2006jbj

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On the World function and the relation between geometries
2006jaz | Garas`ko G. I.  // Electrotechnical Institute of Russia, Moscow, gri9z@mail.ru}

It is shown that the World function can be regarded as a link between the qualitatively different geometries with one and the same congruence of the world lines (geodesics). If the space in which the World function is defined is a polynumber space, then the hypothesis of the analyticity of the vector field of the generalized velocities of the world lines lead to the strict limitations on the structure of the World function. Main result: Minkowskian space and polynumber space correspond to the same physical World.

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Construction of the pseudo Riemannian geometry on the base of the Berwald-Moor geometry
2006jay | Garas`ko G. I., Pavlov D. G.  // gri9z@mail.ru; geom2004@mail.ru

The space of the associative commutative hyper complex numbers, H₄, is a 4-dimensional metric Finsler space with the Berwald-Moor metric. It provides the possibility to construct the tensor fields on the base of the analytical functions of the H₄ variable and also in case when this analyticity is broken. Here we suggest a way to construct the metric tensor of a 4-dimensional pseudo Riemannian space (space-time) using as a base the 4-contravariant tensor of the tangent indicatrix equation of the Berwald-Moor space and the World function. The Berwald-Moor space appears to be closely related to the Minkowski space. The break of the analyticity of the World function leads to the non-trivial curving of the 4-dimensional space-time and, particularly, to the Newtonian potential in the non-relativistic limit

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2006jax

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2006jaw

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2006jav

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2006jau

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2006jat

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2005jbz

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2005jby

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2005jbx | Pavlov D. G.

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2005jbw

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2005jbv

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The generalized Finslerian metric tensors
2005jbu | Lebedev S. V.  // Baumann University's Institute of Applied Math@Mech

The generalized Finslerian metric tensors are proposed. These tensors can have different number of indeces dependent on space dimension as well as space properties. The relationship of these tensors with the Finsler spaces associated with commutative associative algebras is analyzed. Nearest perspectives to research of the tensors of this type are discussed. The generalized differential equations of Finsler geodesics are derived and discussed.

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2005jbt

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2005jbs

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Finsler spinors
2005jbr

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The prolongations of a Finsler metric to the tangent bunde $T^k(M) (k>1)$ of the higher order accelerations
2005jbq | Atanasiu Gh.  // Department of Algebra and Geometry, Transilvania University, Brasov, Romania

An old problem in differential geometry is that of prolongation of a Riemannian structure $g\left( x\right) $ on a real $n-$dimensional $% C^{\infty }$-manifold $M,x\in M,$ to the bundle of $k-$jets $\left( J_{0}^{k}M,\pi ^{k},M\right) $ or, equivalently the tangent bundle $\left( T^{k}M,\pi ^{k},M\right) $ of the higher order accelerations. The problem belongs to so-called geometry of higher order. It was solved in $\left[ 18% \right] $ for $k=1$ and partially in $\left[ 19\right] $ for$\;k=2.$ The same problem of prolongation can be considered for a Finslerian structure $% F\left( x,y^{\left( 1\right) }\right) $. In the paper $\left[ 15\right] $ are given these solutions in the general cases, using the Sasaki-Matsumoto $N-$lift (for $k=2,$ see $\left[ 3\right] $ and $\left[ 6\right] ).

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The 2-Cotangent Bundle with Berwald-Moor Metric
2005jbp | Gheorghe Atanasiu, Vladimir Balan  // Transilvania University, Brasov, Romania; University Politehnica of Bucharest, Department Mathematics I, Romania

On the total space of the dual bundle $(T^{\ast 2}M,\pi ^{\ast 2},M)$ of the $2-$tangent bundle $(T^{2}M, \pi ^{2},M)$, the paper develops results related to the notions: of nonlinear connection, distinguished tensor fields, almost contact structure, Riemannian structures, $N-$linear connections and associated convariant derivations. The Ricci identities are derived and the local expressions of the corresponding $d-$tensors of torsion and curvature are provided. Further, the metric structures and the metric $N-$linear connections are studied, and the obtained results are specialized to the case when the metric tensor field is of Berwald-Moor type.

