THE MIRROR UNIVERSE for Individual Or Non-Commercial Use Are Free of Any Permission Or Charge

L. B. Borissova and D. D. Rabounski This book has first been published by Editorial URSS Publishers, Moscow, 2001 Second revised edition with corrections, CERN Document Server, Geneve,` EXT-2003-025

FIELDS, VACUUM Copyright c L. B. Borissova and D. D. Rabounski 2001

All rights reserved. Any electronic copying and printing of this book THE MIRROR UNIVERSE for individual or non-commercial use are free of any permission or charge. Any part of this book, being mentioned or used in any form in other publications, must be referred to this publication. No part of this book may be reproduced in any form (including stor- ing in any medium) for commercial use without the prior permission of the copyright holder. Applications for the copyright holder’s permission to reproduce any part of this book for commercial use should be addressed to the copyright holder. Second Revised Edition with corrections

This book was typeset using -TEX system BAKOMA

First edition, 2001 Reprinted in CDS, 2003 — 2004 — Second revised edition with corrections, 2004 Typeset and printed in Great Britain 4 Contents

§3.5 Lorentz’ force. The energy-momentum tensor of electromagnetic fields 82 §3.6 Equations of motion of charged particle, obtained using the parallel transfer method 89 Contents §3.7 Equations of motion, obtained using the least action principle as a particular case of the previous 95 §3.8 The geometric structure of the four-dimensional Foreword to the 2nd Edition 6 electromagnetic potential 98 §3.9 Building Minkowski’s equations as a particular case Chapter 1 of the obtained equations of motion 104 INTRODUCTION §3.10 Structure of a space, filled with stationary electromagnetic fields 106 §1.1 Geodesic motion of particles 8 §3.11 Motion in a stationary electric field 109 §1.2 Physical observable quantities 13 §3.12 Motion in a stationary magnetic field 121 §1.3 Dynamic equations of motion of free particles 21 §3.13 Motion in a stationary electromagnetic field 137 §1.4 Introducing the concept of non-geodesic motion of §3.14 Conclusions 147 particles. Problem Statement 28 Chapter 4 Chapter 2 SPIN-PARTICLE IN PSEUDO-RIEMANNIAN SPACE TENSOR ALGEBRA AND THE ANALYSIS §4.1 Problem statement 149 §2.1 Tensors and tensor algebra 32 §4.2 The spin-impulse of a particle in equations of motion 154 §2.2 Scalar product of vectors 37 §4.3 Equations of motion of spin-particle 159 §2.3 Vector product of vectors. Antisymmetric tensors §4.4 Physical conditions of spin-interaction 167 and pseudotensors 40 §4.5 Motion of elementary spin-particles 171 §2.4 Differential and derivative to the direction 46 §4.6 A spin-particle in electromagnetic fields 180 §2.5 Divergence and rotor 48 §4.7 Motion in a stationary magnetic field 186 §2.6 Laplace’s operator and d’Alembert’s operator 57 §4.8 The law of quantization of masses of elementary par- §2.7 Conclusions 61 ticles 198 §4.9 Compton’s wavelength 203 Chapter 3 §4.10 Massless spin-particles 204 CHARGED PARTICLE IN PSEUDO-RIEMANNIAN SPACE §4.11 Conclusions 212

§3.1 Problem statement 62 Chapter 5 §3.2 Observable components of the electromagnetic field PHYSICAL VACUUM AND THE MIRROR UNIVERSE tensor. Invariants of the field 63 §3.3 Maxwell’s equations in chronometrically invariant §5.1 Introduction 214 form. The law of conservation of electric charge. §5.2 The observable density of vacuum. T-classification of Lorentz’ condition 69 matter 223 §3.4 D’Alembert’s equations for the electromagnetic po- §5.3 Physical properties of vacuum. Cosmology 226 tential and their observable components 76 §5.4 The concept of Inversional Explosion of the Universe 234 Contents 5

§5.5 Non-Newtonian gravitational forces 238 §5.6 Gravitational collapse 240 §5.7 Inflational collapse 246 §5.8 The concept of the mirror Universe. Conditions of transition through the membrane from our world Foreword to the 2nd Edition into the mirror Universe 249 §5.9 Conclusions 261

Appendix A Notation 263 The background behind this book is as follows. In 1991 we initiated Appendix B Special expressions 265 a study to find out what kinds of particles may theoretically inhabit Bibliography 268 the space-time of the General Theory of Relativity. As the instru- Index 271 ment, we equipped ourselves with the mathematical apparatus of chronometric invariants (physical observable quantities) developed by Abraham Zelmanov in the beginning of 1940’s. ♦ The study was completed by 1997–2001 to reveal that aside for mass-bearing and massless (light-like) particles, those of the third kind may exist. Their trajectories lay beyond the regular space- time of the General Theory of Relativity. For a regular observer the trajectories are of zero four-dimensional length and zero three- dimensional observable length. Besides, along the trajectories the interval of observable time is also zero. Mathematically, that means that such particles inhabit a fully degenerated space-time with non- Riemannian geometry. We called such space “zero-space” and such particles — “zero-particles”. For a regular observer their motion in zero-space is instant, so zero-particles realize long-range action. Through possible interaction with our-world’s mass-bearing or massless particles, zero- particles may instantly transmit signals to any point in our three- dimensional space. Considering zero-particles in the frames of the wave-particle duality, we have obtained that for a regular observer they are standing waves and the whole zero-space is filled with a system of standing light-like waves (zero-particles), i. e. a standing-light hologram. This result links with known the “stop-light experiment” (Harvard, the last years). Using the mathematical methods of physical observable quantities, we have also showed that in the basic four-dimensional space- time of the General Theory of Relativity a mirror world may exist, where coordinate time has reverse flow in respect of the viewpoint of the regular observer’s time. Foreword 7

All the above results are derived from exclusively application of the Zelmanov mathematical apparatus. When tackling the problem we have to amend the existing theory with some new techniques. In their famous The Classical Theory Chapter 1 of Fields, which has already become a de-facto standard for an university reference book on the General Theory of Relativity, Landau and Lifshitz give an excellent account of the theory of motion of INTRODUCTION particles in gravitational and electromagnetic fields. But the mono- graph does not cover motion of spin-particles. Besides, Landau and Lifshitz employed general covariant methods. The mathematical methods of physical observable quantities (chronometric invariants) §1.1 Geodesic motion of particles has not been yet developed in that time by Zelmanov, that should be also taken into account. Numerous experiments aimed at proving theoretical conclusions of Therefore we faced the necessity to introduce the methods of the General Theory of Relativity have also proven that its basic chronometric invariants into the existing theory of motion of parti- space-time (the four-dimensional pseudo-Riemannian space) is the cles in gravitational and electromagnetic fields. Separate consider- basis of our real world-geometry. So, even by progress of experi- ation has been given to motion of particles with inner mechanical mental physics and astronomy, which will discover new effects, the momentum (spin). We has also added a chapter with an account of four-dimensional pseudo-Riemannian space will remain the corner- tensor algebra and analysis. These all made our book a contempo- stone for further widening of the basic geometry of the General rary supplement to The Classical Theory of Fields. Theory of Relativity and will become one of its particular cases. In conclusion we would like to express our sincere gratitude to Therefore, when building the mathematical theory of motion of Dr. Abraham Zelmanov (1913–1987) and Prof. Kyril Stanyukovich particles, we are considering their motion in the four-dimensional (1916–1989). Many years of acquaintance and hours of friendly pseudo-Riemannian space. conversations with them have planted seeds of fundamental ideas A terminology note should be taken in this point. Generally, the which by now grew up in our minds to be reflected on these pages. basic space-time in the General Theory of Relativity is a Riemann- We are also grateful to Dr. Kyril Dombrovski whose works greatly ian space∗ with four dimensions with Minkowski’s sign-alternating (+ ) ( +++) influenced our outlooks. label −−− or − . The latest implies 3+1 split of coordinate axes in the Riemannian space into three spatial coordinate axes and the time axis. For convenience of calculations, we consider a (+ ) ♦ Riemannian space of the signature −−− , where time is real while spatial coordinates are imaginary. In the same time some theories, ( +++) largely the General Theory of Relativity, employ the label − , so there time is imaginary time while spatial coordinates are real. In general, Riemannian spaces may have non-alternating signature, e. g. (++++). Therefore a Riemannian space with alternating signature label is commonly referred to as a pseudo-Riemannian space, to emphasize the split of coordinate axes into two different types,

2 α β ∗A metric space which geometry is defined by the metric ds = gαβ dx dx known as Riemann’s metric. Bernhard Riemann (1826–1866), a German mathematician, the founder of Riemannian geometry (1854). 1.1 Geodesic motion of particles 9 10 Chapter 1 Introduction

referred to as time and spatial coordinates. But even in this case all allel transfer occurs in the meaning of Levi-Civita∗. Here the abso- its geometric properties are still properties of Riemannian geometry lute derivative of any transferred vector equals zero, in particular and the prefix “pseudo” is not absolutely proper from mathematical it is true for the four-dimensional vector of the particle viewpoint. Nevertheless we are going to use this notation as a long- dQα dxν established and traditionally understood one. + Γα Qμ = 0 , (1.3) dρ μν dρ We consider motion of a particle in the four-dimensional pseudo- Riemannian space. A particle affected by gravitation only falls so the square of the vector being transferred remains unchanged α freely and it moves along the shortest (geodesic) line. Such motion is QαQ = const along all the trajectory. Such equations are referred referred to as free or geodesic motion. If the particle is also affected to as equations of free motion. by additional non-gravitational forces, the latest deviate the particle Kinematic motion of a free particle is characterized by the four- from its geodesic trajectory and the motion becomes non-geodesic. dimensional vector of its acceleration, referred to as the kinematic From geometric viewpoint, motion of a particle in the four- vector dxα dimensional pseudo-Riemannian space is parallel transfer of its own Qα = , (1.4) four-dimensional vector Qα, which is therefore tangential to the dρ trajectory in any of its points. Consequently, equations of motion so the Levi-Civita parallel transfer gives equations of the four- of this particle actually define parallel transfer of the vector Qα dimensional trajectory of the particle (equations of geodesic lines) along its four-dimensional trajectory and they are equations of the 2 α μ ν absolute derivative of this vector with respect to a parameter ρ, d x α dx dx 2 + Γμν = 0 . (1.5) which is non-zero along along all the way dρ dρ dρ

α α ν The necessary condition ρ = 0 along the trajectory implies that DQ dQ α μ dx 6 = + Γμν Q , α, μ, ν = 0, 1, 2, 3. (1.1) the derivation parameters ρ are not the same along trajectories of dρ dρ dρ different kinds. In the pseudo-Riemannian space, three kinds of α α α μ ν trajectories are principally possible, each the kind corresponds to Here DQ = dQ + Γμν Q dx is the absolute differential (the absolute increment in the pseudo-Riemannian space) of the vector its specific kind of particles, namely: Qα. The absolute differential is different from a regular differential 1. Non-isotropic real trajectories, which lay “within” the light α α dQ by the presence of Christoffel’s symbols of the 2nd kind Γμν hyper-cone. Along such trajectories the square of the space- (the coherence coefficients of the given Riemannian space), which time interval is ds2 > 0, while the interval ds itself is real. are calculated through Christoffel’s symbols (the coherence coef- These are trajectories of regular sub-light particles with non- ficients) of the 1st kind Γμν,ρ and they are functions of the first zero rest-masses and real relativistic masses; derivatives of the fundamental metric tensor gαβ∗ 2. Non-isotropic imaginary trajectories, which lay “outside” the light hyper-cone. Along such trajectories the square of the 1 ∂gμρ ∂gνρ ∂gμν Γα = gαρ Γ , Γ = + . (1.2) space-time interval is ds2 < 0, while the ds is imaginary. These μν μν,ρ μν,ρ 2 ∂xν ∂xμ − ∂xρ are trajectories of super-light tachyon particles with imagi- When moving along a geodesic trajectory (free motion) the par- nary relativistic masses [2, 3]; 3. Isotropic trajectories, which lay on the surface of the light ∗Coherence coefficients of a Riemannian space (the Christoffel symbols) are named after German mathematician Elwin Bruno Christoffel (1829–1900), who hyper-cone and are trajectories of particles with zero rest- obtained them in 1869. In the space-time of the Special Theory of Relativity mass (massless light-like particles), which travel at the light (Minkowski’s space) one can always set an inertial reference frame, where the matrix of the fundamental metric tensor becomes unit diagonal, so all the Christoffel ∗Tullio Levi-Civita (1873–1941), an Italian mathematician, who was the first symbols become zeroes. who studied such parallel transfer [1]. 1.1 Geodesic motion of particles 11 12 Chapter 1 Introduction

velocity. Along the isotropic trajectories the space-time inter- Motion of a massless light-like particle (an isotropic geodesic val is zero ds2 = 0, but the three-dimensional interval is not line) is characterized by its own four-dimensional wave vector zero. ω dxα Kα = , (1.11) As a derivation parameter along non-isotropic trajectories the c dσ space-time interval ds is commonly used. But it can not be used in such capacity to trajectories of massless particles, because ds = 0 where ω is a cyclic frequency, specific for this massless particle. α there. For this reason Zelmanov had proposed [4] another variable, Respectively, the Levi-Civita parallel transfer of the vector K gives which does not turn into zero along isotropic trajectories, to be dynamic equations of motion of the massless particle used as the derivation parameter along isotropic trajectories. It is a dKα dxν + Γα Kμ = 0 ,K Kα = 0 . (1.12) three-dimensional (spatial) physical observable interval (see [4]) dσ μν dσ α

2 g0ig0k i k So we have got dynamic equations of motion for free particles. dσ = gik + dx dx , (1.6) − g The equations are presented in four-dimensional general covariant 00 form here. This form has its own advantage as well as a substantial which is different from a three-dimensional regular coordinate drawback. The advantage is their invariance in all transitions from interval. Landau and Lifshitz also arrived to the same conclusion one reference frame to another. The drawback is that, in general (see Section 84 in their The Classical Theory of Fields [5]). covariant form, terms of the equations do not contain actual three- Substituting respective differentiation parameters into the gen- dimensional quantities, which can be measured in experiments eralized equations of geodesic lines (1.5), we arrive to equations of or observations (namely — physical observable quantities). This non-isotropic geodesic lines (trajectories of mass-bearing particles) implies that in general covariant form equations of motion are mere- d2xα dxμ dxν ly an intermediate theoretical result, not applicable to practice. + Γα = 0 , (1.7) ds2 μν ds ds Therefore, in order to make results of any physical mathematical theory applicable in practice, we need to formulate its equations and to equations of isotropic geodesic lines (light-like particles) with physical observable quantities. Namely, to calculate trajecto- d2xα dxμ dxν ries of a particle we have to formulate general covariant equations + Γα = 0 . (1.8) dσ2 μν dσ dσ of its motion through physical observable properties of an actual physical reference frame of the observer. But in order to make the whole picture of motion of a particle, we In the same time, to define physical observable quantities is not have to build dynamic equations of motion, which contain physical a trivial problem. For instance, for a four-dimensional vector Qα properties of this particle (namely — its mass, energy, etc.). (as few as the four components) we may heuristically assume that Motion of a free mass-bearing particle (a non-isotropic geodesic its three spatial components form a three-dimensional observable trajectory) is characterized by its own four-dimensional impulse vector, while the temporal component is observable potential of the vector dxα vector field (which generally does not prove they can be actually P α = m , (1.9) 0 ds observed, though). However a contravariant tensor of the 2nd rank Qαβ (as many as 16 components) makes the problem much more where m is the rest-mass of this particle. From geometric view- 0 indefinite. For tensors of higher rank the problem of heuristic point, parallel transfer in the meaning of Levi-Civita of the vector definition of observable components is more complicated. Besides P α gives dynamic equations of motion of the mass-bearing particle there is an obstacle related to definition of observable components dP α dxν of covariant tensors (in which indices are the lower ones) and of + Γα P μ = 0 ,P P α = m2 = const. (1.10) ds μν ds α 0 mixed kind tensors, which have both lower and upper indices. 1.2 Physical observable quantities 13 14 Chapter 1 Introduction

Therefore the most reasonable way out of the labyrinth of heu- Therefore, physical observable quantities shall be obtained as a ristic guesses is creating a strict mathematical theory to enable cal- result of projecting four-dimensional quantities on time lines and culation of observable components for any tensor quantities. Such the three-dimensional space of the observer’s reference body. theory had been built by Zelmanov in 1944 [4]. It should be noted, From geometric viewpoint, the observer’s three-dimensional many researchers were working on the theory of observable quan- space is the spatial section x0 = ct = const. At any point of the space- tities in 1940’s. For example, Landau and Lifshitz in their famous time a local spatial section (a local space) can be placed orthogonal The Classical Theory of Fields [5] introduced observable time and to the time line. If exists a space-time enveloping curve to such observable three-dimensional interval similar to those introduced local spaces, then it is a spatial section everywhere orthogonal to by Zelmanov. But they limited themselves only to this particular the time lines. Such space is known as holonomic. If no enveloping case and they did not arrive to general mathematical methods to curve exists to such local spaces, so there only exist spatial sections define physical observable quantities in pseudo-Riemannian spaces. locally orthogonal to the time lines, such space is known as non- Over the next decades Zelmanov improved his mathematical holonomic. apparatus of physical observable quantities (the theory of chrono- We assume that the observer is at rest in respect of his physical metric invariants), setting forth the results in the publications [6, 7, references (his reference body). The reference frame of such ob- 8, 9]. Similar results had also been obtained by Cattaneo, an Italian server accompanies the reference body in any displacements, so mathematician, independently from Zelmanov. However Cattaneo such system is called the accompanying reference frame. Any co- published his first study on the theme in 1958 [10, 11, 12, 13]. ordinate net which is at rest in respect of the same reference body In the next §1.2 we will give just a brief overview of the Zelma- are related through the transformation nov theory of physical observable quantities, which is necessary for understanding it and using the mathematical methods in practice. x˜0 =x ˜0 x0, x1, x2, x3 In §1.3 we will present the results of studying geodesic motion i (1.13) i i 1 2 3 ∂x˜ of particles using the mathematical methods. In §1.4 will focus on x˜ =x ˜ x , x , x , = 0  ∂x0  setting the problem of building equations of particles along non- geodesic trajectories, i. e. under action of non-gravitational external where the latest equation implies that spatial coordinates in the forces. tilde-marked net are independent from time of the non-tilded net, which is equivalent to setting a coordinate net of fixed lines of §1.2 Physical observable quantities time xi = const in any point of the net. Transformation of spatial coordinates is nothing but only transition from one coordinate net This §1.2 introduces the Zelmanov mathematical apparatus of chro- to another within the same spatial section. Transformation of time nometric invariants. implies changing the whole set of clocks, so this is transition to To define which components of any four-dimensional quantity another spatial section (to another the three-dimensional reference are physical observable quantities, we consider a real reference space). In practice that means replacement of one reference body frame of a real observer, which includes coordinate nets, spanned with all of its physical references with another reference body over his reference body (which is a real physical body), at each that has its own physical references. But when using different point of which a real clock is installed. The reference body, being references, the observer will obtain different results (other observ- a real physical body possesses a gravitational field, may be rotating able quantities). Therefore physical observable quantities must be and deforming, making the reference space inhomogeneous and invariant in respect of transformations of time, so they shall be anisotropic. Actually, the reference body and attributed to it the chronometrically invariant quantities. reference space may be considered as a set of real physical refer- Because transformations (1.13) define a set of fixed lines of ences, to which observer compares all results of his measurements. time, chronometric invariants (physical observable quantities) are 1.2 Physical observable quantities 15 16 Chapter 1 Introduction

all those quantities, which are invariant in respect of the transfor- 00 00 1 0 h00 = 0 , h = g + , h0 = 0 , mations. − g00 In practice, to obtain physical observable quantities in the ac- 0i 0i i i h0i = 0 , h = g , h = δ = 0 , companying reference frame of a real observer we have to calculate − 0 0 gi0 (1.18) chronometrically invariant projections of four-dimensional quanti- h = 0 , hi0 = gi0, h0 = , i0 − i g ties on time lines and the spatial section of his physical reference 00 g0ig0k ik ik i i i body and formulate them with chronometrically invariant (physical hik = gik + , h = g , h = g = δ . − g − k − k k observable) properties of his reference space. 00 We project four-dimensional quantities using operators, which The tensor hαβ in the three-dimensional space of the accompa- characterize properties of the observer’s reference space. The op- nying reference frame of the observer possesses all properties of erator of projection on the time line bα is a unit vector of the four- the fundamental metric tensor dimensional velocity of the observer in respect of his reference 1 0 0 body, namely — the vector i α i i i i hαhk = δk bkb = δk , δk = 0 1 0 , (1.19) dxα −   bα = , (1.14) 0 0 1 ds   i where δk is the unit three-dimensional tensor∗. For this reason, in which is tangential to the observer’s world-trajectory in its every the accompanying reference frame the three-dimensional chr.inv.- point. Because any reference frame is described by its own tan- α α tensor hik can lift or lower indices in chr.inv.-quantities. gential unit vector b , Zelmanov referred to the b as the monad Projections on time lines and the spatial section of an arbitrary vector. The operator of projection on the spatial section is defined vector Qα in the accompanying reference frame (bi = 0) are as the four-dimensional symmetric tensor

αβ αβ α β α 0 Q0 hαβ = gαβ + bαbβ , h = g + b b , (1.15) T = b Qα = b Q0 = , (1.20) − − √g00 which mixed components are 0 0 β g0k k i i β i k k L = hβQ = Q ,L = hβQ = δkQ = Q . (1.21) β β β − g00 hα = gα + bαb . (1.16) − αβ α Projections of an arbitrary tensor of the 2nd rank Q are As it was shown [4], the vector b and the tensor hαβ possess all necessary properties of the projection operators. Projection of a α β 0 0 Q00 T = b b Qαβ = b b Q00 = , (1.22) tensor quantity on the time line is a result of its contraction with g00 the monad vector bα. Projection on the spatial section is contraction 00 0 0 αβ g0ig0k ik ik i k αβ ik with the tensor hαβ. L = hαhβQ = 2 Q ,L = hαhβQ = Q . (1.23) In the accompanying reference frame, the observer’s three- − g00 dimensional velocity in respect of his reference body is zero bi = 0. After testing the obtained quantities by the transformations The rest components of this monad vector are (1.13) we see that chronometrically invariant (physical observable) quantities are the projection on time lines and spatial components of 1 g b0 = , b = g bα = √g , b = g bα = i0 . (1.17) the projection on the spatial section. We will refer to the observable g 0 0α 00 i iα g √ 00 √ 00 quantities as chr.inv.-projections. i Respectively, in the accompanying reference frame (b = 0) com- i ∗This tensor δk is the three-dimensional part of the four-dimensional unit α ponents of the tensor of projection on the spatial section are tensor δβ , which can be used to replace indices in four-dimensional quantities. 1.2 Physical observable quantities 17 18 Chapter 1 Introduction

So forth, projecting four-dimensional coordinates xα in the ac- to calculate physical observable quantities, based solely on their companying reference frame we obtain the chr.inv.-invariant of property of chronometric invariance. Such method had been devel- physical observable time oped by Zelmanov, who set forth the method as a theorem:

g0i ZELMANOV’S THEOREM τ = √g t + xi, (1.24) 00 c g ik...p √ 00 We assume that Q00...0 are components of a four-dimensional tensor Qμν...ρ of r-th rank, in which all upper indices are not zero, while and the chr.inv.-vector of physical observable coordinates, which 00...0 all m lower indices are zeroes. Then tensor quantities coincide the spatial coordinates xi. In the same way, projection of α m an elementary interval of four-dimensional coordinates dx gives ik...p 2 ik...p T = (g00)− Q00...0 (1.30) an elementary interval of physical observable time, which is the chr.inv.-invariant make up three-dimensional contravariant chr.inv.-tensor of (r m)- th rank. Hence the tensor T ik...p is a result of m-fold projection− on g0i i dτ = √g00 dt + dx , (1.25) time lines by indices α, β . . . σ and also of projection on the spatial c √g00 section by r m indices μ, ν . . . ρ of the initial tensor Qμν...ρ . − αβ...σ and also the chr.inv.-vector of an elementary interval of physical α i An immediate result of this theorem is that for any vector Q two observable coordinates dx . Respectively, the physical observable quantities are physical observable, which were obtained earlier velocity of a particle is the three-dimensional chr.inv.-vector i α Q0 i α i i dx b Qα = , hαQ = Q . (1.31) v = , (1.26) √g00 dτ i αβ which is different from its coordinate velocity ui = dx . For any symmetric tensor of the 2nd rank Q three quantities dt Projecting the fundamental metric tensor, we obtain that the are physical observable, namely hik is the metric chr.inv.-tensor, or, in other word, the observable i α β Q00 iα β Q0 i k αβ ik metric tensor in the accompanying reference frame b b Qαβ = , h b Qαβ = , hαhβQ = Q , (1.32) g00 √g00 i k αβ ik ik α β hαhβ g = g = h , hi hk gαβ = gik bibk = hik . (1.27) in an antisymmetric tensor of the 2nd rank the first quantity is − − − 00 zero, because of Q00 = Q = 0. So, the square of an observable spatial interval dσ is The physical observable quantities (chr.inv.-projections) must be 2 i k compared to the observer’s references — observable properties of dσ = hik dx dx . (1.28) his reference space, which are specific for any particular body of Space-time interval formulated with physical observable quan- reference. Therefore we will now consider the basic properties of tities can be obtained by substituting gαβ from (1.15), namely his accompanying reference space, with which the final equations of theory must be formulated. ds2 = c2dτ 2 dσ2. (1.29) Physical observable properties of the accompanying reference − space can be obtained with the help of chr.inv.-operators of deriva- Aside for their projections on time lines and the spatial section, tion with respect to time and the spatial coordinates. The mentioned four-dimensional quantities of the 2nd rank and above also have operators had been introduced by Zelmanov as follows [4] mixed components which have both upper and lower indices at the same time. How do we find physical observable quantities among ∗∂ 1 ∂ ∗∂ ∂ g0i ∂ = , i = i 0 , (1.33) them, if any? The best approach is to develop a generalized method ∂t √g00 ∂t ∂x ∂x − g00 ∂x 1.2 Physical observable quantities 19 20 Chapter 1 Introduction they are non-commutative, so the difference between the 2nd der- Because the observer’s reference body is a real physical body, ivatives is not zero coordinate nets spanned over it may be deformed. So, his real 2 2 reference space may be deformed as well. Therefore real physical ∗∂ ∗∂ 1 ∗∂ = F , (1.34) references must take the space deformations into account. Namely, ∂xi∂t − ∂t ∂xi c2 i ∂t as a result of the deformations, the observable metric hik of the 2 2 ∗∂ ∗∂ 2 ∗∂ reference space must be non-stationary that can be taken into = A . (1.35) ∂xi∂xk − ∂xk∂xi c2 ik ∂t account by introducing the three-dimensional symmetric chr.inv.- tensor of the rate of the space deformations Here Aik is the three-dimensional antisymmetric chr.inv.- invariant tensor of angular velocities of the space rotation 1 ∂h 1 ∂hik D = ∗ ik ,Dik = ∗ , ik 2 ∂t 2 ∂t 1 ∂vk ∂vi 1 − Aik = + (Fi vk Fk vi) , (1.36) (1.40) 2 ∂xi − ∂xk 2c2 − ∂ ln √h D = hikD = Dn = ∗ , h = det h . ik n ∂t k ikk where vi is the linear velocity of this rotation Given the definitions we can generally formulate any properties g0i vi = c . (1.37) of geometric object located in a space with observable parameters − √g 00 of this space. For instance, the Christoffel symbols, which appear This tensor Aik, being equalized to zero, is the necessary and in equations of motion, are not tensors [14]. Nevertheless, they sufficient condition of holonomity of this space [4]. In this case can be as well formulated with physical observable quantities. The g = 0 and v = 0. In a non-holonomic space A = 0 is true always. formulas obtained by Zelmanov [4] are 0i i ik 6 For this reason, the tensor Aik is also the tensor of the space non- 1 1 ∂w w holonomity∗. Γ0 = + 1 v F k , (1.41) 00 3 w 2 k So forth, Fi is the three-dimensional chr.inv.-vector of gravita- −c 1 ∂t − c  tional inertial force − c2  2  k 1 w k 1 ∂w ∂vi Γ = 1 F , (1.42) F = , w = c2 (1 √g ) , (1.38) 00 2 2 i w i 00 −c − c 1 ∂x − ∂t − − c2 0 1 1 ∂w k k 1 k where w is a gravitational potential, derived from the gravitational Γ0i = 2 i + vk Di + Ai∙ + 2 viF , (1.43) c −1 w ∂x ∙ c  field of the observer’s reference body†. In quasi-Newtonian approxi- − c2 mation, i. e. in a weak gravitational field at velocities much lower   than the light velocity and in the absence of rotations of the space, k 1 w k k 1 k Γ0i = 1 2 Di + Ai∙ + 2 viF , (1.44) the quantity F becomes a non-relativistic gravitational force c − c ∙ c i ∂w 0 1 1 Fi = . (1.39) Γij = Dij + 2 vn ∂xi −c 1 w − c × − c2 ∗The space-time of the Special Theory of Relativity (the Minkowski space) in a 1 Galilean reference frame and also numerous cases in the space-time of the General n n n n n (1.45) vj (Di + Ai∙ ) + vi Dj + Aj∙ + vivj F + × ∙ ∙ c2 Theory of Relativity are examples of holonomic spaces Aik = 0. †The quantities w and vi do not possess the property of chronometric invariance, while the vector of gravitation inertial force and the tensor of angular velocities of 1 ∂vi ∂vj 1 n + + (Fivj + Fj vi) Δij vn , the space rotation, built using them, are chr.inv.-quantities. 2 ∂xj ∂xi − 2c2 − 1.3 Equations of motion of free particles 21 22 Chapter 1 Introduction

k k 1 k k k k 1 k So, projecting general covariant equations of motion of a free Γij = Δij vi Dj + Aj∙ + vj Di + Ai∙ + vivj F , (1.46) − c2 ∙ ∙ c2 mass-bearing particle (1.10) and of a free massless particle (1.12), we obtain chr.inv.-equations of their motion. For the mass-bearing k where Δij are the Christoffel chr.inv.-symbols, which are defined particle the equations are as well as the regular Christoffel symbols (1.2) but through the dm m i m i k metric chr.inv.-tensor hik and chr.inv.-operators of derivation F v + D v v = 0 , (1.51) dτ − c2 i c2 ik i im 1 im ∗∂hjm ∗∂hkm ∗∂hjk d mvi Δjk = h Δjk,m = h + . (1.47) i i k i i n k 2 ∂xk ∂xj − ∂xm + 2m Dk + Ak∙ v mF + mΔnkv v = 0 , (1.52) dτ ∙ − So, we have discussed the basics of the mathematical apparatus while for the massless particle we have of chronometric invariants. Now having any equations obtained dk k k using general covariant methods we can calculate their chr.inv.- F ci + D cick = 0 , (1.53) dτ − c2 i c2 ik projections in any particular reference frame. From here we arrive i to equations containing only quantities measurable in practice. d kc i i k i i n k + 2k Dk + Ak∙ c kF + kΔnk c c = 0 , (1.54) Naturally, the first possible application of this mathematical dτ ∙ − apparatus that comes to our mind is deduction of chr.inv.-equations ω where m is the relativistic mass of the mass-bearing particle, k = c of motion of free particles and studying the results. Particular is the wave number that characterizes the massless particle, and ci solution of this problem had been obtained by Zelmanov [4]. The is the three-dimensional chr.inv.-vector of the light velocity. As it next §1.3 will focus on the general solution of the problem. easy to see, in contrast to general covariant dynamic equations of motion (1.10, 1.12), the chr.inv.-equations have a single derivation parameter for both mass-bearing particles and massless particles. §1.3 Dynamic equations of motion of free particles This universal parameter is physical observable time τ. The chr.inv.-equations had first been obtained by Zelmanov [4]. The absolute derivative of the four-dimensional vector of a particle However the time function dt the equations include is strictly in respect to a non-zero scalar parameter along its trajectory is dτ actually a four-dimensional vector positive [15], so physical time has strictly direct flow dτ > 0 here. The flow of coordinate time dt shows change of time coordinates dQα dxν 0 N α = + Γα Qμ , (1.48) of the particle x = ct in respect of the observer’s clock. Hence the dρ μν dρ sign of the time function shows where the particle travels to in time which chr.inv.-projections are defined as well as for the projections in respect of the observer. The time function dt [15] is derived from the condition that of any four-dimensional vector (1.31) dτ the square of the four-dimensional velocity of the particle remains α N0 g0αN 1 α α β = = g N 0 + g N i , (1.49) unchanged along its world-trajectory uαu =gαβ u u =const. Equa- g g g 00 0i √ 00 √ 00 √ 00 tions in respect of dt are the same for sub-light mass-bearing dτ particles, for massless particles and for super-light mass-bearing i i β i 0 i k N = hβN = h0N + hkN . (1.50) particles. The equations have two solutions

i 2 From geometric viewpoint these are the projection of the vector dt viv c α = ± . (1.55) N on the time line and spatial components of its projection on the w dτ 1,2 c2 1 spatial section in the accompanying reference frame. − c2 1.3 Equations of motion of free particles 23 24 Chapter 1 Introduction

i 2 As it was shown [15], time has direct flow if viv c > 0, time where ψ is the wave phase (eikonal). In the same way, we introduced i 2 ± has reverse flow if viv c < 0, and the flow of time stops if the dynamic vector for a mass-bearing particle i 2 ± viv c = 0. Therefore there exists a whole range of solutions for various± kinds of particles and directions they travel in time in ~ ∂ψ Pα = , (1.61) respect of the observer. For instance, the relativistic mass of a c ∂xα P0 mass-bearing particle∗ = m is positive if this particle travels where is Planck’s constant. The wave phase equation (the eiko- √g00 ± ~ into future, and it is negative if the particle travels into past. The nal equation) in the geometric optics approximation is the condi- α wave number of a massless particle K0 = k is also positive for tion KαK = 0. Hence the eikonal chr.inv.-equation for the massless √g00 ± particle is the motion into future, and it is negative for the motion into past. 1 ∂ψ 2 ∂ψ ∂ψ ∗ + hik ∗ ∗ = 0 , (1.62) As a result for a free mass-bearing particle, which moves into c2 ∂t ∂xi ∂xk past, we obtain chr.inv.-equations of motion and for the mass-bearing particle we have dm m i m i k Fiv + Dikv v = 0 , (1.56) − dτ − c2 c2 1 ∂ψ 2 ∂ψ ∂ψ m2c2 ∗ + hik ∗ ∗ = 0 . (1.63) i c2 ∂t ∂xi ∂xk ~2 d mv i i n k + mF + mΔnkv v = 0 , (1.57) dτ Substituting the wave form of the dynamic vector into general while for a free massless particle we have covariant equations of motion (1.10, 1.12), after their projection in the accompanying reference frame we obtain chr.inv.-equations of dk k i k i k Fi c + Dik c c = 0 , (1.58) motion in their “wave form”. For the mass-bearing particle the −dτ − c2 c2 resulting equations are d kci + kF i + kΔi cnck = 0 . (1.59) dτ nk d ∗∂ψ i ∗∂ψ i k ∗∂ψ + F Dkv = 0 , (1.64) ±dτ ∂t ∂xi − ∂xi For a super-light mass-bearing particle chr.inv.-equations of motion are similar to those for sub-light velocities, save that the d ik ∗∂ψ i i 1 ∗∂ψ k km ∗∂ψ relativistic mass m is multiplied by imaginary unit i. h Dk + Ak∙ v h dτ ∂xk − ∙ ±c2 ∂t − ∂xm ± As it easy to see, chr.inv.-equations of motion into future and (1.65) into past are not symmetric due to different physical conditions in 1 ∗∂ψ ∗∂ψ F i + hmnΔi vk = 0 , the cases of the direct and reverse time flows, so some terms in ± c2 ∂t mk ∂xn equations will be missing. where “plus” in alternating terms stands for motion of the particle Besides motion of mass-bearing and massless particles can be from past into future (the direct flow of time), while “minus” stands considered within the wave-particle concept, assuming their motion for its motion into past (the reverse flow of time). Noteworthy, in propagation of waves in approximation of geometric optics [15]. As contrast to the “corpuscular form” of chr.inv.-equations of motion it is well-known, in the frames of the wave-particle concept the (1.51, 1.52) and (1.56, 1.57), the equations in “wave form” (1.64, dynamic vector of a massless particle is [5] 1.65) are symmetric in respect of the direction of motion in time. ∂ψ For the massless particle chr.inv.-equations of motion in “wave K = , (1.60) α ∂xα form” show the only difference: instead of the particle’s chr.inv.- i ∗The relativistic mass is the projection of the particle’s four-dimensional vector velocity v the equations include the chr.inv.-vector of the light on the observer’s time line. velocity ci. 1.3 Equations of motion of free particles 25 26 Chapter 1 Introduction

The fact that corpuscular equations of motion into past and into The wave form of matter in our world does not affect events in the future are asymmetric leads to the evident conclusion that in the mirror world, while the mirror world’s matter in wave form does four-dimensional inhomogeneous space-time of the General Theory not affect events in our world. To the contrary, the corpuscular form of Relativity there exists a fundamental asymmetry of directions in of matter (particles) in our world may produce significant effect on time. To understand physical sense of this fundamental asymmetry events in the mirror world, while the mirror world’s particles may we had introduced the mirror principle or, in other word — the affect events in our world. Our world is fully isolated from the observable effect of the mirror Universe [15]. mirror world (no mutual effect between particles from two worlds) i k i k Let us imagine a mirror in the four-dimensional space-time under the evident condition Dkv = Ak∙ v , at which the additional which coincides the spatial section, so this mirror separates past term in corpuscular chr.inv.-equations− of∙ motion becomes zero. This i i from future. Then particles and waves traveling from past into becomes true, in particular, when Dk = 0 and Ak∙ = 0, i. e. when no future (bearing positive relativistic masses and frequencies) hit the any deformations and rotation in the space. ∙ mirror and bounce back in time into past. Then their properties So far we have only considered motion of particles along non- take negative values. And vice versa, particles and waves traveling isotropic trajectories, where ds2 = c2dτ 2 dσ2 > 0, and that along into past (the negative relativistic masses and frequencies) bounce isotropic (light-like) trajectories, where ds−2 = 0 and c2dτ 2 = dσ2 = 0. from the mirror to give positive values to their properties and to Besides, we considered trajectories of the third kind [15], which,6 begin traveling into future. When bouncing from the mirror the aside for ds2 = 0, meet even more strict conditions c2dτ 2 = dσ2 = 0 ∗∂ψ quantity changes its sign, so equations of propagation of a 1 dt dτ = 1 w + v ui dt = 0 , (1.67) wave into future become equations of propagation of this wave − c2 i into past (and vice versa). Noteworthy, when reflecting from the 2 i k mirror, chr.inv.-equations of wave propagation transform into each dσ = hik dx dx = 0 . (1.68) other completely without contracting or adding new terms. In other We called such fully degenerated trajectories zero-trajectories word, the wave form of matter undergoes full reflection from the [15], because from viewpoint of a regular sub-light observer both mirror. To the contrary, corpuscular chr.inv.-equations of motion any interval of observable time and also any observable spatial do not transform completely in reflection from the mirror. Spatial interval are zeroes along them. We can as well show that along zero- components of the equations for mass-bearing and massless parti- trajectories the determinant of the fundamental metric tensor is cles, traveling from past into future, have an additional term zero g = 0. In Riemannian spaces, by their definition we have g < 0, i i k i i k so the Riemannian metric is strictly non-degenerated. We called a 2m Dk + Ak∙ v , 2k Dk + Ak∙ c , (1.66) ∙ ∙ space, a metric of which is fully degenerated, zero-space. For the not found in the equations of motion from future into past. The same reason, we called particles, which move along trajectories in equations of motion into past gain the additional term when the such space zero-particles. reflecting. And vice versa, the equations of motion into future lose Actually, formulas (1.67, 1.68) show physical conditions, under the term when the particle hits the mirror. That implies that either which total degeneration of the four-dimensional space-time occurs. in case of motion of a particle-ball (the corpuscular equations) as We can re-write the physical conditions of the degeneration as well as in case of propagation of a wave (the wave equations) we follows w + v ui = c2, (1.69) come across not a simple “bouncing” from the mirror, but rather i 2 passing through the mirror itself into another world — into a world i k 2 w gik u u = c 1 . (1.70) beyond the mirror. − c2 In this mirror world all particles bear negative masses or fre- Respectively, formula for the mass of a zero-particle M, includ- quencies, so they travel (from our viewpoint) from future into past. ing the degeneration conditions, is different from the relativistic 1.3 Equations of motion of free particles 27 28 Chapter 1 Introduction mass m of a regular particle in a non-degenerated area §1.4 Introducing the concept of nongeodesic motion of particles. m Problem statement M = , (1.71) 1 1 (w + v ui) − c2 i So, as it is well-known, free motion of a particle (along its own geodesic line) leaves the absolute derivative of the dynamic world- so that it is a ratio between two quantities, each one equals zero in vector of this particle (its four-dimensional impulse) zero, so the the case where the metric is degenerated, but the ratio is not zero . ∗ square of the vector remains unchanged along all trajectory of the The dynamic vector of a zero-particle, being represented in its motion. In other word, the vector is parallel transferred in the corpuscular and wave forms, is meaning of Levi-Civita. M dxα ~ ∂ψ In non-free (non-geodesic) motion of a particle the absolute P α = ,P = . (1.72) c dt α c ∂xα derivative of its four-dimensional impulse is not zero. But equal to zero is the absolute derivative of the sum of its four-dimensional Then dynamic chr.inv.-equations of motion in the zero-space, α α being taken in their corpuscular form, are impulse P and of an additional impulse vector L this particle gains from interaction with external fields, which deviate its motion i k MDik u u = 0 , (1.73) from geodesic line. Superposition of any number of vectors can be subjected to parallel transfer [14]. Hence, building equations of d i i n k Mu + MΔnk u u = 0 , (1.74) non-geodesic motion first of all requires to define non-gravitational dt perturbation fields. while the wave form of the equations is Naturally, an external field will only interact with the particle ∂ψ and deviate it from geodesic line if the particle has a physical Dmuk ∗ = 0 , (1.75) k ∂xm property of the same kind as the external field does. As of today, we know of three fundamental physical properties of particles, not d ∂ψ ∂ψ hik ∗ + hmnΔi uk ∗ = 0 . (1.76) related to any others. These are mass, electric charge and spin. dt ∂xk mk ∂xn If fundamental character of the former two was under no doubt, The eikonal chr.inv.-equation for the zero-particle is the spin of an electron over a few years after experiments by Stern and Gerlach (1921) and their interpretation by Gaudsmith ∂ψ ∂ψ hik ∗ ∗ = 0 , (1.77) and Ulenbek (1925), was considered as a specific momentum of the ∂xi ∂xk electron caused by its rotation around its own axis. But experiments so it is a standing wave equation, which describe the zero-particle done over the next decades, in particular, discovery of spin in other like as a information ring. Therefore, from viewpoint of a regular elementary particles, proved that views of spin-particles as rotating sub-light observer the whole zero-space is filled with a system of gyroscopes were wrong. Spin proved to be a fundamental property standing light-like waves (zero-particles) — a standing-light holo- of particles just like mass and charge, though it has dimension gram. Besides, in the zero-space observable time has the same value of impulse momentum and in interactions reveals as the specific for any two events (1.67). This implies that from viewpoint of a rotation momentum inside the particle. regular observer the velocity of any zero-particle is infinite, so zero- Gravitational fields by now have received geometric interpreta- particles can instantly transfer information from one point of our tion. In the theory of chronometric invariants, gravitational force regular world to another, performing the long-range action [15]. and the potential (1.38) are obtained as functions of only geomet-

2 2 ric properties of the space itself. Therefore, considering motion of ∗This is similar to the case of massless particles, because given v = c we have 2 2 m0 a particle in a pseudo-Riemannian space, we actually consider its that m0 = 0 and 1 v /c = 0 are zeroes, but their ratio is m = = 0. − 1 v2/c2 6 motion in a gravitational field. p − p 1.4 Nongeodesic motion of particles 29 30 Chapter 1 Introduction

