of a central field, in one spatial dimension, is given by U = Kx, i.e. the force is constant. By this force and the relativistic Newton’s dynamic law, the hyperbolic motion is obtained. In the shown example, the interaction between two bodies is described by the potential function (7), which depends on spatial coordinates as well as on time and, as in General Relativity, the motion is obtained without the Newton dynamic law. This result has been obtained by a simple transformation, using the symmetry preservation in space-time, a well accepted concept of modern physics [17].
It will be shown now that the relationship between the exposed concepts and General Relativity is not a casual one, that r is the proper time, and the obtained hyperbolic motion is a geodesic motion.
Let us start from the following theorem due to Bianchi [18]: “every system of commutative (and, therefore, associative) hypercomplex numbers (which possesses unity) can be associated to a metric element of a space with null Riemann tensor”. These conditions are the same required by Scheffers theorem for hypercomplex functions.The generic metric element of these spaces, allowing its physical interpretation, is now found.
The Riemann tensor is defined by [17]: = Tea^ - P^ + P^P^g - P^P^. If theChristoffel symbols are assumed to be constants and proportional to the structure constants (i.e. = KC"p) of any associative hyper complex number (16), the first two terms of theRiemann tensor are null and the last two cancel each other. Then Rea/3^ = 0 and [18, 19]:. it is possible to obtain the metric tensor gap from P^. by a variable transformation, the metric element assumes its normal form gQp = dz8ap. These results can be obtained by solving a partial differential system of N2(N+l)/2 equations [18, 19]. Making use of the hypercomplex number formalism, the solution is obtained solving only one normal differential equation. In fact, given two expressions of the metric element of the same N dimensional flat space (“space” can stand for “space-time”) [19]:
N-lds2 = ]jP ea(dxa)2; ea = ±1 and ds2 = gapduadu^,
a=0the functions za(u°,... u^-1), transforming the first form in the second one, must satisfy the following linear, partial differential, system with constant coefficients [18, 19]:
- r(u»“ = o (9)
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Let = KC”p and solving system (9), the gap will be found by the transformation functions.Consider equations (9) with the same 7,/?, multiply the equation with xa by ea and sum over a. By means of the definition of hypercomplex number (13) it is obtained:
x,~iP— r = 0 (10)
The conditions (19) give: xnp = e^epxtUU; C^pX<u = C'^3el/xiU, and by (14) equation (10) is reduced to: eyep(x>uu — KxtU) = 0.The solution, independent of the particular indices chosen and valid for every hypercomplex number system, is:
x — xa = g exp K(u — u0) (11)
(g real constant and xQ, u0 arbitrary hypercomplex constants). Equating the terms with the same versors, real transformation functions are obtained.For a physical interpretation of the transformation functions (11), consider the case of complex variables, for which (11) becomes z — zQ = g exp K(w — w0), and by inversion: w — w0 = K~x In (z — z0)/g that represents the complex potential of a point source in the singular point z — z0 and K~x is the field strength. Starting from a flat space, the curvilinear coordinates transformation (11) will describe the deformation of straigth lines (geodesics), induced by a point source.
Consider now an example related to General Relativity and to Einstein ideas, as exposed in the introduction of [20]. In General Relativity, a body in a gravitational field moves on space-time geodesics. In order to find a relation with the above results we have to take a time coordinate. As a conclusive goal, numbers of four or more dimensions have to be studied and the results must be compared with experimental evidence.As a starting point, consider now the two dimensional infinite group of Lorentz functions, representing the symmetries of the time and one spatial coordinate. Solution (11) now becomes (without some arbitrary constants):
x1 + jx2 = x + jt = g exp K(ux + ju2) = g exp Ffu^cosh Ku2 „+ j sinh Ku2) (12)
Geodesics are known to be straight lines in a reference frame for space-time coordinates with metric element in normal form. If = ACjG, where CjG give the symmetries and K depends on the field strength, geodesics can be obtained transforming straight lines through equation (12). Then a body, that on the plane u1, u2 moves on the straight line u1 = const] u2 = t, is seen in the z, t reference frame to move on a hyperbolic motion (8). Because u2 is referred to “time coordinate” in the reference system in which the “spatial coordinate” u1 is constant, r = u2 is the proper time. A motion on a curved geodesic is obtained because the Christoffel symbols are different from zero while the Riemann tensor remains null.The same equation (8) has been obtained in two completely different ways. In the first case, eq. (8) derives from Special Relativity and its “generalization” to all the coordinate transformations that make the speed of light constant. In the second one, eq. (8) is strictly related to General Relativity, i.e. to the description of the gravitational field by Riemann tensor and a “geometrization” of space-time.The generalization of Special Relativity, by means of functional transformations, can be interpreted, in physical fashion, as a change of non inertial and then accelerated frames,
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or in Einstein’s own words [20], as the one describing a field. But this field is a function satisfying the same symmetries of the support space; then, if we limit the General Relativity to the reference system transformations preserving the natural symmetry the same result of the Generalized Special Relativity is obtained. The symmetry preservation is what we do when we transform Laplace equation by conformal mapping.Finally a physical interpretation of Bianchi’s theorem is summarized: if the laws of Nature have the symmetries of a hypercomplex number and, as in the two dimensional case, the fields are described by its functions, then the Riemann tensor is always null. The General Relativity might be “specialized” to symmetry preserving transformations of this group and Einstein equations [17] would be automatically satisfied outside sources by functions of this hypercomplex number system, just like Laplace equation is for functions of complex variable. If this can be demonstrated, a mathematical idea will give rise to physical laws, as Einstein hoped to and as is often happening nowadays.
