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Unification of Two Dimensional Special and General Relativity by Means of Hypercomplex Numbers

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Unification of Two Dimensional Special and General Relativity by Means of Hypercomplex Numbers J-' %0O26q tmmmmmsmmsmmm ISSN/'1120 *5563 Dipartimento Energia ENEA-RT-ERG--95-// UNIFICATION OF TWO DIMENSIONAL SPECIAL AND GENERAL RELATIVITY BY MEANS OF HYPERCOMPLEX NUMBERS FRANCESCO CATONI Centro Ricerche Casaccia, Roma kv -%-* waJ? * Ik Lissa Lia& DiiC 0 41936 USTI ## 8mN IF IMS mWEM IS UKUWfEI FStoH SAU£ nimiTES 'ftg 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 Portions of this document may be illegible in electronic image products. Images are produced from the best available original document 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 sys­ tems define an infinite Lie group. The functional transformations of a particular two dimensional hyper­ complex 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 hyper­ complex 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 Rel­ ativity give the same expected results allowing to speak of unification. [Special Relativity , G eneral Relativity , G raded Lie G roups , Hypercomplex Functions PACS numbers 03.30, 04.20, 02.20.T] RIASSUNTO Unificazione 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. 3 Contents 1 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 5 1 Introduction The 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 Hamil­ ton’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 geo­ metrical 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 groups The 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, 7 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 \yj 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 multiplica­ tive 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 loga­ rithmic 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 6 PEG[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
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