But we still do not know whether Lornetz’ electromagnetic force attention, but it has a significant drawback. Being developed in and the electromagnetic field potential can be expressed through 1940’s it fully relied on the view of spin-particles as swiftly rotating geometric properties of the space. Therefore electromagnetic fields gyroscopes, which does not match experimental data of the recent at the moment have no geometric interpretation. An electromag- decades∗. netic field is introduced into a pseudo-Riemannian space as a sep- There is another way to solve the problem of motion of spin- arate tensor field (the field of Maxwell’s tensor). By now the main particles. In Riemannian spaces the fundamental metric tensor is equations of the theory of electromagnetic fields have been obtained symmetric gαβ = gβα. Nevertheless we can build a space in which in general covariant form∗. In this theory a charged particle gains the metric tensor will have arbitrary form gαβ = gβα (such space will e α 6 a four-dimensional impulse 2 A from the acting electromagnetic have non-Riemannian geometry). Then a non-zero antisymmetric α c fields, where A is the four-dimensional potential of the field, part can be found in the metric tensor†. Appropriate additions will α and e is the electric charge of the particle [5, 16]. Adding this appear in Christoffel’s symbols Γμν and in Riemann-Christoffel’s extra-impulse to the specific impulse vector of the particle and curvature tensor Rαβμν . The additions will be a cause of that a actuating the Levi-Civita parallel transfer, we can obtain general vector transferred along a closed contour not to return into the covariant equations of motion of the particle in the space, filled initial point, so the trajectory becomes twisted like a spiral. Such with gravitational and electromagnetic fields. space is known as twisted space. In such space the spin-rotation of The case of spin-particles is far more complicated. To deduce a particle can be considered as transfer of the rotation vector along an impulse a particle gains due to its spin, we need to define the its surface contour, that generates a local field of the space twist. external field that interacts with the spin as a fundamental property In the same time, this method has got significant drawbacks as of the particle. Initially this problem was approached using methods well. At first, if we have g = g , then functions of the components αβ 6 βα of Quantum Mechanics only (Dirac’s equations, 1928). Geometric gαβ with different order of indices may be varied. The functions methods of the General Theory of Relativity were first used by have been fixed somehow in to order to set a specific field of this Papapetrou and Corinaldesi [17, 18] for studying the problem. Their twist, which dramatically narrows the range of possible solutions, approach relied on general view of particles as mechanical mono- enabling only building equations for a range of specific cases. Sec- poles and the dipoles. From this viewpoint a regular mass-bearing ond, this method fully relies on assumption of the spin-rotation of a particle is a mechanical monopole. If a particle can be represented particle as a local field of a twist, produced by transfer of the vector as two masses co-rotating around a common centre of gravity, then of the particle’s rotation along a contour. This again implies the view the particle is a mechanical dipole. Therefore, proceeding from of spin-particles as rotating gyroscopes with limited radiuses (like representation of a spin-particle as a rotating gyroscope we can Papapetrou’s method), which does not match experimental data. consider it as a mechanical dipole, which centre of gravity lays over Nevertheless, there is not doubts in that an additional impulse, the particle’s surface. Then Papapetrou and Corinaldesi considered which a spin-particle gains, can be represented with methods of motion of such mechanic dipole in a pseudo-Riemannian space with the General Theory of Relativity. Adding it to the specific dynamic Schwarzschild’s metric — a very particular case, where rotation of the space is zero and the metric is stationary (the tensor of the ∗As a matter of fact, considering an electron as a ball with radius of 13 space deformations rate is zero). re= 2.8×10− cm implies that the linear velocity of its rotation on the surface is u = ~ = 2 1011 cm/sec, which is 70 times as high as the light velocity. Exper- No doubt that the method proposed by Papapetrou is worth 2m0re × ∼ iments show there are no such velocities in electrons. ∗Despite this positive fact, due to complicated calculations of the energy- †Generally, in any tensor of the 2nd rank and of high ranks symmetric and momentum tensor for an electromagnetic field in the space-time of the General antisymmetric parts can be distinguished. For instance, in the fundamental metric Theory of Relativity, specific problems are commonly solved either for particular tensor g = 1 (g + g ) + 1 (g g ) = S + N we have the symmetric αβ 2 αβ βα 2 αβ − βα αβ αβ cases of the General Theory of Relativity, or in a Galilean reference frame in the part Sαβ and the antisymmetric part Nαβ . Because the metric tensor of any Minkowski space (the space-time of the Special Theory of Relativity). Riemannian space is symmetric gαβ = gβα, its antisymmetric part is zero. 1.4 Nongeodesic motion of particles 31 vector of this particle (the effect of gravitation) and accomplishing parallel transfer, we can obtain general covariant equations of motion of the particle. Once we have general covariant equations of motion of a spin- Chapter 2 particle and an electric charged particle obtained, we shall project them on time lines and the spatial section in the accompanying TENSOR ALGEBRA AND THE ANALYSIS reference frame, then we shall express their chr.inv.-projections through physical observable properties of the reference space. As a result we shall arrive to chr.inv.-equations of nongeodesic motion. Therefore, the problem we are going to solve in this book falls §2.1 Tensors and tensor algebra apart into few stages. At first stage, we shall build the chr.inv.- theory of an electromagnetic field in the four-dimensional pseudo- We assume a space (not necessarily a metric one) with an arbitrary α Riemannian space, and also arrive to chr.inv.-equations of motion reference frame x located in it. In an area of this space, there α of a charged particle in the field. This problem will be solved in exists an object G defined by n functions fn of the coordinates x . Chapter 3. We know the transformation rule to calculate these n functions in α Then, we shall create the theory of motion of a spin-particle. We any other reference frame x˜ in this space. If the n functions fn will approach this problem in its most general form, assuming spin and also the transformation rule have been given, then the G is α a fundamental property of matter (like mass or electric charge). In a geometric object, which in the system x has axial components α α ˜ α Chapter 4, detailed study will show that the field of non-holonomity fn (x ), while in any other system x˜ it has components fn (˜x ). of the space (the space rotations field) interacts with the spin of a We assume that a tensor object (tensor) of zero rank is any particle, giving the particle additional impulse. geometric object ϕ, transformable according to the rule In Chapter 5 we are going to discuss chr.inv.-projections of Ein- ∂xα ϕ˜ = ϕ , (2.1) stein’s equations. Proceeding from them we will study properties ∂x˜α of physical vacuum and how they are dealt in cosmology. Before turning to these studies, in Chapter 2 we would like where the index one-by-one takes numbers of all coordinate axes to have a look at tensor algebra and the analysis in the terms of (such notation is also known as by-component notation or tensor physical observable quantities (chronometric invariants). Mainly, notation). Any tensor of zero rank has a single component and is we recommend our Chapter 2 to readers who are going to use also known as scalar. From geometric viewpoint, any scalar is a the mathematical apparatus in their own theoretical studies. For point to which a certain number is attributed. general understanding of our book, reading the next Chapter may Consequently, a scalar field∗ is a set of points of the space, which be not necessary. have a common property. For instance, the mass of a point-body is scalar, while a distributed mass (a gas, for instance) makes up scalar field. α ♦ Contravariant tensors of the 1st rank A are geometric objects with components, transformable according to the rule ∂x˜α A˜α = Aμ . (2.2) ∂xμ

∗Algebraic notations for a tensor and a tensor field are the same. The field of a tensor is represented as the tensor in a given point of the space, but its presence in other points in this area of the space is assumed. 2.1 Tensors and tensor algebra 33 34 Chapter 2 Tensor algebra and the analysis

From geometric viewpoint, such object is a n-dimensional vec- Mixed tensors are tensors of the 2nd rank or of higher ranks tor. For instance, the vector of an elementary displacement dxα is with both upper and lower indices. For instance, any mixed sym- α contravariant tensor of the 1st rank. metric tensor Aβ is a geometric object, transformable according to Contravariant tensors of the 2nd rank Aαβ are geometric objects the rule ∂x˜α ∂xν with components, transformable according to the rule A˜α = Aμ . (2.9) β ν ∂xμ ∂x˜β α β αβ μν ∂x˜ ∂x˜ ˜ Tensor objects exist both in metric and non-metric spaces∗. Any A = A μ ν . (2.3) ∂x ∂x tensor has an components, where a is its dimension and n is the From geometric viewpoint such object is an area (parallelogram) rank. For instance, a four-dimensional tensor of zero rank has 1 constructed by two vectors. For this reason, contravariant tensors component, a tensor of the 1st rank has 4 components, a tensor of of the 2nd rank are also known as bivectors. the 2nd rank has 16 components and so on. So forth, contravariant tensors of higher ranks are Indices in a geometric object, marking its axial components, are found not in tensors only, but in other geometric objects as well. For ∂x˜α ∂x˜σ ˜α...σ μ...τ this reason, if we come across a quantity in by-component notation, A = A μ τ . (2.4) ∂x ∙ ∙ ∙ ∂x this is not necessarily a tensor quantity. A vector field or a higher rank tensor field are space distribu- In practice, to know whether a given object is tensor or not, tions of the tensor quantities. For instance, because a mechanical we need to know a formula for this object in a reference frame strength characterizes both its own magnitude and the direction, and to transform it to any other reference frame. For instance, a its distribution in a physical body can be presented by a vector specimen is the classic question: are Christoffel’s symbols, i. e. the field. space coherence coefficients, tensors? Covariant tensors of the 1st rank Aα are geometric objects, To answer this question, we need to calculate the quantities in transformable according to the rule a tilde-marked reference frame μ ∂x α ασ 1 ∂g˜μσ ∂g˜νσ ∂g˜μν A˜ = A . (2.5) Γ =g ˜ Γμν,σ , Γμν,σ = + (2.10) α μ ∂x˜α μν 2 ∂x˜ν ∂x˜μ − ∂x˜σ ∂ϕ So, the gradient of a scalar field ϕ, i. e. the quantity A = , proceedinge frome the quantitiese in a non-marked reference frame. α ∂xα is covariant tensor of the 1st rank. That is, because for a regular We calculate the terms in the brackets (2.10). The fundamental invariant we have ϕ˜ = ϕ, then metric tensor just like as any other covariant tensor of the 2nd rank, is transformable to the tilde-marked reference frame according to ∂ϕ˜ ∂ϕ˜ ∂xμ ∂ϕ ∂xμ = = . (2.6) the rule ε τ α μ α μ α ∂x ∂x ∂x˜ ∂x ∂x˜ ∂x ∂x˜ g˜ = g . (2.11) μσ ετ ∂x˜μ ∂x˜σ Covariant tensors of the 2nd rank A are geometric objects, the αβ Because the g depends on non-tilde-marked coordinates, its transformation rule for which is ετ derivative with respect to tilde-marked coordinates (which are μ ν ˜ ∂x ∂x functions of non-tilded ones) is calculated according to the rule Aαβ = Aμν α β . (2.7) ∂x˜ ∂x˜ ρ ∂gετ ∂gετ ∂x So forth, covariant tensors of higher ranks are = . (2.12) ∂x˜ν ∂xρ ∂x˜ν μ τ ∂x ∂x In non-metric spaces, as it is known, the distance between any two points can A˜α...σ = Aμ...τ . (2.8) ∗ ∂x˜α ∙ ∙ ∙ ∂x˜σ not be measured. 2.1 Tensors and tensor algebra 35 36 Chapter 2 Tensor algebra and the analysis

Then the first term in the brackets (2.10), taking the rule of Multiplication is permitted not only for same-type, but for any transformation of the fundamental metric tensor into account, is tensors of any ranks. External multiplication of tensors of n-rank and m-rank gives a tensor of (n + m)-rank ∂g˜ ∂g ∂xρ ∂xε ∂xτ ∂xτ ∂2xε ∂xε ∂2xτ μσ = ετ +g + . (2.13) ∂x˜ν ∂xρ ∂x˜ν ∂x˜μ ∂x˜σ ετ ∂x˜σ ∂x˜ν ∂x˜μ ∂x˜μ ∂x˜ν ∂x˜σ A B = D ,A Bβγ = Dβγ . (2.19) αβ γ αβγ α α So forth, calculating the rest terms of the tilde-marked Christ- Contraction is multiplication of the same-rank tensors, when offel symbols (2.10), after transposition of free indices we obtain indices are the same. Contraction of tensors by all indices gives scalar ε ρ τ τ 2 ε ∂x ∂x ∂x ∂x ∂ x α γ αβ Γ = Γ + g , (2.14) AαB = C,AαβBγ = D. (2.20) μν,σ ερ,τ ∂x˜μ ∂x˜ν ∂x˜σ ετ ∂x˜σ ∂x˜μ∂x˜ν α ε ρ α 2 γ Often multiplication of tensors implies contraction by not all e α γ ∂x˜ ∂x ∂x ∂x˜ ∂ x indices. Such multiplication is known as internal multiplication, Γμν = Γερ γ μ ν + γ μ ν . (2.15) ∂x ∂x˜ ∂x˜ ∂x ∂x˜ ∂x˜ which implies contraction of some indices inside the multiplication. So, we see thate the Christoffel symbols are transformed not in This is an instance of internal multiplication the way tensors, hence they are not tensors. σ γ βσ β Tensors can be represented as matrixes. But in practice, such AασB = Dα ,AασBγ = Dα . (2.21) form may be illustrative for tensors of the 1st or 2nd rank (single- Using internal multiplication of geometric objects we can find row and flat matrixes, respectively). For instance, the tensor of an whether they are tensors or not. There is a so-called theorem of elementary four-dimensional displacement is fractions, which is given here according to [4]: α 0 1 2 3 dx = dx , dx , dx , dx , (2.16) THEOREM OF FRACTIONS σβ while the four-dimensional fundamental metric tensor is If B is tensor and its internal multiplication with a geometric object A (α, σ) is a tensor D (α, β) g00 g01 g02 g03 A (α, σ) Bσβ = D (α, β) , (2.22) g10 g11 g12 g13 gαβ =   . (2.17) g20 g21 g22 g23 then this object A (α, σ) as well is tensor.    g30 g31 g32 g33    According to the theorem, if internal multiplication of an object Aασ   σβ β Tensors of the 3rd rank are three-dimensional matrixes. Rep- with tensor B gives a tensor Dα resenting tensors of higher ranks as matrixes is more problematic. σβ β Now we turn to tensor algebra — a section of tensor calculus, AασB = Dα , (2.23) which focuses on algebraic operations over tensors. then this object A is tensor. Or, if internal multiplication of an Only same-type tensors of the same rank with indices in the ασ object Aα and a tensor Bσβ gives a tensor Dαβ same position can be added or subtracted. Adding up two same- σ type tensors of the n-rank gives a new tensor of the same type and α σβ αβ A σ∙B = D , (2.24) rank with components being sums of respective components of the ∙ α tensors added up. For instance then the object A σ∙ is tensor. Geometric properties∙ of any metric space are defined by its α α α α α α A + B = D ,Aβ + Bβ = Dβ . (2.18) fundamental metric tensor gαβ, which can lower or lift indices in 2.2 Scalar product of vectors 37 38 Chapter 2 Tensor algebra and the analysis

objects of this metric space∗. For instance, Scalar product is contraction, because multiplication of vectors

β μν σρ ρ at the same time contracts all indices. Therefore scalar product of gαβA = Aα , g g Aμνσ = A . (2.25) two vectors (tensors of the 1st rank) is always scalar (tensor of zero β rank). If the both vectors are the same, their scalar product In Riemannian spaces the mixed fundamental metric tensor gα β σβ β α β α 0 i equals the unit tensor gα = gασg = δα. Diagonal components of the gαβ A A = AαA = A0A + AiA (2.31) unit tensor are units, while the rest are zeroes. Using the unit tensor α we can replace indices in four-dimensional quantities, so that is the square of the given vector A . Consequently the length of this vector Aα is β ν σ μρ νσ δαAβ = Aα , δμδρ A = A . (2.26) A = Aα = g AαAβ. (2.32) | | αβ Contraction of any tensor of the 2nd rank with the fundamental Because the four-dimensionalq pseudo-Riemannian space by its metric tensor gives a scalar quantity, known as the tensor spur or definition has the indefinite metric (the sign-alternating signature its trace + +++ αβ σ −−− or − ), then lengths of four-dimensional vectors may be g Aαβ = Aσ . (2.27) real, imaginary or zero. Vectors with non-zero (real or imaginary) For instance, the spur of the fundamental metric tensor in a lengths are known as non-isotropic. Vectors with zero length are four-dimensional Riemannian space equals the number of coordi- known as isotropic. Isotropic vectors are tangential to trajectories nate axes of light-like particles (isotropic trajectories). αβ σ 0 1 2 3 gαβg = gσ = g0 + g1 + g2 + g3 = 4. (2.28) In three-dimensional Euclidean space scalar product of two vectors is a scalar quantity with module equal to the product of their The metric chr.inv.-tensor hik (1.27) in the observer’s three- lengths, multiplied by cosine of the angle between them dimensional space possesses all properties of the fundamental met- i i i i i ric tensor gαβ. Therefore it can lower, lift or replace indices in AiB = A B cos A\; B . (2.33) chr.inv.-quantities. Respectively, the spur of a three-dimensional chr.inv.-tensor is obtained by means of its contraction with the Theoretically at every point of any Riemannian space a tangen- metric chr.inv.-tensor hik. tial flat space can be set, which basic vectors will be tangential to For instance, the spur of the tensor of the rate of the space basic vectors of the Riemannian space in this point. Then the metric deformations Dik (1.40) is of the tangential flat space will be the metric of the Riemannian space in this point. Therefore this statement is also true in the ik m h Dik = Dm , (2.29) Riemannian space, if we consider the angle between coordinate lines and replace Roman (three-dimensional) indices with Greek that stands for the rate of relative expansion of an elementary (four-dimensional) ones. volume of the space. From here we can see that scalar product of two vectors is zero, if the vectors are orthogonal. In other word, scalar product from §2.2 Scalar product of vectors geometric viewpoint is the projection of one of the vector on the other. If the vectors are the same, then the vector is projected on Scalar product of two vectors Aα and Bα in a four-dimensional itself, so the result of this projection is the square of is length. pseudo-Riemannian space is So forth, we denote chr.inv.-projections of arbitrary vectors Aα α β α 0 i α gαβ A B = AαB = A0B + AiB . (2.30) and B as follows

2 α β A0 i i ∗In Riemannian spaces the metric has square form ds = gαβ dx dx , so the a = , a = A , (2.34) fundamental metric tensor there is the tensor of the 2nd rank gαβ . √g00 2.2 Scalar product of vectors 39 40 Chapter 2 Tensor algebra and the analysis

B §2.3 Vector product of vectors. Antisymmetric tensors and pseu- b = 0 , bi = Bi, (2.35) √g00 dotensors then their rest components are Vector product of two vectors Aα and Bα is a tensor of the 2nd rank V αβ, obtained from their external multiplication according to a + 1 v ai 0 c i a the specific rule A = ,Ai = ai vi , (2.36) 1 w − − c − c2 1 1 Aα Aβ V αβ = Aα; Bβ = AαBβ AβBα = . (2.42) 2 − 2 Bα Bβ b + 1 v bi 0 c i b B = ,Bi = bi vi . (2.37) As it easy to see, here the order in which vectors are multiplied 1 w − − c − c2 does matter, i. e. the order in which we write down tensor indices. α Substituting the chr.inv.-projections into the formulas for AαB Therefore tensors obtained as vector products are antisymmetric. α αβ βα and AαA , we obtain In an antisymmetric tensor V = V indices being moved “re- σβ − β serve” their places as dots gασV = Vα∙ , thus showing from where ∙ A Bα = ab a bi = ab h aibk, (2.38) the index was moved. In symmetric tensors there is no need to α − i − ik “reserve” places for moved indices, because the order in which A Aα = a2 a ai = a2 h aiak. (2.39) α − i − ik they appear does not matter. In particular, the fundamental metric From here we see that the square of any vector’s length is tensor is symmetric gαβ = gβα, while the tensor of the space cur- α vature R βγδ∙∙∙ is symmetric in respect to transposition by pair of its the difference between the squares of the lengths of its time and ∙ spatial chr.inv.-projections. If the both projections are equal, then indices and is antisymmetric inside each pair of the indices. It is the vector’s length is zero, so the vector is isotropic. Hence any evidently, only tensors of the 2nd rank or of higher ranks may be isotropic vector equally belongs to the time line and the spatial symmetric or antisymmetric. section. Equality of the time and spatial chr.inv.-projections also All diagonal components of any antisymmetric tensor by its implies that the vector is orthogonal to itself. If its time projection is definition are zeroes. For instance, in an antisymmetric tensor of “longer”, then the vector is real. If the spatial projection is “longer”, the 2nd rank we have then the vector is imaginary. 1 V αα = [Aα; Bα] = (AαBα AαBα) = 0 . (2.43) Scalar product of any four-dimensional vector with itself can be 2 − instanced by the square of the length of the space-time interval In three-dimensional Euclidean space the numerical value of 2 α β α 0 i ds = gαβ dx dx = dxαdx = dx0dx + dxi dx . (2.40) the vector product of two vectors is defined as the area of the parallelogram they make and equals the product of their modules, In the terms of physical observable quantities, it can be repre- multiplied by sine of the angle between them sented as follows V ik = Ai Bk sin A\i; Bk . (2.44) 2 2 2 i 2 2 i k 2 2 2 ds = c dτ dxidx = c dτ hik dx dx = c dτ dσ . (2.41) − − − This implies that the vector product of two vectors (i. e. an α β Its length ds = gαβ dx dx may be real, imaginary or zero, antisymmetric tensor of the 2nd rank) is a pad, oriented in the depending on whether ds is time-like c2dτ 2 > dσ2 (sub-light real space according to the directions of its forming vectors. p 2 2 2 trajectories), space-like c dτ < dσ (imaginary super-light trajec- Contraction of an antisymmetric tensor Vαβ with any symmetric tories), or isotropic c2dτ 2 = dσ2 (light-like trajectories). tensor Aαβ = AαAβ is zero, because of V = 0 and V = V so αα αβ − βα 2.3 Vector product of vectors. Pseudotensors 41 42 Chapter 2 Tensor algebra and the analysis that we have depending on the number of transpositions of their indices. All the rest components, i. e. those having at least two coinciding indices, α β 0 0 0 i i 0 i k Vαβ A A = V00 A A + V0i A A + Vi0 A A + Vik A A = 0 . (2.45) (+ ) are zeroes. Moreover, for the signature −−− we are using all non- zero components have the sign opposite to their respective covariant According to the theory of chronometric invariants, chr.inv.- components . For instance, in the Minkowski space we have projections of an antisymmetric tensor of the 2nd rank V αβ are ∗ σρτγ 0123 0123 gασ gβρ gμτ gνγ e = g00 g11 g22 g33 e = e , V i V i 1 − (2.49) 0∙ 0∙ i i αβγ 123 123 ∙ = ∙ = ab ba , (2.46) giα gkβ gmγ e = g11 g22 g33 e = e √g00 −√g00 2 − − because of the signature conditions g00 = 1 and g11 = g22 = g33 = 1 1 αβμν − V ik = aibk akbi , (2.47) we have accepted. Therefore, components of the tensor e are 2 − 0123 1023 1203 1230 e = +1, e = 1, e = +1, e = 1, where the third chr.inv.-projection V00 (1.32) is zero, because in − − (2.50) g00 e = 1, e = +1, e = 1, e = +1, any antisymmetric tensor all diagonal components are zeroes. 0123 − 1023 1203 − 1230 Physical observable components V ik (the projections of V αβ on and components of the tensor eikm are the observer’s spatial section) are an analog of a vector product 123 213 231 i e = +1, e = 1, e = +1, V0∙ − in three-dimensional space, while the quantity ∙ , which is the (2.51) √g00 e123 = 1, e213 = +1, e231 = 1. αβ − − space-time (mixed) projection of the tensor V , has no analogs Because we have an arbitrary choice for the sign of the first among components of a regular three-dimensional vector product. component, we assume e0123 = 1 and e123 = 1. Subsequently, the The square of an antisymmetric tensor of the 2nd rank, formu- rest components will be changed.− In general,− the tensor eαβμν is lated with chr.inv.-projections of its forming vectors, is related to the tensor eikm as follows e0ikm = eikm. 1 1 Multiplying the four-dimensional antisymmetric unit tensor by V V αβ = a aib bk a bia bk +aba bi a2b bi b2a ai . (2.48) αβ 2 i k − i k i − 2 i − i itself we obtain a regular tensor of the 8th rank with non-zero components, which are given in the matrix The last two terms in this formula contain quantities a (2.34) and α α α α b (2.35), which are chr.inv.-projections of the multiplied vectors Aα δσ δτ δρ δγ α β β β β and B on the observer’s time line, so the terms have no analogs in αβμν δσ δτ δρ δγ e eστργ =  μ μ μ μ  . (2.52) a vector product in three-dimensional Euclidean space. − δσ δτ δρ δγ Asymmetry of tensor fields is defined by reference antisym-  ν ν ν ν   δσ δτ δρ δγ  metric tensors. In a Galilean reference frame∗ such references    αβμν  are Levi-Civita’s tensors. For four-dimensional quantities this is The rest properties of the tensor e are derived from the the four-dimensional completely antisymmetric unit tensor eαβμν , previous by means of contraction of indices while for three-dimensional quantities this is the three-dimensional δα δα δα ikm σ τ ρ completely antisymmetric unit tensor e . Components of the Levi- αβμν β β β e eστρν =  δσ δτ δρ  , (2.53) Civita tensors, which have all indices different, are either +1 or 1 − μ μ μ − δσ δτ δρ ∗A Galilean frame of reference is the one that does not rotate, is not subject to   ( +++)  deformation and falls freely in the flat space-time of the Special Theory of Relativity ∗If the space-time signature is − , this is true for only the four-dimensional (the Minkowski space). There lines of time are linear and so are three-dimensional tensor eαβμν . Components of the three-dimensional tensor eikm will have the same coordinate axes. signs as well as the respective components of eikm. 2.3 Vector product of vectors. Pseudotensors 43 44 Chapter 2 Tensor algebra and the analysis

α α αβμν δσ δτ α β β α the axes. For instance, being reflected in respect of abscises axis e eστμν = 2 β β = 2 δσ δτ δσ δτ , (2.54) x1 = x˜1, x2 =x ˜2, x3 =x ˜3. The reflected component of an antisym- − δσ δτ − − − 1 metric tensor Vik, orthogonal to x , is V23 = V23, while its dual αβμν α αβμν α − e eσβμν = 6δ , e eαβμν = 6δ = 24. (2.55) component of the pseudovector V i is − σ − α − ∗ e Multiplying the three-dimensional antisymmetric unit tensor 1 1 1km 1 123 132 ikm V ∗ = e V = e V +e V =V , e by itself we obtain a regular tensor of the 6th rank 2 km 2 23 32 23 (2.61) i i i 1 1 1km 1 k1m 1 213 312 δr δs δt V ∗ = e˜ Vkm = e Vkm = e V23 +e V32 =V23 . ikm k k k 2 2 2 e erst =  δr δs δt  . (2.56) δm δm δm eBecause a four-dimensionale e antisymmetrice e tensor of the 2nd  r s t    rank and its dual pseudotensor are of the same rank, their contrac- The rest properties of the tensor eikm are tion is pseudoscalar, so that

i i αβ αβμν αβμν ikm δr δs i k i k VαβV ∗ = Vαβ e Vμν = e Bαβμν = B∗. (2.62) e ersm = k k = δsδr δrδs , (2.57) − δr δs − αβ The square of a pseudotensor V ∗ and the square of a pseudo- ikm i ikm i i e erkm = 2δr , e eikm = 2δi = 6. (2.58) vector V ∗ , expressed through their dual tensors, are

The completely antisymmetric unit tensor defines for a tensor αβ μν αβρσ μν V αβV ∗ = eαβμν V e Vρσ = 24Vμν V , (2.63) object its respective pseudotensor, marked with asterisk. For in- ∗ − stance, any four-dimensional scalar, vector and tensors of the 2nd, i km ipq km V iV ∗ = eikmV e Vpq = 6VkmV . (2.64) 3rd, and 4th ranks have respective four-dimensional pseudotensors ∗ of the following ranks In inhomogeneous anisotropic pseudo-Riemannian spaces we can not set a Galilean reference frame, so references of asymmetry αβμν αβμν αβμ αβμν αβ 1 αβμν V ∗ =e V,V ∗ =e V ,V ∗ = e V , of tensor fields will depend on inhomogeneity and anisotropy of the ν 2 μν (2.59) space itself, which are defined by the fundamental metric tensor. α 1 αβμν 1 αβμν V ∗ = e Vβμν ,V ∗ = e Vαβμν , In this general case, a reference antisymmetric tensor is the four- 6 24 dimensional completely antisymmetric discriminant tensor where pseudotensors of the 1st rank V α are called pseudovectors, ∗ αβμν while pseudotensors of zero rank V are called pseudoscalars. Any αβμν e ∗ E = ,Eαβμν = eαβμν g . (2.65) tensor and its respective pseudotensor are known as dual to each g − − other to emphasize their common genesis. So, three-dimensional p p tensors have respective three-dimensional pseudotensors Here is the proof. Transformation of the unit completely antisymmetric tensor from a Galilean (non-tilde-marked) reference ikm ikm ik ikm V ∗ = e V,V ∗ = e Vm , frame into an arbitrary (tilde-marked) reference frame is (2.60) i 1 ikm 1 ikm ∂xσ ∂xγ ∂xε ∂xτ V ∗ = e Vkm ,V ∗ = e Vikm . e˜ = e = Je , (2.66) 2 6 αβμν ∂x˜α ∂x˜β ∂x˜μ ∂x˜ν σγετ αβμν

Pseudotensors are called such because, in contrast to regular α where J = det ∂x is called the Jacobian of the transformation tensors, they do not change being reflected in respect of one of ∂x˜σ

2.3 Vector product of vectors. Pseudotensors 45 46 Chapter 2 Tensor algebra and the analysis

(the determinant of Jacobi’s matrix) 0 E0ikm εikm = b E0ikm = = eikm√h . (2.74) ∂x0 ∂x0 ∂x0 ∂x0 √g00 ∂x˜0 ∂x˜1 ∂x˜2 ∂x˜3 With its help we can transform chr.inv.-tensors into chr.inv.- ∂x1 ∂x1 ∂x1 ∂x1 pseudotensors. For instance, taking the antisymmetric chr.inv.- ∂x˜0 ∂x˜1 ∂x˜2 ∂x˜3 J = det 2 2 2 2 . (2.67) tensor of angular velocities of the space rotation Aik (1.36), we ∂x ∂x ∂x ∂x i 1 ikm 0 1 2 3 obtain the chr.inv.-pseudovector of this rotation Ω∗ = ε Akm. ∂x˜ ∂x˜ ∂x˜ ∂x˜ 2 ∂x3 ∂x3 ∂x3 ∂x3

∂x˜0 ∂x˜1 ∂x˜2 ∂x˜3 §2.4 Differential and derivative to the direction

Because the fundamental metric tensor g is transformable αβ according to the rule In geometry a differential of a function is its variation between two μ ν ∂x ∂x infinitely close points with coordinates xα and xα + dxα. Respective- g˜αβ = gμν , (2.68) ∂x˜α ∂x˜β ly, the absolute differential in a n-dimensional space is the variation its determinant in the tilde-marked reference frame is of a n-dimensional quantity between two infinitely close points of n- dimensional coordinates in this space. For continuous functions, we ∂xμ ∂xν g˜ = det g = J 2g . (2.69) commonly deal with in practice, their variations between infinitely ∂x˜α ∂x˜β μν close points are infinitesimal. But in order to define infinitesimal

Because in the Galilean (non-tilde-marked) reference frame variation of a tensor quantity we can not use simple “difference” between its values in the points xα and xα + dxα, because tensor 1 0 0 0 algebra does not define the ratio between values of a tensor in 0 1 0 0 g = det gαβ = det − = 1, (2.70) different points in space. This ratio can be defined only using rules k k 0 0 1 0 − − of transformation of tensors from one reference frame into another. 0 0 0 1 − As a consequence, differential operators and the results of their 2 2 application to tensors must be tensors. then J = g˜ . Expressing e˜ αβμν in an arbitrary reference frame as − For instance, the absolute differential of a tensor quantity is a Eαβμν and writing down the metric tensor in a regular non-tilde- marked form, we obtain E = e g (2.65). In the same way, tensor of the same rank as this quantity itself. For a scalar ϕ it is αβμν αβμν − the scalar we obtain transformation rules for the components Eαβμν , because ∂ϕ α p D ϕ = dxα, (2.75) ˜2 ˜ ∂x˜ α for them g =g ˜J , where J = det σ . ∂x ∂x i The discriminant tensor Eαβμν is not a physical observable quan- which in the accompanying reference frame (b = 0) is tity. A physical observable reference of asymmetry of tensor fields ∂ϕ ∂ϕ D ϕ = ∗ dτ + ∗ dxi. (2.76) is the three-dimensional discriminant chr.inv.-tensor ∂t ∂xi As it easy to see, aside for three-dimensional observable differ- εαβγ = hαhβhγ b Eσμνρ = b Eσαβγ , (2.71) μ ν ρ σ σ ential there is an additional term, it takes into account dependence μ ν ρ σ σ εαβγ = hαhβhγ b Eσμνρ = b Eσαβγ , (2.72) of the absolute displacement D ϕ from the flow of physical observable time dτ. i which in the accompanying reference frame (b = 0), taking into The absolute differential of a contravariant vector Aα, formula- account that g = h g00, takes the form ted with the operator of absolute derivation (nabla), is − ∇ ikm α p ikm p p0ikm 0ikm e α α σ ∂A σ α μ σ α α μ σ ε = b0E = √g00 E = , (2.73) DA = σ A dx = dx +ΓμσA dx = dA +ΓμσA dx , (2.77) √h ∇ ∂xσ 2.4 Differential and derivative 47 48 Chapter 2 Tensor algebra and the analysis

α α σ where σ A is the absolute derivative of the A with respect to x , tangential to the trajectory. From geometric viewpoint a derivative and d stands∇ for regular differentials to a given direction of a function is its change in respect of elemen-

α tary displacement along the given direction. The absolute derivative α ∂A α μ to the given direction in a n-dimensional space is a change of a n- σ A = σ + ΓμσA , (2.78) ∇ ∂x dimensional quantity in respect of an elementary n-dimensional ∂ interval along the given direction. For instance, the absolute de- d = dxα. (2.79) ∂xα rivative of a scalar function ϕ to a direction, defined by a curve xα = xα (ρ), where ρ is a non-zero monotone parameter along this Formulate the absolute differential with physical observable curve, shows the “rate” of changes of this function quantities is equivalent to projecting its general covariant form on D ϕ dϕ time lines and the spatial section in the accompanying reference = . (2.86) frame dρ dρ α α g0αDA i i α T = bαDA = ,B = hαDA . (2.80) In the accompanying reference frame it is √g00 i α D ϕ ∂ϕ dτ ∂ϕ dx Denoting chr.inv.-projections of the vector A as follows = ∗ + ∗ . (2.87) dρ ∂t dρ ∂xi dρ A0 i i ϕ = , q = A , (2.81) The absolute derivative of a vector Aα to the given direction of √g00 a curve xα = xα (ρ) is we have its rest components DAα dxσ dAα dxσ = Aα = + Γα Aμ , (2.88) ϕ + 1 v qi dρ ∇σ dρ dρ μσ dρ w 0 c i ϕ A0 = ϕ 1 2 ,A = ,Ai = qi vi . (2.82) − c 1 w − − c its chr.inv.-projections are − c2 Because a regular differential in chr.inv.-form is DAα dϕ 1 dτ dxk b = + F qi + D qi , (2.89) α dρ dρ c − i dρ ik dρ ∗∂ ∗∂ i d = dτ + i dx , (2.83) σ i k ∂t ∂x i DA dq ϕ dx k dτ i i hσ = + + q Dk + Ak∙ dρ dρ c dρ dρ ∙ − after substituting it and also the Christoffel symbols, taken in the (2.90) i ϕ dτ dxk accompanying reference frame (1.41–1.46), into T and B (2.80), F i + Δi qm . we obtain the chr.inv.-projections of the absolute differential of the − c dρ mk dρ vector Aα Actually, the projections are “generic” chr.inv.-equations of mo- 1 T = b DAα = dϕ + F qidτ + D qidxk , (2.84) tion. But once we define a particular vector of motion of a particle, α c i ik − calculate its chr.inv.-projections and substitute them into the given i i σ i ϕ k k i i equations, we immediately obtain chr.inv.-equations of the motion. B = hσDA = dq + dx + q dτ Dk + Ak∙ c ∙ − ϕ (2.85) F idτ + Δi qmdx k. §2.5 Divergence and rotor − c mk To build chr.inv.-equations of motion we will also need chr.inv.- A divergence of a tensor field is its “change” along a coordinate axis. projections of the absolute derivative of a vector to the direction, Respectively, the absolute divergence of a n-dimensional tensor field 2.5 Divergence and rotor 49 50 Chapter 2 Tensor algebra and the analysis

ρσ ρσ is its divergence in a n-dimensional space. A divergence of a tensor then the quantity dg will be dg = a dgρσ = gg dgρσ, or field is a result of contraction of the field tensor with the operator of absolute derivation . The divergence of a vector field is the scalar dg ρσ ∇ = g dgρσ . (2.98) σ g σ ∂A σ μ σ A = + Γ A , (2.91) ∇ ∂xσ σμ Integration of the left part gives ln ( g), because the g is negative − while the divergence of a field of the 2nd rank tensor is the vector while logarithm is defined for only positive functions. Then we have 1 dg 2 1 σα d ln ( g) = g . Taking into account that ( g) = 2 ln ( g), we obtain σα ∂F σ αμ α σμ − − − σ F = σ + ΓσμF + ΓσμF , (2.92) ∇ ∂x 1 ρσ d ln √ g = g dgρσ , (2.99) σ − 2 where, as it can be proven, the Γσμ is σ so Γσμ (2.95) takes the form σ ∂ ln √ g Γσμ = − . (2.93) 1 ∂g ∂ ln √ g ∂xμ Γσ = gρσ ρσ = − , (2.100) σμ 2 ∂xμ ∂xμ To prove this, we will use the definition of the Christoffel sym- σ bols. Then we write down Γσμ in details which has been proven (2.93). Now we are going to deduce chr.inv.-projections of the diver- σ σρ 1 σρ ∂gμρ ∂gσρ ∂gμσ gence of a vector field (2.91) and of a tensor field of the 2nd rank Γ = g Γμσ,ρ = g + . (2.94) σμ 2 ∂xσ ∂xμ − ∂xρ (2.92). The divergence of a vector field Aα is scalar, hence Aσ σ can not be projected on time lines and the spatial section,∇ while Because σ and ρ here are free indices, they can change their this is enough to express is through chr.inv.-projections of the Aα sites. As a result, after contraction with the tensor gρσ the first and and through observable properties of the reference space. Besides, the last terms cancel each other, so Γσ takes the form σμ regular operators of derivation shall be replaced with the chr.inv.-

1 ∂gρσ operators. Γσ = gρσ . (2.95) i σμ 2 ∂xμ Assuming notations ϕ and q for chr.inv.-projections of the vector Aα (2.81), we express the rest components of the vector through ρσ The quantities g are components of a tensor reciprocal to the them (2.82). Then, substituting regular operators of derivations, ρσ tensor gρσ. Therefore each component of the matrix g is expressed through the chr.inv.-operators ρσ ρσ a g = , g = det gρσ , (2.96) 1 ∂ ∗∂ w g k k = , √g00 = 1 2 , (2.101) √g00 ∂t ∂t − c where aρσ is the algebraic cofactor of the matrix’ element with ρ+σ ∂ ∂ 1 ∂ indices ρσ, equal to ( 1) , multiplied by the determinant of ∗ = + v ∗ , (2.102) the matrix obtained by− crossing the row and the column with ∂xi ∂xi c2 i ∂t numbers σ and ρ out of the matrix gρσ. As a result we obtain ρσ ρσ into (2.91), and taking into account that g = h g after some a = gg . Because the determinant of the fundamental metric − 00 tensor g = det g by definition is algebra we obtain k ρσk p p p N(α ...α ) i 0 3 1 ∗∂ϕ ∗∂q ∗∂ ln √h 1 g = ( 1) g0(α0)g1(α1)g2(α2)g3(α3) , (2.97) σ i i − σ A = + ϕD + i + q i 2 Fi q . (2.103) α ...α ∇ c ∂t ∂x ∂x − c 0X 3 2.5 Divergence and rotor 51 52 Chapter 2 Tensor algebra and the analysis

In the third term the quantity vector field qi, because the spur of the chr.inv.-tensor of the rate ik n of the space deformations D = h Dik = Dn is the rate of relative ∗∂ ln √h j expansion of an elementary volume of the space. i = Δji (2.104) σ ∂x Equation σ A = 0, being applied to the four-dimensional vector potential Aα ∇of an electromagnetic field is Lorentz’ condition for the stands for the Christoffel chr.inv.-symbols Δk (1.47), contracted ji field. The Lorentz condition in chr.inv.-form is by two symbols. Hence similarly to the definition of the absolute divergence of a vector field (2.91), the quantity i 1 ∗∂ϕ ∗ q = + ϕD . (2.108) ∇i − c ∂t i i ∗∂q i ∗∂ ln √h ∗∂q i j i + q = + q Δ = ∗ q (2.105) ∂xi ∂xi ∂xi ji ∇i Now we are goinge to deduce chr.inv.-projections of the divergence of an arbitrary antisymmetric tensor F αβ = F βα (later we is the chr.inv.-divergence of a three-dimensional vector field qi. will need them to obtain Maxwell’s equations in chr.inv.-form)− Consequently we call the physical chr.inv.-divergence of the vector i ∂F σα ∂F σα ∂ ln √ g field q the chr.inv.-quantity F σα = +Γσ F αμ +Γα F σμ = + − F αμ, (2.109) ∇σ ∂xσ σμ σμ ∂xσ ∂xμ i i 1 i α σμ ∗ i q = ∗ i q 2 Fi q , (2.106) where the third term ΓσμF is zero, because of contraction of ∇ ∇ − c α the Christoffel symbols Γσμ (which are symmetric by their lower in which the 2nd terme takes into account that the pace of time indices σμ) and an antisymmetric tensor F σμ is zero as well as for is different on the opposite walls of an elementary volume [4]. any symmetric tensor and any antisymmetric tensor. σα As a matter of fact that in calculation of divergence we consider The term σ F is four-dimensional vector, so its chr.inv.- an elementary volume of the space, so we calculate the difference projections are∇ between the amounts of a “substance” which flows in and out of T = b F σα,Bi = hi F σα = F iα. (2.110) the volume over an elementary time interval. But the presence of α∇σ α∇σ ∇σ gravitational inertial force F i (1.38) results in different pace of time We denote chr.inv.-projections of the tensor F αβ as follows at different points in the space. Therefore, if we measure durations of time intervals at the opposite walls of the volume, the beginnings i i F0∙ ik ik and the ends of the interval will not coincide making them invalid E = ∙ ,H = F , (2.111) √g00 for comparison. Synchronization of clocks at the opposite walls of the volume will give the true picture — the measured durations of then the rest non-zero components of the tensor are the intervals will be different. σ 0 1 k The final equation for σ A will be F0∙ = vkE , (2.112) ∇ ∙ c σ 1 ∗∂ϕ i 1 1 1 σ A = + ϕD + ∗ i q . (2.107) 0 n n c ∂t Fk∙ = Ei vnHk∙ 2 vkvnE , (2.113) ∇ ∇ ∙ √g − c ∙ − c 00 The second term in this formula is a physicale observable analog Ei 1 v Hik of a regular divergence in the observer’s three-dimensional space. 0i c k F = − ,F0i = √g00 Ei , (2.114) ∂ϕ √g00 − The first term has no analogs, it falls apart into two parts: ∗ is α ∂t the variation in time of the time projection ϕ of the vector A , while k k 1 k 1 Fi∙ = Hi∙ viE ,Fik = Hik + (viEk vkEi) , (2.115) D ϕ is the variation in time of a volume of the three-dimensional ∙ − ∙ − c c − 2.5 Divergence and rotor 53 54 Chapter 2 Tensor algebra and the analysis and the square of this tensor F αβ is We denote its chr.inv.-projections as follows