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The Berwald-Moor metric in the tangent bundle of the second order
2005jbo | Gheorghe Atanasiu, Nicoleta Brinzei

As an application of the results of the first author obtained in the papers \cite{1} and \cite{2}, the geometry of the second order tangent bundle $% T^{2}M$ (or second order jet bundle $J_{0}^{2}M$) endowed with two special types of metrics compatible with the 2-contact structures is studied. The particularity of these two models is that the horizontal and the $v^{(1)}$-\ part of the metric are both given by the same Riemannian metric (respectively, its horizontal part is Riemannian), while its $v^{(2)}$-part is given by the flag-Finsler Berwald-Moor metric (respectively, the $v^{(1)} $ and $v^{(2)}$- parts are given by the flag-Finsler Berwald-Moor metric, \cite{Mangalia}).

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Berwald-Moor-type $(h,v)$-metric physical models
2005jbn | Balan V., Brinzei N.  // University Politehnica of Bucharest, Department Mathematics I; Department of Mathematics, "Transilvania" University, Brasov, Romania

In the framework of vector bundles endowed with $(h,v)-$metrics several physical models for relativity are presented. A characteristic of these models is that the vertical part is provided by the flag-Finsler Berwald-Moor (fFBM) metric, while the horizontal part is specialized to the conformal and to Synge-relativistic optics metrics. As well, the particular case of $h-$Riemannian $v-$fFBM metric of Riemann-Minkowski type is examined, considering as nonlinear connection both the trivial canonical connection, and the one induced by the Lagrangian of electrodynamics. For all these models, basic properties are described and the extended Einstein and Maxwell equations are determined.

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Invariant frames for a generalized Lagrange space with Berwald-Moor metric
2005jbm | Marius Paun  // Faculty of Mathematics and Informatics, Transilvania, University of Brasov, Romania

The notion of generalized Lagrange space should be geometrically considered as a generalized metric space $M^n=(M,g_{ij}(x,y))$. A theory of invariant Finsler spaces was given by M. Matsumoto and R. Miron with important applications. The notion of non-holonomic space was introduced by Gh. Vranceanu in [VR]. The Vranceanu type invariant frames and the invariant geometry of second order Lagrange spaces was studied by the author in [P3]. The purpose of the present paper is to study the invariant geometry for a generalized Lagrange space endowed with a Berwald-Moor metric. We introduce distinct non-holonomic frames on the two components of the Whitney's decomposition. This will determine a non-holonomic coordinates system on the total space $TM$ and thus its geometry can be studied with methods analogous to the mobile frame. We obtain, in this manner, invariant connections, curvatures and torsions, and the fundamental equations in this theory. Also we can construct the invariant frames so that, with respect to them, the metric of the total space can be written in canonical form and in this case we deduce invariant Einstein equations. We mention that the frames introduced here depend on the metric and all the computations are for this metric.

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2005jbl

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Expansion of Complex Number
2005jbk | Y. A. Furman, A. V. Krevetsky  // Mari state technical university, Yoshkar Ola, Russia

The expanded complex numbers are introduced by means of imaginary unit $i$ replacement by one-dimensional on multivariate $3D$ or $7D$ imaginary unit $r$. It is shown, full quaternions and octaves appear as a result of a turn around the material axis $0Re$ plane where $a+ib$ number is set in $4D$ and $8D$ spaces. Rotor-complanar classes of quaternions and the octaves appearing as a result of similar transformations are considered. They represent commutative-associative algebras.

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Thomas precession by pseudoquaternions
2005jbj | D. E. Burlankov, G. B. Malykin  // Nizhny Novgorod State University; Institute of Applied Physics RAS, Nizhny Novgorod, Russia

When a body moves curvilinearly in a plane with a velocity that is comparable to the velocity of light, only three coordinates of the body undergo Lorentz transformation, and the transformation matrix appears to be three-parametric. This enables description of these transformations by pseudoquaternions, Hamilton quaternions slightly modified for pseudo-Euclidean character of the metrics. Their algebraic properties and relation to the Lorentz transformations in a 2+1-dimensional Minkovsky space were determined. We integrated the pseudoquaternion differential equation of continuous transformations at a body's motion along a circular orbit and, as a result, obtained an expression for the value of the Thomas precession.

