Space-time structure. Algebra and geometry

543 Pages·2007·3.132 MB·English

by  Pavlov D.G., Atanasiu Gh., Balan V. (eds.)

Space-Time Structure.
Algebra and Geometry
D. G. Pavlov, Gh. Atanasiu, V. Balan
Editors
Russian Hypercomplex Society
Lilia Print
Moscow
2007
D.G. Pavlov, Gh. Atanasiu, V. Balan (eds.)
Space-Time Structure. Algebra and Geometry
Moscow: Lilia-Print, 2007. – 528 pp.
Thepresentvolumecontainsselectedarticlespublishedinthesci-
entificjournal“HypercomplexNumbersinGeometryandPhysics”
(numbers 1–6). The topics covered by the included contributions
range from hypercomplex algebras to associated Finsler geome-
tries, focusing on the commutative-associative algebras and on
Berwald-Moor metric structures, applications of polynumber al-
gebras in Physics, and Finslerian extensions of Relativity Theory.
Contacts:
http://www.polynumbers.ru, e-mail: [email protected]
ISBN 5-94205-020-1
(cid:176)c Russian Hypercomplex Society, 2007
(cid:176)c Lilia-Print, 2007
Contents
Preface ......................................................................5
D.G. Pavlov Generalization of scalar product axioms.......................15
D.G. Pavlov Chronometry of the three-dimensional time ................... 32
D.G. Pavlov Four-dimensional time ........................................ 45
S.V. Lebedev Properties of spaces connected with commutative-associative H
3
and H algebras...............................................................56
4
G.I. Garas’ko Generalized-analytical functions of polynumber variable......64
R.V. Mikhailov On some questions of four dimensional topology: a survey of
modern research .............................................................. 80
G.I. Garas’ko, D.G. Pavlov Normal conjugation on the polynumber
manifold .....................................................................85
G.I. Garas’koGeneralizedanalyticalfunctionsandthecongruenceofgeodesics94
G.I. Garas’ko, D.G. Pavlov The notions of distance and velocity modulus in
the linear Finsler spaces .....................................................104
G.I. Garas’ko Generalization of conformal transformations ................118
A.V. Smirnov Some properties of bicomplex numbers .....................128
D.G. Pavlov Philosophical and mathematical reasons for Finsler extensions of
Relativity Theory ...........................................................140
G.I. Garas’ko The relation of elementary generalized conformal transformations
with generalized analytic functions in the polynumber space .................147
G.Yu. Bogoslovsky 4-momentum of a particle and the mass shell equation in
the entirely anisotropic space-time ...........................................156
S.V. Lebedev The generalized Finslerian metric tensors ...................174
S.V. Siparov Hamilton canonical equations and Berwald-Moor metric (on the
formalism of physical theories) ..............................................182
V.M. Chernov On defining equations for the elements of associative and
commutative algebras and on associated metric forms ....................... 189
V.M. Chernov Generalized n-ary composition laws in the algebra H and their
4
relation to associated metric forms ..........................................210
Gh. Atanasiu The prolongations of a Finsler metric to the tangent bundle
Tk(M) (k > 1) of the higher order accelerations .............................224
Gh. Atanasiu, N. Brinzei The Berwald-Moor metric in the tangent bundle of
the second order ............................................................ 234
3
4 CONTENTS
Gh. Atanasiu, V. Balan The 2-Cotangent Bundle with Berwald-Moor
Metric ......................................................................241
V. Balan, N. Brinzei Berwald-Moor-type (h,v)-metric physical models ...260
Gh. Atanasiu, E. Stoica The horizontal and vertical semisymmetric metrical
d-connections in the Relativity Theory ...................................... 270
Gh. Atanasiu, V. Balan, M. Neagu The Pavlov’s 4-polyform of momenta
√
K(p) = 4 p1p2p3p4 and its applications in Hamilton geometry ................279
Gh. Munteanu The Lagrangian-Hamiltonian formalism in gauge complex field
theories .....................................................................286
V. Balan, N. Brinzei, S. Lebedev Geodesics, connections and Jacobi fields for
Berwald-Moor quartic metrics ...............................................300
L. Tam´assy Finsler spaces with polynomial metric ......................... 311
Gh. Atanasiu Pairs of metrical Finsler structures and Finsler connections
compatible to them ......................................................... 320
V. Balan CMC and minimal surfaces in Berwald-Moor spaces ..............327
R.G. Zaripov The definition of a simultaneity in Finsler space-time ........335
G.I. Garas’ko On the World function and the relation between geometries .360
G.I. Garas’ko, D.G. Pavlov Construction of the pseudo-Riemannian geometry
on the base of the Berwald-Moor geometry ..................................376
G.I. Garas’ko On field theory and Finsler spaces ..........................386
A.P. Yefremov Quaternions: algebra, geometry and physical theories......403
V.V. Kassandrov Algebrodynamics: primordial light, particles-caustics and
flow of time..................................................................422
V.V. Kassandrov Quaternionic analysis and the algebrodynamics ........441
A.V. Solov’yov Finslerian 4-spinors as a generalization of twistors ........474
P.D. Suharevsky To the question on quartic geometry ................... 481
S.V. Siparov Theory of the zero order effect suitable to investigate the
space-time geometrical properties ........................................... 495
V.A. Panchelyuga, S.E. Shnoll Experimental investigation of spinning
massive body influence on fine structure of distribution functions of α-decay rate
fluctuations .................................................................515
V.A. Panchelyuga, V.A. Kolombet, M.S. Panchelyuga, S.E.