A Lie groups and hypercomplex number systemsAn important idea due to S. Lie has been to associate a system of differential equations to variable transformations with group properties [6, 7]. Given N real variables x° with a = 0,..., N — 1, a transformation into yu with v = 0,..., N — 1 depending on M parameters an with n = 1,..., M, can be written as: yu = fu{x°...xN~l,a}...aM) = fl/(x,a). This transformation belongs to a group if, given another transformation with different parameters b: zx = fX(y,h), the functional relation between z and x is zx = fx(x,c) with c only depending on a and b. In this case Lie showed that the functions fx have to satisfy a differential system and the number (M) of parameters is given by the initial condition. As the M parameters can continuously change, the group is called continuous with the finite order M.
The relation between continuous groups and systems of differential equations can be inverted as follows [6]: if any composition of system solutions is still a solution, the system defines a continuous group. Two possibilities can occur:1) the solution depends on some arbitrary constants;2) the solution depends on arbitrary functions.In the first case the group is called finite with an order given by the number of constants. In the second one the group is called infinite. From a physical point of view, the infinite groups can describe fields. Examples of both groups are given in sections 2.
Let us now recall something about hypercomplex numbers [8, 9]. Given, as in vector algebra [21]:
x = eQxa = e0x° + etxl (13)
where xa are called components and ea units or versors, this expression defines a hypercomplex number if the versors multiplication rule is given by:
eat($ - (14)
where are real constants, called structure constants, that define the characteristics of the system [8, 9]. Here we will only consider systems with a versor (unity versor), called e0,
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satisfying el = e0 and e0e,- = e,.The definition of versor product is equivalent to define the product of hypercomplex numbers. In fact, the product of two numbers (x = eaxa and y = epy*3) is given by xy =
The product definition makes the difference between vector algebra and hypercomplex systems and allows to relate the second ones to groups.The C2p can be anything but for particular systems they have to satisfy some conditions[8,9]:
commutative systems: C2p = Cja (15)
associative systems: — C^C^. (16)
As in vector algebra, two systems are considered equivalent if they can change into each other through a linear transformation.Finally, it is useful to remind that the matrix with elements asQ = C6apy& is called characteristic matrix and its determinant (characteristic determinant) is the invariant of the system. They depend on the N components of the hypercomplex number, as well as on the structure constants of the system. For the complex number the invariant is the modulus x2 + y2.
As an example, consider the hypercomplex systems with two units, widely used ahead. For defining such a system, we pose z = e0x + exy and eQei = e1-, el = eo = 1 (omitted), and then, from (14), [14] e\ = = e0a + elyS = a + exfi.Let us now consider the dependence on the structure constants.The characteristic determinant is given by:
% yay x + {3y = x2 — ay2 + flxy = p2
If we represent in a Cartesian plane the structure constant a, /?, it can be shown [14] that the parabola A = /32 -f 4o: = 0 divides the plane in two parts, and for a, f3, on the parabola or on its left and right sides we have three equivalent systems that are called [14]:A > 0 Hyperbolic; A = 0 Parabolic; A < 0 Elliptic.Numbers of the same system can be related to each other by a linear transformation, i.e. they are equivalent. Due to this equivalence, special attention is paid to the simplest systems (Canonical systems) [8, 9, 14] obtained with ,5 = 0 and0=1 Hyperbolic; a = 0 Parabolic; a = — 1 Elliptic, or ordinary complex numbers.For canonical hyperbolic system the invariant is given by
= (17)
that is not positive definite, and is equal to zero for x = ±y.For the numbers xdzjx, called null divisor, the division is not allowed. To have null divisor is the main algebraic difference between the Hamilton Quaternions and all other hypercomplex systems.
Consider now the relation between the hypercomplex systems and the finite groups due to Poincare [22].
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A geometric representation of a hypercomplex number can be seen as a point in an N dimensional space. Assuming a hypercomplex constant a = e„a“, the transformation y = eaya = x + a = eQ(xa + a") is a translation that, as can be easily seen, forms a group. This group is the same for all hypercomplex systems.Considering the other transformation, called multiplicative group: e^y1 = y = ax = eaaaepxP = e^C^a01 x® and posing it yields
y1 = A}x0 (18)
the y1 are obtained from x'3 by a linear transformation that, in vector algebra, is called homographic group. As the matrix Al is depending on the N constants aa and on the structure constants, the multiplicative group will change in conformity with the number system.