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Nonclosure of elemenary space-time transformations
2005jbi | Chub V. F.  // Korolev Rocket and Space Corporation "Energia"

The article gives a brief group-theoretical comparative study of three space-time theories: (space-time theory in frames of) classical Newton mechanics, special theory of relativity and author-developed theory based on complex-dual quaternions.

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The notions of distance and velocity modulus in the linear Finsler spaces
2005jaz | Garas'ko G. I., Pavlov D. G.

The formulas for the 3-dimensional distance and the velocity modulus in the 4-dimensional linear space with the Berwald-Moor metrics are obtained. The used algorithm is applicable both for the Minkowski space and for the arbitrary poly-linear Finsler space in which the time-like component could be chosen. The constructed here modulus of the 3-dimensional velocity in the space with the Berwald-Moor metrics coincides with the corresponding expression in the Galilean space at small (non-relativistic) velocities, while at maximal velocities, i.e. for the world lines lying on the surface of the cone of future, this modulus is equal to unity. To construct the 3-dimensional distance, the notion of the surface of the relative simultaneity is used which is analogous to the corresponding speculations in special relativity. The formulas for the velocity transformation when one pass from one inertial frame to another are obtained. In case when both velocities are directed along one of the three selected straight lines, the obtained relations coincide with the analogous relations of special relativity, but they differ in other cases. Besides, the expressions for the transformations that play the same role as Lorentz transformations in the Minkowski space are obtained. It was found that if the three space coordinate axis are straight lines along which the velocities are added as in special relativity, then taking the velocity of the new inertial frame collinear to the one of these coordinate axis, one can see that the transformation of this coordinate and time coordinate coincide with Lorentz transformations, while the transformations of the two transversal coordinates differ from the corresponding Lorentz transformations.

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2005jay

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2005jax

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2005jaw

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2005jav

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2005jau

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2005jat

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2005jas

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Conference “Number, time, relativity”
2004jbz | Gladyshev V.O., Pavlov D.G.

On the 13th august of 2004 the International scientific conference “Number, time, relativity” took place in N. E. Bauman Moscow State Technical University. The purposes of the conference were: to attract the attention of foreign and Russian physicists to Finslerian generalizations of the relativistic theory, to gather the leading specialists in the field of hyper-complex numbers, Finslerian geometry (that generalize the Riemannian manifolds), and the specialists in the field of the relativistic theory. The conference was devoted to 175th anniversary of N. E. Bauman Moscow State Technical University. The conference was performed by: the Bauman University’s cathedra of physics, the theoretical physics cathedra of Moscow State University of M.V. Lomonosov and the United Physics Society of Russian Federation. The main sponsor of the conference was the Fund of 175th anniversary of N. E. Bauman Moscow State Technical University.

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The normal conjugation on the poly-number set.
2004jby | Garasko G.I., Pavlov D.G.

The poly-number set is an example of linear space with several poly-linear forms. The concept of normal conjunction is introduced on the set of non-degenerated n-numbers. The normal conjunction is a (n-1)-nary operation. It is commutative for each argument, but generally not associative. For complex and hyperbolic numbers the generalized conjunction is equivalent to usual one. The normal conjunction may be applied for scrutiny of algebraic and geometric structures of n-number coordinate spaces. It is also useful for introducing such concepts like scalar product and angular characteristics of two and more numbers (vectors)

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Generalized-analytical functions and the congruence of geodetic.
2004jbx | Garas'ko G. I.

Some properties of generalized-analytical functions of poly-number variable are being studied in this job. We can confront many spaces of affine connectedness with the $\{f^i;\Gamma^{i}_{kj}\}$ class of such functions. In each space the congruence of geodetic associated with the given class of general-analytic functions is defined. If the vector field $f^i$ is tangent to one of the geodetic of congruence in each point of space there are certain restrictions for the generalized-analytical function itself.

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Concerning the norm of biquaternions and some other algebras with central conjunction
2004jbw | Eliovich A. A.

The concept of central conjunction is introduced in this article. We apply it to algebras of biquaternions and bioctaves. With the given analysis method of the conjunctions permitted by algebra we derive some new results. Thus the alternative algebras with central conjunction are proven to have the multiplicative norm of second degree (that is in general not real). The consequence of this fact is that these algebras (biquaternions and bioctaves particulary) have the multiplicative real norm of degree higher than 2. This norm has several different but equivalent views. The quadrascalar and quadravector multiplications are introduced. Some results for algebras of biquternions, diquaternions and bioctaves are given in terms of isotropic basis. The developed methods may be useful in the geometrical and physical usage of concerned algebras.