Shnoll Local-time effect on small space-time scale ...........................531
Addenda – The scientific journal ”Hypercomplex Numbers in Geometry and
Physics” ....................................................................538
Space-Time Structure. Algebra and Geometry 5
Preface
A historical perspective
FinslergeometryisanaturalextensionoftheRiemanniangeometry. Asspecific
feature, its metric encompasses information about direction – hence it provides re-
liable models for both isotropic and anisotropic phenomena. Moreover, the Finsler
energy – the square of the fundamental function, does not reduce in general to
a quadratic form in the directional coordinates – as it does in the Riemannian
subcase.
The historical traces of Finsler geometry go back to 1854, when the first Finsler
metric was pointed out by B. Riemann in his famous lecture On Hypotheses Lying
at the Basis of Geometry1. Much later, in 1918 Paul Finsler systematically studied
in his Ph.D. Thesis – having as advisor C. Caratheodory, the spaces which later
received his name.
The Finslerian framework was intensively developed by J.L. Synge, V. Wagner,
L. Berwald, E. Cartan, H. Busemann, H. Rund, M. Matsumoto, S.S. Chern and
others. Finsler geometry has become a separate branch of differential geometry,
being included in the AMS mathematics subject classification (53B40 & 53C60).
The Finsler-type structures have been further extended to fiber spaces (e.g., higher
order tangent and cotangent, osculating and jet spaces).
The applications of Finsler geometry in various branches of theoretical physics
flourished as well, especially in the last two decades. Apart from such conventional
fieldsasthetheoryofanisotropicmediaandLagrangianmechanics,classicalFinsler
geometry and its generalizations have found extensive applications in solving op-
timization problems, in describing systems with chaos, in statistical physics and
thermodynamics, in ecology and in the theory of evolution of biological systems,
in describing the internal symmetry of hadrons, in the theory of space-time and
gravitation as well as in unified gauge field theories.
More recently, Finsler geometry proved to successfully apply to the theory of
deformations of crystalline media, seismic phenomena, Zermelo navigation prob-
lems and interfaces in thermodynamic systems.
It should be noted that between two historical alternative approaches to Finsler
geometryduetoCartanandBusemann, thefirstoneprovidesefficienttoolsindeal-
ing with theories of Kaluza-Klein type, offers new structural possibilities and iden-
tifies new (comparing with Riemann geometry) elements of structure with physical
observables. The physical fields in a Cartan Finsler space, apart from space-time
coordinates, turn out to be functions of supplementary so-called internal variables.
On the other hand, there is a close relation between notion of Number and
fundamental physical categories as Space, Time, matter and field. Usually, this
1B.Riemann, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, habilitation ad-
dress, 1854, translated in: M. Spivak, Differential Geometry, vol. II, Publish or Perish, 1970.
6 Space-Time Structure. Algebra and Geometry
relation is associated with particular numbers as the Real and Complex numbers,
and sometimes quaternions. Apart from denying the fundamental role of these
numbers, the authors of this book point out that there exist others generaliza-
tions of numbers, which have substantial physical and geometrical interpretation.
Emerging from the fact that the progress in physics often was stimulated by novel
geometricalpointsofview, ahighexpectationemergesthattheproperscientificde-
scriptionofthegeometricalbackgroundmightleadtonewqualitativeconsequences
in Physics.
It is a known fact that while creating the Relativity theory, Einstein was forced
to get out of the classic Euclidean geometry, replacing it with the Riemann one.
HenceitisnaturaltoassumethatthefuturedevelopmentofPhysicswillalsoneeda
certain new geometry, which might be Finsler geometry – which naturally extends
Minkowski geometry as well. It is fundamentally important, that the points of
Finsler spaces in some cases may be expressed in terms of hypercomplex numbers,
which are algebras with special properties.