αβ ik i i F F = H H 2E E . (2.116) i F0∗∙ ik ik αβ ik − i H∗ = ∙ ,E∗ = F ∗ , (2.124) √g00 Substituting the components into (2.110) and replacing regular i ik ik i so there are evident relations H∗ H and E∗ E between the operators of derivation with the chr.inv.-operators, after some al- ∼ ∼ gebra we obtain chr.inv.-quantities and chr.inv.-projections of the antisymmetric tensor F αβ (2.111), because of duality of the given quantities F αβ σ i and F αβ. Therefore, given that σ F0∙ ∗∂E i ∗∂ ln √h 1 ik ∗ T = ∇ ∙ = i + E i H Aik , (2.117) √g00 ∂x ∂x − c i F0∗∙ 1 ipq ik ikp ∙ = ε Hpq ,F ∗ = ε Ep , (2.125) √g 2 − ∂Hik ∂ ln √h 1 00 Bi = F σi = ∗ + Hik ∗ F Hik ∇σ ∂xk ∂xk − c2 k − the rest components of the pseudotensor F αβ, formulated with the (2.118) ∗ 1 ∂Ei chr.inv.-projections of its dual tensor F αβ (2.111) are ∗ + DEi , − c ∂t 0 1 kpq 1 F0∗∙ = vk ε Hpq + (vpEq vqEp) , (2.126) where A is the antisymmetric chr.inv.-tensor of non-holonomity ∙ 2c c − ik of the space. Taking into account that 0 1 pq 1 pq Fi∗∙ = εi∙ Hpq + εi∙ (vpEq vqEp) i ∙ 2 g ∙ c ∙ − − ∗∂E i ∗∂ ln √h i √ 00 + E = ∗ E (2.119) (2.127) ∂xi ∂xi ∇i 1 1 εkpqv v H εkpqv v (v E v E ) , −c2 i k pq − c3 i k p q − q p is the chr.inv.-divergence of the vector Ei, and also that 0i 1 ipq 1 F ∗ = ε Hpq + (vpEq vqEp) , (2.128) ik 1 ik ik ∗ H F H = ∗ H (2.120) 2√g00 c − ∇k − c2 k ∇k 1 pq F 0i = √g00 εipqH , (2.129) is the physical chr.inv.-divergence of thee tensor Hik we arrive to ∗ 2 the final equations for chr.inv.-projections of the divergence of an k kp 1 kpq 1 mkp ∙ αβ Fi∗∙ = εi Ep vi ε Hpq 2 vivm ε Ep , (2.130) arbitrary antisymmetric tensor F ∙ ∙ − 2c − c

p 1 pq i 1 ik F ik = εikp E vqH , (2.131) T = ∗ i E H Aik , (2.121) ∗ − c ∇ − c while its square is i i ik 1 ∗∂E i B = ∗ H + DE . (2.122) αβ ipq ∇k − c ∂t F αβF ∗ = ε (EpHiq EiHpq) , (2.132) ∗ − ipq So forth, we calculatee chr.inv.-projections of the divergence of where ε is the three-dimensional discriminant chr.inv.-tensor αβ the pseudotensor F ∗ , which is dual to the given antisymmetric (2.73, 2.74). Then the chr.inv.-projections of the divergence of the αβ tensor F αβ, namely pseudotensor F ∗ are F σ ∂H i ∂ ln √h 1 αβ 1 αβμν 1 μν σ 0∗∙ ∗ ∗ i ∗ ik F ∗ = E Fμν ,F αβ = Eαβμν F . (2.123) ∇ ∙ = i + H∗ i E∗ Aik , (2.133) 2 ∗ 2 √g00 ∂x ∂x − c 2.5 Divergence and rotor 55 56 Chapter 2 Tensor algebra and the analysis

ik α σi ∗∂E∗ ik ∗∂ ln √h 1 ik vector field A is a covariant antisymmetric 2nd rank tensor, de- σ F ∗ = + E∗ FkE∗ i k 2 fined as follows∗ ∇ ∂x ∂x − c − (2.134) i 1 ∗∂H∗ i ∂A ∂A + DH∗ , F = A A = ν μ , (2.140) − c ∂t μν μ ν ν μ μ ν ∇ − ∇ ∂x − ∂x i or, using the chr.inv.-divergence ∗ i H∗ and the physical chr.inv.- where μ Aν is the absolute derivative of the Aα with respect to the ik ∇ divergence ∗ E∗ , as well as (2.119, 2.120), we obtain ∇ μ ∇k coordinate x ∂Aν σ σ μ Aν = Γ Aσ . (2.141) e σ F0∗∙ i 1 ik μ νμ ∇ ∙ = ∗ i H∗ E∗ Aik , (2.135) ∇ ∂x − √g ∇ − c 00 The rotor, contracted with the four-dimensional absolutely anti- αβμν i symmetric discriminant tensor E (2.65), is the pseudotensor σi ik 1 ∗∂H∗ i σ F ∗ = ∗ k E∗ + DH∗ . (2.136) ∇ ∇ − c ∂t αβ αβμν αβμν ∂Aν ∂Aμ F ∗ = E ( A A ) = E . (2.142) ∇μ ν − ∇ν μ ∂xμ − ∂xν Aside for the divergencee of vectors, antisymmetric tensors and pseudotensors of the 2nd rank, we need also to deduce chr.inv.- In electrodynamics F (2.140) is the tensor of a electromagnetic projections of the divergence of a symmetric tensor of the 2nd rank μν field (Maxwell’s tensor), which actually is the rotor of the four- (we will need them to obtain the conservation laws in chr.inv.- dimensional potential Aα of this electromagnetic field. Therefore form). We take them as a whole from Zelmanov [4]. Like as Zel- later we will need formulas for chr.inv.-projections of the four- manov did it in his theory, denoting chr.inv.-projections of a sym- dimensional rotor F and of its dual pseudotensor F αβ, expressed metric tensor T αβ as follows μν ∗ through chr.inv.-projections of the four-dimensional vector poten- T T i tial Aα (2.81), which forms them. 00 = ρ , 0 = Ki,T ik = N ik, (2.137) g00 √g00 Let us calculate components of the rotor Fμν , taking into account 00 that F00 = F = 0 just like for any other antisymmetric tensor. As according to [4] we have a result, after some algebra we obtain

σ σ T0 ∗∂ρ ik i 2 i w ϕ ∗∂ϕ 1 ∗∂qi ∇ = + ρD + D N + c ∗ K F K , (2.138) ik i i F0i = 1 2 2 Fi i , (2.143) √g00 ∂t ∇ − c − c c − ∂x − c ∂t i σi ∗∂K i i i k ∗∂qi ∗∂qk ϕ ∂vi ∂vk σ T = c + cDK + 2c Dk + Ak∙ K + Fik = k i + k i + ∇ ∂t ∙ (2.139) ∂x − ∂x c ∂x − ∂x 2 ik ik i (2.144) + c ∗ N F N ρF . 1 ∂ϕ ∂ϕ 1 ∂q ∂q ∇k − k − + v ∗ v ∗ + v ∗ k v ∗ i , c i ∂xk − k ∂xi c2 i ∂t − k ∂t Among the internal (scalar) product of a tensor with the operator of absolute derivation , which is the divergence of this tensor field, ∇ we can consider a difference between the covariant derivatives of 0 ϕ k 1 k ∗∂ϕ 1 ∗∂qk F0∙ = vkF + v + , (2.145) ∙ −c3 c ∂xk c ∂t the field. This quantity is known as a rotor of the field, because from geometric viewpoint it is the vortex (rotation) of the field. ∗See §98 in Raschewski’s well-known book [14]. Actually, rotor is not the tensor The absolute rotor is the rotor of a n-dimensional tensor field in (2.140), but its dual pseudotensor (2.142), because of the invariance in respect of a n-dimensional space. The rotor of an arbitrary four-dimensional reflection is necessary for any rotations. 2.6 Laplace’s operator and d’Alembert’s operator 57 58 Chapter 2 Tensor algebra and the analysis

0 1 ϕ ∗∂ϕ 1 ∗∂qk Its four-dimensional generalization in a pseudo-Riemannian Fk∙ = Fk + ∙ −√g c2 − ∂xk − c ∂t space is d’Alembert’s general covariant operator 00 αβ 2ϕ m 1 m ∗∂ϕ 1 ∗∂qm = g α β . (2.154) + v Amk + vkv + (2.146) ∇ ∇ c2 c2 ∂xm c ∂t − In the Minkowski space, the operators take the form 1 m ∗∂qm ∗∂qk ϕ m v vkvmF , ∂2 ∂2 ∂2 − c ∂xk − ∂xm − c4 Δ = + + , (2.155) ∂x1∂x1 ∂x2∂x2 ∂x3∂x3 2 2 2 2 2 i im ∗∂qm ∗∂qk 1 im ∗∂ϕ 1 ∂ ∂ ∂ ∂ 1 ∂ Fk∙ = h h vk = = Δ . (2.156) ∙ ∂xk − ∂xm − c ∂xm − c2 ∂t2 ∂x1∂x1 ∂x2∂x2 ∂x3∂x3 c2 ∂t2 (2.147) − − − − 1 im ∗∂qm ϕ i 2ϕ i Our goal is to apply the d’Alembert operator to scalar and vector h vk + vkF + Ak∙ , − c2 ∂t c3 c ∙ fields, located in a pseudo-Riemannian space, and also to present the results in chr.inv.-form. At first, we apply the d’Alembert op- 1 ∂ϕ 1 ∂q ϕ F 0k = hkm ∗ + ∗ m F k+ erator to a four-dimensional scalar field ϕ, because in this case the √g ∂xm c ∂t − c2 00 (2.148) calculations will be much simpler (the absolute derivative of a scalar 1 ∗∂qn ∗∂qm 2ϕ + vnhmk v Amk , field α ϕ does not contain the Christoffel symbols, so it becomes c ∂xm − ∂xn − c2 m ∇ regular derivative) i iα ∂ϕ ∂ϕ ∂2ϕ F0∙ g F0α ik ∗∂ϕ 1 ∗∂qk ϕ i αβ αβ αβ ∙ = = h + F , (2.149) ϕ = g α β ϕ = g = g . (2.157) √g √g ∂xk c ∂t − c2 ∇ ∇ ∂xα ∂xβ ∂xα∂xβ 00 00 So forth we formulate components of the fundamental metric ik iα kβ im kn ∗∂qm ∗∂qn 2ϕ ik ik F = g g Fαβ = h h A , (2.150) tensor in the terms of chronometrically invariants. For g from ∂xn − ∂xm − c ik ik 0i (1.18) we obtain g = h . Components g are obtained from the − i 0i where (2.149, 2.150) are chr.inv.-projections of the rotor Fμν . Re- linear velocity of the space rotation v = c g g00 αβ − spectively, chr.inv.-projections of its dual pseudotensor F ∗ are 1 g0i = vi. p (2.158) i αi −c √g00 F0∗∙ g0αF ∗ ikm 1 ∗∂qk ∗∂qm ϕ ∙ = = ε m k Akm , (2.151) 00 √g00 √g00 2 ∂x − ∂x − c Component g can be obtained from the main property of the βσ β fundamental metric tensor gασg = gα. This property, being taken ik ikm ϕ ∗∂ϕ 1 ∗∂qm under condition α = β = 0, gives F ∗ = ε F , (2.152) c2 m − ∂xm − c ∂t 0σ 00 0i 0 g0σg = g00g + g0ig = δ0 = 1, (2.159) i αi αiμν where F0∗∙ = g0αF ∗ = g0αE Fμν can be calculated using already ∙ then, taking into account that mentioned components of the rotor Fμν (2.143–2.148). w 2 1 w g00 = 1 2 , g0i = vi 1 2 , (2.160) §2.6 Laplace’s operator and d’Alembert’s operator − c − c − c we obtain the formula Laplace’s operator is the three-dimensional operator of derivation 1 1 g00 = 1 v vi , v vi = h vivk = v2. (2.161) w 2 2 i i ik 2 ik 1 − c Δ = = = g i k . (2.153) 2 ∇ ∇ ∇ − ∇ ∇ − c 2.6 Laplace’s operator and d’Alembert’s operator 59 60 Chapter 2 Tensor algebra and the analysis

Substituting the obtained formulas into ϕ (2.157) and replacing made during calculations, the terms of the final equations will not regular operators of derivation with the chr.inv.-operators, we ob- be chronometric invariants”. tain d’Alembertian of the scalar field in chr.inv.-form After some difficult algebra, we obtain required formulas for the α 2 2 chr.inv.-projections of the d’Alembertian of the vector field A in a 1 ∗∂ ϕ ik ∗∂ ϕ ϕ = h = ∗ ϕ , (2.162) pseudo-Riemannian space c2 ∂t2 − ∂xi∂xk i 1 ∗∂ k 1 ∗∂q 1 i ∗∂ϕ where, in contrast to the regular operators, is the d’Alembert T = ∗ ϕ Fk q Fi + F + ∗ − c3 ∂t − c3 ∂t c2 ∂xi chr.inv.-operator, and Δ is the Laplace chr.inv.-operator ∗ ∂ϕ 1 ∂ D ∂ϕ + hikΔm ∗ hik ∗ [(D + A ) qn] + ∗ 1 ∂2 ∂2 ik m i kn kn 2 ∗ ik ∗ ∂x − c ∂x c ∂t − (2.168) ∗ = 2 2 h i k , (2.163) m c ∂t − ∂x ∂x 1 k ∗∂q 2 i k ϕ i ϕ mk Dm k + 3 AikF q + 4 FiF 2 Dmk D 2 − c ∂x c c − c − ik ik ∗∂ ∗Δ = g ∗ i ∗ k = h . (2.164) D 1 1 − ∇ ∇ ∂xi∂xk F qm Δm Dk qn + hikΔm (D + A ) qn, − c3 m − c kn m c ik mn mn Now we apply the d’Alembert operator to an arbitrary four- i dimensional vector field Aα i i 1 ∗∂ i i k D ∗∂q B = ∗ A + Dk + Ak∙ q + + c2 ∂t ∙ c2 ∂t α μν α k A = g μ ν A . (2.165) 1 i i ∗∂q 1∗∂ i 1 i ∗∂ϕ ∇ ∇ + Dk + Ak∙ ϕF F + c2 ∙ ∂t − c3 ∂t − c3 ∂t Because Aα is four-dimensional vector, chr.inv.-projections of 1 ∂qi 1 ∂ϕ 1 this quantity are + F k ∗ Dmi + Ami ∗ + qkF F i + c2 ∂xk − c ∂xm c4 k σ μν σ T = bσ A = bσ g μ ν A , (2.166) ∇ ∇ 1 i m k ϕ i D i i n (2.169) + 2 Δkmq F 3 DF + 2 Dn + An∙ q i i σ i μν σ c − c c ∙ − B = hσ A = hσ g μ ν A . (2.167) ∇ ∇ km ∗∂ i n 1 ∗∂ i i h Δmnq + ϕ Dm + Am∙ + In general, to obtain d’Alembertian in chr.inv.-form for a vector − ∂xk c ∂xk ∙ field in a pseudo-Riemannian space is not a trivial task, because the i n n i p ϕ i n n + ΔknΔmp ΔkmΔnp q + Δkn (Dm + Am∙ ) Christoffel symbols are not zeroes, so formulas for projections of − c ∙ − the second derivatives take dozens of pages∗. The main criterion of n i n i i i ∗∂q n ∗∂q correct calculations is Zelmanov’s rule of chronometric invariance: Δkm Dn + An∙ + Δkn Δkm , − ∙ ∂xm − ∂xn “Correct calculations make all terms in the final equations chro- i nometrically invariant quantities. That is they consist of chr.inv.- where ∗ ϕ and ∗ q are results from application of the d’Alembert quantities themselves, their chr.inv.-derivatives and also chr.inv.- chr.inv.-operator (2.163) to the quantities ϕ = A0 and qi = Ai, properties of the reference space. If any single mistake has been √g00 which are chr.inv.-projections (physical observable components) of α ∗This is one of the reasons why practical applications of the theory of the vector A , so that electromagnetic field are mainly calculated in a Galilean reference frame in the 2 2 2 i 2 i Minkowski space (the space-time of the Special Theory of Relativity), where the 1 ∗∂ ϕ ik ∗∂ ϕ i 1 ∗∂ q km ∗∂ q ∗ ϕ= h , ∗ q = h . (2.170) Christoffel symbols are zeroes. As a matter of fact, general covariant notation c2 ∂t2 − ∂xi∂xk c2 ∂t2 − ∂xk∂xm hardly permits unambiguous interpretation of calculation results, unless they are formulated with physical observable quantities (chronometric invariants) or demoted D’Alembert operator from a tensor field, equalized to zero or to a simple specific case, like one in the Minkowski space, for instance. not zero, gives d’Alembert equations for this field. From physical 2.7 Conclusions 61 viewpoint these are equations of propagation of waves of the field. If d’Alembertian is not zero, these are equations of propagation of waves enforced by the field-inducing sources (the d’Alembert equations with the sources). For instance, the sources in electro- Chapter 3 magnetic fields are electric charges and the currents. If d’Alembert operator for a field is zero, then these are equations of propagation of CHARGED PARTICLE waves of the field not related to any sources. If the space-time area IN PSEUDO-RIEMANNIAN SPACE under our consideration aside for the tensor field in this question is filled with another medium, then the d’Alembert equations will gain an additional term in their right parts to characterize the media, which can be obtained from the equations which define it. §3.1 Problem statement

§2.7 Conclusions In this Chapter we will set forth the theory of electromagnetic field and moving charged particles in a four-dimensional pseudo- We are now ready to outline the results of this Chapter. Aside for Riemannian space. The peculiarity, which differs this theory from general knowledge of tensor and tensor algebra, we have obtained regular relativistic electrodynamics, is that all equations here will some tools to facilitate our calculations in the next Chapters. Equa- be given in chr.inv.-form (expressed through physical observable lity to zero of the absolute derivative of the dynamic vector of quantities, in other word). a particle to that direction the particle moves sets equations of An electromagnetic field is commonly studied as a vector field motion of this particle. Equality to zero of the divergence of a of the electromagnetic four-dimensional potential Aα, located in the vector field sets the Lorentz condition and the continuity equation four-dimensional pseudo-Riemannian space. Its time component for this field. Equality to zero of the divergence of a symmetric is known as the scalar potential ϕ of the field, while its spatial tensor of the 2nd rank sets the conservation law, while equality to components make up so-called the vector-potential Ai. The four- zero of an antisymmetric tensor of the 2nd rank (and of its dual dimensional electromagnetic potential Aα in CGSE and Gaussian pseudotensor) set the Maxwell equations. The rotor of a vector systems of units has the dimension field, being applied to an electromagnetic field, is the field tensor 1 1 α 2 2 (the Maxwell tensor). The d’Alembert equations for a given field A [ gram ×cm /sec]. (3.1) are equations of propagation of the field waves. So, we have a brief list of possible applications of the mathematical apparatus of As it is evidently, its components ϕ and Ai have the same dimen- which we came into possession. So forth, if we now come across sions. Therefore, studying electromagnetic fields is substantially an antisymmetric tensor or a differential operator, then we may different from studying gravitational fields: according to the theory simply use templates already obtained in this Chapter. of chronometric invariants, gravitational inertial force F i and gravitational potential w (1.38) are functions of geometric properties of the space only, while electromagnetic fields (the fields of the ♦ electromagnetic potential Aα) has not been “geometrically interpreted” yet, so we have to study electromagnetic fields just as an external vector fields introduced into the space. Equations of Classical Electrodynamics — Maxwell’s equations, which define the relationship between the electric and magnetic components of the given field, — had been obtained long before 3.2 The electromagnetic field tensor 63 64 Chapter 3 Motion of charged particles theoretical physics accepted the terms of Riemannian geometry and As it easy to see, this formula is a general covariant generaliza- even Minkowski’s space of the Special Theory of Relativity. Later, tion of three-dimensional quantities in Classical Electrodynamics when electrodynamics was set forth in the Minkowski space under 1 ∂A~ the name of relativistic electrodynamics, the Maxwell equations E~ = −→ϕ , H~ = rotA,~ (3.3) had been obtained in four-dimensional form. Then the Maxwell −∇ − c ∂t equations in general covariant form, acceptable for any pseudo- where E~ and H~ are the strength vectors of the electric and magnetic Riemannian space had been obtained. But having accepted general components of the field, respectively. Here ϕ is the scalar potential covariant form, the Maxwell equations became less illustrative, and A~ is the spatial vector-potential of the field, and which used to be an advantage of Classical Electrodynamics. On ∂ ∂ ∂ the other hand, four-dimensional equations in the Minkowski space −→ = ~ı + ~ + ~k (3.4) can be simply presented as their scalar (time) and vector (spatial) ∇ ∂x ∂y ∂z components, because in a Galilean reference frame they are ob- is the gradient operator in three-dimensional Euclidean space. servable quantities by definition. But when we turn to an inhomo- At first, in this §3.2 we are going to discuss which components geneous, anisotropic, curved, and deforming pseudo-Riemannian of the electromagnetic field tensor Fαβ are physical observable space, the problem of comparing the vector and scalar components quantities in a given pseudo-Riemannian space. Then we are going in general covariant equations with equations of Classical Electro- to find a relationship between the observable quantities and the dynamics becomes non-trivial. In other word, a question arises electric strength E~ and the magnetic strength H~ of the field in which quantities in relativistic electrodynamics can be assumed Classical Electrodynamics. The strength vectors will be also ob- physical observable. tained in the pseudo-Riemannian space, which in general is inho- Thus the equations of relativistic electrodynamics in a pseudo- mogeneous, anisotropic, curved, and deformed. Riemannian space shall be formulated with physical observable An important note should be taken. Because in the Minkowski components (chr.inv.-projections) of the electromagnetic field po- space (the space-time of the Special Theory of Relativity) in an tential and also observable properties of the space. We are going inertial reference frame (the one, which moves linearly at a constant to tackle the problem using the mathematical apparatus of chrono- velocity) the metric is metric invariants, namely — projecting general covariant quantities on time lines and the spatial section of an real observer. The results ds2 = c2dt2 dx2 dy2 dz2, (3.5) we are going to obtain in this way will help us to arrive to observable − − − generalization of the basic quantities and the laws of relativistic and components of the fundamental metric tensor are electrodynamics and also Classical Electrodynamics, which will g = 1 , g = 0 , g = g = g = 1 , (3.6) take into account that effects of physical and geometric properties 00 0i 11 22 33 − of the observer’s reference space. no difference exists between covariant and contravariant components of Aα (in particular, this is why all calculations in the Min- §3.2 Observable components of the electromagnetic field tensor. kowski space are much simpler) Invariants of the field 0 i ϕ = A0 = A ,Ai = A . (3.7) By definition, the tensor of a electromagnetic field is the rotor of − its four-dimensional potential Aα. This field tensor is also referred In the pseudo-Riemannian space (and in Riemannian spaces in to as Maxwell’s tensor general) there is a difference, because the metric has more general ∂A ∂A form. Therefore the scalar potential and the vector-potential of F = A A = ν μ . (3.2) μν ∇μ ν − ∇ν μ ∂xμ − ∂xν the electromagnetic field we are considering shall be defined as 3.2 The electromagnetic field tensor 65 66 Chapter 3 Motion of charged particles chr.inv.-projections (physical observable components) of the four- so the covariant (lower-index) chr.inv.-quantities are dimensional potential Aα ∂ϕ 1 ∂q ϕ E = h Ek = ∗ + ∗ i F , (3.14) A i ik i 2 i ϕ = bαA = 0 , qi = hi Aσ = Ai. (3.8) ∂x c ∂t − c α g σ √ 00 ∂q ∂q 2ϕ H = h h Hmn = ∗ i ∗ k A , (3.15) The rest components of the Aα, being not chr.inv.-quantities, are ik im kn ∂xk − ∂xi − c ik i formulated with the ϕ and q as follows m m while the mixed components Hk∙ = H k ∙ are obtained from the ik ∙ − ∙ m im 0 1 1 i ϕ H using the metric chr.inv.-tensor hik, so that Hk∙ = hkiH . In A = ϕ + viq ,Ai = qi vi . (3.9) ∙ w 1 ∗∂hik 1 c − − c this case the space deformations tensor Dik = (1.40) is also − c2 2 ∂t present in the formulas, but in an implicit way and appears when Note, in accordance with the theory of chronometric invariants, m we substitute components qk = hkmq into the time derivatives. the covariant chr.inv.-vector qi is obtained from the contravariant i Besides, we may as well formulate other components of the chr.inv.-vector q as a result of lowering the index using the metric i k electromagnetic field tensor Fαβ with its chr.inv.-projections E and chr.inv.-tensor hik as follows qi = hikq . To the contrary, the regular Hik (3.11) using formulas for components of an arbitrary antisym- covariant vector Ai, which is not chr.inv.-quantity, is obtained as a metric tensor (2.112–2.115). This is possible because the generalized result of lowering the index using the fundamental metric tensor, i ik α formulas (2.112–2.115) contain the E and H in “implicit” form, so that Ai = giαA . irrespective of whether they are components of a rotor or of an According to the general formula for the square of a vector α antisymmetric tensor of any other kind. (2.39), the square of the potential A in the accompanying reference i In the Minkowski space, where no acceleration F , rotation Aik frame is and deformations Dik, that formula for Ei becomes α α β 2 i k 2 2 AαA = gαβ A A = ϕ hik q q = ϕ q , (3.10) − − ∂ϕ 1 ∂Ai Ei = + , (3.16) and the quantity is: a real quantity, if ϕ2 > q2; an imaginary quantity, ∂xi c ∂t 2 2 2 2 if ϕ < q ; zero (isotropic) quantity, if ϕ = q . or in three-dimensional vector form Now using components of the potential Aα (3.8, 3.9) in the 1 ∂A~ definition of the electromagnetic field tensor Fαβ (3.2), formulating E~ = −→ϕ + , (3.17) regular derivatives with chr.inv.-derivatives (1.33), and using for- ∇ c ∂t mulas for components of the rotor of an arbitrary vector field which, up within the sign, matches the formula for E~ in Classical (2.143–2.150), we obtain chr.inv.-projections of the tensor Fαβ Electrodynamics. Now we formulate the electric and magnetic strengthes through F i giαF ∂ϕ 1 ∂q ϕ 0∙ 0α ik ∗ ∗ k i components of the field pseudotensor F αβ, which is dual to the ∙ = = h k + 2 F , (3.11) ∗ √g00 √g00 ∂x c ∂t − c αβ 1 αβμν Maxwell tensor of this field F ∗ = 2 E Fμν (2.123). So, in accordance with (2.124), chr.inv.-projections of this pseudotensor are ik iα kβ im kn ∗∂qm ∗∂qn 2ϕ ik F = g g Fαβ = h h A . (3.12) ∂xn − ∂xm − c i i F0∗∙ ik ik H∗ = ∙ ,E∗ = F ∗ . (3.18) We denote the chr.inv.-projections of the electromagnetic field √g00 tensor in the classic way as follows Using formulas for components of an arbitrary pseudotensor i αβ i F0∙ ik ik F ∗ , we have obtained in Chapter 2 (2.125–2.131), and also for- E = ∙ ,H = F , (3.13) √g00 mulas for Ei and Hik (3.14, 3.15), we obtain expanded formulas for 3.2 The electromagnetic field tensor 67 68 Chapter 3 Motion of charged particles

i ik H∗ and E∗ , namely true in a pseudo-Riemannian space), because the Minkowski space itself does not accelerate the reference frame and neither rotates i 1 imn ∗∂qm ∗∂qn 2ϕ 1 imn H∗ = ε A = ε H , (3.19) nor deforms it. Therefore such effects in the Minkowski space are 2 ∂xn − ∂xm − c mn 2 mn strictly relative. In relativistic electrodynamics we introduce invariants, which ik ikn ϕ ∗∂ϕ 1 ∗∂qn ikn E∗ = ε F = ε E . (3.20) characterize the electromagnetic field we are considering — in other c2 n − ∂xn − c ∂t − k word, the field invariants

It is easy to see, the following pairs of tensors are dual conjugate: μν 0i ik i ik i J1 = Fμν F = 2F0iF + FikF , (3.25) H∗ and Hmn, E∗ and Em. The chr.inv.-pseudovector H∗ (3.19) includes the term μν 0i ik J2 = Fμν F ∗ = 2F0iF ∗ + FikF ∗ . (3.26) 1 imn ∗∂qm ∗∂qn 1 imn ε = ε (∗ q ∗ q ) , (3.21) 2 ∂xn − ∂xm 2 ∇n m − ∇m n The first invariant is scalar, while the second is pseudoscalar. Formulating them with components of the field tensor, we obtain which is the chr.inv.-rotor of the three-dimensional vector field qm. Here is also the term J = H Hik 2E Ei,J = εimn (E H E H ) , (3.27) 1 ik − i 2 m in − i nm 1 imn 2ϕ 2ϕ i ε A = Ω∗ , (3.22) μν 2 c mn c and using formulas for components of the field pseudotensor F ∗ we have obtained in Chapter 2 we write down them as follows i 1 imn where Ω∗ = 2 ε Amn is the chr.inv.-pseudovector of angular ve- i i i locities of the space rotation. In a Galilean reference frame in J1 = 2 EiE H iH∗ ,J2 = 4EiH∗ . (3.28) − − ∗ − the Minkowski space, because no any acceleration, rotations, and deformations in that space, the obtained formula for the magnetic Because the quantities J1 and J2 are invariants, we conclude: i strength chr.inv.-pseudovector H∗ (3.19) takes the form If in a reference frame the squares of the electric and magnetic • 2 2 strengthes are equal E = H∗ , then this equality remains un- i 1 imn ∂qm ∂qn H∗ = ε , (3.23) changed in any other reference frame; 2 ∂xn − ∂xm If in a reference frame the electric and magnetic strengthes • i or in three-dimensional vector form, is are orthogonal EiH∗ = 0, then this orthogonality remains unchanged in any other reference frame. H~ = rotA.~ (3.24) 2 2 An electromagnetic field, where the conditions E = H∗ and i Therefore, the structure of a pseudo-Riemannian space affects EiH∗ = 0 are true, so where one or both of the field invariants electromagnetic fields, located in it, due to the fact that chr.inv.- (3.28) are zeroes, is known as isotropic. Here the term “isotropic” vectors of the electric strength Ei (3.14) and the magnetic strength stands not for location of this field in light-like area of the space i H∗ (3.19) depend on gravitational potential and rotation of this (as it is assumed in geometry), but rather for the field’s property space. of equal emissions at any direction in the three-dimensional space The same will be true as well in the Minkowski space, if a non- (the spatial section). inertial reference frame, which rotates and moves with acceleration, The electromagnetic field invariants can be also formulated with is assumed as the observer’s reference frame. But in the Minkowski chr.inv.-derivatives of the scalar chr.inv.-potential ϕ and the vector space we can always find a Galilean reference frame (that is not chr.inv.-potential qi (3.8) as well as with chr.inv.-properties of the 3.3 Maxwell’s equations 69 70 Chapter 3 Motion of charged particles

1 3 gram 2 2 observer’s reference space. So, we have where ρ [ /cm ×sec] is the electric charge density (the amount e 1 3 3 1 1 [ gram 2 cm 2/sec] of the charge within 1 cm ) and ~j [ gram 2 /cm 2 sec2 ] is ∂q ∂q ∂q ∂ϕ ∂ϕ × × J = 2 himhkn ∗ i ∗ k ∗ m hik ∗ ∗ the current density vector. Equations containing the field-inducing 1 ∂xk − ∂xi ∂xn − ∂xi ∂xk − sources ρ and ~j are known as the 1st group of the Maxwell equa- 2 ∗∂ϕ ∗∂qk 1 ∗∂qi ∗∂qk 8ϕ tions, while equations, which do not contain the sources are known hik hik + Ω Ωi (3.29) − c ∂xi ∂t − c2 ∂t ∂t c2 i − as the 2nd group of the Maxwell equations. 2ϕ ∂q 2ϕ ∂ϕ 2ϕ ∂q ϕ The first equation in the 1st group is Biot-Savart’s law, the εimnΩ ∗ i + ∗ F i + ∗ i F i F F i , − c m ∂xn c2 ∂xi c3 ∂t − c4 i second is Gauss’ theorem, both in differential notation. The first and the second equations in the 2nd group are differential notation of Faraday’s law of electromagnetic induction and the condition 1 imn ∗∂qm ∗∂qn 4ϕ i ∗∂ϕ 1 ∗∂qi ϕ J = ε Ω∗ + F . (3.30) that no magnetic charges exist, respectively. Totally, we have 8 2 2 ∂xn − ∂xm − c ∂xi c ∂t − c2 i equations (two vector ones and two scalar ones) in 10 unknowns: We can know physical conditions in isotropic electromagnetic three components of the E~ , three components of the H~ , three fields, equaling the formulas (3.29, 3.30) to zero. Doing this, we can components of the ~j, and one component of the ρ. see that the conditions of equality of the lengths of the electric and A correlation between the field sources ρ and ~j is set by the law 2 2 i magnetic strengthes E = H∗ and their orthogonality EiH∗ = 0 in of conservation of electric charge a pseudo-Riemannian space depend on not only properties of the ∂ρ field itself (the scalar potential ϕ and the vector-potential qi) but + div~j = 0 , (3.32) i ∂t also on acceleration F , rotation Aik and deformations Dik of the i space itself. In particular, the vectors Ei and H∗ are orthogonal if which is a mathematical notation of an experimental fact that an i the space is holonomic Ω∗ = 0, while the spatial field of the vector- electric charge can not be destroyed, but is merely re-distributed ∂q ∂q potential qi is rotation-free εimn ∗ m ∗ n = 0. between contacting charged bodies. ∂xn − ∂xm Now we have a system of 9 equations in 10 unknowns, so the system defining the field and its sources is still indefinite. The 10th equation, which makes the system definite (the number of equations §3.3 Maxwell’s equations in chronometrically invariant form. shall be the same as that of unknowns), is Lorentz’ condition, which The law of conservation of electric charge. Lorentz’ condition constructs the scalar and vector potentials of the field as follows

In Classical Electrodynamics, correlations of the electric strength of 1 ∂ϕ 1 1 + divA~ = 0 . (3.33) ~ gram 2 cm 2 sec an electromagnetic field E [ / × ] to its magnetic strength c ∂t 1 1 H~ [ gram 2 /cm 2 sec] are set forth in Maxwell’s equations, which had × The Lorentz condition is derived from the fact that the scalar originally been derived from generalization of experimental data. In potential ϕ and the vector-potential A~ of any given electromagnetic the middle 19th century Maxwell had showed that if an electromag- field, related to the strength vectors E~ and H~ with (3.3), are defined netic field is induced in vacuum by given charges and currents, then ambiguously from them: E~ and H~ in (3.3) remain unchanged if we the resulting field is defined by two groups of equations [16] replace ~ ~ 1 ∂Ψ 1 ∂E~ 4π 1 ∂H~ A = A0 + −→Ψ , ϕ = ϕ0 , (3.34) rotH~ = ~j rotE~ + = 0 ∇ − c ∂t − c ∂t c I , c ∂t II , (3.31)   where ψ is an arbitrary scalar. Evidently, ambiguous definitions ~  ~  divE = 4πρ  divH = 0  of ϕ and A~ permit other correlations between the quantities aside   3.3 Maxwell’s equations 71 72 Chapter 3 Motion of charged particles for the Lorentz condition. Nevertheless, it is the Lorentz condition, So forth we are going to consider the Maxwell equations in which enables transformation of the Maxwell equations into wave a pseudo-Riemannian space to obtain them in chr.inv.-form, i. e. equations. formulated with physical observable quantities. Here is how it happens. In a four-dimensional pseudo-Riemannian space the Lorentz The equation divH~ = 0 (3.31) is satisfied completely, if we assume condition has general covariant form H~ = rotA~. In this case the first equation in the 1st group (3.31) takes ∂Aσ the form Aσ = + Γσ Aμ = 0 , (3.41) 1 ∂A~ ∇σ ∂xσ σμ rot E~ + = 0 , (3.35) c ∂t so it is a condition of conservation of the four-dimensional potential which has the solution of a given electromagnetic field we are considering. The law of conservation of electric charge (the continuity equation) is 1 ∂A~ E~ = −→ϕ . (3.36) −∇ − c ∂t jσ = 0 , (3.42) ∇σ ~ ~ ~ Substituting H = rotA and E (3.36) into the 1st group of the where jα is the four-dimensional current vector, also known as the Maxwell equations, we obtain shift current. Chr.inv.-projections of the current vector jα are the electric charge density 1 ∂2A~ 1 ∂ϕ 4π ΔA~ −→ divA~ + = ~j , (3.37) 1 j0 − c2 ∂t2 − ∇ c ∂t − c ρ = , (3.43) c √g00 1 ∂ i Δϕ + divA~ = 4πρ , (3.38) and the spatial current density j . Using the chr.inv.-formula for c ∂t − the divergence of an arbitrary vector field (2.107), we obtain the 2 2 2 Lorentz condition (3.41) in chr.inv.-form where Δ = ∂ + ∂ + ∂ is Laplace’s regular operator. ∂x2 ∂y2 ∂z2 1 ∗∂ϕ ϕ i 1 i Constructing the potentials ϕ and A~ with the Lorentz condition + D + ∗ q F q = 0 , (3.44) c ∂t c i c2 i (3.33), we bring equations in the 1st group to the form ∇ − and also the continuity equation in chr.inv.-form ϕ = 4πρ , (3.39) − ∗∂ρ i 1 i 4π + ρD + ∗ i j Fi j = 0 . (3.45) A~ = ~j , (3.40) ∂t ∇ − c2 − c 2 ik n ∗∂ ln √h where = 1 ∂ Δ is d’Alembert’s regular operator. Here D = h Dik = Dn = is the spur of the tensor of c2 ∂t2 − ∂t Applying the d’Alembert operator to a field results equations the space deformations rate (1.40). Actually the spur is the rate of relative expansion of an elementary volume, while ∗ is the of propagation of waves of this field (see §2.6). For this reason, the ∇i obtained result implies that if the Lorentz condition is true, then the operator of chr.inv.-divergence (2.105). 1st group of the Maxwell equations (3.31) is a system of equations Because Fi (1.38) contains the first derivative of gravitational 2 1 i of propagation of waves of the scalar and vector electromagnetic potential w = c (1 √g00), the term Fi q takes into account that − c2 potentials (in the presence of the field-inducing sources — charges the pace of time is different at the opposite walls of the elementary and currents). The equations will be obtained in the next §3.4. volume. The mentioned formula for gravitational inertial force Fi 3.3 Maxwell’s equations 73 74 Chapter 3 Motion of charged particles

(1.38) takes also non-stationarity of the space rotation into account. of the 2nd rank (2.121, 2.122) and for its dual pseudotensor (2.135, Besides, because the operators of chr.inv.-derivation (1.33) are 2.136), we arrive to the Maxwell equations in chr.inv.-form

∗∂ 1 ∂ ∗∂ ∂ 1 ∗∂ i 1 ik = , i = i 2 vi , (3.46) ∗ i E H Aik = 4πρ ∂t 1 w ∂t ∂x ∂x − c ∂t ∇ − c − c2 i  I, (3.52) ik 1 ik 1 ∗∂E i 4π i α ∗ k H FkH +DE = j  the condition of conservation of the vector field A , namely — the ∇ − c2 − c ∂t c  equations (3.44, 3.45), directly depend on gravitational potential and  the velocity of the space rotation. i 1 ik  ∂ϕ ∂ρ ∗ i H∗ E∗ Aik = 0 Chr.inv.-quantities ∗ and ∗ are observable changes in time ∇ − c ∂t ∂t i  II . (3.53) of the chr.inv.-quantities ϕ and ρ. Chr.inv.-quantities ϕD and ρD ik 1 ik 1 ∗∂H∗ i ∗ k E∗ FkE∗ +DH∗ =0  are observable changes in time of spatial volumes, filled with the ∇ − c2 − c ∂t  quantities ϕ and ρ. The Maxwell equations in this chr.inv.-notation had first been If no any gravitational inertial forces, rotation and deformations  obtained by del Prado and Pavlov [19] independently after as Zel- in the space, then the obtained chr.inv.-formulas for the Lorentz manov asked them to do this. condition (3.44) and the charge conservation law (3.45) become Now, let us transform the Maxwell chr.inv.-equations in a way i i 1 ∂ϕ ∂qi ∂ ln √h that they include E and H∗ as unknowns. Obtaining the quantities + qi = 0 , (3.47) c ∂t ∂xi − ∂xi from their definitions (2.125, 2.124, 2.111) i 1 ∂ρ ∂j ∂ ln √h i mn + j = 0 , (3.48) H i = εimnH , (3.54) ∂t ∂xi − ∂xi ∗ 2 which in a Galilean reference frame in the Minkowski space are ik ikm ϕ ∗∂ϕ 1 ∗∂qm ikm E∗ = ε 2 Fm m = ε Em , (3.55) 1 ∂ϕ ∂qi ∂ρ ∂ji c − ∂x − c ∂t − + = 0 , + = 0 , (3.49) c ∂t ∂xi ∂t ∂xi and multiplying the first equation by εipq, we obtain or, in a regular vector notation ipq 1 ipq mn 1 p q q p mn pq 1 ∂ϕ ∂ρ ε H i = ε εimnH = (δmδn δmδn) H = H . (3.56) + divA~ = 0 , + div~j = 0 , (3.50) ∗ 2 2 − c ∂t ∂t Substituting the result as Hik = εmikH into the first equation which fully matches notations of the Lorentz condition (3.33) and m in the 1st group (3.52) we bring it to the form∗ the charge conservation law (3.32) in Classical Electrodynamics. Let us turn to the Maxwell equations. In a pseudo-Riemannian i 2 m ∗ i E Ω mH∗ = 4πρ , (3.57) space each pair of the equations merge into a single general covar- ∇ − c ∗ iant equation i 1 imn μσ 4π μ μσ where Ω∗ = 2 ε Amn is the chr.inv.-pseudovector of angular ve- σ F = j , σ F ∗ = 0 , (3.51) ik ikm ∇ c ∇ locities of the space rotation. Substituting E∗ = ε Em (3.55) into the first equation of the 2nd group (3.53), we obtain− where F μσ is contravariant (upper-index) form of the electromag- μσ netic field tensor, F ∗ is its dual pseudotensor. Using chr.inv.- i 2 m ∗ i H∗ + Ω mE = 0 . (3.58) formulas for the divergence of an arbitrary antisymmetric tensor ∇ c ∗ 3.3 Maxwell’s equations 75 76 Chapter 3 Motion of charged particles

ik mik Then, substituting H = ε H m into the second equation in 1 ∗∂ϕ ϕ i ∗ + D + ∗ q = 0 the Lorentz condition, (3.65) the 2nd group (3.52) we obtain c ∂t c ∇i

i √ ∗∂ρ i mik 1 mik 1 ∗∂E ∗∂ ln h i 4π i + ρD + ∗ i je = 0 the continuity equation. (3.66) ∗ k ε H m Fk ε H m + E = j (3.59) ∂t ∇ ∇ ∗ −c2 ∗ − c ∂t ∂t c In a Galilean reference frame in the Minkowski space the deter- mik e and, after multiplying both the parts by √h and taking ∗ k ε = 0 minant of the metric chr.inv.-tensor √h = 1, so it is not a subject to into account, we bring this formula (3.59) to the form ∇ deformations Dik = 0, rotation Ω m = 0 or acceleration Fi = 0 in the space. Then the Maxwell chr.inv.-equations∗ (3.63, 3.64), we have ikm 1 ikm 1 ∗∂ i 4π i ε ∗ k H m√h ε FkH m√h E √h = j √h (3.60) ∇ ∗ −c2 ∗ − c ∂t c obtained in the pseudo-Riemannian space of the General Theory of Relativity, bring us directly to the Maxwell equations of Classical or, in the other notation Electrodynamics written in tensor form

ikm 1 ∗∂ i 4π i i ε ∗ k H m√h E √h = j √h , (3.61) ∂E ∇ ∗ − c ∂t c = 4πρ ∂xi i 1  I, (3.67) where j √h is the current’s volume density and ∗ = ∗ F is i e k k 2 k ikm ∂H m ∂H k 1 ∂E 4π i ∇ ∇ − c e ∗ ∗ = j  physical chr.inv.-divergence (2.106), which takes into account that ∂xk − ∂xm − c ∂t c  the pace of time accounts is different at the oppositee walls of the  elementary volume. ∂H i  ∗ = 0 The obtained equation (3.60) is chr.inv.-notation for the Biot- ∂xi II . (3.68) Savart law in the pseudo-Riemannian space. i  ikm ∂Em ∂Ek 1 ∂H∗  ik ikm e = 0  Substituting E∗ = ε Em (3.55) into the second equation in ∂xk − ∂xm − c ∂t − the 2nd group (3.53), after similar transformations we obtain  The same equations, put into vector notation, will be similar ikm 1 ∗∂ i to the Maxwell classic equations in three-dimensional Euclidean ε ∗ k Em√h + H∗ √h = 0 , (3.62) ∇ c ∂t space (3.31). Besides, the obtained Maxwell chr.inv.-equations in which is chr.inv.-notatione for the Faraday law of electromagnetic the four-dimensional pseudo-Riemannian space (3.64) show that induction in the pseudo-Riemannian space. in the absence of the space rotation the chr.inv.-divergence of the i So, the final system of 10 chr.inv.-equations in 10 unknowns magnetic strength is zero ∗ i H∗ = 0. In other word, the field mag- ∇ (two groups of the Maxwell equations, the Lorentz condition, and netic component remains unchanged, if the space is holonomic. In the continuity equation), which define an electromagnetic field and the same time, the divergence of the electric strength in this case i its sources in the pseudo-Riemannian space, is is not zero ∗ i E = 4πρ (3.63), so the electric component is linked directly to the∇ charge density ρ. Hence a conclusion on “magnetic i 2 m ∗ i E Ω mH∗ = 4πρ charge”, if it actually exists, should be linked directly to the field ∇ − c ∗  I, (3.63) of rotation of the space itself. ikm √ 1 ∗∂ i√ 4π i√ ε ∗ k H m h E h = j h  ∇ ∗ − c ∂t c §3.4 D’Alembert’s equations for the electromagnetic potential  ei 2 m  and their observable components ∗ i H∗ + Ω mE = 0 ∇ c ∗  II , (3.64) ikm 1 ∗∂ i As we have already mentioned, d’Alembert’s operator, being applied ε ∗ E √h + H∗ √h = 0  ∇k m c ∂t  to a field, gives equations of propagation of waves of this field. For e  3.4 D’Alembert’s equations 77 78 Chapter 3 Motion of charged particles this reason, d’Alembert’s equations for the scalar electromagnetic and the space rotation are weak) we obtain potential ϕ are equations of propagation of waves of this scalar 1 ∂ 1 ∂qi 1 ∂ϕ ∂ϕ field, while for the spatial vector-potential A~ these are equations ∗ k ∗ i ∗ ik m ∗ ∗ ϕ 3 Fkq 3 Fi + 2 F i +h Δik m of propagation of waves of this vector field A~. − c ∂t − c ∂t c ∂x ∂x − (3.71) General covariant form of the d’Alembert equations for the elec- ik1 ∗∂ n 1 ik m n h i (Aknq ) + h ΔikAmnq = 4πρ, tromagnetic field potential Aα had been obtained by Stanyuko- − c ∂x c vich [20]. He did it, using the 1st group of the Maxwell general k i i 1 ∗∂ i k 1 i ∗∂q 1 ∗∂ ϕF μσ 4π μ ∗ A + Ak∙ q + Ak∙ covariant equations σ F = c j (3.51) and the Lorentz condition c2 ∂t ∙ c2 ∙ ∂t − c3 ∂t − σ ∇ σ A = 0 (3.41). His equations are 1 ∂ϕ 1 ∂qi 1 ∂ϕ 1 ∇ F i ∗ + F k ∗ Ami ∗ + Δi qmF k − c3 ∂t c2 ∂xk − c ∂xm c2 km − α α β 4π α A Rβ A = j , (3.69) − − c km ∗∂ i n 1 ∗∂ i (3.72) h k Δmnq + k ϕAm∙ + α αμ σ α − ∂x c ∂x ∙ where Rβ = g R μβσ is Ricci’s tensor, while R μβσ is Riemann- ∙ ∙ i n n i p ϕ i n n i Christoffel’s tensor of the space curvature. The term RαAβ is absent + ΔknΔmp ΔkmΔnp q + ΔknAm∙ ΔkmAn∙ + β − c ∙ − ∙ in the left part, if the Ricci tensor is zero, so the space metric n i ∗∂q ∗∂q 4π + Δi Δn = ji. satisfies Einstein’s equations away from gravitating masses (in vac- kn ∂xm − km ∂xn c uum, in other word). This term can be neglected in that case, where the space curvature is not significant. But even in the Minkowski We see that physical observable properties of the reference i i space this problem can be considered in the presence of acceleration space, namely — the quantities F , Aik, Dik, and Δkm make some i and rotation. Even this approximation may reveal, for instance, additional “sources” that among with the sources ϕ and j induce effects of acceleration and rotation of the observer’s reference body waves traveling through the given electromagnetic field. on the observable velocity of propagation of electromagnetic waves. Let us now analyze the results. At first we consider the obtained The reason for the above discussion is that obtaining chr.inv.- equations (3.71, 3.72) in a Galilean reference frame in the Minkow- projections of the d’Alembert equations in full is a very difficult ski space. Here the metric takes the form as in formula (3.5) and task. The resulting equations will be so bulky to make any unam- therefore the d’Alembert chr.inv.-operator ∗ (2.163) transforms 1 ∂2 biguous conclusions. Therefore we will limit the scope of our work into the d’Alembert regular operator ∗ = Δ = . Then the c2 ∂t2 − to fill the d’Alembert equations into chr.inv.-tensor form for an obtained equations (3.71, 3.72) will be electromagnetic field in a non-inertial reference frame in the Min- 4π kowski space. But that does not affect other Paragraphs in this ϕ = 4πρ , qi = ji, (3.73) Chapter, where we will be back to the pseudo-Riemannian space of − c the General Theory of Relativity. which fully matches the equations of Classical Electrodynamics So forth, calculating chr.inv.-projections of the d’Alembert equa- (3.39, 3.40). tions Now we return to the obtained d’Alembert chr.inv.-equations 4π Aα = jα (3.70) (3.39, 3.40). To make their analysis easier we denote all terms in − c the left parts as T in the scalar equation (3.39) and as Bi in the using general formulas (2.168, 2.169) and taking into account that vector equation (3.40). Transpositioning the variables into the right 1 α the observable charge density is ρ = g0αj , in the space out of parts and expanding the formulas for ∗ (2.173) we obtain c √g00 2 dynamic deformations and in the linear approximation (to within of 1 ∗∂ ϕ ik h ∗ ∗ ϕ = T + 4πρ , (3.74) higher order terms withheld — we assume that fields of gravitation c2 ∂t2 − ∇i ∇k 3.4 D’Alembert’s equations 79 80 Chapter 3 Motion of charged particles