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On some distributive algebras
2004jbv | Solovey L.G.

The examined type of sets, that are not rings, but in some sense are close to them. These sets are called 'Hyper-rings'. They consist of several additive groups, that intersect each over at the zero only. Yet, they are multiplicative groupoids (or groups, excluding the zero). The distributive laws are fulfilled.
Rings (and in particular the bodies and the fields) are the special case of the concerned sets. The given examples witness that such sets are highly disseminated. So, the idea that the real physical values may be "layed" in the ring is wrong, because they are subset of hyper-ring.
The real hyper-rings with unity can not be reduced to rings. Their additive groups are vector spaces, and they may be treated as a generalized Hyper-complex systems, in which we include the real binary (provided with summation and multiplication) distributive algebraic structures with neutral element, where the number of included vector spaces is more than one and finite.
The example of hyper-rings, suggesting that scrutiny of them is worth-able, are the second order matrices, that are mostly like unitary matrixes, but normalized not by unity. They are normalized by an arbitrary non-negative number. The complex numbers and the quaternions may be represented with such matrixes while they are the ones subspace.

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Deformation principle as foundation of physical geometry and its application to space-time geometry
2004jbu | Rylov Yuri A.

Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function $% d$, or by the world function $\sigma =d^{2}/2$. One suggests a new general method of the physical geometry construction. The proper Euclidean geometry is described in terms of its world function $\sigma _{\mathrm{E}}$. Any physical geometry $\mathcal{G}$ is obtained from the Euclidean geometry as a result of replacement of the Euclidean world function $\sigma _{\mathrm{E}}$ by the world function $\sigma $ of $\mathcal{G}$. This method is very simple and effective. It introduces a new geometric property: nondegeneracy of geometry. Using this method, one can construct deterministic space-time geometries with primordially stochastic motion of free particles and geometrized particle mass. Such a space-time geometry defined properly (with quantum constant as an attribute of geometry) allows one to explain quantum effects as a result of the statistical description of the stochastic particle motion (without a use of quantum principles).

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The Nilpotent Vacuum
2004jbt | Rowlands Peter

A fermionic state vector which is a nilpotent or square root of zero appears to be the most convenient packaging of the fundamental physical parameters space, time, mass and charge into a single unit. It also has the advantage of being a supersymmetric quantum field operator, which uniquely and simultaneously specifies both amplitude and phase for any fermionic state, and incorporates all the specific aspects required in BRST field quantization into a single package. The mathematical structure of the state vector immediately generates vacuum terms relevant to all four fundamental interactions, and explains the symmetry-breaking between them. By incorporating the vacuum aspects into our understanding of the fermion, we generate a ‘string theory without strings’. The nilpotent vacuum operators suggest links with many well-known vacuum phenomena, including the Casimir effect and zero-point energy.

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Division Algebra, Generalized Supersymmetries and Octonionic M-Theory
2004jbs | Toppan Francesco

This is the report of the talk given at the conference ``Number, Time and Relativity", held at the Bauman University, Moscow, August 2004, concerning the recent research activity of the author and his collaborators about the inter-relation of the concepts of division algebras, representations of Clifford algebras, generalized supersymmetries with the introduction of an alternative description of the M-algebra in terms of the non-associative structure of the octonions.

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From Editorialboard
2004jaz | Editorialboard of Journal

Number is one of the most fundamental concepts not only in mathematics, but in general natural science as well. It may be primary even in comparison with such global categories as time, space, substance, matter, and field. That is why editing the first issue of the journal "Hypercomlex numbers in geometry and physics" the editorial board sincerely hopes that articles not only on numbers in general, but primarily the works that reveal their organic connection with the real world will find here their true scope.

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Generalization jf Scalar Product Axioms
2004jay | Pavlov D. G.