The wide background Finsler spaces provides simple and wonderful particular
caseslinkedwithhypercomplexnumbers–whichpossesstheusualassociativityand
commutativity properties. Unfortunately, today there exist few attempts of classi-
fyingFinslerSpacesemergingfromthisperspective. SuchFinslerrelatedalgebras–
having their roots in the applied aspects of geometry with modern physics are, e.g.,
the algebras of the quaternions over the field of complex numbers (biquaternions)
and over the ring of duals (diquaternions), and also algebras of complex numbers
over complex numbers (bicomplex numbers) and dual numbers abovedual numbers
(quadranumbers). All this spaces possess multiplicative norms of fourth order, and
prove to be tightly connected with the fundamental for physicists Lorenz group.
In Finsler geometry the class of new and classic invariants is much wider than
in the Riemannian case, which makes it very tempting to use this type of geometry
for modeling different physical phenomena, instead of using habitual Riemannian
geometry. The study of Finslerian invariants reveals the existence of interesting
special nonlinear transformations, whose Euclidean analogues are the conformal
transformations. Moreover, inFinslerspaceswithfundamentalformofhigherorder
than quadratic, due to the existence of new metric properties (besides conservation
of lengths and angles), such new invariants to (usually nonlinear) equiform-type
transformations are numerous and play a fundamental role.
The development of Finsler Geometry in parallel with Algebra of numbers had
led to a notable benefit to both these fields of mathematics. Thus, for the first
time has appeared the possibility to solve one of the key problems of the Geometry
– to find natural and simple means to generalize the concept of angle, based not
on the classic scalar product, but on a symmetric polyscalar form. This approach
shows that the concept of Finsler metric tensor – as introduced by Cartan, lacks
from being completely effective in applied models and requires its replacement by
another tensor with rank grater than two.
Preface 7
From a physical perspective, as a characteristic feature, Finsler geometry is
able to provide the formal grounds for posing the problem of local anisotropy of
therealspace-time, sustainedbythefactthatwithintheframeworkofthemodelof
locally isotropic (Riemann) space-time it is, in principle, impossible to embody the
Mach principle for a test body. Physical evidence leads to the conclusion that the
inertial body mass, should be a tensor rather than a scalar. Thus the discovery of
an anisotropy of inertia would be a direct evidence for a local anisotropy of space.
At the same time it has long been pointed out that the conventional experimen-
tal estimate of 3D anisotropy at the level 10−22 is not correct (S.T. Epstein and
G.Yu. Bogoslovsky), and a reliable upper bound of anisotropy should be recon-
sidered. The experimental findings of breaking of discrete space-time symmetries,
anisotropy of background radiation and the absence of the GZK effect have revived
interest in the problem of anisotropy of space-time.
On the other hand, very recent measurements of the Compton edge (made
by French, Russian, Italian and Armenian scientists) of the scattered electrons in
GRAAL facility in European Synchrotron Radiation Facility (ESRF) in Grenoble
with respect to the Cosmic Microwave Background dipole has revealed up to 10
σ variations larger than the statistical errors. It was shown2 that the variations
were not due to the frequency variations of the accelerator, and since the nature
of Compton edge variations remains unclear, follows the imperative of dedicated
studies of light speed anisotropy. The team of scientists has conjectured that the
variations could be an effect of existence in the Universe of vector fields which
provide a certain vector distribution.
As well, recent experiments developed in Canada have shown3 that the Hubble
constant is affected by anisotropy, reinforcing thus the idea of having a direction
dependent Finsler encompassing mathematical model for General Relativity.
Recent investigations on the Finslerian generalization of relativity aim to find
a relativistically invariant Finsler metric with partially or even entirely broken
3D isotropy (e.g., G.Yu. Bogoslovsky and H.F. Goenner). All three relativistically
invariantmetrics, i.e. theisotropicRiemannandtwoFinslermetrics(withapartial
andfullanisotropy), satisfythecorrespondenceprinciple, whichleadstoa”hybrid”
geometric model in terms of which spacetime may be not only in a state which is
described by Riemann geometry but also in states which are described by Finsler
geometry.
The mathematical Finslerian framework has a well-established wide basic back-
ground of monographs and a significant variety of both theoretical and applicative
research papers. Significant trends in Finslerian geometry have been developed in
2V.G. Gurzadyan, J.-P. Bocquet, A. Kashin, A. Margarian..., On the light speed anisotropy
vs cosmic microwave background dipole: European synchrotron radiation facility measurements;
WSPC – Proceedings Compton MG11-4, arXiv:astro-ph/0701127v1, 5 Jan 2007.