2 i 2 i 1 ∗∂ q mk i i 4π i 1 ∗∂ q i 4π i h ∗ ∗ q = B + j , (3.75) = B + j . (3.81) c2 ∂t2 − ∇m ∇k c c2 ∂t2 c ik where h ∗ i ∗ k = ∗Δ is Laplace’s chr.inv.-operator. As it easy to In an inertial reference frame (the Christoffel symbols are zero) ∇ ∇ i see, if the potentials ϕ and q are stationary (they are not depend on ∗∂ϕ general covariant derivative equals to the regular one ∗ i ϕ = , time), the d’Alembert wave equations become the Laplace equations ∇ ∂t so the d’Alembert scalar chr.inv.-equation (3.74) is

∗Δϕ = T + 4πρ , (3.76) 2 2 1 ∗∂ ϕ ik ∗∂ ϕ 2 2 h i k = T + 4πρ . (3.82) i i 4π i c ∂t − ∂x ∂x ∗Δq = B + j , (3.77) c Here the left part takes the most simple form, which facilitates which characterize static states of this field. more detailed study of it. As it is known from the theory of oscilla- A field is homogeneous along a direction, if its regular derivative tions in mathematical physics, in the d’Alembert equations in their with respect to this direction is zero. In Riemannian spaces, a regular form field is homogeneous if its general covariant derivative is zero. If 1 ∂2ϕ ∂2ϕ ϕ = + gik (3.83) a tensor field located in a Riemannian space is considered in the a2 ∂t2 ∂xi∂xk accompanying reference frame, then observable inhomogeneity of the term a is the absolute value of the three-dimensional velocity this field is characterized by the difference of chr.inv.-operator ∗ i of elastic oscillations which spread across the field ϕ. taken from the field potential from zero [4, 7]. In other word, if for∇ Expanding chr.inv.-derivatives by spatial coordinates (3.46) we a scalar value A the condition A = 0 is true, then the field A is ∗ i bring the d’Alembert scalar equation (3.82) to the form observed as homogeneous. ∇ Therefore, the d’Alembert chr.inv.-operator ∗ is the difference 2 2 2 k 2 1 v ∗∂ ϕ ik ∂ ϕ 2v ∂ ϕ 1 ∗∂ 1 h + + between the 2nd derivatives of the operator , which charac- c2 − c2 ∂t2 − ∂xi∂xk c2 w ∂xk∂t c ∂t (3.84) terizes observable non-stationarity of the field, and the operator ik − h ∂vk ∂ϕ 1 k ∂ϕ ∗ , which characterizes its observable spatial inhomogeneity. If + + v Fk = T + 4πρ , ∇i c2 w ∂xi ∂t c2 ∂t the field is stationary and homogeneous, then the left parts in the − 2 i k d’Alembert equations (3.74, 3.75) are zeroes, so this field does not where v = hikv v and the second chr.inv.-derivative with respect generate waves — it is not a wave field. to time formulates with regular derivatives as follows 1 ∗∂ In an inhomogeneous stationary field (∗ i = 0 , = 0) the 2 2 ∇ 6 c ∂t ∗∂ ϕ 1 ∂ ϕ 1 ∂w ∂ϕ d’Alembert equations (3.74, 3.75) characterize a standing wave = + . (3.85) ∂t2 2 ∂t2 3 ∂t ∂t w c2 1 w ik 1 2 2 h ∗ i ∗ k ϕ = T + 4πρ , (3.78) − c − c − ∇ ∇ We can now see that the larger is the square of the linear velocity mk i i 4π i 2 h ∗ m∗ k q = B + j . (3.79) of the space rotation v , the lesser is the effect of observable non- − ∇ ∇ c ∗∂ϕ 1 ∗∂ stationarity of the field (the term ) on propagation of the waves. In a homogeneous non-stationary field (∗ = 0 , = 0) the ∂t ∇i c ∂t 6 In the ultimate case, where v c, the d’Alembert operator becomes d’Alembert equations describe changes of the field in time depend- the Laplace operator, so the d’Alembert→ wave equations becomes the ing on the field-inducing sources (charges and currents) Laplace stationary equations. At low velocities of the space rotation 1 ∂2ϕ v c ones assume that observable waves of electromagnetic waves ∗ = T + 4πρ , (3.80) c2 ∂t2 propagate at the light velocity. 3.4 D’Alembert’s equations 81 82 Chapter 3 Motion of charged particles

In general case the absolute value of the observable velocity of §3.5 Lorentz’ force. The energy-momentum tensor of electromag- waves of the scalar electromagnetic potential v(ϕ) becomes netic fields c v(ϕ) = . (3.86) In this §3.5 we are going to deduce chr.inv.-projections (physical 1 v2 observable components) of the four-dimensional force, which is the − c2 q result of that electromagnetic fields affects an electric charge in It is evidently that the chr.inv.-quantity (3.85), which is the a pseudo-Riemannian space. This problem will be solved for two observable acceleration of the scalar potential ϕ, is the more differ- cases of: (a) a point charge; (b) a charge distributed in the space. In ent from the analogous “coordinate” quantity the higher is gravita- addition, we are going to deduce chr.inv.-projections of the energy- tional potential and the higher is the rate of change of the gravita- momentum tensor for an electromagnetic field. tional potential in time In three-dimensional Euclidean space of Classical Electrodyna- mics, motion of a charged particle is characterized by the vector ∂2ϕ w 2 ∂2ϕ 1 ∂w ∂ϕ = 1 ∗ + . (3.87) equation ∂t2 c2 ∂t2 c2 w ∂t ∂t d~p e − = eE~ + ~u; H~ , (3.90) − dt c In the ultimate case, where w c2 (approaching gravitational collapse like as on the surface of a→ black hole), observable accel- where ~p = m~u is the particle’s three-dimensional impulse vector and erations of the scalar potential become infinitesimal, while the m is its relativistic mass. The right part of this equation is referred ∂ϕ coordinate rate of growth of the potential , to the contrary, to as Lorentz’ force. ∂t The equation, characterizing the change of the kinetic (relati- becomes infinitely large. But under regular conditions, gravitational vistic) energy of the particle potential w needs only smaller corrections into the acceleration and 2 the velocity of growth of the potential ϕ. m0c E = mc2 = (3.91) All what has been said the above about the chr.inv.-scalar quan- u2 i 1 2 ∗∂2ϕ ∗∂2q − c tity 2 is also true for the chr.inv.-vector 2 , because the q ∂t ∂t due to work accomplished by the field’s electric strength to displace 1 ∗∂2 ik ∗∂2 d’Alembert chr.inv.-operator ∗ = h shows differ- it within an unit of time, takes the vector form c2 ∂t2 − ∂xi∂xk ence from the scalar and vector functions in only the second term dE — the Laplace operator, in which chr.inv.-derivatives of the scalar = eE~ ~u, (3.92) dt and vector quantities are different from each other and is also known as the live forces theorem. k ∗∂ϕ k ∗∂q k m In four-dimensional form, thanks to unification of energy and ∗ ϕ = , ∗ q = + Δ q . (3.88) ∇i ∂xi ∇i ∂xi im momentum, in a Galilean reference frame in the Minkowski space If the space rotation and gravitational potential are infinitesimal, the both equations (3.90, 3.92) take the form α α the d’Alembert chr.inv.-operator for the scalar potential becomes dU e α σ α dx m0 c = F σ∙U ,U = , (3.93) the d’Alembert regular operator ds c ∙ ds α 2 2 and are known as the Minkowski equations (F ∙ is the electromag- 1 ∂ ϕ ik ∂ ϕ σ ∗ ϕ = h , (3.89) ∙ c2 ∂t2 − ∂xi∂xk netic field tensor). Because the metric here is diagonal (3.5), hence so in this case electromagnetic waves, produced by the scalar po- u2 dx 2 dy 2 dz 2 ds = cdt 1 , u2 = + + , (3.94) tential ϕ, propagate at the light velocity. − c2 dt dt dt r 3.5 Lorentz’ force. The energy-momentum tensor 83 84 Chapter 3 Motion of charged particles and components of the particle’s four-dimensional velocity U α are Given that components of U α are

i 1 i 0 1 i u viv 1 i U = ,U = , (3.95) 0 c2 i v 2 2 U = ± ,U = , (3.102) 1 u c 1 u 2 w 2 c2 c2 1 v 1 c 1 v − − − c2 − c2 − c2 i q q where ui = dx is its three-dimensional coordinate velocity. Once q q dt then, taking into account formulas for components of an arbitrary e α σ components of F σ∙U in the Galilean reference frame are rotor (2.143–2.159), we arrive to c ∙ i e ∂ϕ 1 ∂q ϕ e 0 σ e Eiu ∗ ∗ i i F ∙U = , (3.96) T = i + 2 Fi v , (3.103) σ 2 2 − v2 ∂x c ∂t − c c ∙ −c 1 u c2 1 − c2 − c2 q q e i σ 1 i e ikm i e ∗∂ϕ 1 ∗∂qk ϕ ik F σ∙ U = eE + e ukH m , (3.97) B = k + 2 Fk h + c ∙ − 2 c ∗ − 2 v2 ± ∂x c ∂t − c c 1 u c 1 − c2 − c2 (3.104) q q ∗∂q ∗∂q 2ϕ then, in the Galilean reference frame as well, the time and spatial + himhkn m n Aik v . ∂xn − ∂xm − c k components of the Minkowski equations (3.93) are 1 Chr.inv.-scalar quantity T , to within the multiplier 2 , is the dE i −c = eE u , (3.98) i dt − i field’s work to displace this charge e. Chr.inv.-vector quantity B , to within the multiplier 1, in a non-relativistic case is a force which i c dp i e ikm i i acts the particle from the electromagnetic field = eE + e ukH m , p = mu . (3.99) dt − c ∗ i i i 1 ikm The above relativistic equations, save for the sign in the right Φ = cB = e E + ε H mvk , (3.105) − c ∗ parts, match the live forces theorem and the equations of motion of a charged particle in Classical Electrodynamics (3.90, 3.91). Note and it is the Lorentz observable force. Note that alternating sign is that difference in signs in the right parts is conditioned only by derived here from the fact that in pseudo-Riemannian spaces the (+ ) dt choice of the space signature. We use the signature −−− , but if square equation with respect to has two roots (1.55). Respec- we accept the signature ( +++), then the sign in the right parts of dτ − tively, “plus” in the Lorentz force stands for the particle’s motion the equations will be the opposite. into future (in respect of the observer), while “minus” denotes the We now turn to this problem not in the Minkowski space, but in motion into past. In a Galilean reference frame in the Minkowski the pseudo-Riemannian space of the General Theory of Relativity. space there is not difference between physical observable time τ So forth, chr.inv.-projections of the four-dimensional impulse vec- and coordinate time t. So, the Lorentz force (3.99) obtained from tor Φα = eF α U σ, which the charged particle gains from interaction c σ∙ the Minkowski equations will have no alternating signs. of its charge∙ e with the electromagnetic field, are If the charge is not a point but it is spatially distributed matter, σ α e α σ e F0σU then the Lorentz force Φ = c F σ∙U in the Minkowski equations T = , (3.100) (3.93) will be replaced by the four-dimensional∙ vector of the Lorentz c √g00 force density i e i σ e i 0 i k α 1 α σ B = F σ∙ U = F 0∙U + F k∙U . (3.101) f = F σ∙j , (3.106) c ∙ c ∙ ∙ c ∙ 3.5 Lorentz’ force. The energy-momentum tensor 85 86 Chapter 3 Motion of charged particles where the four-dimensional current density jσ = cρ; ji is defined Denoting the term by the 1st group of the Maxwell equations (3.51) 1 μσ ν 1 μν αβ μν c F F σ∙ + g F Fαβ = T , (3.116) jσ = F σμ. (3.107) 4π − ∙ 4 4π ∇μ we obtain the expression Chr.inv.-projections of the Lorentz force density f α f ν = T μν , (3.117) ∇μ f0 1 i = Ei j , (3.108) so the four-dimensional vector of the Lorentz force density f ν √g00 − c equals the absolute divergence of a quantity T μν , referred to as the

i i 1 i k i 1 ikm energy-momentum tensor of the electromagnetic field. Its structure f = ρE + H k∙ j = ρE + ε H m jk . (3.109) μν νμ − c ∙ − c ∗ shows that it is symmetric T = T , while its spur (given that the μν ν spur of the fundamental metric tensor is gμν g = δν = 4) is zero in three-dimensional Euclidean space the projections are ν μν 1 μσ 1 μν αβ f q 1 T = gμν T = F Fμσ + gμν g F Fαβ = 0 = = E~ ~j , (3.110) ν 4π − 4 √g c c (3.118) 00 1 = F μσF + F αβF = 0 . 1 4π − μσ αβ f~ = ρE~ + ~j; H~ , (3.111) c Chr.inv.-projections of the energy-momentum tensor are where q is the density of a heat power released into a current T cT i conductor. q = 00 ,J i = 0 ,U ik = c2T ik, (3.119) Now we transform the Lorentz force density (3.106), using the g00 √g00 σ Maxwell equations. Substituting j (3.107) we arrive to where the chr.inv.-scalar q is of the observable density of the field, i 1 1 1 the chr.inv.-vector J is the observable density of the field’s mo- σ σμ σμ σμ ik fν = Fνσj = Fνσ μ F = μ (FνσF ) F μ Fνσ . (3.112) mentum, and the chr.inv.-tensor U is the observable density of c 4π ∇ 4π ∇ − ∇ h i the field’s momentum flux. For the energy-momentum tensor of Transpositioning the mute indices μ and σ, by which we add-up, the electromagnetic field (3.116) we obtain the expressions and taking into account that the Maxwell tensor Fαβ is antisym- E2 + H 2 metric, we transform the second term to the form q = ∗ , (3.120) 8π σμ 1 σμ F μ Fνσ = F ( μ Fνσ + σ Fμν ) = i c ikm ∇ 2 ∇ ∇ J = ε EkH m , (3.121) (3.113) 4π ∗ 1 σμ 1 σμ = F ν Fμσ = F ν Fσμ . 2 −2 ∇ 2 ∇ ik 2 ik c i k i k U = qc h E E + H∗ H∗ , (3.122) − 4π As a result, for fν (3.112) and its contravariant form we obtain 2 i k 2 i k where E = hikE E and H∗ = hikH∗ H∗ . Comparing the obtained 1 1 f = F μσF + δμF αβF , (3.114) formula for q (3.120) with that for the energy density of the electro- ν 4π ∇μ − νσ 4 ν αβ magnetic field from Classical Electrodynamics we have 2 2 ν 1 μσ ν 1 μν αβ E + H f = μ F F σ∙ + g F Fαβ . (3.115) W = , (3.123) ∇ 4π − ∙ 4 8π 3.5 Lorentz’ force. The energy-momentum tensor 87 88 Chapter 3 Motion of charged particles where E2 = (E~ ; E~ ) and H2 = (H~ ; H~ ), we can see that the chr.inv.- 3. Effect of gravitational inertial force on the electromagnetic i quantity q is the observable energy density of the electromagnetic field momentum density (the term FiJ ); field in the pseudo-Riemannian space. Comparing the obtained 4. The observable “spatial variation” (physical divergence) of the i formula for the chr.inv.-vector J (3.121) with that for Poynting’s i electromagnetic field momentum density (the term ∗ i J ); vector in Classical Electrodynamics we have ∇ 5. Magnitudes and mutual orientation of the current density c vector ji and the electric strength vector Ei (the righte part). S~ = E~ ; H~ , (3.124) 4π The second chr.inv.-identity (3.126) shows that observable we can see that the J i is the Poynting observable vector in the change of the electromagnetic field momentum density in time (the k pseudo-Riemannian space. Correspondence of the third observable quantity ∗∂J ) depends on the following factors: component U ik (3.122) to quantities in Classical Electrodynamics ∂t can be established using similarities with mechanics of continuous 1. The rate of changes of the observable volume of the space, k medias, where the three-dimensional tensor of similar structure is filled with the electromagnetic field (the term DJ ); the stress tensor for an elementary volume of a media. Therefore, 2. Forces of the space deformation and Coriolis’ forces, which ik k k i U is the observable stress tensor of the electromagnetic field in are accounted by the term 2 Di + A i∙ J ; the pseudo-Riemannian space. ∙ 3. Effect of gravitational inertial force on the observable density Now we can obtain identities for the chr.inv.-projections of the of the electromagnetic field (the term qF k); Lorentz force density (3.108, 3.109), formulating them with chr.inv.- ik 4. The observable “spatial variation” of the field stress ∗ U ; components of the energy-momentum tensor of this field (3.120– ∇i 3.122). Taking the equation f ν = T μν and using ready formulas 5. Effect of the Lorentz force observable density — the right part, ∇μ for chr.inv.-components of the absolute divergence of an arbitrary k k 1 kim e defined by the quantity f = ρE + c ε H i jm . symmetric tensor of the 2nd rank (2.138, 2.139), we obtain − ∗ In conclusion we consider a particular case, where the electro- ∗∂q 1 ij i 1 i 1 i magnetic field is isotropic. A formal definition of isotropic fields + qD + D U + ∗ J F J = E j , (3.125) ∂t c2 ij ∇i − c2 i − c i made with the Maxwell tensor [16] is a set of two conditions k e μν μν ∗∂J k k k i ik k Fμν F = 0 ,Fμν F ∗ = 0 , (3.127) + DJ + 2 Di + A i∙ J + ∗ i U qF = ∂t ∙ ∇ − (3.126) μν k 1 kim which implies that the both field invariants J1 = Fμν F (3.25) and = ρE +e ε H i jm . μν − c ∗ J2 = Fμν F ∗ (3.26) are zeroes. In chr.inv.-notation, taking (3.28) into account, the conditions take the form The first chr.inv.-identity (3.125) shows that if the observable i vector of the current density j is orthogonal to the observable 2 2 i E = H∗ ,E H∗ = 0 . (3.128) electric strength of the field Ei, the right part turns to zero. In i general case, i. e. in the case of an arbitrary orientation of the We see that an electromagnetic field in a pseudo-Riemannian i i vectors j and E , observable change of the electromagnetic field space is observed as isotropic, if the observable lengths of its electric ∂q density in time (the quantity ∗ ) depends on the following factors: and magnetic strength vectors are equal, while the Poynting vector ∂t J i (3.121) is zero 1. The rate of changes of the observable volume of the space,

filled with the electromagnetic field (the term qD); i c ikm c ikm k ij J = ε EkH m = ε hmkEkH∗ = 0 . (3.129) 2. Effect of forces of the space deformations (the term Dij U ); 4π ∗ 4π 3.6 Equations of motion (the parallel transfer method) 89 90 Chapter 3 Motion of charged particles

In the terms of chr.inv.-components of the energy-momentum using the least action principle. Extremum length lines are also tensor (3.120, 3.121) the obtained conditions (3.128) also imply that lines of constant direction. But, for instance, in spaces with non- metric geometry, length is not defined as category. Therefore lines J = cq , (3.130) of extremum lengths are neither defined and we can not use the least action method to obtain the equations. Nevertheless, even in where J = √J 2 and J 2 = h J iJ k. In other word, the length J of the ik non-metric spaces we can define lines of constant direction and momentum density chr.inv.-vector of any isotropic electromagnetic non-zero derivation parameter along them. Hence ones can assume field depends only on the field density q. that in metric spaces, to which Riemannian spaces belong, lines of extremum length are merely a particular case of constant direction §3.6 Equations of motion of charged particle, obtained using the lines. parallel transfer method In accordance with general formulas we have obtained in Chapter 2, chr.inv.-projections of the parallel transfer equations In this §3.6 we will obtain chr.inv.-equations of motion of a charged (3.132) are as follows mass-bearing test-particle in an electromagnetic field, located in a dϕ˜ 1 dτ dxk four-dimensional pseudo-Riemannian space∗. + F q˜i + D q˜i = 0 , (3.133) ds c − i ds ik ds The equations are chr.inv.-projections of parallel transfer equations of the four-dimensional summary vector i k k dq˜ ϕ˜ dx k dτ i i ϕ˜ i dτ i m dx + +˜q Dk +Ak∙ F +Δmk q˜ =0 , (3.134) α α e α ds c ds ds ∙ − c ds ds Q = P + A , (3.131) c2 α where the space-time interval s is assumed as the derivation pa- α dx i where P = m0 is the four-dimensional impulse vector of the rameter along the trajectory, ϕ˜ and q˜ are chr.inv.-projections of ds α particle, and e Aα is a part of the previous — an additional four- the dynamic vector Q (3.131) of this particle c2 dimensional impulse which the particle gains from interaction of Q0 1 e α α its charge e with the electromagnetic field potential A deviating ϕ˜ = bαQ = = P0 + 2 A0 , (3.135) √g00 √g00 c its trajectory from a geodesic line. Given this problem statement, e parallel transfer of superposition on the non-geodesic impulse of q˜i = hi Qα = Qi = P i + Ai. (3.136) the particle and the deviating vector is also geodesic, so that α c2 ν Chr.inv.-projections of the impulse vector are d α e α α μ e μ dx P + A + Γμν P + A = 0 . (3.132) ds c2 c2 ds P 1 1 0 = m , P i = mvi = pi, (3.137) By definition, a geodesic line is a line of constant direction, √g00 ± c c so the one for which any vector tangential to it in a given point will remain tangential along the line being subjected to parallel where “plus” stands for motions into future (in respect of the observer), while “minus” appears if the particle moves into past, transfer [4]. i Equations of motion may be obtained in another way, namely — and pi = mdx is the three-dimensional impulse chr.inv.-vector of dτ by considering motion along a line of the least (extremum) length the particle. Chr.inv.-projections of the additional impulse vector e Aα are as follows ∗Generally, using the method described herein we can also obtain equations of c2 motion for a particle, which is not a test one. A test particle is one with charge e A0 e e e and mass so small that they do not affect electromagnetic and gravitational fields in i i 2 = 2 ϕ , 2 A = 2 q , (3.138) which it moves. c √g00 c c c 3.6 Equations of motion (the parallel transfer method) 91 92 Chapter 3 Motion of charged particles

i i where ϕ is the scalar potential and q is the vector-potential of effect from physical and geometric properties of the space (F , Aik, i the acting electromagnetic field — these are chr.inv.-components of Dik, Δnk). It is evidently, if the particle becomes charge-free e = 0 the four-dimensional field potential Aα (3.8). Then the quantities ϕ˜ the right parts turn to zero and the resulting equations fully match (3.135) and q˜i (3.136), which actually are chr.inv.-projections of the the chr.inv.-equations of motion of a free mass-bearing particle (see summary vector Qα, take the form formulas 1.51, 1.52 and also 1.56, 1.57). Let us consider the right parts in details. The obtained equations e are absolutely symmetric for motions either into future or past and ϕ˜ = m + 2 ϕ , (3.139) ± c they change their sign once the sign of the charge changes. We 1 e denote the right parts of the scalar chr.inv.-equations of motion q˜i = pi + qi . (3.140) c c2 (3.141, 3.143) as T . Given that We now substitute the quantities ϕ˜ and q˜i into general formulas dϕ ∂ϕ ∂ϕ = ∗ + vi ∗ , (3.145) for chr.inv.-equations of motion (3.133, 3.134). Moving the terms, dτ ∂t ∂xi which characterize electromagnetic interaction, into the right parts then using the formula for the electric strength in covariant form we arrive to the chr.inv.-equations of motion for the our-world E (3.14), we can represent T as follows charged particle (the particle moves into future in respect of a i regular observer) e i e ∗∂ϕ e ∗∂qi k i e i ϕ i T = 2 Eiv 2 + 3 Dik q v + 3 q v Fi . (3.146) dm m i m i k e dϕ e i i k −c − c ∂t c ∂t − c − c Fiv + Dik v v = + Fiq Dik q v , (3.141) dτ c2 c2 c2 dτ c3 − − − Substituting this formula into (3.141, 3.143) and multiplying the 2 d mvi results by c , we obtain the equation for the relativistic energy i i i k i n k 2 mF + 2m Dk + Ak∙ v + mΔnkv v = E = mc of the charged particle, which moves into future and into dτ − ∙ (3.142) i past,± respectively e dq e ϕ k k i i eϕ i e i n k = v + q Dk +Ak∙ + F Δnk q v , − c dτ − c c ∙ c2 − c dE ∗∂ϕ mF vi + mD vivk = eE vi e + while for the analogous particle located in the mirror-world (it dτ − i ik − i − ∂t moves into past in respect of the observer) the equations are (3.147) e ∗∂qi k i e i ϕ i + Dik q v + q v Fi , dm m m e dϕ e c ∂t − c − c F vi + D vivk = + F qi D qivk , (3.143) − dτ − c2 i c2 ik −c2 dτ c3 i − ik dE i i k i ∗∂ϕ mFiv + mDikv v = eEiv e + i − dτ − − − ∂t d mv i i n k (3.148) + mF + mΔ v v = e ∂q e ϕ dτ nk ∗ i k i i i i (3.144) + Dik q v + q v Fi , e dq e ϕ k k i i eϕ i e i n k c ∂t − c − c = v + q Dk +Ak∙ + F Δnk q v . − c dτ − c c ∙ c2 − c i where eEiv is a work done by the electric component of the field As it is easy to see, the left parts of the equations fully match to displace the particle in an unit of time. those of the chr.inv.-equations of motion of this particle, being it The scalar chr.inv.-equations of motion of a charged particle would be free. The only difference is that the equations include (3.147, 3.148) make the theorem of live forces in the pseudo- terms, which characterize its non-geodesic motion. Therefore the Riemannian space, represented in chr.inv.-form. As it is easy to right parts here are not zeroes. The right parts account for the effect see, in a Galilean reference frame the scalar equation for the particle that the electromagnetic field makes on the particle, as well as the which moves into future (3.147) matches the time component of the 3.6 Equations of motion (the parallel transfer method) 93 94 Chapter 3 Motion of charged particles

Minkowski equations (3.98). In three-dimensional Euclidean space and the sum of the latter three terms in the M i equals the equation (3.147) transforms into the theorem of live forces from e ∂q ∂q e ∂qi e Classical Electrodynamics dE = eE~u~ (3.92). himvk ∗ m ∗ k vk ∗ Δi qnvk = dt 2c ∂xk − ∂xm − c ∂xk − c nk Let us turn to the right parts of the vector chr.inv.-equations of (3.156) i e ∂qk e ∂qi e ∂h motion (3.142, 3.144). We denote them as M . Because of = himv ∗ vk ∗ himqn vk ∗ km . −2c k ∂xm − 2c ∂xk − 2c ∂xn dqi ∂qi ∂qi ∗ k ∗ At last, the vector chr.inv.-equations of motion of the charged = + v k , (3.149) dτ ∂t ∂x particle (3.142, 3.144) which moves into future and into past take ik and in it, taking into account, that ∗∂h = 2Dik (1.40) the form, respectively ∂t − i i ∗∂q ∗∂ ik i k ik ∗∂qk d mv i i i k i n k = h qk = 2Dk q + h , (3.150) mF + 2m Dk + Ak∙ v + mΔnkv v = ∂t ∂t − ∂t dτ − ∙ then M i takes the form i 1 ikm = e E + ε vkH m + − 2c ∗ i e ik ∗∂qk eϕ i ik e ik M = h + F + A vk + A qk + e ϕ ∂ϕ e c ∂t c2 c k k i ik ∗ ik − (3.151) + q v Dk + eh + A qk e ϕ e ∂qi e c − c ∂xk c − + qk vk Di vk ∗ Δi qn vk. k k nk e ∂qk e ∂qi e ∂h c − c − c ∂x − c himv ∗ vk ∗ himqn vk ∗ km , 2c k ∂xm 2c ∂xk 2c ∂xn i ik − − − Using formulas for chr.inv.-components E (3.11) and H (3.12) (3.157) of the Maxwell tensor Fαβ, we write down the first two terms from i i d mv i i n k the M (3.151) and the third term as follows + mF + mΔnkv v = dτ 1 e ik ∗∂qk eϕ i i ik ∗∂ϕ i ikm h + F = eE + eh , (3.152) = e E + ε vkH m + 2 k − 2c ∗ − c ∂t c − ∂x e k ϕ k i ik ∗∂ϕ e ik eϕ ik e im n ∗∂qm ∗∂qn e ik + q v D + eh + A q A v = h v H v . (3.153) k k k c2 k 2c ∂xn − ∂xm − 2c k c − c ∂x c − k i e im ∗∂q e k ∗∂q e im n k ∗∂hkm ik ik mik h vk v h q v . We write down the quantity H as H = ε H m (3.56). Then − 2c ∂xm − 2c ∂xk − 2c ∂xn we have follows ∗ i 1 ikm From here we see that the first term e E + ε vkH m 2c ∗ eϕ ik e im n ∗∂qm ∗∂qn e ikm − A vk = h v ε H mvk , (3.154) in their right parts is different from the Lorentz chr.inv.-force c2 2c ∂xn − ∂xm − 2c ∗ i i 1 ikm 1 Φ = e E + c ε vkH m by the coefficient 2 at the term that stands− for the magnetic∗ component of the force. This fact is very i i 1 ikm e k ϕ k i M = e E + ε vkH m + q v Dk + surprise, because regular equations of motion of a charged particle, − 2c ∗ c − c being three-dimensional components of the general covariant equa- ik ∗∂ϕ e ik e im k ∗∂qm ∗∂qk tions, contain the Lorentz force in full form. In §3.9 we are going + eh + A qk + h v (3.155) ∂xk c 2c ∂xk − ∂xm − to show the structure of the electromagnetic field potential Aα at i i e k ∗∂q e i n k which the other terms in the M fully compensate this coefficient v k Δnk q v , 1 − c ∂x − c 2 so that only the Lorentz force is left. 3.7 Equations of motion (the least action principle) 95 96 Chapter 3 Motion of charged particles

§3.7 Equations of motion, obtained using the least action principle We represent the variation of the second integral from the initial as a particular case of the previous equations formula (3.160) as the sum

b b b In this §3.7 we are going to deduce chr.inv.-equations of motion of e α e α α δ Aαdx = δAαdx + Aαdδx . (3.162) a mass-bearing charged particle, using the least action principle. − c − c Za Za Za The principle says that an action S to displace a particle along the Integrating the second term, we obtain shortest trajectory is the least, so the variation of the action is zero b b b b α α α Aαdδx = Aαδx dAαδx . (3.163) δ dS = 0 . (3.158) a a − a Z Z Za Here the first term is zero, as the integral is variated with Therefore, equations of motion, obtained from the least action the given numerical values of coordinates of the integration limits. principle are equations of the shortest lines. Taking into account that the variation of any covariant vector is The elementary action of gravitational and electromagnetic fields to displace a charged particle at an elementary space-time ∂Aα β ∂Aα β δAα = β δx , dAα = β dx , (3.164) interval ds is [5] ∂x ∂x e dS = m cds A dxα. (3.159) we obtain the variation of the electromagnetic part of the action − 0 − c α b b We see that this quantity is only applicable to characterize par- e α e ∂Aα α β ∂Aα α β δ Aαdx = dx δx δx dx . (3.165) ticles which move along non-isotropic trajectories (ds = 0). On the − c − c ∂xβ − ∂xβ Za Za other hand, obtaining equations of motion through6 the parallel Transpositioning free indices α and β in the first term of this transfer method (constant direction lines) is equally applicable to formula and accounting for the variation of the gravitational part both non-isotropic (ds = 0) and isotropic trajectories (ds = 0). More- of the action (3.161) we arrive to the variation of the total action over, parallel transfer is6 as well applicable to non-metric geometries, (3.160) as follows in particular, to obtain equations of motion of particles in a fully degenerated space-time (zero-space). Therefore equations of the b b μ ν e β α least length lines, because they are obtained through the least action δ dS = m0c (dUα Γα,μν U dx ) Fαβdx δx , (3.166) a a − − c method, are merely a narrow particular case of constant direction Z Z h i ∂A μ lines, which result from parallel transfer. where F = ∂Aβ α is the Maxwell tensor, and U μ = dx is the αβ ∂xα − ∂xβ ds But we are returning to the least action principle (3.158). For the four-dimensional velocity of the particle. Because the quantity δxα charged particle we are considering the condition takes the form is arbitrary, the formula under the integral is always zero. Hence, we arrive to general covariant equations of motion of the charged b b b e δ dS = δ m cds δ A dxα = 0 , (3.160) particle in their covariant (lower-index) form − 0 − c α Za Za Za dUα μ ν e β where the first term can be denoted as follows m0c Γα,μν U U = FαβU , (3.167) ds − c b b α or, lifting the index α, we arrive to the contravariant form of the δ m0 cds = m0 c DUαδx = − a − a equations Z Z b (3.161) α μ ν α dU α μ ν e α β = m0 c (dUαds Γα,μν U dx ) δx . m0c + Γμν U U = F β∙U . (3.168) − ds c ∙ Za 3.7 Equations of motion (the least action principle) 97 98 Chapter 3 Motion of charged particles

The equations (3.168) actually are the Minkowski equations in of constant direction lines, defined by parallel transfer. Therefore the pseudo-Riemannian space. In a Galilean reference frame in the there is little surprise in that the equations of parallel transfer, as Minkowski space (the Special Theory of Relativity), the obtained more general ones, have additional terms, which account for the equations transform into regular relativistic equations (3.93). structure of the acting electromagnetic field and of the space. Therefore chr.inv.-projections of the obtained equations (3.168) may be called the Minkowski chr.inv.-equations in the pseudo- §3.8 The geometric structure of the four-dimensional electro- Riemannian space. For an our-world charged particle (it moves magnetic potential into future in respect of a regular observer) the Minkowski chr.inv.- equations are In this §3.8 we are going to find that structure of the acting electro- α dE magnetic field potential A , under which the length of any charged mF vi + mD vivk = eE vi, (3.169) particle’s summary vector Qα = P α + e Aα remains unchanged in dτ − i ik − i c2 its parallel transfer in the Levi-Civita meaning (so, a pseudo- i d mv i i i k i n k Riemannian space is assumed). mF + 2m Dk + Ak∙ v + mΔnkv v = dτ − ∙ As it is known, the Levi-Civita parallel transfer conserves the (3.170) length of any transferred vector Qα, so the condition Q Qα = const i 1 ikm α = e E + ε vk H m , − c ∗ is true. Given that the square of the length of any n-dimensional vector is invariant in that n-dimensional pseudo-Riemannian space and for the analogous particle in the mirror world (it moves into where the vector is located, this condition must be true in any past) the equations are reference frame, including the case of any observer who accompanies his reference body. Hence we can analyze the condition dE mF vi + mD vivk = eE vi, (3.171) Q Qα = const, formulating it with physical observable quantities in dτ i ik i α − − − the accompanying reference frame, in chr.inv.-form in other word. i Components of the vector Qα in the accompanying reference d mv i i n k i 1 ikm + mF + mΔnkv v = e E + ε vk H m . (3.172) frame are dτ − c ∗ w eϕ Q = 1 m + , (3.173) The scalar chr.inv.-equations of motion, both in our world and 0 − c2 ± c2 the mirror world, represent the live forces theorem. The vector 1 eϕ 1 e chr.inv.-equations in their right parts the Lorentz chr.inv.-force in Q0 = m + + v mvi + qi , (3.174) w 2 2 i the pseudo-Riemannian space. As it is easy to see, in a Galilean 1 ± c c c − c2 reference frame in the Minkowski space the obtained equations 1 e 1 eϕ become the regular theorem of live forces (3.92) and the regular Qi = mvi + qi m + 2 vi , (3.175) three-dimensional equations of motion (3.90) accepted in Classical − c c − c ± c Electrodynamics. 1 e Qi = mvi + qi , (3.176) It is evidently, the right parts of the equations of motion (3.169– c c 3.172), obtained through the least action method, are different from and its square is the right parts of the equations (3.146, 3.157), obtained by the e2 2me 1 parallel transfer method. The difference is in the absence here Q Qα = m2 + ϕ2 q qi + ϕ v qi . (3.177) α 0 c4 − i c2 ± − c i (3.169–3.172) of numerous terms, which characterize the structure of the acting electromagnetic field and the space itself. But as we From here we can see that the square of the charged particle’s have already mentioned, least length lines are only a particular case summary vector falls apart into the following quantities: 3.8 The electromagnetic potential 99 100 Chapter 3 Motion of charged particles

The square of the four-dimensional impulse of the particle So, the length of the summary vector Qα remains unchanged in • α 2 i PαP = m0; its parallel transfer, if the observable potentials ϕ and q of the field α The square of the four-dimensional additional impulse e Aα are related to its four-dimensional potential A as follows • c2 which the particle gains from the acting electromagnetic field A0 ϕ0 i i ϕ i (the second term); = ϕ = ,A = q = v . (3.183) √g00 1 v2 c 2me 1 i 2 The term ( ϕ vi q ), which describes interaction be- − c • c2 ± − c q tween the mass of this particle m and its electric charge e. e α Then for the vector 2 A , which characterizes interaction of the α 2 c In the formula for QαQ (3.177), the first term m0 remains particle’s charge with the electromagnetic field we have unchanged anyhow. In other word, it is an invariant, which does not i depend on reference frame. Our goal is to deduce that conditions, e A0 eϕ0 e i eϕ0 v 2 = , 2 A = 3 . (3.184) under which the whole formula (3.177) remains unchanged. c √g00 c2 1 v2 c c 1 v2 So forth, let us propose that the field vector-potential has the − c2 − c2 q q α structure e α α dx ϕ Dimensions of the vectors A and P = m0 in CGSE and qi = vi. (3.178) c2 ds c Gaussian systems of units are the same and equal a mass m [ gram ]. Comparing chr.inv.-projections of the both vectors we can see In this case the second term of (3.177) is ∗ that an analog for the relativistic mass m in interactions between 2 2 2 2 the particle’s charge and the acting electromagnetic field is the e α e ϕ v AαA = 1 . (3.179) quantity c4 c4 − c2 eϕ eϕ0 2 = , (3.185) c 2 v2 Transforming the third term in the same way, we obtain the c 1 2 α − c square of the vector Q (3.177) in the form q where eϕ is the potential energy of the particle moving at the i 2 2 2 2 i dx α 2 e ϕ v 2m0 e v observable velocity v = in respect of the acting electromagnetic QαQ = m + 1 + ϕ 1 . (3.180) dτ 0 c4 − c2 c2 − c2 field (this particle is at rest in respect of the observer and his r reference body). In general, the scalar potential ϕ is the potential Then introducing notation for the field scalar potential energy of the field itself, divided by an unit of charge. Then eϕ is the potential relativistic-energy of the particle with charge e in this ϕ0 ϕ = , (3.181) electromagnetic field, while eϕ0 is the particle’s rest-energy in the 1 v2 − c2 field. When the particle is at rest in respect of the field, its potential q rest-energy equals the potential relativistic-energy. we can represent the obtained formula (3.180) as follows Comparing E = mc2 and W = eϕ we arrive to the same con- W0 eϕ0 2 2 clusion. Respectively, = is an electromagnetic analog for e ϕ 2m0 eϕ0 c2 c2 Q Qα = m2 + 0 + = const. (3.182) eϕ α 0 4 2 the rest-mass m . Then the chr.inv.-quantity eAi = vi is similar c c 0 c c2 i i i ϕ i to the observable impulse chr.inv.-vector p = mv . Therefore, when ∗A similar problem could be solved, assuming that q = c v . But in compar- ± i ϕ i the particle is at rest in respect of the field, its “electromagnetic ative analysis of two groups of the equations only positive values of q = c v will be important, because the observer’s physical time τ, by definition, flows from past projection” on the observer’s spatial section (the chr.inv.-vector) into future only, so the interval of physical observable time dτ is always positive. is zero, while only the time projection (the potential rest-energy 3.8 The electromagnetic potential 101 102 Chapter 3 Motion of charged particles eϕ = const) is observable. But if the particle moves in the field Therefore the calculated chr.inv.-projections of the vector e Aα 0 c2 at the velocity vi, its observable “electromagnetic projections” will have the form be the potential relativistic-energy eϕ and the three-dimensional e A eϕ eϕ e eϕ eϕ i 0 = = 0 , Ai = vi, (3.190) impulse 2 v . 2 2 2 3 c c √g00 ± c ± 2 v2 c c e α c 1 2 Having obtained chr.inv.-projections of the vector 2 A calcu- − c c q lated for the given its structure (3.183), we can restore the vector where “plus” stands if the particle is located in our world, so it α A in general covariant form. Taking into account that its spatial travels from past into future, while “minus” stands if the particle i component A is is located in the mirror world, traveling into past in respect of us. i i The square of the vector’s length is i i ϕ i ϕ dx dx A = q = v = = ϕ0 , (3.186) 2 2 2 2 2 2 2 c v dτ ds e α e ϕ v e ϕ0 c 1 2 A A = 1 = = const . (3.191) − c c4 α c4 − c2 c4 q we obtain the desired general covariant notation for Aα This vector has real length at v2 < c2, zero length at v2 = c2 and α α 2 2 α dx e α eϕ0 dx imaginary length at v > c . But here we limit our study to real form A = ϕ0 , A = . (3.187) ds c2 c2 ds of the vector (sub-light velocities), because light-like or super-light In the same time, taking chr.inv.-projections the final formula charged particles are unknown. α eϕ α for the Aα (3.187) Comparing formulas for P α = m dx and e Aα = 0 dx we 0 ds c2 c2 ds can see that the both vectors are collinear, so they are tangential A0 ϕ0 i i ϕ i = ϕ = ,A = q = v , (3.188) to the same non-isotropic trajectory, to which the derivation pa- √g00 ± ± 1 v2 c − c2 rameter s is assumed. Hence in this case the impulse vector of the particle P α is co-directed with the acting electromagnetic field, so we obtain alternating signsq in the time chr.inv.-projection, which the particle moves “along” the field. was not in the case in the initial formula (3.183). Naturally, a We are going to consider the general case, where the vectors are question arises: how did the scalar observable component of the not co-directed. From the square of the summary vector Q Qα vector Aα, initially defined as ϕ, at the given structure of the Aα α (3.177) we see that the third term there is the doubled scalar (3.187) accept the alternating sign? The answer is that in the first product of the vectors P α and e Aα. Parallel transfer of the vectors case ϕ and qi were defined proceeding from the general rule of c2 building chr.inv.-quantities. But without knowing the structure of remains their scalar product unchanged the projected vector Aα itself, we can not calculate them. Therefore D(P Aα) = AαDP + P DAα = 0 , (3.192) in formulas for the time and spatial projections (3.183) the symbols α α α ϕ and qi merely denote the quantities without revealing their struc- because the absolute increment of each the vector is zero. Hence ture. On the contrary, in the formulas (3.188) the quantities were we obtain g calculated using formulas ϕ = √g A0 + 0i Ai and qi = Ai, where 2e 2me 1 00 √g P Aα = ϕ v qi = const, (3.193) 00 c2 α c2 ± − c i detailed formulas for the components A0 and Ai were given. Hence in the second case the quantity ϕ results from calculation and sets that is, the scalar product of P α and e Aα remains unchanged. ± c2 forth the specific formula Consequently the lengths of the both vectors remain unchanged as ϕ0 well. In particular, we have ϕ = . (3.189) ± v2 1 A Aα = ϕ2 q qi = const. (3.194) − c2 α − i q 3.8 The electromagnetic potential 103 104 Chapter 3 Motion of charged particles

