The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

Solution of equations in Boolean algebra

The explicit form of solutions of Boolean equations with one unknown is obtained. The effectiveness of the method is demonstrated for a number of equations whose solution previously has been found only in “tabular” form. The proposed approach leads to a method for solving systems of equations in Boolean set algebra. We use it to analyze the famous paradoxes of set theory, such as the barber paradox and the liar paradox, as well as Russell's and Cantor's paradoxes. Translated from Nelineinaya Dinamika i Upravlenie, pp. 119–132, 1999.     

Certain questions of Lobachevskii geometry,connected with physics

An historical survey of the development of Lobachevskii geometry as a typical representative of a geometry of negative curvature, Friedman cosmology, Lobachevskii geometry, and an interpretation of the velocity space in the special theory of relativity as a Lobachevskii space are presented.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 13, pp. 157–188, 1982.     

On Parabolic Trigonometry in a Degenerate Minkowski Plane

A Minkowski plane is considered having a parabola as indicatrix curve and taking for its center the vertex of the parabola causing by this a degeneration of the Minkowski geometry. After a short discussion of the concept of the length and angle, the trigonometrical functions of this geometry has been calculated and their basic properties explained. A two-parameter pseudogroup of generalized rotations has been found as an analogon of the Lorentz transformation group in special relativity. The theory has applications to thermodynamics.     

双标量-张量几何与标量-张量引力论变分原理

<正> §1.引言 自从1915年A.Einstein奠定了广义相对论的基础以来,曾出现过各种各样的引力理论;但似乎只有标量-张量引力理论可同广义相对论媲美.看来标量-张量理论同Einstein的广义相对论一样,是一种具有生命力的引力理论. 如所周知,Einstein的广义相对论实质上是引力现象的几何化理论,即是一种引力的度规张量理论.Einstein与Weyl的物理学之几何化思想对物理学的发展曾起过、并且将     

Thermal molecule and atom test of the modified special relativity theory

The correction to Maxwell–Boltzmann's velocity distribution law is obtained in the framework of the modified special relativity theory. The detection of velocity and velocity rate distributions for thermal molecules or atoms can serve as a test of the modified special relativity theory.     

Topology and a lorentz-invariant pseudo-riemannian metric of the manifold of directions in the physical space

In the mathematical model of the special relativity theory, a two-dimensional Minkowski subspace is treated as a one-dimensional direction in the physical space. The manifold of such planes is naturally endowed with the structure of a pseudo-Riemannian manifold on which the group of isochronous Lorentz transformations acts transitively by isometries. In this paper, the topology and the metric geometry of this manifold are studied. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminar POMI, Vol. 246, 1997, pp. 141–151. Translated by S. Yu. Pilyugin.     

The Einstein Relativistic Velocity Model of Hyperbolic Geometry and Its Plane Separation Axiom

The relativistically admissible velocities of Einstein’s special theory of relativity are regulated by the Beltrami–Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. It is shown in this expository article that the Einstein velocity addition law of relativistically admissible velocities enables Cartesian coordinates to be introduced into hyperbolic geometry, resulting in the Cartesian–Beltrami-Klein ball model of hyperbolic geometry. Suggestively, the latter is increasingly becoming known as the Einstein Relativistic Velocity Model of hyperbolic geometry. Möbius addition is a transformation of the ball linked to Clifford algebra. Einstein addition and Möbius addition in the ball of the Euclidean n-space are isomorphic to each other, and they share remarkable analogies with vector addition. Thus, in particular, Einstein (Möbius) addition admits scalar multiplication, giving rise to gyrovector spaces, just as vector addition admits scalar multiplication, giving rise to vector spaces. Moreover, the resulting Einstein (Möbius) gyrovector spaces form the algebraic setting for the Beltrami-Klein (Poincaré) ball model of n-dimensional hyperbolic geometry, just as vector spaces form the algebraic setting for the standard Cartesian model of n-dimensional Euclidean geometry. As an illustrative novel example special attention is paid to the study of the plane separation axiom (PSA) in Euclidean and hyperbolic geometry.     

Modification of special relativity and local structures of gravity-free space and time

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the one-way speed of light in all inertial frames of reference, the special theory of relativity uses the assumption about the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the so-called primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the flat structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine this assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite speed c^′ in the primed inertial coordinate system and finite speed c in the usual inertial coordinate system. It also involves the c^′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between any two usual inertial coordinate systems. The physical principle is: the c^′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed to the usual inertial coordinate systems. Evidently, the modified special relativity theory and the quantum mechanics theory together found a convergent and invariant quantum field theory.     