3M.L. McClure, C.C. Dyer, Anisotropy in the Hubble constant as observed in the HST extra-
galactic distance. Scale key project results, arXiv:astro-ph/0703556v1, 21 Mar 2007.
8 Space-Time Structure. Algebra and Geometry
recent years in many countries4.
AnewsignificanttrendinmodernFinslerianframeworkemergesfromtheworks
of P.K. Rashevski dated 1935–1936, who stated the problem of developing such a
geometry, whoseemergingobjectswouldincludenotonlytheanaloguesofthebasic
notions of point and straight line, but planes and n-planes as well (his hypothetical
geometry was called at the time poly-metrical). In the last decade, the goal of
developing this theory at a superior level within the framework of the relativistic
Berwald-Moor Finslerian model has been proposed by the Russian specialists, who
have joined their efforts around Dr. D.G. Pavlov to organize two dedicated to the
subject Conferences in Cairo (2005 and 2006) and one in Moscow (2007), and have
issued the scientific journal ”Hypercomplex Numbers in Geometry and Physics”
(Moscow, Russia) – whose submission topics confine to the goal of founding a new
Finslerian-based model of Relativity.
The layout of the book
The present volume outlines achievements adjacent to this concern, and em-
braces papers written by Russian and Romanian physicists and mathematicians.
The first three papers, ”Generalization of scalar product axioms”, ”Chronom-
etry of three-dimensional Time” and ”Four-dimensional time” are authored by
D.G. Pavlov. The first article describes the generalization of the classical (pseudo-)
scalar products, by introducing the ”poly-linear” product associated to a given
Finsler structure; this leads to the new notions of related fundamental metric poly-
form and various generalized geometric objects as vector length, angle between
vectors and orthogonality. The second paper proposes a Finslerian model, switch-
ing thus from the traditional quadratic (Riemannian) metrics to the Finslerian
ones; the discussed topics are: light pyramids, specific groups of transformations,
planes of relative simultaneity, physical distance and speed. It is emphasized that
neither Euclidean, nor pseudo-Euclidean spaces with three or more dimensions do
possess analogous qualities to those exhibited in the studied Finslerian models.
The third paper specializes the study to the main properties of the Finsler met-
ric space with four-dimensional time, based on the Berwald-Moor Finsler metric
function; among the basic introduced physical objects, we mention: event, world
lines, reference frames, set of multiple relatively simultaneous events, proper time,
three-dimensional distance, speed, etc.
Thepaper”Properties of spaces connected with commutative-associative H and
3
H algebras” (S.V. Lebedev) provides physical significance to the algebraic specific
4
structure H , discussing further world lines of resting particles and properties of
3
4E.g., Russia: G.S. Asanov, G. Bogoslovski, G.I. Garas’ko, S.V. Lebedev, D.G. Pavlov;
Romania: R. Miron, Gh. Atanasiu, A. Bejancu; USA: D. Bao, R.G. Beil, R.L. Bryant,
S.S.Chern,Z.Shen;Canada: P.L.Antonelli; Germany: H.-B.Rademacher; Italy: G.Bellettini,
M. Paolini; Japan: T. Aikou, S. Ikeda, T. Kawaguchi, M. Matsumoto, H. Shimada; Hungary:
L. Tamassy, L. Kozma; Serbia and Montenegro: I. Comic, etc.
Preface 9
the surface of simultaneity which to defines the distance between the real axis
and a parallel to it, line of universe; as well are described Lorentz transformations
attached to the structure.
Further, in ”Generalized-analytical functions of poly-number variable” (G.I.
Garas’ko) the reader will find details on the new notion of generalized-analytic
function of polynumber variable, a primary step towards constructing a relevant
theory able to develop theoretical-physical models.
Though the next paper, ”On some questions of four dimensional topology. A
survey of modern research” (R.V. Mikhailov), has a rather abstract algebraic
character, it aims to justify the fact that 4-dimensionality plays special role in
almost all modern physical theories and provides a brief survey of some problems
of 4-dimensional topology.
In ”Normal conjugation on the polynumber manifold” (G.I. Garas’ko and
D.G.Pavlov)isdefinedthenormalconjugationonthemanifoldofnon-degenerated
n-numbers. In this context, the polynumber space endowed with the introduced
specific (n − 1)-ary operation appears as an ideal support for multi-dimensional
fractal sets, one of the perspective directions of applying multi-linear geometries.