As it is known, the scalar product of two vectors is the product Multiplying the both parts of the equation by c4 and denoting of their lengths multiplied by cosine of the angle between them. the relativistic energy of the particle as E = mc2, we obtain Therefore parallel transfer also remains the angle between the transferred vectors unchanged E2 c2p2 + e2ϕ2 e2q qi = E2 + e2ϕ2 . (3.201) − − i 0 0 P Aα cos P\α; Aα = α = const. (3.195) 2 i m0 ϕ qi q §3.9 Building Minkowski’s equations as a particular case of the − Taking into account the formulap for relativistic mass m, we can obtained equations of motion re-write the condition (3.193) as follows i In §3.6 we considered a mass-bearing charged particle in a pseudo- 2e α 2m0 e ϕ 2m0 e vi q PαA = = const, (3.196) Riemannian space. There general covariant equations of its motion c2 ± c2 v2 − c2 v2 1 c 1 were obtained applying the parallel transfer method. So, we have − c2 − c2 obtained chr.inv.-projections of the general covariant equations. or as the relationshipq between the scalarq and vector potentials We showed that their time chr.inv.-projection (3.147) in a Ga- ϕ v qi i = const. (3.197) lilean reference frame takes the form of the time component of ± 1 v2 − c 1 v2 the Minkowski equations (3.98), becoming the live forces theorem − c2 − c2 of Classical Electrodynamics (3.92) in three-dimensional Euclidean q q For instance, we can find the relationship between the potentials space. But the right parts of the spatial chr.inv.-projections, instead i i i 1 ikm ϕ and q for that case, where the impulse vector of the particle of the Lorentz chr.inv.-force Φ = e E + ε vkH m , have the α e α − c ∗ P is orthogonal to the additional impulse 2 A , gone from the i 1 ikm c term e E + 2c ε vkH m and numerous other additional terms electromagnetic field. Because parallel transfer remains the angle which− depend on observable∗ characteristics of the acting electro- between transferred vectors unchanged (3.195), then cosine of the magnetic field and the space itself. Therefore for the spatial chr.inv.- angle between transferred orthogonal vectors is zero. So, we have projections the principle of correspondence with three-dimensional α 1 i components of the Minkowski equations is set non-trivially. PαA = ϕ vi q = 0 . (3.198) ± − c On the other hand, equations of constant direction lines, ob- Consequently, if the particle travels in the electromagnetic field tained through parallel transfer in a pseudo-Riemannian space, are so that the vectors P α and Aα are orthogonal, then the scalar a more general case of the least length lines, obtained with the least potential of the field is action method. Equations of motion, obtained from the least action 1 principle in §3.7, have the structure matching that of the Minkowski ϕ = v qi, (3.199) ± c i equations. Hence we can suppose that chr.inv.-projections of the equations of motion in §3.6, as more general ones, in a particular so it is the scalar product of the particle’s observable velocity vi case can be transformed into chr.inv.-projections of the equations and the spatial observable vector-potential of the field qi. of motion, obtained from the least action principle in §3.7. Now we are going to obtain the formula for the square of the summary vector Qα, assuming that the structure of the electro- To find out exactly under what conditions this can be true, we α α dx are going to consider the spatial chr.inv.-projections of the equations magnetic field potential is A = ϕ0 (3.187), so the field vector α ds α of motion (3.157), which contain the mismatch with the Lorentz A is collinear to the particle’s impulse vector P . Then force. For analyzes convenience we considered the right parts in 2 2 2 i α 2 m i e 2 i 2 e 2 (3.157) as separate formulas up denoted as M . Substituting the Q Q = m v v + ϕ q q = m + ϕ . (3.200) ik eϕ ik i α 2 i 4 i 0 4 0 magnetic strength H (3.12) into the term A v from M , we − c c − c c2 k 3.9 Minkowski’s equations 105 106 Chapter 3 Motion of charged particles write down the term as follows Substituting the relativistic definition of ϕ (3.181) into the first derivative and after derivation returning to the ϕ again, we obtain eϕ ik e im n ∗∂qm ∗∂qn e ikm A vk = h v ε H mvk , (3.202) c2 2c ∂xn − ∂xm − 2c ∗ eϕ ∂ eϕ 2 i ∗ i k i k c T = eEiv 2 hikv v + 2 Dikv v + ikm ik − − 2c ∂t c where ε H m = H . Now we substitute chr.inv.-components of k k ∗ α eϕ ∗∂v i eϕ ∗∂hik i k ∗∂v the electromagnetic field potential A as of (3.188) into (3.157). + vk = eEi v v v +2vk + (3.207) e α c2 ∂t − − 2c2 ∂t ∂t With this potential, the impulse vector 2 A which the charged c k particle gains from this electromagnetic field is tangential to the eϕ i k eϕ ∗∂v i ϕ + 2 Dikv v + 2 vk = eEi v , particle’s trajectory. Using the first formula qm = c vm we arrive c c ∂t − to dependence of the right part under consideration from only the because we took into account that ∗∂hik = 2D by definition of the scalar potential of the field ∂t ik tensor of the space deformations rate Dik (1.40). i i 1 ikm M = e E + ε vkH m + So, chr.inv.-equations of motion of a charged particle, obtained − c ∗ (3.203) using the parallel transfer method in a pseudo-Riemannian space, v2 ∂ϕ eϕ ∂ v2 match the equations, obtained using the least action principle in a + ehik 1 ∗ + hik ∗ 1 . − c2 ∂xk 2 ∂xk − c2 particular case, where: The electromagnetic field potential Aα has the structure as Substituting the relativistic formula of the scalar potential ϕ • α Aα = ϕ dx (3.187); (3.181) into this formula we see that the sum of the last two terms 0 ds becomes zero The field potential Aα is tangential to the four-dimensional • eϕ ∂ v2 eϕ ∂ v2 trajectory of the moved particle. hik ∗ 1 + hik ∗ 1 = 0 . (3.204) − 2 ∂xk − c2 2 ∂xk − c2 Consequently, given such electromagnetic potential in a Galilean reference frame in the Minkowski space, the obtained chr.inv.- Then M i takes the form of the Lorentz chr.inv.-force equations of motion fully match the live force theorem (the scalar chr.inv.-equation) and the Minkowski equations (the vector chr.inv.- i i 1 ikm M = e E + ε vkH m , (3.205) equations) in three-dimensional Euclidean space, taking the well- − c ∗ known form in Classical Electrodynamics. which is exactly what we had to prove. Noteworthy, this is another illustration of the geometric fact that Now we are going to consider the right part c2T of the scalar the least length lines, obtained from the least action principle, are chr.inv.-equation of motion (3.147) under the condition that the merely a particular case of constant direction lines, which result vector Aα has the structure as mentioned in the above and the from the parallel transfer method. vector is tangential to the particle’s trajectory. Substituting chr.inv.- projections ϕ and qi of the vector Aα of the given structure into §3.10 Structure of a space, filled with stationary electromagnetic (3.146), we transform the quantity T to the form fields

2 i ∗∂ϕ e ∗∂ k k i It is evidently, setting a particular structure of electromagnetic c T = eEi v e + ϕhik v ϕDik q v = − − ∂t c2 ∂t − fields imposes certain limits on motion of charges, which, in their (3.206) ∂ϕ v2 eϕ eϕ ∂vk turn, imposes limitations on the structure of a pseudo-Riemannian = eE vi e ∗ 1 + D vivk + v ∗ . − i − ∂t − c2 c2 ik c2 k ∂t space where the motions take place. We are going to find out what 3.10 A space, filled with stationary electromagnetic fields 107 108 Chapter 3 Motion of charged particles kind of the structure the pseudo-Riemannian space should have so Because chr.inv.-derivative with respect to time is different from that a charged particle moved in a stationary electromagnetic field. ∂ w ∗∂ the regular derivative only down to multiplier = 1 2 , the Chr.inv.-equations of motion of a charged mass-bearing particle ∂t − c ∂t regular derivative of stationary quantity is zero as well. in our world have the form For the tensor of the space deformations rate Dik under a sta- dE dϕ e tionary rotation of the space we have follows mF vi + mD vivk = e + F qi D qivk , (3.208) dτ − i ik − dτ c i − ik ∗∂Dik 1 ∗∂hik 1 ∗∂ 1 1 ∗∂gik i = = gik + 2 vivk = . (3.214) d mv i i i k i n k ∂t 2 ∂t 2 ∂t − c −2 ∂t mF + 2m Dk + Ak∙ v + mΔnk v v = dτ − ∙ i (3.209) Because in the case under consideration the right parts of equa- e dq e ϕ k k i i eϕ i e i n k = v + q Dk +Ak∙ + F Δnk q v . tions of motion are stationary, the left parts should be the same − c dτ − c c ∙ c2 − c as well. This implies, that the space does not deformations. Then Because we assume the electromagnetic field to be stationary, according to (3.124) the three-dimensional coordinate metric gik the field potentials ϕ and qi depend on spatial coordinates, but not i does not depend on time, so the Christoffel chr.inv.-symbols Δjk time. In this case chr.inv.-components of the electromagnetic field (1.47) are stationary as well. tensor are Using chr.inv.-components of the Maxwell tensor (3.210, 3.211), ∂ϕ ϕ ∂ϕ ∂ w we transform the Maxwell equations (3.63, 3.64) for the stationary E = ∗ F = ϕ ln 1 , (3.210) i ∂xi − c2 i ∂xi − ∂xi − c2 electromagnetic field. As a result we have i i 1 imn 1 imn ∂qm ∂qn 2ϕ ∂E ∂ ln √h 2 H∗ = ε Hmn = ε Amn . (3.211) + Ei Ω H m = 4πρ 2 2 ∂xn − ∂xm − c i i m ∗ ∂x ∂x − c ∗  I, (3.215) From here we can arrive to limitations on the space metric, ikm 4π i  ε ∗ k H m√h = j √h  imposed by the stationary state of the acting electromagnetic field. ∇ ∗ c i The formulas for Ei and H∗ , among with chr.inv.-derivatives  i e √  of the scalar and vector electromagnetic potentials, also include ∂H∗ ∂ ln h i 2 m i + i H∗ + Ω mE = 0 properties of the space, namely — the chr.inv.-vector of gravitational ∂x ∂x c ∗  II . (3.216) inertial force Fi and the chr.inv.-tensor of the space non-holonomity ikm  ε ∗ k Em√h = 0  Aik. It is evidently, in stationary electromagnetic fields the men- ∇ tioned properties of the space should be stationary as well  Then the Lorentze condition (3.65) and the continuity equation ∂F ∂F i ∂A ∂Aik (3.66), respectively, take the form as down ∗ i = 0 , ∗ = 0 , ∗ ik = 0 , ∗ = 0 . (3.212) ∂t ∂t ∂t ∂t i i ∗ q = 0 , ∗ j = 0 . (3.217) ∇i ∇i From these definitions we see that the quantities Fi and Aik are stationary (do not depend on time), if the linear velocity of the space So, we have founde the way that anye stationary state of an elec- rotation is as well stationary ∂vi = 0. So, the condition ∂vi = 0, tromagnetic field, located in a pseudo-Riemannian space, affects ∂t ∂t physical observable properties of the space itself and hence the namely — stationary rotation of the space, turns chr.inv.-derivative main equations of electrodynamics. with respect to spatial coordinates into the regular derivative In the next §3.11–§3.13 we will use the results for solving ∂ ∂ 1 ∂ ∂ equations of motion of a charged particle (3.208, 3.209) in stationary ∗ = ∗ = . (3.213) ∂xi ∂xi − c2 ∂t ∂xi electromagnetic fields of three kinds: 3.11 Motion in a stationary electric field 109 110 Chapter 3 Motion of charged particles

(a) A stationary electric field (the magnetic strength is zero); electromagnetic field has non-zero electric component and zero (b) A stationary magnetic field (the electric strength is zero); magnetic component, then the pseudo-Riemannian space where the field is located should satisfy the conditions as: (c) A stationary electromagnetic field (the both components are 1. Potential w of the acting gravitational field is negligible w 0; non-zeroes). ≈ 2. The space does not rotate Aik = 0;

§3.11 Motion in a stationary electric field 3. The space does not deformations Dik = 0. To make further calculations easier we assume that our three- We are going to consider motion of a charged mass-bearing particle i dimensional space is close to Euclidean one, so we assume Δnk 0. in a pseudo-Riemannian space, filled with a stationary electromag- Then chr.inv.-equations of motion of a particle bearing an elec-≈ netic field of strictly electric kind. The magnetic component of the tric charge e (3.208, 3.209) take the form field does not reveal itself for the observer, so the component is absent, in other word. dm e dϕ = , (3.221) What conditions should the space satisfy to allow existence of a dτ −c2 dτ stationary electromagnetic field of strictly electric kind? From the d e dqi formula for a stationary state of the magnetic strength mvi = . (3.222) dτ − c dτ ∂qi ∂qk 2ϕ Hik = Aik (3.218) From the scalar chr.inv.-equation of motion (the live forces the- ∂xk − ∂xi − c orem) we can see that change of the particle’s relativistic energy 2 we see that Hik = 0 in this case provided two conditions: E = mc is due to work done by the field electric component Ei. ∂q ∂q From the vector chr.inv.-equations of motion we can see that 1. The vector-potential qi is irrotational i = k ; ∂xk ∂xi the particle’s observable impulse has change under change of the i 2. The space is holonomic Aik = 0. field vector-potential q . Assuming that the field four-dimensional potential is tangential to the four-dimensional trajectory of the The stationary electric strength E (3.210) is the sum of the i ϕ particle, we get the Lorentz three-dimensional force spatial derivative of the scalar potential ϕ and of the term F . c2 i But on the Earth surface the ratio of the gravitational potential and Φi = eEi (3.223) the square of the light velocity is nothing but only − in the right part. That is, in this case the particle’s observable im- w GM 10 pulse has change under action of the electric strength of the field. = ⊕ 10− . (3.219) c2 c2R ≈ Both groups of the Maxwell chr.inv.-equations for a stationary ⊕ field (3.215, 3.216) in this case become very simple Therefore in a real Earth laboratory the second term in (3.210) may be neglected so that the Ei will only depend on spatial distri- ∂Ei bution of the scalar potential = 4πρ ∂E ∂xi I, εikm m = 0 II . (3.224)  ∂xk  ∂ϕ i E = . (3.220) j = 0   i ∂xi Because the right parts of the equations of motion that stand for Integrating the scalar chr.inv.-equation of motion (the live forces the Lorentz force are stationary, the left parts should be stationary theorem) we arrive to so-called the live forces integral too. Under the conditions we are considering this is true if the eϕ m + = B = const, (3.225) tensor of the space deformations rate is zero. So, if a stationary c2 3.11 Motion in a stationary electric field 111 112 Chapter 3 Motion of charged particles where B is integration constant. Integrating the scalar chr.inv.-equation of motion (the live forces Another consequence from the Maxwell chr.inv.-equations is theorem), we arrive to the live forces integral that in this case the scalar potential of the field satisfies either: eE ∂2ϕ ∂2ϕ ∂2ϕ m = x + B,B = const. (3.230) 1. Poisson’s equation + + = 4πρ, if ρ = 0; c2 ∂x2 ∂y2 ∂z2 6 This constant B can be obtained from the initial conditions of ∂2ϕ ∂2ϕ ∂2ϕ 2. Laplace’s equation + + = 0, if ρ = 0. integration m = m and x = x ∂x2 ∂y2 ∂z2 |τ=0 (0) |τ=0 (0) So, we have found out the properties of the pseudo-Riemannian eE B = m x , (3.231) space that allows motion of charged particles in a stationary electric (0) − c2 (0) field. It would be natural now to obtain exact solutions of chr.inv.- so the solution (3.230) takes the form equations of motion for a such particle, namely — the equations (3.221, 3.222). But unless a particular structure of the field itself is eE m = x x + m . (3.232) set by the Maxwell equations this can not be done. For this reason to c2 − (0) (0) simplify the calculations we assume the electric field homogenous. Substituting the obtained integral of live forces into the vector We assume that the covariant chr.inv.-vector of the electric chr.inv.-equations of motion (3.229), we bring them to the form∗ strength Ei is directed along the axis x. Following Landau and Lifshitz (see §20 of The Classical Theory of Fields [5]) we are going eE eE x˙ 2 + B + x x¨ = eE , to consider the case of a charged particle repulsed by the field — c2 c2 the case of a negative value of the electric strength and increasing eE eE x˙ y˙ + B + x y¨ = 0 , (3.233) coordinate x of the particle∗. Then components of the vector Ei are c2 c2 eE eE E1 = Ex = E = const, E2 = E3 = 0 . (3.226) x˙ z˙ + B + x z¨ = 0 . − c2 c2 ∂ϕ Because the field homogeneity implies E = = const, the sca- i ∂xi From here the last two are equations with separable variables lar potential ϕ is a function of x that satisfies the Laplace equation eE x˙ eE x˙ ∂2ϕ ∂E y¨ − c2 z¨ − c2 = = 0 . (3.227) = , = , (3.234) ∂x2 ∂x y B + eE x z B + eE x c2 c2 This implies that the homogeneous stationary electric field sat- which can be integrated. Their solutions are isfies the condition of the absence of charges ρ = 0. C C We assume that the particle moves along the electric strength Ei, y˙ = 1 , z˙ = 2 , (3.235) so it is directed along x. Then chr.inv.-equations of its motion are B + eE x B + eE x c2 c2 dm e dϕ e dϕ i e dx where C1 and C2 are integration constants which can be found = 2 = 2 i v = 2 E , (3.228) dτ −c dτ −c dx c dτ setting the initial conditions y˙ =y ˙ and x˙ =x ˙ and using |τ=0 (0) |τ=0 (0) d dx d dy d dz the formula for B (3.121). As a result we obtain m = eE , m = 0 , m = 0 . (3.229) dτ dτ dτ dτ dτ dτ C1 = m(0) y˙(0) ,C2 = m(0) z˙(0) . (3.236) ∗Naturally, in the case of the particle attracted by the field the electric strength is positive while the coordinate of the particle decreases. ∗Dot stands for derivation with respect to physical observable time τ. 3.11 Motion in a stationary electric field 113 114 Chapter 3 Motion of charged particles

Let us solve the equation of motion along x — the first equation Substituting the variable p = dx into the formula (3.239) we dt from (3.233). So, we replace x˙ = dx = p. Then arrive to the last equation with separable variables dτ

2 2 d x dp dp dx eE 2 x¨ = = = = pp0, (3.237) B + x C dt2 dt dx dt dx c2 − 3 = c r , (3.242) dt B + eE x and the above equation of motion along x transforms into an equa- c2 tion with separable variables which solution is the function pdp eEdx = , (3.238) p2 eE 2 2 1 B + x c eE 2 2 c2 ct = B + x C3 + C4 ,C4 = const, (3.243) − c eE c2 − s which is a table integral. After integration we arrive to the solution where the integration constant C4, taking into account the initial 2 conditions at the moment t = 0, is p C3 1 2 = ,C3 = const. (3.239) − c B + eE x m(0)c r c2 C = x˙ . (3.244) 4 − eE (0) Assuming p =x ˙ =x ˙ and substituting B from (3.231) we τ=0 (0) Now formulating coordinate x explicitly from (3.243) with t we find the integration| constant obtain the final solution of the spatial chr.inv.-equations of motion

2 of the charged particle along x x˙ (0) C3 = m(0) 1 . (3.240) s − c2 2 2 2 c e E 2 2 x = 4 (ct C4) + C3 B , (3.245) In the case under consideration we can replace the interval of eE "r c − − # physical observable time dτ with the interval of coordinate time dt. Here is why. or, after substituting integration constants In The Classical Theory of Fields Landau and Lifshitz solved 2 2 2 2 2 equations of motion of a charged particle in a Galilean reference m(0)cx˙ (0) m(0)c x˙ (0) m(0)c x= ct+ + 1 +x(0) . (3.246) frame in the Minkowski space of the Special Theory of Relativi- s eE eE − c2 − eE ty [5]. Naturally, to be able to compare our solutions with theirs we consider the same particular case — motion in a homogeneous If the field attracts the particle (the electric strength is positive stationary electric field (see §20 in The Classical Theory of Fields). E1 = Ex = E = const), we will obtain the same solution for x but But in this case, as we showed earlier in this §3.11, using the bearing the opposite sign methods of chronometric invariants, we have Fi = 0 and Aik = 0, c2 e2E2 hence we obtain that in this case x = B (ct C )2 + C2 . (3.247) eE c4 4 3 w 1 " − r − # dτ = 1 dt v dxi = dt . (3.241) − c2 − c2 i In The Classical Theory of Fields [5] a similar problem is con- In other word, in the four-dimensional area under this study sidered, but Landau and Lifshitz solved it through integration of where the particle travels the metric is Galilean one. three-dimensional components of general covariant equations of 3.11 Motion in a stationary electric field 115 116 Chapter 3 Motion of charged particles

motion (the Minkowski three-dimensional equations) without ac- where C5 is integration constant. From y = y(0) at t = 0 we find counting for the live forces theorem. Their formula for x is m(0) y˙(0) c x˙ (0) C5 = y arc sinh . (3.251) (0) 2 1 2 2 2 − eE x˙ x = (m0c ) + (ceEt) . (3.248) (0) eE c 1 2 q 2 r − c m(0)c This formula matches our solution (3.245) if x(0) = 0 and Substituting the constant into y (3.250) we finally have − eE the initial velocity of the particle is zero x˙ (0) = 0. The latter stands m(0) y˙(0) c for significant simplifications accepted in The Classical Theory of y = y(0) + Fields, according to which some integration constants are assumed eE × zeroes. (3.252) eEt + m(0) x˙ (0) x˙ (0) As it easy to see, even when solving equations of motion in a arc sinh arc sinh  . Galilean reference frame in the Minkowski space the mathematical ×  x˙ 2 − x˙ 2   (0) (0)  methods of chronometric invariants give certain advantages reveal- m(0) c 1 2 c 1 2 r − c r − c ing hidden factors which are left unnoticed when solving regular   Formulating from here t with y and y(0) and taking into account three-dimensional components of general covariant equations of that a = arc sinh b if b = sinh a, after substituting formula arc sinh b = motion. That means that even when physical observable quantities = ln (b + √b2 + 1) into the second term we have coincide coordinate quantities, this is geometrically correct to solve a system of chr.inv.-equations of motion, because the live forces theorem, being their scalar part, inevitably affects the solution of 1 x˙ 2 t = m c 1 (0) the vector equations.  (0) 2 eE  s − c × Of course in the case of an inhomogeneous non-stationary elec-  tric field some additional terms will appear in our solution to reflect (3.253)  more complicated and varying in time the field structure.  y y(0) x˙ (0) + c Now, let us calculate three-dimensional trajectory of the particle sinh  − eE + ln  m(0) x˙ (0) . in the homogeneous stationary electric field we are considering. To × m(0) y˙(0) c x˙ 2 −  (0)   c 1 2  obtain it, we integrate the equations of motion along the axes y  r − c  and z (3.235), formulate time from there and substitute it into the    Now we substitute it into our solution for x (3.246). As a result solution for x we have obtained.  we obtain the desired equation for the three-dimensional trajectory First, substituting the obtained solution for x (3.245) into the of the particle equation for y˙, we obtain the equation with separable variables 2 x˙ 2 dy C1 m(0) c (0) = , (3.249) x = x(0) + 1 2 dt 2 2 eE s − c × e E 2 2 (ct C4) + C c4 − 3 (3.254) r 2 y y(0) x˙ (0) + c m(0) c integrating which we have cosh  − eE + ln  . × m(0) y˙(0) c x˙ 2  − eE  c 1 (0)  m(0) y˙(0) c eEt + m(0) x˙ (0) − c2 y = arc sinh + C5 , (3.250) r eE 2   x˙ (0) The obtained formula implies that a charged particle in a homo- m(0) c 1 2 r − c geneous stationary electric field, located in our world, travels along 3.11 Motion in a stationary electric field 117 118 Chapter 3 Motion of charged particles a curve based on chain line, while factors which deviate it from we obtain the energy of the particle “pure” chain line are functions of the initial conditions. 2 2 Our formula (3.254) fully matches the result from The Classical m c m(0) c + eE x x(0) E = (0) = − , (3.259) Theory of Fields v2 2 2 2 1 x˙ (0) +y ˙(0) +z ˙(0) 2 2 m c eEy − c 1 2 x = (0) cosh (3.255) q r − c eE m(0) y˙(0) c which at the velocity much lower than the light velocity is m c2 (formula 20.5 in [5]) once we assume that x (0) = 0, and the (0) eE − E = m c2 + eE x x . (3.260) initial velocity of the particle x˙ (0) = 0 as well. The latter condition (0) − (0) suggests that the integration constant in the scalar chr.inv.-equation of motion (the live forces theorem) is zero, which is not always true. The relativistic impulse of the particle is obtained in the same At low velocities after equaling relativistic terms to zero and ex- way, but the formula being bulky is withheld here. 2 4 6 panding hyperbolic cosine into series cosh b = 1 + b + b + b + ... So, we have studied motion of a charged particle in a homo- 2! 4! 6! geneous stationary electric field, located in our world. Now we our formula for the three-dimensional trajectory of the particle consider motion of an analogous particle of the mirror world under (3.254), having higher order terms withheld, takes the form the same conditions. 2 eE y y Chr.inv.-equations of motion of the mirror-world particle, taking − (0) x = x(0) + 2 , (3.256) into account the constraints imposed here on the geometric struc- 2m(0) y˙(0) ture of the space, are so the particle travels along parabola. This conclusion, once the dm e dϕ = , (3.261) initial coordinates of the particle are assumed zeroes, also matches dτ c2 dτ the result from The Classical Theory of Fields d e dqi eEy2 mvi = . (3.262) x = . (3.257) dτ − c dτ 2m y˙2 (0) (0) In other word, the only difference from the equations in our Integration of the equation of motion along the axis z gives world (3.221, 3.222) is the sign in the live forces theorem. the same results. This is because the only difference between the We assume that the electric strength is negative (i. e. the field equations with respect to y˙ and z˙ (3.235) is a fixed coefficient — repulses the particle) and that the particle moves along the field the integration constant (3.236), which equals to the initial impulse strength, so it is co-directed with the axis x. of the particle along y (in the equation for y˙) and along z (in the Then integrating the live forces theorem for the mirror-world equation for z˙). particle (3.261) we obtain the live forces integral Let us find properties of the particle (its energy and impulse) affected by the acting homogeneous stationary electric field. Calcu- eE lating the relativistic square root (accounting for the assumptions m = x + B, (3.263) − c2 we made) 2 2 2 where the integration constant, calculated from the initial condi- x˙ (0)+y ˙(0)+z ˙(0) v2 x˙ 2+y ˙2+z ˙2 m(0) 1 2 tions, is 1 = 1 = r − c , (3.258) − c2 − c2 eE eE r r m(0)+ x x(0) B = m(0) + x(0) . (3.264) c2 − c2 3.11 Motion in a stationary electric field 119 120 Chapter 3 Motion of charged particles

Substituting the results into the vector chr.inv.-equations of mirror world. Calculating y in the same manner as for the our- motion (3.262), we have (compare them with 3.233) world particle, we have

eE eE m(0) y˙(0) c x˙ 2 + B x x¨ = eE , y = y(0) + − c2 − c2 eE × eE eE (3.268) x˙ y˙ + B x y¨ = 0 , (3.265) eEt + m(0) x˙ (0) x˙ (0) − c2 − c2 arcsin arcsin  . ×  x˙ 2 − x˙ 2  eE eE  (0) (0)  x˙ z˙ + B x z¨ = 0 . m(0) c 1 + 2 c 1 + 2 − c2 − c2 r c r c   In contrast to the formula for the our-world particle (3.252) this After some algebra similar to that done to obtain the trajectory formula has regular arcsine and “plus” sign under the square route. of the our-world charged particle, we arrive to Formulating time t from here with the coordinates y and y(0)

c2 e2E2 x = B C2 (ct C )2 , (3.266) x˙ 2 eE 3 c4 4 1 (0) " − r − − # t = m(0) c 1 + eE  s c2 ×  x˙ 2 cm x (3.269) where C = m 1 + (0) and C = (0) (0) . Or,  3 (0) c2 4 − eE  r y y(0) x˙ (0) + c sin  − eE + ln  m(0) x˙ (0) , 2 2 2 × m y˙ c 2 − 2 x˙ (0) (0) x˙ (0)  m(0)c (0) m(0)cx˙ (0) c 1 +  x = 1 + ct + +  c2  − s eE c2 − eE  r  (3.267)    m c2 and substituting it into our formula for x (3.267), we obtain the final + (0) + x . eE (0) formula for the trajectory

The obtained coordinate x of the mirror-world charged particle, m c2 x˙ 2 x = x (0) 1 + (0) repulsed by the field, is similar to that for the our-world particle (0) − eE s c2 × attracted by the field (3.247) when the electric strength is positive (3.270) E1 = Ex = E = const. Hence an interesting conclusion: transition of 2 y y(0) x˙ (0) m(0) c a charged particle from our world into the mirror world (where is cos  − eE + arcsin  . × m(0) y˙(0) c x˙ 2  − eE the reverse flow of time) is the same as changing the sign of its  (0)  c 1 + 2 charge. r c   Noteworthy, the similar conclusion can be done in respect of In other word, motion of the particle is harmonic oscillation. particles’ masses: purported transition of a particle from our world Once we assume the initial coordinates of the particle equal to zero, into the mirror world is the same as changing the sign of its mass. as well as its initial velocity x˙ (0) = 0 and the integration constant Hence our-world particles and mirror-world particles are mass and B = 0, the obtained equation of the trajectory takes a simpler form charge complementary. m c2 eEy Let us find the three-dimensional trajectory of the charged x = (0) cos . (3.271) particle in the homogeneous stationary electric field, located in the − eE m(0) y˙(0) c 3.12 Motion in a stationary magnetic field 121 122 Chapter 3 Motion of charged particles

At low velocities, after equaling relativistic terms to zero and ex- will use later as the basic space for spin-particles). As we did it in 2 4 6 2 panding into the cosine series cos b = 1 b + b b + ... 1 b the previous §3.11, we assume deformations of the space to be zero − 2! 4! − 6! ≈ − 2! (which is always possible within a smaller part of the trajectory), and the three-dimensional metric to be Euclidean gik = δik. The observable metric h = g + 1 v v in this case is not Galilean we bring our formula (3.270) as follows ik − ik c2 i k one, because in non-holonomic spaces we have h = g . 2 ik 6 − ik eE y y(0) We assume that the space rotates around the axis z at the x = x + − , (3.272) (0) 2 constant angular velocity Ω = Ω = Ω. Then the linear velocity 2m(0) y˙(0) 12 21 k − of this rotation vi = Ωik x has two non-zero components v1 = Ωy and which is the equation of parabola. So, the charged particle in the v = Ωx, while the non-holonomity tensor has the only non-zero 2 − mirror world at low velocity travels along a parabola, as does the components A12 = A21 = Ω. In this case the metric takes the form our-world particle in the same conditions in the field. − − ds2 = c2dt2 2Ωydtdx + 2Ωxdtdy dx2 dy2 dz2. (3.275) Therefore, a charged particle of our world travels in homoge- − − − − neous stationary electric fields along a chain line, which at low In this space we have Fi = 0 and Dik = 0. In the previous §3.11, velocities becomes a parabola. An analogous mirror-world particle which focused on a charged particle in a stationary electric field, travels along a harmonic trajectory, smaller parts of which at low we assumed the Christoffel symbols to be zeroes. In other word, velocities becomes a parabola (as is the case for the our-world we considered its motion in a Galilean reference frame in the particle). Minkowski space. But in this §3.12 the three-dimensional observable metric hik is not Euclidean one, because the space rotation and i §3.12 Motion in a stationary magnetic field the Christoffel symbols Δjk (1.47) are not zeroes. If the linear velocity of the space rotation is not infinitesimal Let us consider motion of a charged particle when the electric compared to the light velocity, components of the metric chr.inv.- component of the electromagnetic field is absent, while the magnetic tensor hik are component is present and it is stationary. In this case chr.inv.- Ω2y2 Ω2x2 Ω2xy h = 1 + , h = 1 + , h = , h = 1 , (3.276) vectors of the electric and magnetic strengthes are 11 c2 22 c2 12 − c2 33 ik ∗∂ϕ ϕ ∂ϕ ϕ 1 ∂w so its determinant and components of h are Ei = i 2 Fi = i 2 i = 0 , (3.273) ∂x − c ∂x − c 1 w ∂x Ω2 x2 + y2 − c2 2 h = det hik = h11h22 h12 = 1 + 2 , (3.277) k k − c i 1 imn 1 imn ∂qm ∂qn 2ϕ H∗ = ε Hmn = ε Amn = 0 (3.274) 1 Ω2x2 1 Ω2y2 2 2 ∂xn − ∂xm − c 6 h11 = 1 + , h22 = 1 + , 2 2 h c h c (3.278) because if the field is strictly magnetic ϕ = const (E = 0), then i Ω2xy gravitational effect can be neglected. From (3.274) we can see that h12 = , h33 = 1 . i hc2 the magnetic strength H∗ is not zero, if at least one of the following conditions is true: Respectively, from here we obtain non-zero components of the Δi 1. The potential qi is rotational; Christoffel chr.inv.-symbols jk (1.47), namely 4 2 2. The space is non-holonomic Aik = 0. 1 2Ω xy 6 Δ11 = , (3.279) Ω2 x2 + y2 We are going to consider motion of the particle in general case, c4 1 + c2 when the both conditions are true (the non-holonomic space we 3.12 Motion in a stationary magnetic field 123 124 Chapter 3 Motion of charged particles

2 2 2 2Ω x d i i n k e ikm Ω y 1 + 2 mv + mΔnkv v = ε vkH m . (3.288) 1 c dτ − c ∗ Δ12 = , (3.280) Ω2 x2 + y2 c2 1 + Integrating the live forces theorem for the our-world particle c2 and the mirror-world particle we obtain, respectively Ω2x2 2 1 + m0 m0 1 2Ω x c2 m = = const = B, m = = const = B, (3.289) Δ22 = , (3.281) − c2 Ω2 x2 + y2 1 v2 − 1 v2 1 + − c2 − c2 c2 q q e 2 Ω2y2 where B and B are integration constants. That implies v = const, so 2 1 + 2 2Ω y c2 the module of the particle’s observable velocity remains unchanged Δ11 = , (3.282) − c2 Ω2 x2 + y2 in the absencee of the electric component of the electromagnetic 1 + c2 field. Then the vector chr.inv.-equations of motion for the our-world particle (3.286) are 2Ω2y2 Ω2x 1 + 2 c2 dvi e Δ12 = , (3.283) i k i n k ikm Ω2 x2 + y2 + 2Ak∙ v + Δnkv v = ε vkH m , (3.290) c2 1 + dτ ∙ −mc ∗ c2 while for the mirror-world particle (3.288) we have the same equa- 2Ω4x2y i k Δ2 = . (3.284) tion but without the term 2Ak∙ v , namely 22 − 2 2 2 ∙ 4 Ω x + y i c 1 + 2 dv i n k e ikm c + Δnkv v = ε vkH m . (3.291) dτ −mc ∗ We are going to solve chr.inv.-equations of motion of a charged particle in the stationary magnetic field, located in the pseudo- The magnetic strength here is defined by the Maxwell equations Riemannian space. To make the calculations easier we assume that for stationary fields (3.215, 3.216), which in the absence of the the field four-dimensional potential Aα is tangential to the four- electric strength and under the constraints we assumed in this §3.12, are dimensional trajectory of the particle. Because the field electric m Ω mH∗ = 2πcρ component is zero Ei = 0, it does not perform any work, so the right ∗ − I, (3.292) parts of the scalar chr.inv.-equation of motion turn into zeroes. ikm 4π i  ε ∗ k H m√h = j √h Applying chr.inv.-equations of motion of a charged particle (3.208, ∇ ∗ c  3.209) to the particle in the stationary magnetic field located in our i  world, we obtain i ∂H∗ ∂ ln √h i dm ∗ i H∗ = i + i H∗ = 0 II . (3.293) = 0 , (3.285) ∇ ∂x ∂x  dτ  From the first equation of the 1st group we see that the scalar d i i k i n k e ikm  mv + 2mAk∙ v + mΔnkv v = ε vkH m , (3.286) dτ ∙ − c ∗ product of the space non-holonomity pseudovector and the magnetic strength pseudovector is a function of the charge density. while for the analogous charged particle which moves in the same Hence, if the charge density is ρ = 0, then the pseudovectors Ω i stationary magnetic field located in the mirror world, we have i ∗ and H∗ are orthogonal. dm So forth we consider two possible orientations of the magnetic = 0 , (3.287) − dτ strength in respect of the space non-holonomity pseudovector. 3.12 Motion in a stationary magnetic field 125 126 Chapter 3 Motion of charged particles

Ω2(x2 + y2) A Magnetic field is co-directed with non-holonomity field Because of h = 1 + (3.277), this implies: the current c2 i We assume that the magnetic strength pseudovector H∗ is directed vector in the homogeneous stationary magnetic field is non-zero in along the axis z, i. e. in the same direction that the pseudovector of only the strong field of the space non-holonomity, i. e. where the i 1 ikm angular velocities of the space rotation Ω∗ = 2 ε Akm. Then the space rotation velocity is comparable to the light velocity. In a weak 3 1 2 space rotation pseudovector has one non-zero component Ω∗ = Ω, field of the space non-holonomity we have h = 1, hence j = j = 0. while the magnetic strength pseudovector has Now expressing the magnetic strength from the Maxwell equations (3.295) we write down the vector chr.inv.-equations of motion 3 1 3mn 1 312 321 H∗ = ε H = ε H + ε H = H = for the our-world particle (3.290, 3.291) in the form 2 mn 2 12 21 12 (3.294) 2 2 2 ϕ ∂v1 ∂v2 2ϕ 2Ω Ω xyx˙ Ω x = + Ω . x¨ + + 1 + y˙ + Δ1 x˙ 2 + 2Δ1 x˙ y˙ + c ∂x − ∂y c h c2 c2 11 12 The condition ϕ = const is derived from the absence of the field eH Ω2xyx˙ Ω2x2 + Δ1 y˙2 = + 1 + y˙ , electric component. Hence the 1st group of the Maxwell equations 22 − mc − c2 c2 (2.392) in this case are 2 2 2 (3.298) Ωϕ ∂v ∂v 2ϕ Ω2 2Ω Ω xyy˙ Ω y 2 2 2 3 1 2 y¨ + 1 + x˙ + Δ11 x˙ + 2Δ12 x˙ y˙ + Ω 3H∗ = + = 2πcρ − h c2 c2 ∗ c ∂x − ∂y c −  2 2 2  2 2 eH Ω xyy˙ Ω y ∂ 4π 1  + Δ y˙ = + 1 + x˙ , H 3√h = j √h  22 2 2 ∂y ∗ c  mc − c c  . (3.295)  ∂ 4π 2  z¨ = 0 , H 3√h = j √h −∂x ∗ c  while those for the mirror-world particle they are 3  j = 0   1 2 1 1 2  x¨ + Δ11x˙ + 2Δ12 x˙ y˙ + Δ22 y˙ = The 2nd group of the equations (3.293) will be trivial turning 3 eH Ω2xyx˙ Ω2x2 into simple relationship ∂H∗ = 0, so that H 3 = const. Actually this = + 1 + y˙ , ∂z ∗ − mc − c2 c2 implies that the stationary magnetic field we are considering is homogneous along the z. In the below we assume the stationary 2 2 2 2 2 (3.299) i y¨ + Δ x˙ + 2Δ x˙ y˙ + Δ y˙ = magnetic field to be strictly homogeneous H∗ = const. Then from 11 12 22 the first equation of the 1st group (3.295) we see that the field is eH Ω2xyy˙ Ω2y2 = + 1 + x˙ , homogeneous provided that mc − c2 c2 ∂v ∂v ϕ Ω2 1 2 = const, ρ = = const. (3.296) z¨ = 0 . ∂x − ∂y − πc2 2 The terms in the right parts which contain Ω appear, because in Hence the charge density is ρ > 0, if the field scalar potential is c2 ϕ < 0. Then the other equations from the 1st group (3.295) are the space rotation the observable chr.inv.-metric hik is not Euclid- ean. Hence in the case under consideration there is a difference c ∂ ln √h c ∂ ln √h between the contravariant form of the observable velocity and its j1 = , j2 = . (3.297) 4π ∂y 4π ∂x covariant form. The right parts include the covariant components 3.12 Motion in a stationary magnetic field 127 128 Chapter 3 Motion of charged particles

Ω2xy Ω2x2 First we approach the equations for the our-world particle. The v = h v1 + h v2 = x˙ + 1 + y˙ , (3.300) 2 21 22 − c2 c2 equation along z can be integrated straightaway. The solution is Ω2xy Ω2y2 z =z ˙(0)τ + z(0) . (3.306) v = h v1 + h v2 = y˙ + 1 + x˙ . (3.301) 1 11 12 − c2 c2 From here we see that if at the initial moment of time the If no the space rotation here Ω = 0, then the chr.inv.-equations of particle’s velocity along z is zero, so the particle moves within x y motion of the our-world particle (3.298) to within their sign match plane only. The rest two equations of the (3.304) we re-write as the equations of motion in a homogeneous stationary magnetic field follows dx˙ dy˙ given by Landau and Lifshitz (see formula 21.2 in The Classical = (2Ω + ω)y ˙ , = (2Ω + ω)x ˙ , (3.307) Theory of Fields) dτ − dτ eH eH eH where we denote ω = mc for convenience. The same notation was x¨ = y˙ , y¨ = x˙ , z¨ = 0 , (3.302) used in §21 of The Classical Theory of Fields. Then, formulating x˙ mc − mc from the second equation, we derive it to the observable time x˙ and while our equations (3.298) imply that substitute the result into the first equation. So, we obtain eH eH d2y˙ x¨ = y˙ , y¨ = x˙ , z¨ = 0 . (3.303) + (2Ω + ω)2 y˙ = 0 , (3.308) − mc mc dτ 2 The difference is derived from the fact that Landau and Lifshitz which is the equation of oscillations; it solves as follows assumed the magnetic strength in the Lorentz force to bear “plus” sign, while in our equations it bears “minus”, which is not that y˙ = C1 cos (2Ω + ω) τ + C2 sin (2Ω + ω) τ , (3.309) important though, because only depends on choice of the space y¨ where C =y ˙ and C = (0) are integration constants. Substi- signature. 1 (0) 2 2Ω + ω If the space rotates (it is non-holonomic), the equations of motion tuting y˙ (3.309) into the first equation (3.307) we obtain Ω2 Ω4 will include the terms that contain Ω, 2 , and 4 . dx˙ c c = (2Ω + ω)y ˙ cos (2Ω + ω) τ y¨ sin (2Ω + ω) τ , (3.310) In a strong field of the space non-holonomity solving the equa- dτ − (0) − (0) tions we have obtained is a non-trivial task, which is likely to be tackled in future with computer-aided numerical methods. Hope- or, after integration, fully, the results will be quite interesting. y¨(0) Let us find their exact solutions in a weak field of the space x˙ =y ˙(0) sin (2Ω + ω) τ cos (2Ω + ω) τ + C3 , (3.311) − 2Ω + ω non-holonomity, namely — truncating terms of the second order of smallness and below. In this case the equations of motion we have y¨(0) where the integration constant is C3 =x ˙ (0) + . obtained (3.298, 3.299) for the our-world particle are 2Ω + ω Having all the constants substituted, the obtained formulas for eH eH x˙ (3.311) and y˙ (3.309) finally transform into x¨ + 2Ωy ˙ = y˙ , y¨ 2Ωx ˙ = x˙ , z¨ = 0 , (3.304) − mc − mc y¨ y¨ x˙ =y ˙ sin (2Ω+ω) τ (0) cos (2Ω+ω) τ +x ˙ + (0) , (3.312) and for the mirror-world particle are (0) − 2Ω+ω (0) 2Ω+ω eH eH y¨ x¨ = y˙ , y¨ = x˙ , z¨ = 0 . (3.305) y˙ =y ˙ cos (2Ω + ω) τ + (0) sin (2Ω + ω) τ . (3.313) − mc mc (0) 2Ω + ω 3.12 Motion in a stationary magnetic field 129 130 Chapter 3 Motion of charged particles