The concept of scalar product is vital in studying basic properties of either Euclidean or pseudo-Euclidean spaces. A generalizing of a special sub-class of Finslerian spaces, that we will call the polylinear, is presented in the work. The idea of scalar polyproduct and of related fundamental metric polyform has been introduced axiomatically. The definition of different metric parameters such as the vector length and the angle between vectors are founded on the idea. The concept of orthogonallity is also generalized. Some peculiarities of the geometry of the four-dimensional linear Finslerian space related to the algebra of commutative-associative hypercomplex numbers, that are called Quadranumerical, are proved in the concrete polyform.

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Chronometry of the three-dimensional time
2004jax | Pavlov D. G.

The concept of the multi-dimensional time has tried not once to take its place in natural science, but every time under the pressure of some paradox was rejected. Meanwhile a philosophical question: why the space admits quite a number of dimensions and the time dos not, still preserves. In this work a new attempt has been made to resolve the matter, by switching from the traditional quadratic metrics to the Finslerian one, which may admit an arbitrary degree of the vector component that is included into the metric function. Though the offered method enables us to build continuums of time of any natural dimensionality, in order to demonstrate the specificity of the raised topic this study will focus on a simple (after rather trivial two-dimensional case) example of three temporial dimensions.

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Four-dimensional time
2004jaw | Pavlov D. G.

The generalized metric space, that can be called the flat four-dimensional time, is based on the Berwald-Moore's Finslerianview of metric function. This variety let us introduce physical notions: the event, the world lines, the reference frames, the multitude of relatively simultaneous events, the proper time, the three-dimensional distance, the speed, etc. It is demonstrated how from the point of the physical observer, associated with the world line, in absolutely symmetrical four-dimensional time the contraposition of the coordinate takes place, that defines its proper time, with the ones that appear as the result of the measurements made with the help of sample signals. When the signals correspond with lines, which are practically parallel to the world line of the observer, he starts to see the three-dimensional space which at the limit is the Euclidean space.

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Finsleroid -- space supplemented by angle and scalar product
2004jav | Asanov G. S.

The science of the past century has achieved great success on the basis of the geometrical quadratic concepts that were followed as logical and mathematical primaries. More profound ideas will imply using a more capacious class of geometries, for example the Finsler one which inscribes structures because the Finslerian indicatrices are no more isotropic in all directions. In the present work an attempt is made to resolve the respective difficulties of Finsler generalization by choosing the particular Finsleroid--type metric that implies one preferred direction, admitting the total axial symmetry around it. In this case, interesting constructive methods of introducing the concept of the angle and scalar product outside the frame of the Euclidean Geometry can conveniently be opened up.

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Properties of spaces associated with commutative-associative H₃ and H₄ algebras
2004jau | Lebedev S. V.

In the first part of this work a real axis of the space associated with the H₃ algebra and the lines parallel to this axis are interpreted as the world lines of resting particles; surface of simultaneity is used for introduction of a distance between the real axis and a line parallel thereto. The coordinate system similar to a polar one can be introduced on this surface such that this allows us to reveal its simplest invariant transformations. In the second part of this paper the Lorentz transformations in form of special kind of rotations in the space associated with H₄ algebra are presented.

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Generalized-analytical functions of poly-number variable
2004jat | Garas`ko G. I.

We introduce the notion of the generalized-analytical function of the poly--number variable, which is a non--trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in theoretical physical constructions. As an example we consider in detail the associative-commutative hypercomplex numbers H₄ and an interesting class of corresponding functions.

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Algebrodinamics: Primodial Light, Particles-Caustics and Flow of Time
2004jas | Kassandrov V. V.

In the field theories with twistor structure particles can be identified with (spacially bounded) caustics of null geodesic congruences defined by the twistor field. As a realization, we consider the ``algebrodynamical'' approach based on the field equations which originate from noncommutative analysis (over the algebra of biquaternions) and lead to the complex eikonal field and to the set of gauge fields associated with solutions of the eikonal equation. Particle-like formations represented by singularities of these fields possess ``elementary'' electric charge and other realistic ``quantum numbers'' and manifest self-consistent time evolution including transmutations. Related concepts of generating ``World Function'' and of multivalued physical fields are discussed. The picture of Lorentz invariant light-formed aether and of matter born from light arises then quite naturally. The notion of the Time Flow identified with the flow of primodial light (``pre-Light'') is introduced in the context.

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