Several properties of the generalized-analytic functions of polynumber variable
are studied in ”Generalized analytical functions and the congruence of geodesics”
(G.I.Garas’ko). Itisshownthateachsuchclassoffunctionsisnaturallyassociated
to a space of congruences of geodesics.
In ”The notions of distance and velocity modulus in the linear Finsler spaces”
(G.I. Garas’ko and D.G. Pavlov), the authors determine in 4-dimensional spaces
with Berwald-Moor metrics, formulas for both the 3-dimensional distance – by
means of the surface of relative simultaneity, and for the velocity modulus – which
coincides with the corresponding expression of the Galilean space for small (non-
relativistic) velocities, while at maximal velocities (i.e., for world lines lying on the
surface of the cone), it equals unity. They obtain as well the expressions for the
transformations which play the same role as the Lorentz transformations in the
Minkowski space.
A generalization of conformal transformations to the case of Finsler spaces is
provided in ”Generalization of conformal transformations” (G.I. Garas’ko); exam-
ples are provided as well for the case of complex and hypercomplex numbersH .
4
An adjacent issue to the main flow, is the study of the commutative algebra of
bi-complex numbers endowed with a metric of signature (+,−,−,+), performed
in ”Some properties of bicomplex numbers” (A.V. Smirnov).
Further, in ”Philosophical and mathematical reasons for Finsler extensions of
Relativity Theory” (D.G. Pavlov), the author provides an overview of arguments
to show that Finsler Geometry (and in particular the 4-dimensional quadratic
Berwald-Moore Finsler model) is by far the best candidate to provide relevant
models for the further development of Physics (in general) and Relativity Theory
(inparticular). Theassociatedcommutativeandassociativealgebra(quad-numbers
10 Space-Time Structure. Algebra and Geometry
algebra) is essentially described, and it is suggested to generalize the notion of
”symmetry” and to widen the classic case, based on isometric and conformal trans-
formations, by introducing generalized conformal transformations.
Elementary generalized conformal transformations in the space of nondegener-
atepoly-numbersandgeneralizedanalyticalfunctionsofthesamepolynumbervari-
able are further provided, with concrete examples for complex and hypercomplex
numbers H , in ”The relation of elementary generalized conformal transformations
4
with generalized analytic functions in the polynumber space” (G.I. Garas’ko).
In ”4-momentum of a particle and the mass shell equation in the entirely
anisotropic Space-Time” (G.Yu. Bogoslovsky), the author motivates the use
of Finsler geometry models in physics, especially of those, whose metric is of
Berwald-Moor type, and studies a certain Finslerian model which involves a pref-
fered direction in the 3D isotropy space, and a dimensionless parameter r which
determinesthedeviationofthemetricfromthemetricofisotropicMinkowskispace.
Further it is investigated the model for the entirely anisotropic flat space-time,
which generalizes of the Finslerian Berwald-Moor one. The variational principle
is used to obtain the formulas that relate the 4-momentum of a particle to the
3-velocity of the latter, and the invariants with respect to the relativistic symmetry
group of the entirely anisotropic space-time, are determined.
The way in which a Finslerian metric function provides a 3- and 4-rank gener-
alized Finslerian metric tensors is described further in ”The generalized Finslerian
metric tensors” (S.V. Lebedev). For these tensors the author determines the
generalized rank five Christoffel symbols and the generalized differential equations
of Finsler geodesics.
In ”Hamilton canonical equations and the Berwald-Moor metric (on the formal-
ism of physical theories)” S.V.Siparovpresentsanoverviewofdifferentapproaches
used to investigate a Finsler space: the pure mathematical approach, the approach
which belongs to theoretical Physics and the approach characteristic for philosophy
or for some meta-theory. As effective results, the Hamilton canonical equations are
obtained, on the base of the function related to the Berwald-Moor metric, and it
is pointed out that these equations can be used to construct the physical theory in
a Finsler space.
The theory of finite-dimensional algebras and its methods can be successfully
employed in geometry, physics and computer science. This fact is illustrated by
the paper ”On defining equations for the elements of associative and commutative
algebras and on associated metric forms” (V.M. Chernov), where the author stud-
ies the three non-isomorphic 4D algebras H ,H ⊕C,C ⊕C, their automorphisms
4 2
and metric forms, and provides a generalization for algebras of higher dimension
2d, constructed using the Grassmann-Clifford algebra, emphasizing that the coef-
ficients of the defining equations of an automorphism of algebras are associated
with the Minkowski and Berwald-Moor metrics. Authored by V.M. Chernov as
well, the paper ”Generalized n-ary composition laws in the algebra H and their
4