Hence the formulas for components of the particle’s velocity trajectory within x y plane x˙ and y˙ in the homogeneous stationary magnetic field are the 1 y¨2 2C equations of harmonic oscillations. The frequency in a weak field of x2 + y2 = y˙2 + (0) 4 eH (2Ω + ω)2 (0) (2Ω + ω)2 − 2Ω + ω × the space non-holonomity is 2Ω + ω = 2Ω + mc . From the live forces integral in the stationary magnetic field y¨ y˙ cos (2Ω + ω) τ + (0) sin (2Ω + ω) τ + (3.289) we see that the square of the particle’s velocity is a constant (0) × 2Ω + ω (3.318) quantity. Calculating v2 =x ˙ 2 +y ˙2 +z ˙2 for the our-world particle we y¨(0) obtain that this quantity + y˙(0) sin (2Ω + ω) τ + cos (2Ω + ω) τ 2Ω + ω × 2C5 2 2 2 2 y¨(0) 2 2 v =x ˙ +y ˙ +z ˙ + 2 x˙ + + C4 + C5 . (0) (0) (0) (0) 2Ω + ω × × 2Ω + ω (3.314) y¨(0) y¨(0) Assuming for the initial moment of time y¨(0) = 0 and the integra- +y ˙(0) sin (2Ω+ω) τ cos (2Ω+ω) τ × 2Ω+ω − 2Ω+ω tion constants C4 and C5 to be zeroes, we can simplify the obtained formulas (3.315, 3.316), namely

2 y¨(0) 1 is constant v = const, provided that C3 =x ˙ (0) + = 0. x = y˙ cos (2Ω + ω) τ , (3.319) 2Ω + ω −2Ω + ω (0) Integrating x˙ and y˙ to τ (namely — integrating the equations 3.312, 3.313), we obtain coordinates of the our-world particle which 1 y = y˙ sin (2Ω + ω) τ . (3.320) moves in the homogeneous stationary magnetic field 2Ω + ω (0) Given the formulas, our equation of the trajectory (3.318) trans- y¨ 1 x = (0) sin (2Ω + ω) τ y˙ cos (2Ω + ω) τ + forms into a simple equation of the circle 2Ω + ω − (0) 2Ω + ω (3.315) y˙2 y¨ 2 2 (0) + x˙ + (0) τ + C , x + y = . (3.321) (0) 2Ω + ω 4 (2Ω + ω)2 Hence, if the initial velocity of the our-world charged particle y¨ 1 y = y˙ sin (2Ω+ω) τ + (0) cos (2Ω+ω) τ +C , (3.316) in respect of the direction of the homogeneous magnetic field (the (0) 2Ω+ω 2Ω+ω 5 axis z) is zero, then the particle moves within x y plane along a circle of the radius where the integration constants are y˙ y˙ r = (0) = (0) , (3.322) y˙(0) y¨(0) 2Ω + ω 2Ω + eH C4 = x(0) + ,C5 = y(0) + . (3.317) mc 2Ω + ω (2Ω + ω)2 which depends on the field strength and the angular velocity of the From (3.315) we see that the particle performs harmonic oscilla- space rotation. y¨ If the initial velocity of the particle along the magnetic field tions along x provided that the equation x˙ + (0) = 0 is true. (0) 2Ω + ω direction is not zero, then it moves along a spiral line of the radius This is also the condition for the constant square of the particle’s r along the field. In general case the particle moves along an ellipse velocity (3.314), i. e. it satisfies the live forces integral. Taking within x y plane (3.318), which shape deviates from that of a circle this result into account we arrive to the equation of the particle’s depending on the initial conditions of this motion. 3.12 Motion in a stationary magnetic field 131 132 Chapter 3 Motion of charged particles

As it is easy to see, our results match those in §21 of The Clas- is true and hence the argument of trigonometric functions in the sical Theory of Fields equations of motion becomes zero. 1 1 We consider the observer’s reference frame, whose reference x = y˙(0) cos ωτ , y = y˙(0) sin ωτ , (3.323) space is attributed to the nucleus in an atom. Then the ratio in the −ω ω question (in CGSE and Gaussian systems of units) for an electron once we assume Ω = 0, i. e. in the absence of the space rotation. in this atom is y˙(0) mc In this particular case the radius r = ω = y˙(0) of the particle’s 10 eH Ω e 4.8×10− 6 1 1 cm 2 gram 2 trajectory does not depend on the velocity of the space rotation. If = = 28 10 = 8.8×10 / , (3.325) H −2me c − − Ω = 0, then the non-holonomity field disturbs the particle to move 18.2×10− 3.0×10 6 in the magnetic field adding up with the magnetic strength, due to where their “minus” sign is derived from the fact that Ω and H in eH which the correction quantity 2Ω to the term ω = mc appears in the (3.324) are oppositely directed. equations. In a strong field of the space non-holonomity, where Ω Now, let us solve the equations of motion of the mirror-world can not be neglected compared to the light velocity, the disturbance particle in the homogeneous stationary magnetic field (3.305), which is even stronger. match the equations in the absence of the space non-holonomity On the other hand, in a non-holonomic space the argument of trigonometric functions in our equations contains a sum of two x¨ = ωy˙ , y¨ = ωx˙ , z¨ = 0 . (3.326) − terms, one of which is derived from interaction of the particle’s charge with the magnetic strength, while the other is a result of The solution of the third equation of motion (along z) is a simpler the space rotation, which depends neither on the electric charge of integral z =z ˙(0)τ + z(0). this particle, nor on the presence of the magnetic field. This allows The equations of motion along x and y are similar to those considering two special cases of motion of a charged particle in a for the our-world particle, save for the fact that the argument of homogeneous stationary magnetic field, located in a non-holonomic trigonometric functions has ω instead of ω + 2Ω space. y¨(0) y¨(0) In the first case, where the particle is electrically neutral or the x˙ =y ˙ sin ωτ cos ωτ +x ˙ + , (3.327) (0) − ω (0) ω magnetic field is absent, its motion will be the same as that under action of the magnetic component of the Lorentz force, save that y¨ y˙ =y ˙ cos ωτ + (0) sin ωτ . (3.328) this motion will be caused by the space rotation 2Ω, comparable to (0) ω ω = eH . mc Hence the formulas for components of the velocity of the mirror- How real this case may be? To answer this question we need at world particle x˙ and y˙ are the equations of harmonic oscillations at least an approximate assessment of the ratio between the angular eH velocity of the space rotation Ω and the magnetic strength H in a the frequency ω = mc . special case. The best example may be an atom, because on the Consequently, their solutions, namely — formulas for coordinates scales of electronic orbits electromagnetic interactions are a few of the mirror-world particle moving in the homogeneous stationary orders of the magnitude stronger than the others and beside, orbital magnetic field are velocities of electrons are relatively high. 1 y¨ y¨ Such assessment can be made proceeding from the second case x = (0) sin ωτ y˙ cos ωτ + x˙ + (0) τ + C , (3.329) ω ω − (0) (0) ω 4 of the special motions, where eH 1 y¨ = 2Ω , (3.324) y = y˙ sin ωτ + (0) cos ωτ + C , (3.330) mc − ω (0) ω 5 3.12 Motion in a stationary magnetic field 133 134 Chapter 3 Motion of charged particles where the integration constants are In such simplified case the mirror-world particle which is at rest in respect of the field direction makes a circle within x y plane y˙ y¨ C = x + (0) ,C = y + (0) . (3.331) 4 (0) ω 5 (0) ω2 y˙2 x2 + y2 = (0) (3.336) As we have already mentioned, the live forces integral in sta- ω2 tionary magnetic fields (3.289) implies the constant relativistic mass y˙ with radius r = (0) = mc y˙ . Consequently, if the initial velocity of of a particle moved in the fields and hence the constant square of ω eH (0) its observable velocity. Then putting the solutions for the velocities the particle along the the magnetic field direction (the axis z) is not of the mirror-world particle, namely — the quantities x˙, y˙, z˙, in the zero, then the particle moves along a spiral line around the magnetic power of two and adding them up we obtain that field direction. Hence motion of mirror-world charged particles in homogeneous stationary magnetic fields is the same as that of our- 2 2 2 2 v =x ˙ (0) +y ˙(0) +z ˙(0) + world charged particles in the absence of the space non-holonomity. y¨ y¨ y¨ (3.332) + 2 x˙ + (0) (0) +y ˙ sin ωτ (0) cos ωτ (0) ω ω (0) − ω is constant v2 = const provided that B Magnetic field is orthogonal to non-holonomity field

y¨(0) We are going to consider the case, where the magnetic strength x˙ (0) + = 0 . (3.333) i ω pseudovector H∗ is orthogonal to the pseudovector of the space non-holonomity field Ω i = 1 εikmA . Then the first equation from From the formula for x (3.329) we see that the particle performs ∗ 2 km strictly harmonic oscillations along x provided that the same con- the 1st group of the Maxwell equations we have obtained for sta- dition (3.333) is true. Taking the fact into account, putting in the tionary magnetic fields (3.292) implies that the charges density is power of two and adding up x (3.329) and y (3.330) for the mirror- zero ρ = 0. We assume that the magnetic strength is directed along y (only world particle in the homogeneous stationary magnetic field, we 2 the component H∗ = H is not zero), while the non-holonomity field obtain its trajectory within x y plane 3 is directed along z (only the component Ω∗ = Ω is not zero). We 1 y¨2 2C y¨ also assume that the magnetic field is stationary and homogeneous. x2 +y2 = y˙2 + (0) 4 y˙ cos ωτ + (0) sin ωτ + ω2 (0) ω2 − ω (0) ω Hence the non-zero component of the magnetic strength is (3.334) y¨(0) 2C5 2 2 + y˙ sin ωτ + cos ωτ + C + C , 2 ϕ ∂v3 ∂v1 (0) 4 5 H∗ = H = = const . (3.337) ω ω 31 c ∂x − ∂z which only differs from the our-world particle trajectory (3.318) by ω + 2Ω replaced with ω and by numerical values of integration Then, if the non-holonomity field is weak, the equations of constants (3.331). Therefore a mirror-world charged particle of zero motion of the our-world particle are initial velocity along z (the direction of the magnetic strength), eH eH moves along an ellipse within x y plane. x¨ + 2Ωy ˙ = z˙ , y¨ 2Ωx ˙ = 0 , z¨ = x˙ , (3.338) mc − − mc Once we assume y¨(0), as well as the constants C4 and C5 to be zeroes, the solutions become simpler eH or, denoting ω = mc , 1 1 x = y˙(0) cos ωτ , y = y˙(0) sin ωτ . (3.335) x¨ + 2Ωy ˙ = ωz˙ , y¨ 2Ωx ˙ = 0 , z¨ = ωx˙ . (3.339) −ω ω − − 3.12 Motion in a stationary magnetic field 135 136 Chapter 3 Motion of charged particles

2Ωx ˙ ωx˙ Derivating the first equation with respect to τ and substituting where y + (0) = C and z (0) = C . (0) ω2 6 (0) − ω2 7 y¨ and z¨ into it from the second and the third equations we have Provided that Ω = 0 (the space rotation is absent), and that some ... x + 4Ω2 + ω2 x˙ = 0 . (3.340) integration constants are zeroes, the above equations fully match well-known formulase of relativistic electrodynamicse for the case, Replacing x˙ = p we arrive to the equation of oscillations where a stationary magnetic field directed along the axis z

x˙ (0) x˙ (0) eH 2 x = sin ωτ , y = y +y ˙ τ , z = cos ωτ . (3.348) p¨ + ω2 p = 0 , ω = 4Ω2 + ω2 = 4Ω2 + , (3.341) ω (0) (0) ω s mc Because the live forces integral implies that the square of the p e e which solvese as followse observable velocity of a charged particle in stationary magnetic fields is constant, we have a possibility to calculate v2 =x ˙ 2 +y ˙2 +z ˙2. p = C1 cos ωτ + C2 sin ωτ , (3.342) Substituting into here the obtained formulas for the velocity components, we obtain x¨(0) where C1 =x ˙ (0) and C2 = 2 aree integratione constants. Integrating 2 2 2 2 2 ω v =x ˙ +y ˙ +z ˙ + x¨(0) + 2Ωy ˙(0) ωz˙(0) x˙ = p with respect to τ we obtain the expression for x as follows (0) (0) (0) ω − × x¨(0) x¨(0) (3.349) x˙ (0) ex¨(0) x¨(0) +x ˙ (0) sin ωτ cos ωτ , x = sin ωτ cos ωτ + x + , (3.343) × ω e − ω ω − ω2 (0) ω2 2 x¨(0) so v = const, provided that e e where x(0) + 2 = C3eis integratione constant. e e ωe e e x¨ + 2Ωy ˙ ωz˙ = 0 . (3.350) Substituting x˙ = p (3.342) into the equations of motion in respect (0) (0) − (0) of y and z (3.339) after integration we obtain e The spatial trajectory of the particle can be found, calculating 2Ω 2Ω 2Ω x2 + y2 + z2, so that we obtain the equation y˙ = x˙ (0) sin ωτ 2 x¨(0) cos ωτ +y ˙(0) + 2 x¨(0) , (3.344) ω − ω ω x¨2 2 2 2 1 2 (0) 2 2 2 ω ω ω x + y + z = x˙ + + C3 + C6 + C7 + z˙ = x¨ coseωτ x˙ sin ωτe +z ˙ x¨ , (3.345) ω2 (0) ω2 2 (0) (0) (0) 2 (0) eω −eω − ωe 2 2 2 + C4 + C5 τ + 2 (C4C6 + C5C7) τ + (ωC7 2ΩC6) + 2Ω¨x(0) ωx¨(0) − where y˙(0) + =eC4 and z˙(0) e = C5 are new integration e e (3.351) e ω2 e − ω2 e x¨(0)h 1 constants. Then integrating these equations (3.344, 3.345) with re- + 2 (ωC5 2ΩC6) τ x˙ (0) cos ωτ + sin ωτ + − ω ω2 spect to τ we obtain final formulas for y and z i e e 2C3 x¨(0) + x˙ (0) cos ωτ sin ωτe , e 2Ω x¨(0) ω2 − ω e e y = x˙ (0) cos ωτ + sin ωτ +y ˙(0)τ+ − ω2 ω (3.346) which includes a linear (with respect to time) term and a square 2Ω 2Ω e e + x¨ τ + y + x˙ , term, ase well as a parametrice term and two harmonic terms. In a e ω2 (0) e (0) ω2 (0) e e particular case, if we assume integration constants to be zeroes, the ω x¨ obtained formula (3.351) takes the form of a regular equation of a z = x˙ cos ωτ + (0) sin ωτ +z ˙ τ ω2 (0) eω (0)e− sphere (3.347) x¨2 ω ω 2 2 2 1 2 (0) x¨(0)τ + z(0) x˙ (0) , x + y + z = x˙ + , (3.352) e − ω2 e − ω2 ω2 (0) ω2 e e

e e e e 3.13 Motion in a electromagnetic field 137 138 Chapter 3 Motion of charged particles which radius is the Euclidean metric. Here the Maxwell equations for stationary x¨2 fields (3.215, 3.216) are 1 2 (0) r = x˙ (0) + 2 , (3.353) ω s ω m Ω mH∗ = 2πcρ ∗ − 2 4π I, (3.356) 2 2 2 eH ikm √ i√  where ω = 4Ω + ω = 4Ωe + mc .e ε k H m h = j h = 0 r ∇ ∗ c  So, an our-worldp charged particle in a homogeneous stationary e m  magnetic field, orthogonal to the space non-holonomity field, moves Ω mE = 0 on a surface of a sphere which radius depends on the magnetic ∗ ikm II , (3.357) strength and the angular velocity of the space rotation. ε k Em√h = 0  ∇  In a particular case, where the non-holonomity field is absent and the initial acceleration is zero, our equation of the trajectory because the condition of observable homogeneity of the field is simplifies significantly becoming an equation of the sphere the equality to zero of its chr.inv.-derivative [4, 7], while in the particular case under this consideration the Christoffel chr.inv.- 2 2 2 1 2 1 mc symbols equal zero (the metric is Galilean one) so the chr.inv.- x + y + z = x˙ , r = x˙ (0) = x˙ (0) (3.354) ω2 (0) ω eH derivative is the same as regular one. Hence the Maxwell equations with radius which depends only on interaction of the particle’s imply that the following conditions will be true here: charge with the magnetic field — the result, well-known in electro- The space non-holonomity pseudovector and the electric • m dynamics (see §21 in The Classical Theory of Fields). strength are orthogonal to each other Ω mE = 0; ∗ For a mirror-world charged particle which moves in a homoge- The space non-holonomity pseudovector and the magnetic neous stationary magnetic field, orthogonal to the non-holonomity • strength are orthogonal to each other in that case, where the field, the equations of motion are charge density is ρ = 0 ; eH eH The electromagnetic field current is absent ji = 0. x¨ = z,˙ y¨ = 0, z¨ = x˙ . (3.355) • mc − mc The last condition implies that the presence of the electromag- These are only different from the equations for the our-world netic field currents ji = 0 is derived from inhomogeneity of the 6 particle (3.338) by the absence of the terms which include the acting magnetic strength. angular velocity Ω of the space rotation. Given that the non-holonomity pseudovector is orthogonal to the electric strength, we can consider motion of the particle in two §3.13 Motion in a stationary electromagnetic field cases of mutual orientation of the fields: 1. H~ E~ and H~ Ω~ ; In this §3.13 we are going to focus on motion of a charged particle ⊥ k 2. H~ E~ and H~ Ω~ . under action of the both magnetic and electric components of a k ⊥ stationary electromagnetic field. In either case we assume that the electric strength is co-directed As a “background” we will consider a non-holonomic space with the axis x. In the background metric (3.275) the space rotation which rotates around the axis z at a constant angular velocity pseudovector is co-directed with z. Hence in the first case the Ω = Ω = Ω, so the space is of the metric (3.275). In such space magnetic strength is co-directed with z, while in the second case it 12 − 21 we have Fi = 0 and Dik = 0. is co-directed with x. We will solve the problem assuming that the non-holonomity Chr.inv.-equations of motion of a charged particle in the station- field is weak and hence the three-dimensional observable space has ary electromagnetic field, where the electric strength co-directed 3.13 Motion in a electromagnetic field 139 140 Chapter 3 Motion of charged particles with x are as follows. For the our-world particle Note that the vector E~ can be also directed along y, but can not be directed along z. It is true, because in the space with such dm eE1 dx metric co-directed with z is the non-holonomity pseudovector Ω~ , = 2 , (3.358) dτ − c dτ while the 2nd group of the Maxwell equations require the E~ to be orthogonal to the Ω~ . d i i k i 1 ikm mv + 2mAk∙ v = e E + ε vkH m , (3.359) Now taking into account the integration results from the live dτ ∙ − c ∗ forces theorem (3.363) we will write down the vector chr.inv.- and for the mirror-world particle equations for all three cases under this study.

dm eE1 dx 1. We assume that H~ E~ and H~ Ω~ , so the magnetic strength H~ = , (3.360) k dτ c2 dτ is directed along z ⊥(parallel to the non-holonomity field).

d i i 1 ikm Then out of all components of the magnetic strength the only non- mv = e E + ε vkH m . (3.361) dτ − c ∗ zero one is, namely As we did it before, we consider the case of a particle repulsed 3 ϕ ∂v1 ∂v2 2ϕ H∗ = H = + A = const = H. (3.365) by the filed. Then components of the electric strength Ei, co- 12 c ∂y − ∂x c 12 directed with x, are (in a Galilean reference frame covariant and contravariant indices of tensor quantities are the same) Consequently, the vector chr.inv.-equations of motion for the our-world particle are ∂ϕ E1 = Ex = = const = E,E2 = E3 = 0 . (3.362) ∂x − eE eE eH x˙ 2 + B + x (¨x + 2Ωy ˙) = eE y˙ , c2 c2 − c Integration of the live forces theorem gives the live forces integral for our world and the mirror world, respectively eE eE eH x˙ y˙ + B + x (¨y 2Ωx ˙) = x˙ , (3.366) c2 c2 − c eE eE m = x + B , m = x + B. (3.363) eE eE c2 − c2 x˙z˙ + B + x z¨ = 0 , c2 c2 Here B is our-world integration constant andeB is the mirror- world integration constant. Calculated them from the initial condi- while for the mirror-world particle we have tions at the moment τ = 0, these are e eE eE eH x˙ 2 + B x x¨ = eE y˙ , eE eE c2 − c2 − c B = m(0) x(0) , B = m(0) + x(0) , (3.364) − c2 c2 eE eE eH x˙ y˙ + Be x y¨ = x˙ , (3.367) where m is the relativistic mass of the particle and x is its c2 − c2 c (0) e (0) displacement at the initial moment of time. eE eE x˙z˙ + Be x z¨ = 0 . From the obtained integrals of live forces (3.363) we see that c2 − c2 the differences between the three case under this study, due to different orientations of the magnetic strength H~ , will only reveal Besides, the 1st groupe of the Maxwell equations require that in themselves in the vector chr.inv.-equations of motion, while the the case under this study the next condition must be true scalar chr.inv.-equations (3.358, 3.360) and their solutions (3.363) 3 will be the same. Ω 3H∗ = 2πcρ , (3.368) ∗ − 3.13 Motion in a electromagnetic field 141 142 Chapter 3 Motion of charged particles

3 where Ω 3 = Ω = const and H∗ = H = const. Hence this mutual ori- ing the absolute value of its observable velocity negligible compared entation∗ of the space non-holonomity pseudovector and the mag- to the light velocity. Hence we can also assume the particle’s mass netic strength is only possible in that case, where the charge density at the initial moment of time equal to its rest-mass is ρ = 0. 6 m0 2. H~ E~ , H~ Ω~ , and E~ Ω~ , so the magnetic and electric strengthes m(0) = ∼= m0 . (3.372) k ⊥ ⊥ 1 v2 are co-directed with x, while the non-holonomity field is still − c2 directed along z. q We further assume the electric strength E to be negligible as Here out of all components of the magnetic strength only the first well, thus the term eEx can be truncated. Given that, the vector component is non-zero c2 chr.inv.-equations of motion will be transformed as follows. For the 1 ϕ ∂v2 ∂v3 our-world particle they become H∗ = H = = const = H, (3.369) 23 c ∂z − ∂y eH eH while the vector chr.inv.-equations of motion for the our-world m (¨x+2Ωy ˙) = eE y˙ , m (¨y 2Ωx ˙) = x˙ , m z¨ = 0 , (3.373) 0 − c 0 − c 0 particle become eE eE while for the mirror-world particle we have x˙ 2 + B + x (¨x + 2Ωy ˙) = eE , c2 c2 eH eH m x¨ = eE y˙ , m y¨ = x˙ , m z¨ = 0 . (3.374) eE eE eH 0 − c 0 c 0 x˙ y˙ + B + x (¨y 2Ωx ˙) = z˙ , (3.370) c2 c2 − − c These equations match those obtained in §22 in The Classical eE eE eH Theory of Fields [5] in the case, where the space non-holonomity x˙z˙ + B + x z¨ = y˙ , c2 c2 c is absent Ω = 0 and the electric strength is co-directed with x. while for the mirror-world particle we have The obtained equations for the mirror-world particle are a particular case of the our-world equations at Ω = 0. Therefore we can eE eE x˙ 2 + B x x¨ = eE , only integrate the our-world equations, while the mirror-world c2 − c2 solutions are obtained automatically by assuming Ω = 0. Integrating eE eE eH the equation of motion along z we arrive to x˙ y˙ + Be x y¨ = z˙ , (3.371) c2 − c2 − c eE eE eH z =z ˙(0)τ + z(0) . (3.375) x˙z˙ + Be x z¨ = y˙ , c2 − c2 c Integrating the second one (along y) we arrive to Now that we have equationse of motion of the charged particle for eH all three cases of mutual orientation of the acting stationary fields y˙ = 2Ω + x + C , (3.376) m c 1 (the electromagnetic field and the space non-holonomity field) we 0 can turn to solving them. eH where the integration constant is C1 =y ˙(0) 2Ω + x(0). − m0c A Magnetic field is orthogonal to electric field and is parallel Substituting y˙ into the first equation (3.373) we obtain second- to non-holonomity field order differential equation with respect to x

Let us solve the vector chr.inv.-equations of motion of the charged 2 eE 2 x¨ + ω x = + ω x(0) ωy˙(0) , (3.377) particle (3.366, 3.367) in non-relativistic approximation, i. e. assum- m0 − 3.13 Motion in a electromagnetic field 143 144 Chapter 3 Motion of charged particles where ω = 2Ω + eH . Introducing a new variable while in the mirror world we have m0c 1 eE A eE 2 p = m0 y˙(0) sin ωτ + m0 x˙ (0) cos ωτ , u = x 2 ,A = + ω x(0) ωy˙(0) , (3.378) ω − − ω m0 − 2 eE eE (3.384) we obtain the equation of harmonic oscillations p = + m0 y˙(0) cos ωτ +x ˙ (0) sin ωτ , ω − m ω 0 2 3 u¨ + ω u = 0 , (3.379) p = m0 z˙(0) , which solves as follows so in contrast to our world the frequency is ω = eH . m0c u = C2 cos ωτ + C3 sin ωτ , (3.380) From here we see that the impulse of an our-world charged particle in the given configuration of the acting fields performs u˙ (0) harmonic oscillations along x and y, while along z it is a linear where the integration constants are C2 = u(0), C3 = ω . Returning to the variable x by reverse substitution of variables we finally function of the observable time τ (if the initial velocity is z˙ = 0). 6 obtain a formula for x Within x y plane the oscillation frequency is ω = 2Ω + eH . m0c It should be noted that obtaining exact solutions of the equations 1 eE x˙ eE y˙ x = y˙ cos ωτ + (0) sin ωτ + +x (0) . (3.381) of motion in the presence of the both electric and magnetic compo- ω (0) − m ω ω m ω2 (0)− ω 0 0 nents is problematic, because we need to solve elliptic integrals in Substituting the formula into the obtained equation for y˙ (3.376), the process. It may be possible to solve them in future, when the after integration we arrive to a formula for y solutions will be obtained on computers, but this problem evidently stays beyond the goal of this book. Presumably Landau and Lifshitz 1 eE x˙ (0) eE x˙ (0) faced a similar problem, because in §22 of The Classical Theory of y = y˙(0) sin ωτ cos ωτ + 2 +y(0)+ . (3.382) ω − m0ω − ω m0ω ω Fields where considering a similar problem∗ they obtained equations of motion and solved them assuming the velocity to be non- The vector chr.inv.-equations in the mirror world have the same relativistic and the electric strength to be weak eEx 0. c2 solutions, but because for them Ω = 0, the frequency is ω = eH . ≈ m0c Energies of our-world and mirror-world particles are E = mc2 B Magnetic field is parallel to electric field and is orthogonal and E = mc2, respectively. to non-holonomity field At last,− we obtain the three-dimensional impulse of the our- world particle Let us solve the vector chr.inv.-equations of motion of the charged particle (3.370, 3.371) in the same approximation as we did it in the eE first case. Then for our world and for the mirror world the equation p1 = m x˙ = m y˙ sin ωτ + m x˙ cos ωτ , 0 ω − 0 (0) 0 (0) are, respectively 2Ωm eH eE eE eH eH p2 = m y˙ = 0 + y˙ + m y˙ + x¨ + 2Ωy ˙ = , y¨ 2Ωx ˙ = z˙ , z¨ = y˙ , (3.385) 0 ω ωc m ω − (0) 0 (0) m0 − −m0c m0c 0 (3.383) 2Ωm0 eH eE eE eH eH + + y˙(0) cos ωτ +x ˙ (0) sin ωτ , x¨ = , y¨ = z˙ , z¨ = y˙ . (3.386) ω ωc − m ω m0 −m0c m0c 0 3 ∗But in contrast to this book, they used general covariant methods and did not p = m0 z˙ = m0 z˙(0) , account for the space non-holonomity. 3.13 Motion in a electromagnetic field 145 146 Chapter 3 Motion of charged particles

Integrating the first equation of motion in our world (3.385) we Then the general solution of the initial inhomogeneous equation obtain (3.391) becomes eE 2ΩeE x˙ = τ 2Ωy + C1 ,C1 = const =x ˙ (0) + 2Ωy(0) . (3.387) u = C3 cos ωτ + C4 sin ωτ + 2 τ , (3.395) m0 − m0ω Integrating the third equation (along z) we have where the integration constants can be obtained by substituting the initial conditions at τ = 0 into the obtained formula. As a result we u˙ (0) eH eH have C3 = u(0) and C4 = . z˙ = y + C2 ,C2 = const =z ˙(0) y(0) . (3.388) ω m0c − m0c Returning to the old variable y (3.390) we find the final solution for this coordinate Substituting the obtained formulas for x˙ and z˙ into the second equation of motion (3.385) we obtain the linear differential equation 1 eH y = y(0) + 2 C2 + 2ΩC1 cos ωτ + of the 2nd order with respect to y ω m0c (3.396) y˙(0) 1 eH 2ΩeE 2 2 + sin ωτ C + 2ΩC + τ . 2 e H 2ΩeE eH 2 2 1 2 y¨ + 4Ω + y = τ + 2ΩC C . (3.389) ω − ω m0c m0ω 2 2 1 2 m0c m0 − m0c Then substituting this formula into equations for x˙ and z˙ after We are going to solve it, using the method of replacement of integration we arrive to their solutions for x and z variables. Introducing a new variable u eE 4Ω2 2Ω x = 1 τ 2 y + A sin ωτ + 2 2 2 (0) 1 eH e H 2m0 − ω − ω u = y + C 2ΩC , ω2 = 4Ω2 + , (3.390) (3.397) 2 2 1 2 2 2Ωy ˙(0) ω m0c − m0c + cos ωτ + (C + 2ΩA) τ + C , ω 1 5 we obtain an equation of forced oscillations eH y˙ z = y + A sin ωτ (0) cos ωτ 2ΩeE m c ω (0) − ω − u¨ + ω2u = τ , (3.391) 0 (3.398) m0 eH A C2 τ + C6 , − m0c − which solution is the sum of the general solution of the equation of where a convenience notation was introduced free oscillations u¨ + ω2u = 0 , (3.392) 1 eH A = 2 C2 2ΩC1 , (3.399) ω m0 c − and of a particular solution of the inhomogeneous equation while the new integration constants are u˜ = Mτ + N, (3.393) 2Ωy ˙(0) eHy˙(0) C5 = x0 ,C6 = z(0) + 2 . (3.400) where M = const and N = const. Derivating u˜ twice with respect − ω m0 c ω to τ and substituting the results into the inhomogeneous equation If we assume Ω = 0, then from coordinates of the our-world (3.391) and then equalizing the obtained coefficients for τ we obtain charged particle (3.396–3.398) we immediately obtain the solutions the linear coefficients for the analogous charged particle in the mirror world

2ΩeE eE 2 M = 2 ,N = 0 . (3.394) x = τ +x ˙ (0)τ + x(0) , (3.401) m0ω 2m0 3.14 Conclusions 147 148 Chapter 3 Motion of charged particles

z˙(0) y˙(0) z˙(0) General Theory of Relativity, simply as the chronometrically in- y = cos ωτ + sin ωτ + y , (3.402) ω ω − ω (0) variant electrodynamics (CED). Here we have obtained only the z˙ y˙ y˙ basics of this theory: z = (0) sin ωτ (0) cos ωτ + (0) + z . (3.403) ω − ω ω (0) Chr.inv.-components of the electromagnetic field tensor (the • Consequently, components of the three-dimensional impulse of Maxwell tensor); the our-world particle under considered configuration of the acting The Maxwell equations in chr.inv.-form; • fields take the form The law of conservation of electric charge in chr.inv.-form; • 4Ω2 The Lorentz condition in chr.inv.-form; p1 = m x˙ + eE 1 τ • 0 (0) − ω2 − The d’Alembert equations in chr.inv.-form (the wave propaga- • y˙ y˙ tion equations) for the scalar potential and the vector-potential 2m Ω (0) sin ωτ + y + A cos ωτ (0) A , − 0 ω (0) − ω − of the electromagnetic field; 2ΩeE The Lorentz force in chr.inv.-form; p2 = m y˙ cos ωτ ω y + A sin ωτ + , (3.404) • 0 (0) − (0) ω2 The energy-momentum tensor of the electromagnetic field h i • and its chr.inv.-components; 3 p = m0 z˙(0) + Chr.inv.-equations of motion of a charged test-particle; eH y˙ 2ΩeE • + y +A cos ωτ + (0) sin ωτ A+ τ y , The geometric structure of the four-dimensional potential of c (0) ω − m ω2 − (0) • 0 the electromagnetic field. 2 2 It is evidently, the whole scope of the chr.inv.-electrodynamics 2 e H where the frequency is ω = 4Ω + 2 . is much wider. In addition to what has been said we could obtain m0 c In the mirror world, givenr this configuration of the acting fields, chr.inv.-equations of motion of a spatially distributed charge or components of the three-dimensional impulse of the analogous study motion of a particle which bears its own electromagnetic charged particle are emission, interacting the field or, at last, deduce equations of motion for a particle which travels at an arbitrary angle to the field strength 1 p = m0 x˙ (0) + 2eEτ , (either for an individual particle or a distributed charge), or tackle scores of other interesting problems. p2 = m y˙ cos ωτ z˙ sin ωτ , (3.405) 0 (0) − (0) 3 p = m0 z˙(0) cos ωτ y˙(0) sin ωτ , − ♦ where in contrast to our world the frequency is ω = eH . m(0) c

§3.14 Conclusions

In fact the theory we have built in this Chapter can be more precisely referred to as the chronometrically invariant representation of electrodynamics in a pseudo-Riemannian space. In other word, because the mathematical apparatus of physical observable quantities initially assumes the four-dimensional space-time of the 150 Chapter 4 Motion of spin-particle

2 dimension of impulse momentum [ gram×cm /sec]. This alone hints that spin’s tensor by its geometric structure should be similar to the tensor of impulse momentum, i. e. should be an antisymmetric tensor of the 2nd rank. We are going to check if other source prove Chapter 4 that. Bohr’s second postulate says that the length of the orbit of an SPIN-PARTICLE electron should comprise the integer number of de Broglie wave- IN PSEUDO-RIEMANNIAN SPACE h lengths λ = p , which stands for the electron according to the wave- particle concept. In other word, the length of the electron orbit 2πr comprises k de Broglie wavelengths h §4.1 Problem statement 2πr = kλ = k , (4.2) p In this Chapter we are going to obtain equation of motion of a where p is the orbital impulse of the electron. Taking into account particle bearing an inner mechanical momentum (spin). As we that Planck’s constant is = h , this equation (4.2) should be mentioned in Chapter 1, these are equations of parallel transfer ~ 2π α of the four-dimensional dynamic vector of the particle Q , which rp = k~ . (4.3) is the sum of vectors i Qα = P α + Sα, (4.1) Because the radius-vector of the electron orbit r is orthogonal to the vector of its orbital impulse pk, this formula in tensor notation α where P α = m dx is the four-dimensional impulse vector of this is vector product, namely 0 ds particle. The four-dimensional vector Sα is an additional impulse i k ik r ; p = k~ . (4.4) which this particle gains from its inner momentum (spin), so this impulse makes motion of the particle non-geodesic. Therefore we From here we see that the Planck constant deduced from the will refer to Sα as the spin-impulse. Because we know components Bohr second postulate in tensor notation is antisymmetric 2nd rank of the impulse vector P α, to define summary dynamic vector Qα tensor. we only need to obtain components of the spin-impulse vector Sα. But this representation of the Planck constant is linked to orbital Hence our first step will be defining a particle’s spin as geometric model of atom — of the system more complicated than electron or quantity in the four-dimensional pseudo-Riemannian space of the any other elementary particle. Nevertheless spin, also defined by General Theory of Relativity. Then in §4.2 herein we are going to this constant, is an inner property of elementary particles them- deduce the spin-impulse vector Sα itself. In §4.3 our goal will be selves. Therefore according to the Bohr second postulate we have to equations of motion of a spin-particle in the pseudo-Riemannian consider the geometric structure of the Planck constant proceeding space and their chr.inv.-projections. Other Paragraphs will focus from another experimental relationship which is related to inner on motion of elementary particles. structure of electron only. The numerical value of spin is n~, measured in the fractions of We have such opportunity thanks to classical experiments by Planck’s constant, where n is so-called± the spin quantum number. O. Stern and W. Gerlach (1921). One of their results is that any As of today it is known [21] that for various kinds of elementary electron bears inner magnetic momentum Lm, which is proportional particles this number may be n = 0, 1 , 1, 3 , 2. Alternating sign to its inner mechanical momentum (spin) 2 2 ± stands for possible right-wise or left-wise inner rotation of the me Lm = n~ , (4.5) particle under consideration. Besides, the Planck constant ~ has e 4.1 Problem statement 151 152 Chapter 4 Motion of spin-particle

αβ where e is the charge of the electron, me is its mass and n is the spin This antisymmetric tensor ~ corresponds to dual the Planck 1 αβ 1 αβμν quantum number (for electron n = 2 ). The magnetic momentum pseudotensor ~∗ = 2 E ~μν . Subsequently, spin of a particle in of a contour with an area S = πr2, which conducts a current I, is the four-dimensional pseudo-Riemannian space is characterized by αβ αβ Lm = IS. The current equals to the charge e divided by its period the antisymmetric tensor n ~ , or by its dual pseudotensor n ~∗ . 2πr Note that physical nature of spin does not matter here, this is of circulation T = u along this contour enough that this fundamental property of particles is characterized eu I = , (4.6) by a tensor (or a pseudotensor) of a certain kind. Thanks to this 2πr approach we can solve the problem of motion of spin-particles where u is the linear velocity of the charge circulation. Hence the without any preliminary assumption on their inner structure, i. e. inner magnetic momentum of the electron is using strictly formal mathematical method. Hence from geometric viewpoint the Planck constant is an anti- 1 Lm = eur , (4.7) symmetric tensor of the 2nd rank, which dimension is impulse 2 momentum irrespective of through what quantities it was obtained: or in tensor notation∗ mechanical or electromagnetic ones. The latter also implies that the 1 1 Planck tensor does not characterize rotation of masses inside atoms Lik = e ri; uk = ri; pk , (4.8) m 2 2 m or any masses inside elementary particles, but it is derived from some fundamental quantum rotation of the space itself and sets all i where r is the radius-vector of the inner current circulation pro- “elementary” rotations in the space irrespective of their nature. k vided by the electron, and u is the vector of the circulation velocity. Own rotation of the space is characterized by the chr.inv.-tensor From here we see that the Planck constant, being calculated mn Aik (1.36), which results from lowering indices Aik = himhknA in from the inner magnetic momentum of an electron (4.5), is also the the components Amn of the contravariant four-dimensional tensor vector product of two vectors. So it is an antisymmetric tensor of 1 ∂b ∂b the 2nd rank, namely Aαβ = chαμhβν a , a = ν μ . (4.11) μν μν 2 ∂xμ − ∂xν me i k ik r ; p = n~ , (4.9) 2e m In the accompanying reference frame (bi = 0) the auxiliary quan- which proves similar conclusion based on the Bohr second postulate. tity aμν has the components Subsequently, considering inter-electronic quantum relation- 1 ∂w ∂v 1 ∂v ∂v a = 0 , a = i , a = i k , (4.12) ships in the four-dimensional pseudo-Riemannian space, we arrive 00 0i 2c2 ∂xi − ∂t ik 2c ∂xk − ∂xi to the Planck four-dimensional antisymmetric tensor ~αβ, which ik so we have spatial components are three-dimensional quantities ~ A00 = 0 ,A0i = Ai0 = 0 , ~00 ~01 ~02 ~03 − 10 11 12 13 (4.13) αβ ~ ~ ~ ~ 1 ∂vk ∂vi 1 ~ =  20 21 22 23  . (4.10) Aik = + (Fi vk Fk vi) . ~ ~ ~ ~ 2 ∂xi − ∂xk 2c2 −  ~30 ~31 ~32 ~33    In the absence of gravitational fields the tensor of angular ve-   locities of the space rotation formulates with the linear velocity of ∗Equations (4.8) and (4.9) are given for the Minkowski space, which is quite acceptable for the above experiments. In Riemannian spaces the result of integration this rotation vi only, hence we denote it as Aαβ = Ωαβ depends on the integration path. Therefore the radius-vector of a finite length is not 1 ∂v ∂v defined in Riemannian spaces, because its length depends on constantly varying Ω = 0 , Ω = Ω = 0 , Ω = k i . (4.14) direction. 00 0i − i0 ik 2 ∂xi − ∂xk 4.1 Problem statement 153 154 Chapter 4 Motion of spin-particle

On the other hand, according to the wave-particle concept, any and in the Minkowski space, when the reference frame is Galilean 2 αβ 2 particle corresponds to a wave with the energy E = mc = ~ω, where one and the metric is diagonal (2.70), it equals ~αβ~ = 6~ . In the αβ m is the relativistic mass of the particle and ω is its specific frequen- pseudo-Riemannian space the quantity ~αβ~ can be deduced by cy. In other word, from geometric viewpoint any particle can be substitution of dependency of spatial components of the fundamen- considered as a wave defined within infinite proximity of geometric tal metric tensor from the metric chr.inv.-tensor h = g + 1 v v ik − ik c2 i k location of the particle, which specific frequency depends on certain and the space rotation velocity into (4.17). Hence, though the phys- distribution of the angular velocities ωαβ, also defined within this ical observable components ~ik of the Planck tensor are constant proximity. Then the above quantum relationship in tensor notation (bear opposite signs for left and right-wise rotations), its square 2 αβ becomes mc = ~ ωαβ. in general case depends from the angular velocity of the space Because the Planck tensor is antisymmetric, all of its diagonal rotation. elements are zeroes. Its space-time (mixed) components in the Now having components of the Planck tensor defined, we can accompanying reference frame also should be zero similar to respec- approach deduction of an impulse that a particle gains from its spin tive components of the four-dimensional tensor of the space rotation as well as equations of motion of the spin-particle in the pseudo- (4.14). Numerical values of spatial (three-dimensional) components Riemannian space. This will be the focus of the next §4.2. of the Planck tensor, observable in experiments, are ~ depending on the rotational direction and make the Planck three-dimensional± §4.2 The spin-impulse of a particle in equations of motion chr.inv.-tensor ~ik. In the case of left-wise rotations the components 12 23 31 13 32 21 ~ , ~ , ~ are positive, while the components ~ , ~ , ~ are The additional impulse Sα that a particle gains from its spin can be negative. obtained from considering action for this spin-particle. Then the geometric structure of the Planck four-dimensional Action S for a particle that bears an inner scalar field k, with tensor, represented as matrix, becomes which an external scalar field A interacts and thus displaces the 0 0 0 0 particle at an elementary interval ds, is αβ 0 0 ~ ~ b ~ =  −  . (4.15) 0 ~ 0 ~ S = α(kA) kAds , (4.18) −  0 ~ ~ 0  Za  −    where α is a scalar constant, which characterizes properties of In the case of right-wise rotations the components ~12, ~23, ~31 (kA) the particle in a given interaction and equalizes dimensions [5, 16]. change their sign to become negative, while the components ~13, If the inner scalar field of the particle k corresponds to an external ~32, ~21 become positive filed of the tensor of the 1st rank Aα, then action to displace the 0 0 0 0 particle by that field is αβ 0 0 ~ ~ ~ =  −  . (4.16) b 0 ~ 0 ~ S = α kA dxα. (4.19) − (kAα) α  0 ~ ~ 0  a  −  Z   The square of the Planck four-dimensional tensor can be de- In interaction of the particle’s inner scalar field k with an ex- duced as follows ternal field of the tensor of the 2nd rank Aαβ, action to displace the particle by that field is αβ 2 2 2 2 ~αβ ~ = 2~ g11g22 g + g11g33 g + g22g33 g + − 12 − 13 − 23 b (4.17) α β h S = α(kA ) kAαβ dx dx . (4.20) + 2 (g12g23 g22g13 g12g33 +g13g23 g11 g23 +g12g13) , αβ − − − Za i 4.2 The spin-impulse of particle 155 156 Chapter 4 Motion of spin-particle