B The functions of hypercomplex variablesIn order to define the functions of a hypercomplex variable, Riemann approch to functions of ordinary complex variable [2] can be followed. The conditions to be satisfied can be determined as follows [23]. Be x = e0x° -f e,z' a hypercomplex variable and w = e0w° + &iWl a hypercomplex function where, in the domain we are considering, the wa have the regularity conditions we need. Let them be represented in a 2N dimensional Cartesian space. Consider a curve, through the point chosen for the derivative calculus, defined by a curvilinear coordinate s. Let (p = dx1 /ds be the direction parameter of the tangent curve through the same point. The difference quotient, taking dx different from a null divisor, is:
dw w^dx1 _ wiy(p dx e^dx^ en<p
This quotient must be the same for every curve, in particular for the N straight lines in the coordinate axis directions. For these straight lines the <p are all zero but one. The straight line with <j> = (1, 0,..., 0) gives dw/dx = wiX = w<0, while the other ones give:
= e„to0 = evW'X. (19)
Then, as in complex analysis:. The derivative of a hypercomplex function is given by the partial derivative in respect of the variable associated to unity versor.. ty is a function of x, if and only if [23] the N — 1 differential conditions (19) are satisfied. The (19) relations among hypercomplex functions, are equivalent to N(N — 1) real partial differential equations. Therefore, among the N2 first derivatives of the components of a hypercomplex function, there are only N2 — N(N — 1) = N indipendent ones. N is also the number of the partial derivatives of each component to'9 of to. Every partial derivative can be expressed through the ones related to any single component. Relations (19) can be defined as GCR (Generalized Cauchy-Riemann conditions).
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References[1] Encyclopaedia of Mathematical Sciences, vol. 7, 8, 9, 10, 54, 69, Several Complex Vari
ables, (Springer-Verlag, Berlin, 1989-1994)
[2] B. Chabat, Introduction a l ’Analyse Complexe (Mir, Moscou, 1990)
[3] W. R. Hamilton, Elements of Quaternions, (Chelsea publishing Company, New York,1969)
[4] B. Doubrovine, S. Novikov, A. Fomenko Geometrie Contemporaine, Vol. I, (Mir, Moscou, 1985)
[5] C. A. Deavours, Amer. Math. Mon., 80, 995 (1973)
[6] L. Bianchi, Lezioni sui gruppi continui finiti di trasformazioni (Spoerri, Pisa, 1918)
[7] L.P. Eisenhart, Continuous Groups of Transformations (Princeton University Press, Princeton, 1933)
[8] S. Lie and M.G. Scheffers, Vorlesungen iiber continuierliche Gruppen, Kap. 21 (Teubner, Leipzig, 1893)
[9] E. Cartan, Oeuvres Completes, II, 107, (Gauthier Villars,Paris, 1953); L.E. Dickson, Trans. Am. Math. Soc., 6, 344 (1905)
[10] P.M. Morse and H. Feshbach Methods of Theoretical Physics (Me Graw-Hill, New York,1953)
[11] N. Efimov, Geometrie Superieure, (Mir, Moscou, 1985)
[12] P. Fjelstad, Am. J. Phys. , 54, 416 (1986)
[13] M.G. Scheffers, Comptes-rendus Acad. Sc., 116, 1114 (1893)
[14] M. Lavrentiev and B. Chabat, Effets Hydrodynamiques et modeles mathematiques (Mir, Moscou, 1980)
[15] Y.V. Sidorov, M.V. Fednyuk and M.I. Shabunin, Lectures on the Tkeory of Functions of a Complex Variable (Mir, Moscou, 1985)
[16] C.M. Misner,K.S. Thorne and J.C. Wheeler Gravitation (W.H. Freeman and Company, S. Francisco 1970)
[17] H. C. Ohanian, Gravitation and Spacetime, (W.W. Norton e Company Inc., New York, 1976)
[18] L. Bianchi, Rend. Acc. Lincei, (5), 25, 177 (1916); Opere, IX, 237 (Cremonese, Roma,1958)
[19] L.P. Heisenhart, Riemannian Geometry, (Princeton University Press, Princeton, 1949)
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[20] A. Einstein, Annalen der Physik, sez. 4, 49, 769 (1916)
[21] As in tensor calculus the sum symbol on repeated indices is omitted, the indices in Greek letters run from 0 to N — 1, and the Latin one, if not already stated, from 1 to N-l.
[22] H. Poincare, Comptes-rendus Acad. Sc., 99, 740 (1884)
[23] G. B. Rizza, Comm. Pont. Ac. Sc., 14, 163 (1950)
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