Modification of special relativity and formulation of convergent and invariant quantum field theory

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the speed of light in all inertial frames of reference, the special theory of relativity uses another assumption. This other assumption concerns the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the Euclidean structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine the alternative assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite c′ in the primed inertial coordinate system and finite c in the usual inertial coordinate system. It also involves the c′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between two corresponding usual inertial coordinate systems. Since all our experimental data are collected and expressed in the usual inertial coordinate system, the physical principle is: the c′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed inertial coordinate system to the usual inertial coordinate system. This principle is applied to a reformulation of mechanics, field theory and quantum field theory. Relativistic mechanics in the usual inertial coordinate system is unchanged, while field theory is developed and divergence-free. Any c′-type Galilean-invariant field system can be quantized by using the canonical quantization method in the primed inertial coordinate system. We establish a transformation law for quantized field systems as they are transformed from the primed to the usual inertial coordinate system. It is shown that the modified special relativity theory, together with quantum mechanics, leads to a convergent and invariant quantum field theory, in full agreement with experimental facts. The formulation of this quantum field theory does not demand departures from the concepts such as local Lorentz invariance in the usual inertial coordinate system, locality of interactions, and local or global gauge symmetries.     

狭义相对论数学基础的探讨

狭义相对论的变革点就是相对时空观,而相对论时空与非欧几何学有着密切的联系.在介绍了传统的Minkowski空间后,引入双曲虚单位,其所构造的双曲复数对应双曲Minkowski复空间.利用双曲Minkowski空间复数运算规则,可以使高速运动客体的物理规律与复数的性质结合起来,为解决狭义相对论的普遍形式提供新的数学工具.     

Group-Like Structure Underlying the Unit Ball in Real Inner Product Spaces

Abstraction of the relativistic velocity addition law and of the Thomas rotation of the special theory of relativity yields a means of endowing the unit ball in any real inner product space with a group- like structure, in which the standard associative- commutative laws are relaxed by means of the Thomas rotation. The resulting group- like object is called a complete weakly associative- commutative groupoid. Any complete WACG can be extended to a group analogous to the Lorentz group of the special theory of relativity.     

The vacuum structure,special relativity theory,and quantum mechanics: A return to the field theory approach without geometry

We formulate the main fundamental principles characterizing the vacuum field structure and also analyze the model of the related vacuum medium and charged point particle dynamics using the developed field theory methods. We consider a new approach to Maxwell’s theory of electrodynamics, newly deriving the basic equations of that theory from the suggested vacuum field structure principles; we obtain the classical special relativity theory relation between the energy and the corresponding point particle mass. We reconsider and analyze the expression for the Lorentz force in arbitrary noninertial reference frames. We also present some new interpretations of the relations between special relativity theory and quantum mechanics. We obtain the famous quantum mechanical Schrödinger-type equations for a relativistic point particle in external potential and magnetic fields in the semiclassical approximation as the Planck constant ? → 0 and the speed of light c→ ∞.     

On the notion of space in geometry

The development of the notion of space in geometry is traced from the early axiomatization in Euclid’s Elements over the discovery of non-Euclidean geometries to geometry of manifolds in relativity theory and in gauge and string theories in contemporary physics. The notion of space is considered in a historic-philosophical perspective including a short discussion of the contributions of artists to visualization of spatial objects.     

Mathematical paradoxes as pathways into beliefs and polymathy: an experimental inquiry

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.     

On Algebraic Structures Related to Beltrami–Klein Model of Hyperbolic Geometry

A new approach to the algebraic structures related to hyperbolic geometry comes from Einstein’s special theory of relativity in 1988 (cf. Ungar, in Found Phys Lett 1:57–89, 1988). Ungar employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry (cf. Ungar, in Math Appl 49:187–221, 2005). Another approach is from Karzel for algebraization of absolute planes in the sense of Karzel et al. (Einführung in die Geometrie, 1973). In this paper we are going to develop a formulary for the Beltrami–Klein model of hyperbolic plane inside the unit circle ${\mathbb D}$ of the complex numbers ${\mathbb C}$ with geometric approach of Karzel.     

Fantappiè's group as an extension of special relativity on ε Cantorian space–time

In this paper, we will analyze the Fantappiè group and its properties in connection with Cantorian space–time. Our attention will be focused on the possibility of extending special relativity. The cosmological consequences of such extension appear relevant, since thanks to the Fantappiè group, the model of the Big Bang and that of stationary state become compatible. In particular, if we abandon the idea of the existence of only one time gauge, since we do not see the whole Universe but only a projection, the two models become compatible. In the end we will see the effects of the projective fractal geometry also on the galactic and extra-galactic dynamics.