And so forth. For instance, if the specific vector potential of the the space-time interval is α particle k corresponds to an external vector field Aα, then action u2 of this interaction to displace the particle is ds = g dxαdxβ = cdt 1 , (4.25) αβ − c2 b r α q α S = α(k Aα) k Aαds . (4.21) and hence the action (4.24) becomes Za b t2 2 Besides, the action can be represented as follows irrespective of 2 u S = m0cds = m0c 1 dt . (4.26) nature of inner properties of particles and external fields − c2 Za Zt1 r t2 Therefore the Lagrange function of the free particle in a Galilean S = Ldt , (4.22) reference frame in the Minkowski space is Zt1 2 where L is so-called Lagrange’s function. Because the dimension of 2 u L = m0c 1 . (4.27) gram cm2 2 action S is [ erg sec = × /sec ], then the Lagrange function has − c × r gram cm2 2 dimension of energy [ erg = × /sec ]. And the derivative of the Derivating it with respect to the particle’s coordinate velocity Lagrange function with respect to the three-dimensional coordinate i we arrive to covariant form of its three-dimensional impulse velocity ui = dx of the particle dt ∂ 1 u2 ∂L 2 c2 m0ui ∂L pi = = m0c − = , (4.28) = pi (4.23) i q i 2 ∂ui ∂u ∂u − 1 u − c2 i i q is covariant notation of its three-dimensional impulse p = cP which from which, having indices lifted, we arrive to the four-dimensional can be used to restore full notation for the four-dimensional impulse impulse vector of the free particle as follows vector of the particle P α. Hence having action for the particle, α α having the Lagrange function outlined and derivated with respect α m0 dx dx P = = m0 . (4.29) to the coordinate velocity of particle, we can calculate the additional 2 c 1 v dt ds impulse which the particle gains from its spin. − c2 As it is known, action to displace a free particle in the pseudo- q dxα In the final formula both multipliers, m0 and , are general Riemannian space is∗ ds b covariant quantities, so they do not depend on choice of a particular S = m0cds . (4.24) reference frame. For this reason, this formula obtained in a Galilean Za reference frame in the Minkowski space is also true in any other In a Galilean reference frame in the Minkowski space, because arbitrary reference frame in any pseudo-Riemannian space. non-diagonal terms of the fundamental metric tensor are zeroes, Now let us consider motion of a particle that possesses inner structure, which in experiments reveals itself like its spin. Inner ∗In The Classical Theory of Fields [5] Landau and Lifshitz put “minus” before αβ the action, while we always have “plus” before the integral of the action and also rotation (spin) of the particle n~ in the four-dimensional pseudo- before the Lagrange function. This is because the sign of action depends on the Riemannian space corresponds to the external field Aαβ of the space signature of the pseudo-Riemannian space. Landau and Lifshitz use the signature rotation. Therefore summary action to displace this spin-particle is ( +++) − , where time is imaginary, spatial coordinates are real and three-dimensional coordinate impulse is positive (see in the below). To the contrary, we stick to b (+ ) αβ Zelmanov’s [4] signature −−− , where time is real and spatial coordinates are S = m0cds + α(s)~ Aαβ ds , (4.30) imaginary, because in this case three-dimensional observable impulse is positive. Za 4.2 The spin-impulse of particle 157 158 Chapter 4 Motion of spin-particle

i where α(s) [ sec/cm ] is a scalar constant, which characterizes the pressed with its observable velocity vi = dx as follows particle in spin-interaction. Because action constants may include dτ only this particle’s properties or fundamental physical constants, ui h uiuk vi = , v2 = ik . (4.33) α(s) is evidently the spin quantum number n, which is the function i i 2 w + viu w + v u of inner properties of the particle, divided by the light velocity 1 1 i n − c2 − c2 α(s) = c . Then the action to displace the particle, produced by interaction of its spin with the space non-holonomity field Aαβ is Then the additional action (4.31), produced by interaction of spin with the space non-holonomity field, becomes b b αβ n αβ S = α(s) ~ Aαβ ds = ~ Aαβ ds . (4.31) c t2 w + v ui 2 u2 Za Za αβ i S = n ~ Aαβ 1 2 2 dt . (4.34) t s − c − c A note should be taken that building the four-dimensional im- Z 1 pulse vector for a spin-particle using the same method as for a Therefore the Lagrange function for this action is free particle is impossible. As it is known, we first obtained the i 2 2 impulse of a free particle in a Galilean reference frame in the αβ w + viu u L = n~ Aαβ 1 . (4.35) Minkowski space, where a formula for ds expressed through dt and s − c2 − c2 substituted into the action had simple form (4.25). It was shown that the obtained formula (4.29) due to its property of general covariance Now to deduce the spin-impulse we only have to derivate the was true in any reference frame in the pseudo-Riemannian space. Lagrange function (4.35) with respect to the coordinate velocity But as we can see from the formula of the action for a spin- of the particle. Taking into account that ~αβ, being the tensor of particle, spin affects motion of the particle in the non-holonomic inner rotations of the particle, and Aαβ (4.13), being the tensor of space Aαβ = 0 only, i. e. where non-diagonal terms g0i of the funda- the space rotation, are not functions of the particle’s velocity, after mental metric6 tensor are not zeroes. In a Galilean reference frame, derivation we obtain by definition, all non-diagonal terms in the metric tensor are zeroes, i 2 2 hence zeroes are components of the linear velocity of the space ∂L mn ∂ w + viu u g pi = = n~ Amn 1 = rotation v = c 0i and also components of the non-holonomity ∂ui ∂ui s − c2 − c2 i g − √ 00 mn (4.36) tensor Aαβ. Therefore this is pointless to deduce the desired formula n~ Amn = (vi + vi) , for the spin-particle impulse in a Galilean reference frame in the − c2 1 v2 Minkowski space (where it is zero), instead we should deduce it − c2 q directly in the pseudo-Riemannian space. k where vi = hikv . We compare (4.36) with the spatial covariant com- In an arbitrary accompanying reference frame in the pseudo- α ponent p = cP of the four-dimensional impulse vector P α = m dx Riemannian space the interval ds is i i 0 ds of the particle in pseudo-Riemannian space∗. If the particle is located in our world, so it travels from past into future in respect of v2 w + v ui u2 ds = cdτ 1 = cdt 1 i 1 , (4.32) us, its three-dimensional covariant impulse is − c2 − c2 − i 2 r v 2 w+viu u c 1 2 α m0 u − c pi = cPi = cgiαP = m (vi + vi) = (vi + vi) . (4.37) u − − v2 t 1 2 i − c where the coordinate velocity of the particle ui = dx can be ex- q dt ∗In this comparing we mean a mass-bearing particle. 4.3 Equations of motion of spin-particle 159 160 Chapter 4 Motion of spin-particle

α i k k From here we see that the four-dimensional impulse S , which dq ϕ dx k dτ i i ϕ i dτ i m dx + +q Dk +Ak∙ F +Δmk q = 0 , (4.43) the particle gains from its spin (the spin-impulse) is ds c ds ds ∙ − c ds ds α α 1 μν dx where ϕ is the projection of the summary vector Qα on the observ- S = n~ Aμν , (4.38) i c2 ds er’s time line and q is its projection on the spatial section μν mn or, introducing notation η0 = n~ Aμν = n~ Amn to make the for- α Q0 P0 S0 ϕ = bαQ = = + , (4.44) mula simpler, we obtain √g00 √g00 √g00 α α 1 dx S = 2 η0 . (4.39) i i α i i i c ds q = hαQ = Q = P + S . (4.45) Then the summary vector Qα (4.1), which characterizes motion Therefore attaining the goal requires deducing ϕ and qi, substi- of the spin-particle is tuting them into (4.42, 4.43) and canceling similar terms. Chr.inv.- α α dx α α projections of the impulse vector P = m0 are α α α dx 1 μν dx ds Q = P + S = m0 + n~ Aμν . (4.40) ds c2 ds P 1 0 = m , P i = mvi, (4.46) So, any spin-particle in a non-holonomic space (Aμν = 0) actually √g00 ± c gains an additional impulse, which deviates its motion from6 geodesic line and makes it non-geodesic. In the absence of the space rotation, and now we have to deduce chr.inv.-projections of the spin-impulse α α i. e. where the space is holonomic, we have Aμν = 0, so the particle’s vector S . Taking into account in the formula for S (4.39) that the spin does not affect its motion. But there is hardly an area in the space-time interval, formulated with physical observable quantities, space where rotation is fully absent. Therefore spin most often is ds = cdτ 1 v2/c2, we obtain components of the Sα, which are − affects motion of particles in the subject domain of atomic physics, p mn i 2 1 n~ Amn viv c where rotations are especially strong. S0 = ± , (4.47) 2 2 w c 1 v c2 1 − c2 − c2 §4.3 Equations of motion of spin-particle q mn 1 n~ Amn Equations of motion of a spin-particle are equations of parallel Si = vi, (4.48) c3 v2 transfer of the summary vector Qα = P α + Sα (4.40) along the tra- 1 − c2 jectory of the particle (its parallel transfer in the four-dimensional q mn pseudo-Riemannian space), namely 1 w n~ Amn S0 = 2 1 2 , (4.49) ± c − c 1 v2 d dxν − c2 (P α + Sα) + Γα (P μ + Sμ) = 0 , (4.41) μν mn q ds ds 1 n~ Amn α Si = 3 (vi vi) , (4.50) where the square of the vector remains unchanged QαQ = const in −c 1 v2 ± its parallel transfer along the trajectory. − c2 Our goal is to deduce chr.inv.-projections of the equations. The also formulated with physicalq observable quantities. So, chr.inv.- projections in general notation, as obtained in Chapter 2, should be projections of the particle’s spin-impulse vector are

k dϕ 1 i dτ 1 i dx S0 1 i 1 i Fi q + Dik q = 0 , (4.42) = 2 η , S = 3 ηv , (4.51) ds − c ds c ds √g00 ± c c 4.3 Equations of motion of spin-particle 161 162 Chapter 4 Motion of spin-particle

mn where the quantity η is Substituting here η0 = n~ Amn we have mn n Amn ~ mn ∗∂Amn k ∗∂Amn η = , (4.52) n~ + v = 0 . (4.57) 1 v2 ∂t ∂xk − c2 q while alternating signs, which results from substituting the time To illustrate the result we formulate the space non-holonomity function dt (1.55) indicate motion of the particle into future (the tensor Aik, which is actually the tensor of angular velocities of the dτ space rotation, with the pseudovector of angular velocities of this upper sign) or into past (the lower sign). Then the square of the rotation spin-impulse vector is i 1 imn Ω∗ = ε Amn , (4.58) α β 2 α α β 1 2 dx dx 1 2 SαS = gαβ S S = η0 gαβ = η0 , (4.53) i c4 ds2 c4 which is also chr.inv.-quantity. Multiplying Ω∗ by εipq and the square of the summary vector Qα is i 1 imn 1 m n n m Ω∗ εipq = ε εipqAmn = δp δq δp δq Amn = Apq , (4.59) α α β 2 2 1 2 2 2 − QαQ = gαβQ Q = m0 + 2 m0η0 + 4 η0 . (4.54) c c we obtain (4.57) as follows Therefore the square of the summary vector of any spin-particle falls apart into three parts, namely: mn ∗∂ i k ∗∂ i n~ εimnΩ∗ + v k εimnΩ∗ = α 2 ∂t ∂x The square of the impulse vector of the particle PαP = m0; • (4.60) α 1 2 mn 1 ∗∂ i k 1 ∗∂ i The square of its spin-impulse vector SαS = η0; = n~ εimn √h Ω∗ +v √h Ω∗ = 0 . • c4 √h ∂t √h ∂xk 2 The term m0η0, describing spin-gravitational interactions. • c2 The vector of gravitational inertial force and the space non- To effect the parallel transfer (4.41) it is necessary that the holonomity tensor are related through Zelmanov’s identities, one of square of the transferred summary vector remains unchanged along which (see formula 13.20 in [4]) is the entire path. But the obtained formula (4.54) implies that because α m0 = const the square of the spin-particle’s summary vector Q 2 ∗∂ i ijk √h Ω∗ + ε ∗ j Fk = 0 , (4.61) remains unchanged only provided that η0 = const, i. e. √h ∂t ∇ ∂η dη = 0 dxα = 0 . (4.55) or, in the other notation 0 ∂xα ∗∂Aik 1 ∗∂Aik 1 ∗∂Fk ∗∂Fi Dividing the both parts of the equation by dτ, which is always + ∗ F ∗ F = + = 0 , (4.62) ∂t 2 ∇k i − ∇i k ∂t 2 ∂xi − ∂xk possible because an elementary interval of the observer’s physical time is greater than zero∗, we obtain the chr.inv.-condition of con- ijk where ε ∗ j Fk is the chr.inv.-rotor of the gravitational inertial servation of the square of the spin-particle’s summary vector ∇ force field Fk. From here we see that non-stationarity of the space dη ∂η ∂η rotation A is due to rotor character of the acting field of gravita- 0 = ∗ 0 + vk ∗ 0 = 0 . (4.56) ik dτ ∂t ∂xk tional inertial force Fik. Hence taking into account equation (4.61), our formula (4.60) becomes ∗The condition dτ = 0 only has sense in a generalized space-time, where degeneration of the fundamental metric tensor gαβ is possible. In this case the above condition defines fully degenerated domain (zero-space) that hosts zero-particles, mn mn k 1 ∗∂ i n~ ∗ m Fn + n~ εimnv √h Ω∗ = 0 , (4.63) which are capable of instant displacement, so they are carriers of long-range action. − ∇ √h ∂xk 4.3 Equations of motion of spin-particle 163 164 Chapter 4 Motion of spin-particle or in the other notation For mass-bearing particles this is the case, for instance, where k i v = 0, so where they are at rest in respect of the observer and his mn mn k i ∗∂ ln √h ∗∂Ω∗ n~ ∗ m Fn = n~ εimnv Ω∗ + . (4.64) reference body. In this case equality to zero of derivatives in (4.68) ∇ ∂xk ∂xk is not essential. But massless particles travel at the light velocity, Now we should recall that this formula is nothing but expanded hence for them in the vortexless field of force Fi the derivatives ∂ i chr.inv.-notation of the conservation condition of the summary vec- (√h Ω∗ ) must be zeroes. ∂xk tor (4.57). The left part in (4.64) equals Let us obtain chr.inv.-equations of motion of a spin-particle in the pseudo-Riemannian space. Substituting (4.46) and (4.51) into 2n~ ∗ 1 F2 ∗ 2 F1 + ∗ 1 F3 ∗ 3 F1 + ∗ 2 F3 ∗ 3 F2 , (4.65) ± ∇ − ∇ ∇ − ∇ ∇ − ∇ (4.44) and (4.45) we arrive to chr.inv.-projections of the summary where “plus” and “minus” stand for right-rotating and left-rotating vector of the spin-particle reference frames, respectively. Therefore the left part of the for- 1 1 1 ϕ = m + η , qi = mvi + ηvi. (4.69) mula (4.64) is the chr.inv.-rotor of gravitational inertial force. The ± c2 c c3 right part (4.64) depends on the spatial orientation of the space Having the quantities substituted for ϕ > 0 into (4.42, 4.43) we rotation pseudovector Ω i. ∗ obtain chr.inv.-equations of motion for a mass-bearing spin-particle Hence to conserve the square of the spin-particle’s transferred located in our world (the particle travels from past into future) vector it is necessary that the right and the left parts of (4.64) are equal to each other along the trajectory. In general case, where dm m m 1 dη η η F vi + D vivk = + F vi D vivk, (4.70) no any additional assumptions on the geometric structure of the dτ − c2 i c2 ik −c2 dτ c4 i − c4 ik space, this requires balance between the vortical field of the acting gravitational inertial force and the spatial distribution of the space d i i i k i i n k mv + 2m Dk + Ak∙ v mF + mΔnkv v = rotation pseudovector. dτ ∙ − (4.71) 1 d 2η η η If the field of gravitational inertial force is vortexless, then i i i k i i n k = 2 ηv 2 Dk + Ak∙ v + 2 F 2 Δnkv v , the left part of the conservation condition (4.64) is zero and this −c dτ − c ∙ c − c condition becomes while for the mirror-world particle (which moves into past), having the quantities (4.69) substituted for ϕ < 0, we have mn k 1 ∗∂ √ i n~ εimnv k h Ω∗ = 0 . (4.66) √h ∂x dm m i m i k 1 dη η i η i k Fi v + Dikv v = + Fi v Dikv v , (4.72) − dτ − c2 c2 c2 dτ c4 − c4 Introducing chr.inv.-derivative ∗∂ = ∂ + 1 v ∗∂ , we have ∂xk ∂xk c2 k ∂t d i i i n k mv + mF + mΔnkv v = 1 ∂ 1 ∗∂ dτ n mnε vk √h Ω i v √h Ω i = 0 . (4.67) (4.73) ~ imn k ∗ 2 k ∗ 1 d η η √h ∂x − c ∂t = ηvi F i Δi vnvk, −c2 dτ − c2 − c2 nk Since the force field Fi is vortexless, because of (4.66) the second term in this formula is zero. Therefore the square of the summary We write down the obtained equations in a way that their left vector of the spin-particle remains unchanged in the vortexless parts have the geodesic part, which describes free (geodesic) motion of this particle, while the right parts have the terms produced by force field Fi, provided that chr.inv.-formula (4.66) and the formula with regular derivatives are zeroes the particle’s spin, which makes the motion non-geodesic (the non- geodesic part). Hence for a spin-free particle the right parts become mn k 1 ∂ i zeroes and we obtain chr.inv.-equations of free motion. Such form n~ εimnv √h Ω∗ = 0 . (4.68) √h ∂xk of the equations will facilitate their analysis. 4.3 Equations of motion of spin-particle 165 166 Chapter 4 Motion of spin-particle

Within the wave-particle concept a massless particle is described It is evidently that in the form applicable to massless particles α α by the four-dimensional wave vector Kα = ω dx , where the quan- (i. e. along isotropic trajectories) the square of P (4.75) is zero c dσ 2 i k tity dσ = hikdx dx is the square of the spatial physical observable dxα dxβ ds2 P P α = g P αP β = m2g = m2 = 0 . (4.76) interval, not equal to zero along isotropic trajectories. Because mass- α αβ αβ dσ dσ dσ2 less particles travel along isotropic trajectories (the light propagation α Chr.inv.-projections of the four-dimensional impulse vector of a trajectories), the vector K is also isotropic one: its square is zero. α α dx α 1 massless particle P = m are But because the vector’s dimension K is [ /cm ], the equations dσ have the dimension different from that of equations of motion of P 1 mass-bearing particles. Besides, this fact does not permit building 0 = m , P i = mci, (4.77) an uniform formula of action for both massless and mass-bearing √g00 ± c particles [4]. where ci is the three-dimensional chr.inv.-vector of the light veloc- On the other hand, spin is a physical property, possessed by ity. In this case the spin-impulse vector of this particle (4.39) is as both mass-bearing and massless particles (photons, for instance). well isotropic Therefore deduction of equations of motion for spin-particles re- α α α α 1 dx 1 dx 1 dx quire using uniform vector for both kinds of particles. Such vector S = η0 = η = η , (4.78) c2 ds c2 cdτ c2 dσ can be obtained by applying physical conditions which are true along isotropic trajectories, because its square is zero α β 2 2 2 2 2 1 dx dx 1 ds ds = c dτ dσ = 0 , cdτ = dσ = 0 , (4.74) S Sα = g SαSβ = η2g = η2 = 0 , (4.79) − 6 α αβ c4 αβ dσ2 c4 dσ2 to the four-dimensional impulse vector of a mass-bearing particle so the square of the particle’s summary vector Qα = P α + Sα is also dxα m dxα dxα zero. Chr.inv.-projections of the isotropic spin-impulse are P α = m = = m . (4.75) 0 ds c dτ dσ S0 1 i 1 i = 2 η , S = 3 ηc , (4.80) As a result the observable spatial interval, not equal to zero along √g00 ±c c isotropic trajectories, becomes the derivation parameter, while the dimension of the formula, in contrast to the four-dimensional wave so its spatial observable projection coincides the one for a mass- α bearing particle (4.51), which instead of the particle’s observable 1 cm vector K [ / ], coincides that of the four-dimensional impulse i i vector P α [ gram]. Relativistic mass m, not equal to zero for massless velocity v (4.51) has the light velocity chr.inv.-vector c . Subse- particles, can be obtained from the energy equivalent using E = mc2 quently, cr.inv.-projections of the summary vector of the massless 6 spin-particle are formula. For instance, a photon energy of E = 1 MeV = 1.6×10− erg 28 corresponds to its relativistic mass of m = 1.8 10− gram. 1 1 1 × ϕ = m + η , qi = mci + ηci. (4.81) Therefore the four-dimensional impulse vector (4.75), depending ± c2 c c3 on its form, may describe motion of either mass-bearing particles Having these quantities substituted for the positive ϕ into the (non-isotropic trajectories) or massless ones (isotropic trajectories). initial formulas (4.42, 4.43), we arrive to chr.inv.-equations of mo- As a matter of fact, for massless particles m = 0 and ds = 0, there- 0 tion of the massless spin-particle located in our world (the particle fore their ratio in (4.75) is a 0 indeterminacy. However the transi- 0 travels from past into future), namely tion (4.75) m0 to m solves the indeterminacy, because the relativ- ds dσ istic mass of any massless particle is m = 0, so along their trajectory dm m i m i k 1 dη η i η i k 6 Fi c + Dik c c = + Fi c Dik c c , (4.82) we have dσ = 0. dτ − c2 c2 −c2 dτ c4 − c4 6 4.4 Physical conditions of spin-interaction 167 168 Chapter 4 Motion of spin-particle

d i i i k i i n k On the other hand, chr.inv.-projections of the non-holonomity mc + 2m Dk + Ak∙ c mF + mΔnk c c = αβ 1 αβμν dτ ∙ − (4.83) tensor Aαβ (4.11) and of its dual pseudotensor A∗ = 2 E Aμν are 1 d i 2η i i k η i η i n k = ηc D + A∙ c + F Δ c c , i 2 2 k k 2 2 nk A0∙ ik im kn −c dτ − c ∙ c − c ∙ = 0 ,A = h h Amn , √g00 while for the analogous particle in the mirror world (the particle (4.87) i travels from future into past), having the quantities (4.81) substi- A0∗∙ ik ∙ = 0 ,A∗ = 0 . tuted for ϕ < 0, the chr.inv.-equations of motion are √g00 dm m m 1 dη η η Comparing the formulas we see that spin-interaction gives an i i k i i k ik ik im kn Fi c + Dik c c = + Fi c Dik c c , (4.84) analogy for only the “magnetic” component = A = h h A − dτ − c2 c2 c2 dτ c4 − c4 mn of the space non-holonomity field. The “electric”H component of the d A i i i i n k field in spin-interaction turns to be zero i = 0∙ = 0. This is no mc + mF + mΔnk c c = √g ∙ dτ (4.85) E 00 1 d η η surprise, because the inner rotational field of a particle (its spin, = ηci F i Δi cnck. −c2 dτ − c2 − c2 nk in other word) interacts the space non-holonomity field like as an external field, and the both fields are produced by motion. §4.4 Physical conditions of spin-interaction Besides, for the “magnetic” component of the non-holonomity i ik ik i A0∗∙ field = A = 0 can not be dual to zero value ∗ = ∙ = 0. So, As we have shown, spin of a particle (its inner mechanical momen- H 6 H √g00 tum) interacts an external field of the space rotation, described similarity with electromagnetic fields turns out to be incomplete. ∂b by the space non-holonomity tensor Aαβ = 1 chαμhβν ν ∂bμ , But full matching could not even be expected, because the space 2 ∂xμ ∂xν − α non-holonomity tensor and the electromagnetic field tensor have depending on the rotor of the four-dimensional velocity vector b somewhat different structures: the Maxwell tensor is a “pure” rotor of the observer in respect of his reference body. In electromagnetic ∂A F = ∂Aβ α , while the non-holonomity tensor is an “add-on” phenomena a particle’s charge interacts an external electromagnetic αβ ∂xα − ∂xβ ∂Aβ ∂A ∂b μ field — the field of Maxwell’s tensor F = α . Therefore it rotor Aαβ = 1 chαμhβν ν ∂b . On the other hand we have αβ ∂xα − ∂xβ 2 ∂xμ − ∂xν seems natural to compare chr.inv.-projections of the Maxwell tensor no doubts that in future comparative analysis of these fields will Fαβ to chr.inv.-projections of the space non-holonomity tensor Aαβ. produce a theory of spin-interactions, similar electrodynamics. In Chapter 3 we have obtained that the electromagnetic field Incomplete similarity of the space non-holonomity field and tensor, known also as the Maxwell tensor, results two groups of electromagnetic fields also leads to another result. If we define chr.inv.-projections, produced by the tensor Fαβ itself and by its the force in spin-interaction in the same way that we define the αβ 1 αβμν α e α σ α η0 α σ dual pseudotensor F ∗ = 2 E Fμν , namely∗ Lorentz force Φ = F σ∙U , the obtained formula Φ = 2 A σ∙U c ∙ c ∙ i will include not all the terms from the right parts of equations of F0∙ i ik ik ∙ = E ,F = H , motion of a spin-particle. But an external force acting the particle, √g00 by definition, must include all those factors which deviate the (4.86) i particle from geodesic line, i. e. all terms in the right parts of the F0∗∙ i ik ik ∙ = H∗ ,F ∗ = E∗ . equations of motion. In other word, the four-dimensional force of √g00 α spin-interaction Φ [ gram/sec ] is defined by the formula Here Eαβμν is the four-dimensional completely antisymmetric discriminant ∗ DSα dSα dxν tensor, which produce pseudotensors in the four-dimensional pseudo-Riemannian Φα = = + Γα Sμ , (4.88) space, see §2.3 in Chapter 2 for details. ds ds μν ds 4.4 Physical conditions of spin-interaction 169 170 Chapter 4 Motion of spin-particle which chr.inv.-projection on the spatial section (after being divided electromagnetic interactions, weak (spin) interactions, or strong by c) gives the three-dimensional observable force of the interaction interactions. i Φ [ gram cm/sec2 ]. For instance, for a mass-bearing particle located Keeping this fact in mind, we can assume that w 0 for spin- × → in our world, having (4.71) as a base, we have interaction in the formula for Aik (4.93). Then in the microscopic scales of elementary particles the tensor Aik is the physical observ- i 1 d i 2η i i k η i η i n k able rotor in “strict” notation Φ = ηv Dk + Ak∙ v + F Δnkv v . (4.89) −c2 dτ − c2 ∙ c2 − c2 1 ∂v ∂v A = ∗ k ∗ i , (4.94) From further comparison between electromagnetic interaction ik 2 ∂xi − ∂xk and spin-interaction, using similarity with the electromagnetic field invariants (3.25, 3.26) we deduced the invariants of the space non- while the acting gravitational inertial force (4.92) have only its holonomity field inertial part ∂v 1 ∂v ∂v F = ∗ i = i = i . (4.95) αβ ik ikn m i i w J1 = AαβA = AikA = εikm ε Ω∗ Ω n = 2Ω i Ω∗ , (4.90) − ∂t −1 ∂t − ∂t ∗ ∗ − c2 αβ J2 = AαβA∗ = 0 . (4.91) Zelmanov’s identities, which link the space rotation to the gravitational inertial force, acting in it, i Hence the scalar invariant J1 = 2Ω i Ω∗ is always non-zero, be- ∗ cause otherwise the space would be holonomic (not rotating) and 2 ∗∂ i ijk k 1 k √h Ω∗ + ε ∗ j Fk = 0 , ∗ k Ω∗ + Fk Ω∗ = 0 (4.96) spin-interaction would be absent. √h ∂t ∇ ∇ c2 Now we are approaching physical conditions of motion of ele- for elementary particles (w 0) become mentary spin-particles. Using the definition of the chr.inv.-vector → of gravitational inertial force 2 2 1 ∂ √ i 1 ijk ∗∂ vk ∗∂ vj h Ω∗ + ε j k = 0 , ∂ ln 1 w √h ∂t 2 ∂x ∂t − ∂x ∂t 1 ∂w ∂vi 2 ∗∂vi (4.97) F = = c2 − c (4.92) i w i i k 1 ∗∂vk k 1 ∂x − ∂t − ∂x − ∂t ∗ k Ω∗ Ω∗ = 0 . − c2 ∇ − c2 ∂t we formulate the non-holonomity tensor Aik with the gravitational If we substitute here ∗∂vk = 0, so assume that the observable ∂t potential w and the linear velocity vi of the space rotation k rotation of the space is stationary, we obtain ∗ Ω∗ = 0, so the spa- ∇k ∂ ln 1 w ∂ ln 1 w ce rotation pseudovector remains unchanged. Then the Zelmanov 1 ∗∂vk ∗∂vi − c2 − c2 1st identity become Aik = + vi vk . (4.93) 2 ∂xi − ∂xk q∂xk − q∂xi i i ∗∂Ω∗ Ω∗ D + = 0 , (4.98) ∂t From here we see that the non-holonomity tensor Aik is the √ three-dimensional observable rotor of the linear velocity of the from which we see that D = det Dn = ∗∂ ln h, so the rate of space rotation with two additional terms, produced by both the || n|| ∂t relative expansion of the space elementary volume is zero D = 0. gravitational potential w and the space rotation. Therefore, the equations suggest that for elementary particles On the other hand, because of the small numerical value of the (w 0) at stationary rotations of the space ∗∂vk = 0 the tensor of Planck constant, spin-interaction only affects elementary particles. → ∂t k And as it is known, in the scales of such small masses and distances angular velocities of this rotation remains unchanged ∗ k Ω∗ = 0 gravitational interaction is a few orders of magnitude weaker than and the space relative expansions (deformations) are absent∇ D = 0. 4.5 Motion of elementary spin-particles 171 172 Chapter 4 Motion of spin-particle

It is possible, that stationarity of the space non-holonomity field applied for elementary particles (4.97) imply that (the external field in spin-interaction) is the necessary condition of k √ stability of the elementary particle under this action. Out of this we k ∂Ω∗ ∂ h k 1 ∂ k ∗ k Ω∗ = + Ω∗ = √h Ω∗ = 0 . (4.100) may conclude that long-living spin-particles should possess stable ∇ ∂xk ∂xk √h ∂xk inner rotations, while short-living particles must be unstable spatial ∂ k vortexes. The first condition is true provided that (√h Ω∗ ) = 0. This ∂xk To study motion of short-living particles is pretty problematic is true if the space rotation pseudovector is as we do not have experimental data on the structure of unstable Ω i i (0)∗ i vortexes, which may produce them. In the same time the study for Ω∗ = , Ω(0)∗ = const , (4.101) long-living ones, i. e. in the stationary field of the space rotation, √h can give exact solutions of their equations of motion. We will focus in this case the second condition (4.100) is true too. on these issues in the next §4.5. Taking what has been said in the above into account, from the formulas (4.70, 4.71) after some algebra we obtain chr.inv.- equations of motion of a mass-bearing elementary particle. For the §4.5 Motion of elementary spin-particles particle, which is located in our world so it travels into future in respect of an regular observer, the equations are As we have mentioned, the Planck constant, being a small absolute dm 1 dη = , (4.102) value, only “works” on the scales of elementary particles, where dτ −c2 dτ gravitational interactions is a few orders of magnitude weaker than d i i k i n k electromagnetic, weak and strong ones. Hence assuming w 0 in mv + 2mAk∙ v + mΔnkv v = dτ ∙ chr.inv.-equations of motion of spin-particles (4.70–4.73) and→ (4.82– (4.103) 1 d i 2η i k η i n k 4.85), we will arrive to chr.inv.-equations of motion of elementary = ηv Ak∙ v Δnkv v , −c2 dτ − c2 ∙ − c2 particles. while for the particle, which is located in the mirror world so it Besides, as we have obtained in the previous §4.4, under sta- travels into past, from (4.72, 4.73) we obtain tionary rotations of the space ∗∂vk = 0 in the scale of elementary ∂t dm 1 dη particles the spur of the space deformations tensor is zero D = 0. Of = 2 , (4.104) course, zero spur of a tensor does not necessarily imply the tensor − dτ c dτ itself is zero. On the other hand, the space deformation is a rare d 1 d η mvi + mΔi vnvk = ηvi Δi vnvk. (4.105) phenomenon, so for our study of motion of elementary particles we dτ nk −c2 dτ − c2 nk will assume D = 0. ik As it is easy to see, the scalar chr.inv.-equations of motion are In §4.3 we have showed that under stationary rotations of the the same for both our-world particles and mirror-world particles. space the conservation condition for the spin-impulse vector an Integrating the scalar equation for an our-world particle (the direct α arbitrary spin-particle S becomes (4.68) so that flow of time), namely — taking the integral τ 1 ∂ 2 d η n mnε vk √h Ω i = 0 . (4.99) m + dτ = 0 , (4.106) ~ imn k ∗ dτ c2 √h ∂x τ1=0 Z we obtain η ∗∂v m + = const = B, (4.107) On the other hand, under k = 0 the Zelmanov identities we 2 ∂t c 4.5 Motion of elementary spin-particles 173 174 Chapter 4 Motion of spin-particle

where B is integration constant that can be defined from the initial while the other are hik = 0 if i = k. Noteworthy, the four-dimensional conditions. metric can not be Galilean here,6 because the spatial section rotates To illustrate physical sense of the obtained live forces integral, in respect of time. In other word, though the observable three- we use analogy between chr.inv.-projections dimensional space (the spatial section) in this case is a flat Euclidean one, the four-dimensional space-time is not the Minkowski space P 1 1 0 = m , P i = mvi = pi, but it is a pseudo-Riemannian space which metric is √g00 ± c c 2 0 0 0 i i k (4.108) ds = g00dx dx + 2g0idx dx + gikdx dx = S0 1 1 (4.111) i i 2 2 i 1 2 2 2 3 2 = 2 η , S = 3 ηv . = c dt + 2g0i cdtdx dx dx dx . √g00 ±c c − − − We assume that the space rotates at a constant angular velocity of the particle’s four-dimensional impulse vector and those of its 3 α η α Ω = const around the axis x , for instance. Then the linear velocity spin-impulse vector, i. e. of P α = m dx and Sα = 0 dx . k 0 ds c2 ds of the space rotation vi = Ωik x becomes Using analogy with relativistic mass m we will refer to the ± 2 1 quantity 1 η as relativistic spin-mass, so the quantity 1 η is rest v1 = Ω12 x = Ω y , v2 = Ω21 x = Ω x , (4.112) ±c2 c2 0 − spin-mass. Further, live forces theorem for the elementary spin- where Aik = Ωik. Then the space non-holonomity tensor Aik has particle (4.107) implies that with the assumptions we made the only two non-zero components sum of the particle’s relativistic mass and of its spin-mass remains A = A = Ω . (4.113) unchanged along the trajectory. 12 − 21 − Now using the live forces integral∗, we approach the vector Taking this into account the vector chr.inv.-equations of motion chr.inv.-equations of motion of the mass-bearing elementary par- (4.109) become ticle, located in our world, namely — the equations (4.103). Substi- dv1 dv2 dv3 tuting the live force integral (4.107) into (4.103), having the constant + 2Ωv2 = 0 , 2Ωv1 = 0 , = 0 , (4.114) canceled, we obtain pure kinematic equations of motion dτ dτ − dτ

i where the third equation solves immediately as follows dv i k i n k + 2Ak∙ v + Δnkv v = 0 , (4.109) 3 3 dτ ∙ v = v(0) = const. (4.115)

i n k 3 which are non-geodesic, in this case. The term Δnkv v , which is Taking into account that v3 = dx , we represent x3 as follows contraction of the Christoffel chr.inv.-symbols with the particle’s dτ observable velocity, is relativistic in the sense that it is a square 3 3 3 x = v(0)τ + x(0) , (4.116) function of the velocity. Therefore it can be neglected, provided that the observable metric h = g + 1 v v along the trajectory where x3 is the numerical value of the coordinate x3 at the initial ik − ik c2 i k (0) is close to Euclidean one. Such case is possible, if the linear velocity moment τ = 0. We formulate v2 from the first equation (4.114) of the space rotation is much lower than the light velocity, while the 1 dv1 three-dimensional coordinate metric g is Euclidean one as well. v2 = , (4.117) ik −2Ω dτ Then diagonal components of the metric chr.inv.-tensor are having this formula derivated with respect to dτ h = h = h = +1 , (4.110) 11 22 33 dv2 1 d2 v1 = , (4.118) ∗The solution of the scalar chr.inv.-equation of motion. dτ −2Ω dτ 2 4.5 Motion of elementary spin-particles 175 176 Chapter 4 Motion of spin-particle and having it substituted into the second equation (4.114) we obtain Now having the obtained v1 (4.122) substituted into the second equation (4.114), we arrive to 2 1 d v 2 1 + 4Ω v = 0 , (4.119) dv2 dτ 2 = 2Ωv1 cos (2Ωτ) + ˙v1 sin (2Ωτ) , (4.125) dτ (0) (0) i. e. the equation of free oscillations. It solves as follows which after integration gives v2 1 1 v = C1 cos (2Ωτ) + C2 sin (2Ωτ) , (4.120) ˙v v2 = v1 sin (2Ωτ) (0) cos (2Ωτ) + C . (4.126) (0) − 2Ω 4 where C1 and C2 are integration constants (4.119), which can be 2 2 defined from the conditions at the moment τ = 0 Assuming for the moment τ = 0 the value v = v(0), we obtain 1 2 ˙v(0) 1 the constant C3 = v + . Then v(0) = C1 , (0) 2Ω (4.121) ˙v1 ˙v1 dv1 2 1 (0) 2 (0) v = v(0) sin (2Ωτ) cos (2Ωτ) + v(0) + . (4.127) = 2ΩC1 sin (2Ωτ) τ=0 + 2ΩC2 cos (2Ωτ) τ=0 . − 2Ω 2Ω dτ τ=0 − | | 2 2 dx 1 Taking into account that v = , we integrate the formula with ˙v 1 dτ 1 (0) 1 dv 2 Thus C1 = v(0), C2 = , where ˙v(0) = . Then the equa- respect to dτ. Then we obtain the formula for the coordinate x of 2Ω dτ τ=0 tion for v1 finally solves as follows this particle, namely

1 1 1 1 ˙v v ˙v τ ˙v 2 (0) (0) 2 (0) 1 1 (0) x = sin (2Ωτ) cos (2Ωτ) + v τ + + C . (4.128) v = v cos (2Ωτ) + sin (2Ωτ) , (4.122) 2 (0) 5 (0) 2Ω −4Ω − 2Ω 2Ω Integration constant can be found from the conditions x2 = x2 so the velocity of the mass-bearing elementary spin-particle along (0) 1 1 x performs sinusoidal oscillations at the frequency equal to the v(0) at τ = 0 as C = x2 + . Then the coordinate x2 finally is double angular velocity of the space rotation. 5 (0) 2Ω 1 1 1 1 1 Taking into account that v1 = dx , we integrate the obtained ˙v τ ˙v v v dτ x2 = v2 τ+ (0) (0) sin (2Ωτ) (0) cos (2Ωτ)+x2 + (0) . (4.129) formula (4.122) with respect to dτ. We obtain (0) 2Ω −4Ω2 − 2Ω (0) 2Ω v1 ˙v1 From this formula we see: if at the initial moment of observable 1 (0) (0) x = sin (2Ωτ) cos (2Ωτ) + C3 . (4.123) time τ = 0 the mass-bearing elementary spin-particle had the ve- 2Ω − 4Ω2 2 2 1 1 locity v(0) along x and the acceleration ˙v(0) along x , then this 1 1 particle, among with free oscillations of the coordinate x2 at the Assuming that at the initial moment τ = 0 we have x = x(0), we 1 frequency, equal to the double angular velocity of the space rotation 1 ˙v(0) 1 obtain the integration constant C3 = x + . Then we have ˙v τ (0) 4Ω2 Ω, is subjected to a linear displacement at Δx2 = v2 τ + (0) . (0) 2Ω 1 1 1 η v ˙v ˙v Looking back at the live forces integral m + =B =const (4.107), x1 = (0) sin (2Ωτ) (0) cos (2Ωτ) + x1 + (0) , (4.124) c2 2Ω − 4Ω2 0 4Ω2 we find the integration constant B. Writing (4.107) in the form so the coordinate x1 of the elementary particle also performs free η v2 m + 0 = B 1 , (4.130) oscillations at the frequency 2Ω. 0 2 2 c r − c 4.5 Motion of elementary spin-particles 177 178 Chapter 4 Motion of spin-particle so the square of the particle’s observable velocity is v2 = const. vector chr.inv.-equations of motion of a mass-bearing spin-particle, Because components of the velocity have been already defined, we located in our world, solve as follows can present the formula for its square, which, because the three- v1 dimensional metric in this question is Euclidean one, becomes v1 = v1 cos (2Ωτ) , x1 = (0) sin (2Ωτ) + x1 , (0) 2Ω (0) 2 1 1 1 2 2 2 2 2 2 ˙v(0) v v v1 + v2 + v3 = v1 + v2 + v3 + + 2 2 2 (0) (0) 2 (4.135) (0) (0) (0) 2Ω2 v = v(0) sin (2Ωτ) , x = cos (2Ωτ) + + x(0) , (4.131) − 2Ω 2Ω ˙v1 ˙v2 ˙v1 ˙v1 (0) (0) 2 (0) 1 (0) 3 3 3 3 3 + + 2 v + v sin (2Ωτ) cos (2Ωτ) . v = v(0), x = v(0)τ + x(0) . Ω (0) 2Ω (0) − 2Ω Let us look at the form of a spatial curve along which the particle We see that the square of the velocity remains unchanged, if 2 1 moves. We set the observer’s reference frame so that the initial ˙v(0) = 0 and ˙v(0) = 0. The constant B from the live forces integral is 1 2 3 displacement of the particle is zero x(0) = x(0) = x(0) = 0. Now all its η0 spatial coordinates at an arbitrary moment of time are m0 + c2 2 2 1 2 3 2 B = , v = v + v = const , (4.132) 1 2 3 2 (0) (0) (0) x =x=a sin (2Ωτ) , x =y=a [1 cos (2Ωτ)] , x =z=bτ , (4.136) v(0) − 1 r − c2 v1 where a = (0) , b = v3 . The obtained solutions for coordinates are while the live forces integral itself (4.170) becomes 2Ω (0) parametric equations of a surface, along which the particle travels. η0 To illustrate what kind of surface it is, we switch from parametric η m0 + 2 c notation to coordinate one, removing the parameter τ from the m + 2 = , (4.133) c v2 equations. Putting formulas for x and y in the power of two we 1 (0) r − c2 obtain so it is the conservation condition for the sum of the relativistic zΩ η x2 + y2 = 2a2 [1 cos (2Ωτ)] = 4a2 sin2 (Ωτ) = 4a2 sin2 . (4.137) mass of the particle m and of its spin-mass . c2 − b A note should be taken here concerning all that has been said The obtained result reminds a spiral line equation x2 + y2 = a2, in the above on elementary particles. Taking into account in the mn k z = bτ. However the similarity is not complete — the particle travels definition η0 = n~ Amn that Amn = εmnkΩ∗ , we obtain 3 along the surface of a cylinder at a constant velocity b = v(0) along mn k its axis z, while its radius oscillates at a frequency Ω within the η0 = n~ Amn = 2n~ k Ω∗ . (4.134) ∗ 1 v(0) πkb 1 mn range∗ from zero up to the maximum 2a = at z = . where ~ k = 2 εnmk~ . Here ~ k is the three-dimensional pseudo- Ω 2Ω vector of∗ the inner momentum∗ of the elementary particle. Hence So the trajectory of the mass-bearing elementary spin-particle, located in our world, reminds a spiral line “wound” over an oscil- η0 is the scalar product of three-dimensional pseudovectors: that of lating cylinder. The particle’s life span is the length of the cylinder the particle’s inner momentum ~ k and that of the angular velocity k ∗ divided by its velocity along z (the cylinder’s axis). Oscillations of the space rotation Ω∗ . Hence spin-interaction is absent if pseudovectors of the particle’s inner rotation and the external rotation of of the cylinder are energy “breath ins” and “breath outs” of the the space are collinear. particle. Now we refer back to equations of motion of spin-particles. 3 ∗Where k = 0, 1, 2, 3,... If v(0) = 0, the particle simply oscillates within x y plane Taking into account the integration constants we have obtained, the (the plane of the cylinder’s section). 4.5 Motion of elementary spin-particles 179 180 Chapter 4 Motion of spin-particle

This means that the cylinder we have obtained is the cylinder On the other hand, from viewpoint of a hypothetical observer, of events of the particle from its birth in our world (the act of who is located in the mirror world, motion of mass-bearing spin- materialization) through its death (the dematerialization). But even particles in our world will be linear and even, while mirror-world after the death, its event cylinder does not disappear completely, particles will travel along oscillating “spiral” lines. but the cylinder splits into few event cylinders of other particles, We could also get an analysis of motion of massless (light-like) produced by this decay either in our world or in the mirror world. spin-particles in a similar way, but we don’t know how adequate in Therefore analysis of births and decays of elementary particles such case would be our assumption that the linear velocity of the in the General Theory of Relativity implies analysis of branch points space rotation is much smaller compared to the light velocity. And of event cylinders of the particles, taking into account possible it was this assumption thanks to which we were able to obtain branches that lead into the mirror world. exact solutions of chr.inv.-equations of motion for mass-bearing If we consider motion of two linked spin-particles, which rotate elementary spin-particles. Though in general, the methods to solve around a common centre of masses, for instance, that of positro- chr.inv.-equations of motion are the same for mass-bearing and nium (dumb-bell shaped system of an electron and a positron), we massless particles. obtain a double DNA-like spiral — a twisted “rope ladder” with a number of steps (links of the particles), wound over an oscillating §4.6 A spin-particle in electromagnetic fields cylinder of their events. Let us solve chr.inv.-equations of motion of a mass-bearing spin- In this §4.6 we are going to deduce chr.inv.-equations of motion particle, which moves in the mirror world, a world with the reverse of a particle which bears both spin and an electric charge, and flow of time. Under physical conditions we are considering∗ the it travels in an external electromagnetic field located in the four- mentioned equations (4.104, 4.105) become dimensional pseudo-Riemannian space. As a matter of fact that the particle’s summary vector is dm 1 dη = 2 , (4.138) e − dτ c dτ Qα = P α + Aα + Sα, (4.141) c2 d i 1 d i mv = ηv . (4.139) were P α is the four-dimensional impulse vector of the particle. dτ −c2 dτ The rest two four-dimensional vectors are an additional impulse, Solution of the scalar chr.inv.-equation is the live forces integral η which the particle gains from interaction of its charge with the m + = B = const, as it was in the case for the analogous our-world c2 electromagnetic field, and an additional impulse gained from inter- particle (4.107). Substituting it into the vector chr.inv.-equations action of the particle’s spin with the space non-holonomity field. (4.139) we solve them as follows Note, because the vectors P α and Sα are tangential to the four- dimensional trajectory of the particle, we assume that the vector dvi = 0 , (4.140) Aα (the electromagnetic field potential) is also tangential to the α dτ α dx trajectory. In this case the vector is A = ϕ0 , while the formula i i ds hence v = v = const. This result implies that from viewpoint of ϕ (0) qi = vi (see §3.8) sets the relationship between the scalar potential a regular observer the mass-bearing spin-particle travels in the c i mirror world linearly at a constant velocity in contrast to observable ϕ and the vector potential q of the electromagnetic field. i motion of the analogous our-world particle, which travels along an Then chr.inv.-projections ϕ˜ and q˜ of the particle’s summary α oscillating “spiral” line. vector Q (4.141) we are considering are eϕ η 1 1 ∗Namely — stationary rotation of the space at a low velocity, the absence of the ϕ˜ = m + + , q˜i = mvi + (η + eϕ) vi, (4.142) space deformation, and the Euclidean three-dimensional metric. ± c2 c2 c2 c3 4.6 A spin-particle in electromagnetic fields 181 182 Chapter 4 Motion of spin-particle where m is the relativistic mass of the particle, ϕ is the scalar any reference frame. In particular, in the accompanying reference potential of the acting electromagnetic field, while η describes inter- frame it is constant as well action of the particle’s spin with the space non-holonomity field α α e α α β e β β QαQ = gαβ P + A + S P + A + S = m ϕ η c2 c2 m = 0 , ϕ = 0 , η = 0 . (4.143) (4.148) 2 α β 2 v2 v2 v2 eϕ0 η0 dx dx eϕ0 η0 1 1 1 = g m + + = m + + . − c2 − c2 − c2 αβ 0 c2 c2 ds ds 0 c2 c2 q q q Generally speaking, the equations can be deduced in the same In §3.9 we have showed that introducing a specific direction of way as those for a charged particle and a charge-free spin-particle the four-dimensional electromagnetic potential Aα in respect of the severally, save that now we have to project the absolute derivative trajectory of a charged particle the field moves, we substantially of the sum of the three vectors. Using formulas for ϕ˜ and q˜i (4.142), simplify the right parts of chr.inv.-equations of its motion. The we obtain chr.inv.-equations of motion of the charged mass-bearing right part of the vector chr.inv.-equations of motion becomes the spin-particle located in our world (it travels from past into future) i i 1 ikm Lorentz chr.inv.-force Φ = e E + c ε vkH m , while the right part of the scalar chr.inv.-equation− is obtained∗ as scalar product of dm m m i i k the electric strength vector E and the observable velocity of the 2 Fi v + 2 Dikv v = i dτ − c c (4.144) particle. Keeping this in mind, we present the obtained chr.inv.- 1 d η + eϕ η + eϕ i i k equations (4.144–4.147) in a more specific form. For the particle = 2 (η + eϕ) + 4 Fi v 4 Dikv v , −c dτ c − c located in our world we obtain

d i i i k i i n k d η 1 η i 1 η i k e i mv + 2m Dk + Ak∙ v mF + mΔnkv v = m+ m+ Fi v + m+ Dikv v = Eiv , (4.149) dτ ∙ − dτ c2 −c2 c2 c2 c2 −c2 1 d 2 (η + eϕ) i i i k (4.145) = 2 (η + eϕ) v 2 Dk + Ak∙ v + d η i η i i k η i −c dτ − c ∙ m+ v + 2 m+ Dk +Ak∙ v m+ F + dτ c2 c2 ∙ − c2 η + eϕ i η+ eϕ i n k h i (4.150) + 2 F 2 Δnkv v , η i n k i 1 ikm c − c + m + Δnkv v = e E + ε vkH m , c2 − c ∗ while for the analogous particle located in the mirror world (it tra- while for the analogous particle in the mirror world we have vels from future into past) the equations are d η 1 η 1 η e dm m m i i k i i i k m+ 2 2 m+ 2 Fiv + 2 m+ 2 Dikv v = 2 Eiv , (4.151) 2 Fi v + 2 Dikv v = −dτ c −c c c c −c − dτ − c c (4.146) 1 d η + eϕ η + eϕ d η η η = (η + eϕ) + F vi D vivk, m+ vi + m+ F i + m+ Δi vnvk = c2 dτ c4 i − c4 ik dτ c2 c2 c2 nk h i (4.152) i 1 ikm d = e E + ε vkH m . i i i n k − c ∗ mv + mF + mΔnkv v = dτ (4.147) 1 d η + eϕ η + eϕ To make exact conclusions on motion of charged mass-bearing = (η + eϕ) vi F i Δi vnvk. spin-particles in the pseudo-Riemannian space we have to set a con- −c2 dτ − c2 − c2 nk crete geometric structure of the space. As we did it in the previous Parallel transfer in Riemannian spaces remains the length of §4.5, where we analyzed motion of charge-free spin-particles, we any transferred vector unchanged. Hence its square is invariant in assume that: 4.6 A spin-particle in electromagnetic fields 183 184 Chapter 4 Motion of spin-particle

Because gravitational interactions in the scales of elementary while for the mirror-world particle these are • particles are infinitesimal, we can assume w 0; → d η e dxi ∗∂v m + = E , (4.157) The space rotation is stationary, so k = 0; 2 2 i ∂t dτ c c dτ • The space does not deformations, so Dik = 0; • η 1 i k d m + 2 v The three-dimensional coordinate metric gikdx dx is Euclid- c 1 1 1km • = e E + ε vkH m , 1, i = k dτ − c ∗ ean one, so gik = − ; η 0, i = k d m + v2 6 c2 2 1 2km (4.158) The space rotates at a constant angular velocity Ω around = e E + ε vkH m , dτ − c ∗ • 3 x = z, so components of the linear velocity of the rotation are η 3 2 1 d m + v v1 = Ω12x = Ωy, v2 = Ω21x = Ωx. c2 3 1 3km − = e E + ε vkH m . dτ − c ∗ Keeping in mind these constraints, we obtain a formula for ds2 in the scales of elementary particles becomes Let us look at the scalar chr.inv.-equation of motion in our world (4.155) and in the mirror world (4.157). From here we see that the ds2 = c2dt2 2Ωydtdx + 2Ωxdtdy dx2 dy2 dz2, (4.153) − − − − sum of the particle’s relativistic mass and of its spin-mass equals a work made by the electric component of the acting electromagnetic while physical observable characteristics of the reference space field to displace this charged particle at an elementary interval dxi. under this metric are From the vector chr.inv.-equations of motion we see that in our world (4.156) as well as in the mirror world (4.158) the sum of the F = 0 ,D = 0 ,A = A = Ω ,A = A = 0 . (4.154) i ik 12 − 21 − 23 31 spatial impulse vector of the particle and its spin-impulse vector 3 As we did it in the previous §4.5 looking at motion of elementary along x = z is defined only by the Lorentz force’s component along spin-particles, we assume that the linear velocity of the space the same axis. rotation is much less than the light velocity (a weak field of the Now our goal is to obtain the trajectory of an elementary charged spin-particle in an electromagnetic field of the given particular space non-holonomity). In such case the metric chr.inv.-tensor hik is Euclidean one and all the Christoffel chr.inv.-symbols Δi become properties. As we did it in Chapter 3, we assume the field constant, jk i zeroes, which dramatically simplifies the involved algebra. Then so its electric and magnetic strengths Ei and H∗ are chr.inv.-equations of motion of the particle located in our world ∂ϕ E = , (4.159) become i dxi d η e dxi m + 2 = 2 Ei , (4.155) dτ c −c dτ i 1 imn 1 imn ∂ (ϕvm) ∂ (ϕvn) H∗ = ε Hmn = ε n m 2ϕAmn . (4.160) η 2 2c dx − dx − d m+ v1 c2 η 2 1 1 1km + 2 m+ Ωv = e E + ε vkH m , In Chapter 3 we tackled a similar problem — solving chr.inv.- dτ c2 − c ∗ equations of motion for a charged mass-bearing particle, but with- η d m+ v2 out taking its spin into account. It is evidently that in a particular c2 η 1 2 1 2km (4.156) 2 m+ Ωv = e E + ε vkH m , case of a spin-free charged particle, so that where spin is zero, dτ − c2 − c ∗ η solutions of chr.inv.-equations of motion of a charged spin-particle, d m+ v3 c2 3 1 3km as more general ones, should coincide those obtained in Chapter 3 = e E + ε vkH m , dτ − c ∗ within “pure” electrodynamics. 4.6 A spin-particle in electromagnetic fields 185 186 Chapter 4 Motion of spin-particle

To compare our results with those obtained in electrodynamics, which implies that the field scalar potential ϕ = const, so that it would be reasonable to analyze motion of the mass-bearing spin- particle in three typical kinds of electromagnetic fields, which were i ϕ imn ∂vm ∂vn ∂vm ∂vn H∗ = ε 2 . (4.164) 2c ∂xn − ∂xm − ∂xn − ∂xm under study in Chapter 3 as well as in The Classical Theory of Fields by Landau and Lifshitz [5]: For a relativistic case the electric component reveals itself (it (a) A homogeneous stationary electric field (the field magnetic performs a work to displace the particle), provided that the absolute strength is zero); value of the particle’s velocity is not stationary (b) A homogeneous stationary magnetic field (the field electric 2 strength is zero); 1 η0 dv e i 3 m0 + 2 = 2 Ei v = 0 . (4.165) 2 2 c dτ c (c) A homogeneous stationary electromagnetic field (the both 2 v − 6 2c 1 2 components are non-zeroes). − c On the other hand, electrodynamics studies motion of regular Hence the electric component of the acting electromagnetic field, macro-particles and it is not evidently that all three cases men- given the constraints on the metric, typical for elementary particles, tioned in the above are applicable, given the metric constraints, reveals itself only for relativistic particles, which velocity is not typical for micro-world. Here is why. constant along the trajectory. Hence all “slow-moving” particles First, spin of an elementary particle affects its motion only if an fall out of our consideration in the field of strictly electric kind. external field of the space non-holonomity exists, hence the non- Therefore, the general case∗ should be only studied for a sta- holonomity tensor is Aik = 0. But from the formulas for the electric tionary electromagnetic field of strictly magnetic kind, where the 6 i and magnetic strengths Ei and H∗ (4.159, 4.160) we see that the electric component is absent. This will be done in the next §4.7. space non-holonomity only affects the magnetic strength. Hence we will largely focus on motion of the elementary spin-particle in an §4.7 Motion in a stationary magnetic field electromagnetic field of strictly magnetic kind. Second, the scalar chr.inv.-equation of motion of a mass-bearing In this §4.7 we are going to look at motion of a charged spin-particle charged spin-particle (4.155) in a homogeneous stationary electromagnetic field of strictly magnetic kind. η0 d 1 e i m0 + 2 = 2 Ei v (4.161) As we did it in the previous §4.6, we assume that the space- c dτ 1 v2 −c − c2 time has the metric (4.153), so Fi = 0 and Dik = 0 there. The non- q holonomity field is stationary. In the space rotation around z out of in a non-relativistic case, where the particle’s velocity is much less all components of the non-holonomity tensor only the components than the light velocity, becomes A = A = Ω = const are not zeroes, so the space rotates within 12 − 21 − i x y plane at a constant velocity Ω. Ei v = 0 , (4.162) mn Under the considered conditions the quantity η0 = n~ Amn, so the electric component of the field does not perform work to which describes interaction between the particle’s spin (its inner displace the particle under constraints on the metric, typical for the rotation) and an external field of the space non-holonomity, is world of elementary particles. Because we are looking at stationary mn 12 21 fields, the obtained condition (4.162) can be presented as follows η0 = n~ Amn = n ~ A12 + ~ A21 = 2n~ Ω , (4.166) − ∂ϕ ∂ϕ dxi dϕ E vi = vi = = = 0 , (4.163) ∗Motion of an elementary charged spin-particle at an arbitrary velocity, either i ∂xi ∂xi dτ dτ low or relativistic one. 4.7 Motion in a stationary magnetic field 187 188 Chapter 4 Motion of spin-particle where the sign before the product ~ Ω depends only on mutual Having the formulas for the live forces integrals substituted into orientation of the ~ and Ω. “Plus” stands for co-directed ~ and Ω. (4.168, 4.170), we arrive to the vector chr.inv.-equations of motion “Minus” implies that they are oppositely directed. in our world and in the mirror world, respectively

In this case∗ chr.inv.-equations of motion of the particle located i in our world become dv i k i n k e ikm + 2Ak∙ v + Δnkv v = ε vkH m , (4.174) d η dτ ∙ −cB ∗ m + = 0 , (4.167) 2 i dτ c dv i n k e ikm + Δnkv v = ε vkH m . (4.175) dτ −cB ∗ d η i η i k η i n k m+ v + 2 m+ Ak∙ v + m+ Δnkv v = dτ c2 c2 ∙ c2 These are similar to chr.inv.-equations of motion of a charged h i (4.168) e e ikm macro-particle (that is a spin-free charged particle) in a homoge- = ε vkH m , − c ∗ neous stationary magnetic field (3.290, 3.291), save that here the while for the analogous particle located in the mirror world we integration constant from the live forces integral, found in the right obtain part, is not equal to the relativistic mass m of the particle, as it was d η in electrodynamics (3.290, 3.291), but to the formula (4.171), which m + = 0 , (4.169) −dτ c2 accounts for interaction of the particle’s spin with the space non- holonomity field. The same is true for the vector chr.inv.-equations d η i η i n k e ikm m + v + m + Δnkv v = ε vkH m . (4.170) (3.298, 3.299). dτ c2 c2 − c ∗ h i For those of our readers with special interest in the method of Having the live forces theorem (the scalar chr.inv.-equation of chronometric invariants we will make a note related to this notation motion) integrated, we obtain the live forces integral. In our world of chr.inv.-equations of motion. When obtaining components of the and in the mirror world it is, respectively 1 k term Ak∙ v , found only in the our-world equations, we have, for ∙ η η instance, for i = 1 m + 2 = B = const , m + 2 = B = const , (4.171) c c − 1 k 1 1 1 2 12 1 11 2 Ak∙ v = A1∙ v + A2∙ v = h A12v + h A21v , (4.176) ∙ ∙ ∙ where B and B are integration constants ine our world and in the 1 1 where A12 = A21 = Ω. Then obtaining A1∙ and A2∙ we have mirror world, respectively. We can obtain these constants having − − ∙ ∙ the initial conditionse at τ = 0 substituted into (4.171). As a result, 1 1m 11 12 12 we obtain A1∙ = h A1m = h A11 + h A12 = h A12 , (4.177) mn ∙ η0 n~ Amn B = m0 + = m0 + , (4.172) 1 1m 11 12 11 2 2 A2∙ = h A2m = h A21 + h A22 = h A21 , (4.178) c c ∙ mn ik η0 n~ Amn where h are elements of a matrix reciprocal to the matrix h B = m = m . (4.173) ik − 0 − c2 − 0 − c2 h22 h12 Formulas for the live forces integrals (4.171) imply that, in the h11 = , h12 = . (4.179) e h − h absence of the electric component of the acting electromagnetic field, the square of the velocity of the charged spin-particle remains Then, because the determinant of the metric chr.inv.-tensor (see 2 i k unchanged v = hikv v = const. §3.12 for details) is 2 2 2 α Ω x + y ∗Provided that the electromagnetic field potential A is directed along the four- h = det h = 1 + , (4.180) dimensional trajectory of the particle. ik 2 k k c 4.7 Motion in a stationary magnetic field 189 190 Chapter 4 Motion of spin-particle

1 k the unknown quantity Ak∙ v (4.176) is and those for the mirror-world particle ∙ 2 2 2 1 y˙ 1 k Ω Ω Ω x (0) Ak∙ v = xyx˙ + 1 + y˙ . (4.181) x = y˙(0) cos ωτ +x ˙ (0) sin ωτ + x(0) + , (4.186) ∙ h c2 c2 −ω ω 2 k 1 x˙ (0) The component Ak∙ v , found in the equation of motion along y, y = y˙(0) sin ωτ x˙ (0) cos ωτ + y(0) , (4.187) can be found in a similar∙ way. ω − − ω Let us get back to the vector chr.inv.-equations of motion of the which are different from solutions for a charged particle in electro- charged spin-particle in the homogeneous stationary magnetic field. dynamics only by the fact that the frequency ω accounts for inter- We approach them in two possible cases of mutual orientation of the action of the particle’s spin with the space non-holonomity field. magnetic strength and the space non-holonomity pseudovector. In our world masses of particles are positive, so ω is

2 2 A Magnetic field is co-directed with non-holonomity field v(0) v(0) eH eH 1 2 eH 1 2 ω = = r − c = r − c , (4.188) We assume that the space non-holonomity field is directed along z η η0 2n Ω mc + m0c + m c ~ and it is weak. Then the vector chr.inv.-equations of motion of the c c 0 ∓ c mass-bearing charged spin-particle located in our world are where the sign in the denominator depends on mutual orientation eH eH of the ~ and Ω — “minus” stands for the co-directed ~ and Ω x¨ + 2Ωy ˙ = y˙ , y¨ 2Ωx ˙ = x˙ , z¨ = 0 , (4.182) − cB − − cB (their scalar product is positive), while “plus” implies that they are oppositely directed, irrespective of our choice of right or left-hand while for the analogous particle located in the mirror world we have reference frames. eH eH Masses of particles, which inhabit the mirror world, are always x¨ = y˙ , y¨ = x˙ , z¨ = 0 . (4.183) negative − cB − cB m m = 0 < 0 , (4.189) The equations are different from those for a spin-free charged − v2 e e (0) particle under the same conditions (3.104, 3.305) only by having in 1 2 r − c the right part the integration constant from the live forces integral, so in the mirror world ω is which describes interaction of the particle’s spin with the space non-holonomity field, instead of the relativistic mass of the particle. 2 2 v(0) v(0) Using ready solutions from §3.12 we can immediately obtain the eH eH 1 2 eH 1 2 ω = = r − c = r − c . (4.190) formulas for coordinates of the our-world charged spin-particle η η0 2n Ω mc + m0c + m c ~ c − c − 0 ∓ c 1 x = y˙(0) cos (2Ω + ω) τ +x ˙ (0) sin (2Ω + ω) τ + Note that the obtained formulas for coordinates (4.184–4.187) − 2Ω + ω (4.184) already took account of the fact that the square of the particle’s y˙(0) + x + , velocity remains unchanged both in our world and in the mirror (0) 2Ω + ω world (respectively) 1 y = y˙(0) sin (2Ω + ω) τ x˙ (0) cos (2Ω + ω) τ + y¨0 y¨0 − 2Ω + ω x˙ (0) + = 0 , x˙ (0) + = 0 , (4.191) (4.185) 2Ω + ω ω x˙ + y (0) , (0) − 2Ω + ω which results from the live forces integral (§3.12). 4.7 Motion in a stationary magnetic field 191 192 Chapter 4 Motion of spin-particle

The third equation of motion (along z) solves simply as follows will depend on the absolute value and the orientation of the spin. If the initial velocity of a charged particle with spin, directed along the z =z ˙(0)τ + z(0) . (4.192) magnetic strength (along z), is not zero, the particle travels along the magnetic strength along a spiral line with the same radius r. The obtained formulas for coordinates (4.184–4.187) say that An analogous mirror-world particle, provided its displacement a mass-bearing charged spin-particle in a homogeneous stationary and the velocity at the initial moment of time satisfy the conditions magnetic field, parallel to a weak field of the space non-holonomity, performs harmonic oscillations along x and y. In our world the y˙ x˙ x + 0 = 0 , y 0 = 0 , (4.198) frequency of the oscillations is (0) ω (0) − ω

2 will also travel along a circle eH v ω = 2Ω + ω = 2Ω + 1 (0) , (4.193) 2n~ Ω s − c2 y˙2 m0c c x2 + y2 = 0 , (4.199) ∓ ω2 while ine the mirror world the analogous particle performs similar with the radius oscillations at a frequency ω we have obtained in (4.190). In a weak field of the space non-holonomity the quantity n~ Ω y˙0 y˙0 2 r = = . (4.200) is much less than the energy m0c , because for any small quantity ω v2 1 eH 1 (0) α it is true that = 1 α, for low velocities we have 2n Ω c2 1 α ∼ ± m0c ~ r − ∓ − ∓ c eH 2n~ Ω In general case, where no additional conditions (4.195, 4.198) ω = 2Ω + 1 2 . (4.194) ∼ m0c ± m0c imposed, the trajectory within x y plane will be not circle. Let us obtain the energy and the impulse of the particle. Using If at the initial momente of time the displacement and the velocity formulas for the live forces integrals, we find the quantity η0, of the our-world particle satisfy the conditions mn 12 21 which is η0 = n~ Amn = n (~ A12 + ~ A21) = 2n~ Ω. Then for the − y˙0 x˙ 0 particle located in our world we have x(0) + = 0 , y(0) = 0 , (4.195) 2Ω + ω − 2Ω + ω 2 2 m0c 2n~ Ω Etot = Bc = ∓ = const , (4.201) it will travel, like a charged spin-free particle, within x y plane along 2 v(0) a circle∗ 1 2 y˙2 r − c x2 + y2 = 0 . (4.196) (2Ω + ω)2 while in the mirror-world we have 2 But in this case its radius, which is equal to 2 m0c 2n~ Ω Etot = Bc = − ∓ = const . (4.202) v2 y˙ y˙ (0) 0 0 1 2 r = = , (4.197) e r − c 2Ω + ω v2 2Ω + eH 1 (0) Because in this §4.7 we assumed that the electric component 2n Ω 2 m0c ~ r − c ∓ c of the acting electromagnetic field is absent, the field does not contribute into the total energy of the particle (as it is known, the ∗We set the axis y along the initial impulse of the particle, which is always possible. Then all formulas for coordinates will have zero initial velocity of the magnetic component of the field does not perform work to displace particle along x. electric charges). 4.7 Motion in a stationary magnetic field 193 194 Chapter 4 Motion of spin-particle

From the obtained formulas (4.201, 4.202) we see that the total greater than its relativistic impulse, if ~ and Ω are co-directed, and energy of the particle remains unchanged along the trajectory, it is less then the relativistic impulse otherwise. while its numerical value depends on mutual orientation of the In the case of opposite mutual orientation of the ~ and Ω the particle’s inner momentum ~ and the angular velocity of the space total impulse becomes zero (so it does the total energy), provided 2 rotation Ω. that the condition m0c = 2n~ Ω is true. i The latter statement requires some comments to be made. By For the mirror-world particle the quantity ptot is definition the scalar quantity n (the absolute value of spin in the 2 ~ units) is always positive, while ~ and Ω are numerical values of i m0c 2n~ Ω i i 2n~ Ω i ik ptot = − ∓ v = mv v , (4.205) components of the antisymmetric tensors h and Ωik, which take 2 − ∓ 2 v(0) v(0) opposite signs in right or left-handed reference frames. But because c2 1 c2 1 − c2 − c2 we are dealing with the product of the quantities, only their mutual r r orientation matters, which does not depend on our choice of a right so the particle moves more slowly if the ~ and Ω are co-directed, or left-handed reference frames. and it is faster otherwise. Components of the velocity of the charged spin-particle in the If ~ and Ω are co-directed, then the total energy of the our-world 2 magnetic field co-directed with the space non-holonomity field, particle Etot (4.201) is the sum of its relativistic energy E = mc and its “spin-energy” taking into account the conditions (4.191), in our world are 2n~ Ω Es = , (4.203) 2 x˙ =y ˙(0) sin (2Ω + ω) τ x˙ (0) cos (2Ω + ω) τ , (4.206) v(0) − 1 2 r − c y˙ =y ˙(0) cos (2Ω + ω) τ +x ˙ (0) sin (2Ω + ω) τ , (4.207) so the total energy becomes greater than E = mc2. while for the analogous particle located in the mirror world we have If ~ and Ω are oppositely directed, then Etot is the difference between the relativistic energy and the spin-energy. Such orienta- 2 x˙ =y ˙(0) sin ωτ x˙ (0) cos ωτ , (4.208) tion permits a specific case, where m0c = 2n~ Ω and therefore the − total energy becomes zero (this case will be discussed in the next y˙ =y ˙ cos ωτ +x ˙ sin ωτ . (4.209) §4.8). (0) (0) For charged spin-particles bearing negative masses, which in- Then components of the total impulse of the particle∗ in our habit the mirror world, the situation is different. The total energy world are 2 Etot (4.202) is negative and by its absolute value is greater than the 1 m0c 2n~ Ω p = ∓ y˙(0) sin (2Ω + ω) τ , (4.210) relativistic energy E = mc2, provided that and Ω are oppositely tot 2 ~ v(0) directed. − c2 1 − c2 So forth, for the total spatial observable impulse of the our-world r 2 particle we have 2 m0c 2n~ Ω ptot = ∓ y˙(0) cos (2Ω + ω) τ , (4.211) v2 m c2 2n Ω 2n Ω 2 (0) i 0 ~ i i ~ i c 1 2 ptot = ∓ v = mv v , (4.204) r − c v2 ∓ v2 2 (0) 2 (0) 2 c 1 2 c 1 2 3 m0c 2n~ Ω r − c r − c ptot = ∓ z˙(0) , (4.212) v2 so it is an algebraic sum of the particle’s relativistic observable 2 (0) c 1 2 impulse pi = mvi and of its spin-impulse that the particle gains r − c from the space non-holonomity field. The particle’s total impulse is ∗The initial impulse of the particle within x y plane is directed along y. 4.7 Motion in a stationary magnetic field 195 196 Chapter 4 Motion of spin-particle

where ω is as (4.188). In the mirror world we have ω x¨(0) z = x˙ (0) cos ωτ + sin ωτ +z ˙(0)τ 2 ω2 ω − 1 m0c 2n~ Ω (4.219) p = − ∓ y˙(0) sin ωτ , (4.213) tot 2 ω ω v e x¨(0)τ +ez(0) x˙ (0) , c2 1 (0) e − ω2 e − ω2 − c2 r which are different from the respective solutions for a charged spin- 2 2 m0c 2n~ Ω free particle by the fact thate the frequency ωe here depends on the ptot = − ∓ y˙(0) cos ωτ , (4.214) v2 spin and its mutual orientation with the non-holonomity field c2 1 (0) − c2 e r 2 2 v2 3 m0c 2n~ Ω e2H2 1 (0) ptot = − ∓ z˙(0) , (4.215) c2 2 2 2 v 2 − v(0) ω = 4Ω + ω = u4Ω + . (4.220) c2 1 u 2 c2 u m c2 2n~ Ω r − p u 0 c u ∓ where ω is as of (4.190). Noteworthy, though the magnetic strength e t does not appear in the total energy Etot, it appears in that for the Subsequently, an equation of the trajectory of the charged spin- total impulse, being a term of the formula for ω (4.190). particle is similar to that of the spin-free one. In a particular case, namely — under certain initial conditions, the trajectory equation B Magnetic field is orthogonal to non-holonomity field is that of a sphere 1 x2 + y2 + z2 = x˙ 2 , (4.221) Now we are going to approach motion of the mass-bearing charged ω2 (0) spin-particle in a magnetic field, which is orthogonal to the space which radius, in contrast to the radius of the trajectory of the spin- non-holonomity field. So, the magnetic field is homogeneous and free particle, depends on the particle’se and its orientation in respect stationary. The non-holonomity field is directed along z and it is of the non-holonomity field weak, so the magnetic one is directed along y. Then vector chr.inv.- equations of motion will be similar to those for a charged spin-free 1 r = x˙ (0) . (4.222) particle under the above conditions in our world (3.338), namely 2 2 2 2 v(0) eH eH e H 1 v − c2 x¨ + 2Ωy ˙ = z˙ , y¨ 2Ωx ˙ = 0 , z¨ = x˙ . (4.216) u4Ω2 + cB − − cB u 2 u m c2 2n~ Ω The difference from (3.338) is that here the denominator of the u 0 c u ∓ right part instead of the relativistic mass contains the integration t constant from the live forces integral, which accounts for interaction Let us look an analogous particle which, being located in the between the particle’s spin and the non-holonomity field. After mirror world, moves in a weak field of the space non-holonomity, integration the equations solve as follows directed along y orthogonal to the magnetic field. For the particle the vector chr.inv.-equations of motion are x˙ (0) x¨(0) x¨(0) x = sin ωτ 2 cos ωτ + x(0) + 2 , (4.217) eH eH ω − ω ω x¨ = z˙ , y¨ = 0 , z¨ = x˙ , (4.223) 2Ω x¨ cB − cB y = x˙ ecos ωτ + (0)esin ωτ +y ˙ τ + − ωe2 (0) e ω e(0) so they are different from the equations for the our-world particle (4.218) 2Ω 2Ω (4.216) by the absencee of the terms which containe the angular +e x¨(0)τ + ye(0) + x˙ (0) , e ω2 e ω2 velocity of the space rotation Ω. As a result their solutions can

e e 4.7 Motion in a stationary magnetic field 197 198 Chapter 4 Motion of spin-particle

2 be obtained from the solutions for our world (4.217–4.219), if we 2 m0c 2n~ Ω 2Ω assume ω = ω. Subsequently, an equation of the trajectory of the ptot = ∓ x˙ (0) sin ωτ , (4.232) v2 ω charged spin-particle located in the mirror world is c2 1 (0) − c2 e r e m c2 2n~ Ω 2 e 2 2 2 1 2 0 c 3 m0c 2n~ Ω ω x + y + z = x˙ (0) , r = − ∓ x˙ (0) . (4.224) ptot = ∓ x˙ (0) sin ωτ , (4.233) ω2 v2 v2 ω (0) 2 (0) eH 1 2 c 1 2 r − c r − c e e The total energy of the particle Etot in this case, where the and for the analogous particle located in the mirror world magnetic field is orthogonal to the space non-holonomity field, 2 is the same as it was for the case of parallel orientation of the 1 m0c 2n~ Ω ptot = − ∓ x˙ (0) cos ωτ , (4.234) fields. But the formulas for components of the total impulse (4.201, v2 2 (0) 4.205) are different, because they include the particle’s velocity c 1 2 r − c e which depends on mutual orientation of the magnetic field and 2 the non-holonomity field. In the particular case, where the fields 2 m0c 2n~ Ω ptot = − ∓ y˙(0) = 0 , (4.235) are orthogonal to each other, components of the particle’s velocity v2 2 (0) (obtained by derivation of the formulas for 4.217–4.219) in our c 1 2 r − c world are 2 x¨(0) 3 m0c 2n~ Ω x˙ =x ˙ (0) cos ωτ + sin ωτ , (4.225) ptot = − ∓ x˙ (0) sin ωτ . (4.236) ω v2 c2 1 (0) 2Ω 2Ω 2Ω − c2 y˙ = x˙ (0) sin ωτ x¨(0)ecos ωτ +y ˙(0)e+ x¨(0) , (4.226) r e ω − ω2 e ω2 As it easy to see, the obtained solutions can be transformed into ω ω ω z˙ = x¨ cos ωτ x˙ sin ωτ +z ˙ x¨ , (4.227) respective ones from electrodynamics (§3.12) by assuming ~ 0. 2 (0) e (0) e (0) 2 (0) → eω −eω − ωe while in the mirror worlde we obtaine §4.8 The law of quantization of masses of elementary particles e e e x¨(0) x˙ =x ˙ (0) cos ωτ + sin ωτ , (4.228) As we have obtained it before, scalar chr.inv.-equations of motion ω of a charged spin-particle in an electromagnetic field, located in our

y˙ =y ˙(0) , (4.229) world and in the mirror world, are

1 1 d η e i d η e i z˙ = x¨(0) cos ωτ x˙ (0) sin ωτ +z ˙(0) x¨(0) . (4.230) m + = E v , m + = E v . (4.237) ω − − ω dτ c2 −c2 i −dτ c2 −c2 i Now we assume thate the initial acceleration of the particle and The equations can be easily integrated to produce the live forces the integration constants are zeroes. We also set the axis x along the integrals η η initial impulse of the particle. In the frames of this consideration m + = B, m + = B, (4.238) we obtain components of the total impulse for the particle located c2 − c2 in our world where B is integration constant in our world and B is that in the 2 e 1 m0c 2n~ Ω mirror world. The constants depend only on the initial conditions. ptot = ∓ x˙ (0) cos ωτ , (4.231) v2 Hence it is possible to choose them as to make the integration 2 (0) e c 1 2 constants zeroes. r − c e 4.8 The law of quantization of masses 199 200 Chapter 4 Motion of spin-particle

mn We find out under what initial conditions the integration con- obtain numerical values of the quantity∗ η0 = n~ Amn as follows. stants become zeroes. For charged spin-particles, located in our We formulate the tensor of angular velocities of the space rotation i 1 imn world and in the mirror world (4.238), we obtain, respectively Amn with the pseudovector Ω∗ = 2 ε Amn η η i 1 ipq 1 p q p q m + = 0 , m + = 0 , (4.239) Ω∗ ε = ε ε A = (δ δ δ δ ) A = A , (4.242) c2 − c2 imn 2 imn pq 2 m n − n m pq mn i while the right parts of the vector chr.inv.-equations of motion so we have Amn = εimnΩ∗ . Then because (4.150, 4.152), which contain the Lorentz chr.inv.-force, also become zeroes. In other word, with the integration constants in the scalar 1 mn εimn~ = ~ i (4.243) chr.inv.-equations equal to zero the acting electromagnetic field 2 ∗ does not a work to displace the particles. is the Planck pseudovector, the quantity η = n mnε Ω i is Having relativistic square root cancelled in (4.239), which is 0 ~ imn ∗ always possible for any particle bearing non-zero rest-masses, we i η0 = 2n~ i Ω∗ , (4.244) can present these formulas in a notation that does not depend on ∗ the particle’s velocity. Then for mass-bearing particles located in so it is the double scalar product of the Planck three-dimensional our world we have pseudovector and the three-dimensional pseudovector of angular 2 mn m0c = n~ Amn , (4.240) velocities of the space rotation, multiplied by the particle’s spin − i quantum number. If the ~ i and Ω∗ are co-directed, then the cosine while for mirror-world mass-bearing particles we have is positive, hence ∗ 2 mn m0c = n Amn . (4.241) i ~ ~ η0 = 2n~ i Ω∗ = 2n~ Ω cos ~ ; Ω~ > 0 , (4.245) ∗ We will refer to the formulas (4.240, 4.241) as the law of quan- while if they are oppositely directed, then tization of masses of elementary particles: i ~ η0 = 2n~ i Ω∗ = 2n~ Ω cos ~ ; Ω~ < 0 . (4.246) The rest-mass of any spin-particle is proportional to the energy ∗ of interaction between its spin and the field of the space non- holonomity, taken with the opposite sign. Therefore for any mass-bearing elementary particle, located in our world, the integration constant from the live forces integral Or, in other word: i becomes zero, provided that the pseudovectors ~ i and Ω∗ are ∗ The rest-energy of any mass-bearing spin-particle equals the oppositely directed. For any mass-bearing elementary particle, lo- energy of interaction between its spin and the field of the space cated in the mirror world, the constant becomes zero if the pseudo- i non-holonomity, taken with the opposite sign. vectors ~ i and Ω∗ are co-directed. ∗ Because in the mirror world the energy of any particle is negative, This implies that if the energy of interaction of a mass-bearing “plus” in the right part of (4.241) stands for the energy of interaction elementary particle with the space non-holonomity field becomes 2 in the mirror world taken with the opposite sign. The same is true equal to its rest-energy E = m0c , then the impulse of the particle for “minus” in (4.240) for our world. reveals itself neither in our world nor in the mirror world. It is evidently, these quantum formulas are not applicable to We assume that the axis z is co-directed with the pseudovector i non-spin particles. of angular velocities of the space rotation Ω∗ . Then out of all three

Let us make some quantitative estimates, which are derived ∗This quantity characterize the energy of interaction between the particle’s spin from the obtained law. Considering an elementary particle, we will and the space non-holonomity field — the “spin-energy”, in other word. 4.8 The law of quantization of masses 201 202 Chapter 4 Motion of spin-particle

i components of the Ω∗ the only non-zero one is The results, proceeding from the calculations for elementary particles of known kinds, are given in Table 1. 312 3 1 3mn 1 312 321 312 e These results show that in the scales of elementary particles the Ω∗ = ε Amn = ε A12 +ε A21 = ε A12 = A12 . (4.247) 2 2 √h observer’s space is always non-holonomic. So forth for instance, in observation of an electron r = 2.8 10 13 cm the linear velocity To simplify the algebra we assume that the three-dimensional e × − of rotation of the observer’s space is v = Ωr = 2200 km/sec. Because coordinate metric g is Euclidean one, while the space rotates at a ∗ ik other elementary particles are even smaller, this linear velocity constant angular velocity Ω. Then components of the linear velocity seems to be the upper limit . of the space rotation are v = Ωx, v = Ωy, and A = Ω. Hence † 1 2 − 12 − 312 3 e A12 Ω Elementary particles Rest-mass Spin Ω, 1/sec Ω∗ = A12 = = . (4.248) √h √h −√h LEPTONS + 20 The square root of the determinant of the metric chr.inv.-tensor, electron e−, positron e 1 1/2 7.782×10 electron neutrino νe and as defined from (4.180) is 4 17 electron anti-neutrino ν˜e < 4×10− 1/2 < 3×10 Ω2 (x2 + y2) μ-meson neutrino νμ and √ μ-meson anti-neutrino ν˜ < 8 1/2 < 6 1021 h = det hik = 1 + 2 . (4.249) μ × k k c + 23 r μ−-meson, μ -meson 206.766 1/2 1.609 10 p × Because we are dealing with very small coordinate values in the BARIONS scales of elementary particles, we can assume √h 1 and, according nuclons 3 ≈ 24 to (4.248) also Ω∗ = Ω = const. Then the law of quantization of proton p, anti-proton p˜ 1836.09 1/2 1.429×10 − 24 masses of elementary particles (4.240), taken in our world and in neutron n, anti-neutron n˜ 1838.63 1/2 1.431×10 hyperons the mirror world, respectively, becomes 0 0 24 Λ -hyperon, anti-Λ -hyperon 2182.75 1/2 1.699×10 + + 24 2n~ Ω 2n~ Ω Σ -hyperon, anti-Σ -hyperon 2327.6 1/2 1.811×10 m0 = , m0 = . (4.250) 24 2 2 Σ−-hyperon, anti-Σ−-hyperon 2342.6 1/2 1.823×10 c − c 0 0 24 Σ -hyperon, anti-Σ -hyperon 2333.4 1/2 1.816×10 24 Hence for any mass-bearing elementary particle, located in our Ξ−-hyperon, anti-Ξ−-hyperon 2584.7 1/2 2.011×10 0 0 24 world, the following relationship between its rest-mass m0 and the Ξ -hyperon, anti-Ξ -hyperon 2572 1/2 2.00×10 23 angular velocity of the space rotation Ω is true Ω−-hyperon, anti-Ω−-hyperon 3278 3/2 8.50×10

2 m0c Table 1. Frequencies of rotation of the observer’s reference space, Ω = . (4.251) which correspond to mass-bearing elementary particles 2n~ This implies that the rest-mass (the true mass) of an observable object, under regular conditions not dependent from properties of ∗This value of v equals the velocity of an electron in the Bohr 1st orbit, though when calculating the velocity of the space rotation (see Table 1) we considered a free the observer’s reference space, in the scales of elementary particles electron, i. e. the one not related to an atomic nucleus and quantization of orbits in becomes strictly dependent from such, in particular, from the an- an atom of hydrogen. The reason is that the “genetic” quantum non-holonomity of gular velocity of the space rotation. the space seems not only to define rest-masses of elementary particles, but to be Hence, proceeding from the quantization law, we can calculate the reason of rotation of electrons in atoms. †It is interesting, the angular velocities of the space rotation in barions (see frequencies of rotation of the observer’s space, corresponding to Table 1) up within the order of the magnitude match the frequency 1023 1/sec rest-masses of elementary particles, located in our world. which characterizes elementary particles as oscillators [22]. ∼ 4.9 Compton’s wavelength 203 204 Chapter 4 Motion of spin-particle

So, what have we got? Generally observer compares results of his λΩ = c, we have c ~ measurements with special standards located in his reference body. λ = = 2n . (4.252) But the body and himself are not related to the observed object and Ω m0c do not affect it during observations. Hence in macro-world there In other word, if we observe a mass-bearing particle with spin is no dependence of the true properties of observed bodies (rest- 1 n = 2 the length of the space non-holonomity wave equals Com- mass, rest-energy, ect.) from properties of the reference body and pton’s wavelength of this particle λ–c = ~ . the reference space — these are properties of objects non-related to m0c What does that mean? Compton effect, named after A. Compton each other. who discovered it in 1922, is “diffraction” of a photon on a free In other word, though observed images are distorted by influ- electron, which results in decrease of its own frequency ence from physical properties of the observer’s reference frame, the observer himself and his reference body in macro-world do not h e Δλ = λ2 λ1 = (1 cos ϑ) = λc (1 cos ϑ) , (4.253) affect measured objects anyhow. − mec − − But the world of elementary particles presents a big difference. where λ and λ are the photon’s wavelengths before and after the In this §4.8 we have seen that once we reach the scale of elementary 1 2 encounter, ϑ is the angle of “diffraction”. The multiplier λe, specific particles, where spin, a quantum property of the particles, signif- c to the electron, at first was called the Compton wavelength of the icantly affects their motion, physical properties of the reference electron. Later it was found out that other elementary particles body (the reference space) and those of the particles become tightly during “diffraction” of photons reveal as well their specific wave- linked to each other, so the reference body affects the observed h lengths λc = , or, respectively, λ–c = ~ . That is, elementary particles. In other word, the observer does not just compare prop- m0c m0c erties of the observed particles to those of his references any longer, particles of every kind (electrons, protons, neutrons etc.) have their but instead directly affects the observed particles. The observer own Compton wavelengths. The physical sense behind the quantity shapes their properties in a tight quantum relationship with prop- had been explained later. It was obtained, within an area smaller erties of the references he possesses. than λ–c, any elementary particle is no longer a point object and its This means follows. When looking at effects in the world of interaction with other particles (and with the observer) is described elementary particles, there is no border between the observer (his by Quantum Mechanics. Hence the λ–c-sized area is sometimes reference body and the reference space) and the observed particle. interpreted as the “size” of the elementary particle. Hence we get an opportunity to define a relationship between the As for the results we have obtained in the previous §4.8, these space non-holonomity field, linked to the observer, and rest-masses can be interpreted as follows. In observation of a mass-bearing of the observed particles — objects of his observations, which in elementary particle the observer’s space rotates so fast that the macro-world are not related to the reference body. So, the obtained angular velocity of this rotation makes a specific wavelength equal law of quantization of masses is only true for elementary particles. to the Compton wavelength of the observed particle, so to the “size” inside which the particle is no longer a point object. In other word, it is the angular velocity of the space rotation (the wavelength in the §4.9 Compton’s wavelength space non-holonomity field), which defines the Compton observable wavelength (the specific “size”) of the particle. So, we have obtained that in observation of an elementary particle with rest-mass m0 the rotation frequency of the observer’s space 2 §4.10 Massless spin-particles is Ω = m0c (4.251). We are going to find the wavelength which 2n~ corresponds to that frequency. Assuming that this wave, i. e. the Because massless particles do not bear electric charge, their scalar wave of the space non-holonomity, propagates at the light velocity chr.inv.-equations of motion in our world and in the mirror world 4.10 Massless spin-particles 205 206 Chapter 4 Motion of spin-particle are, respectively, velocity (a non-accompanying reference frame). As a result we obtain d η d η 1 ∂˜b ∂˜b m + 2 = 0 , m + 2 = 0 . (4.254) ˜αβ ˜αμ˜βμ ν μ dτ c −dτ c A = ch h a˜μν , a˜μν = μ ν , (4.257) 2 ∂x − ∂x Their integration always gives the constant equal zero, hence where ˜bα is the four-dimensional velocity of a light-like reference we always obtain the formulas as of (4.239). Hence for massless frame in respect of the observer and particles in our world and in the mirror world, respectively h˜αμ = gαμ + ˜bα˜bμ (4.258) mc2 = η , mc2 = η . (4.255) − − is the four-dimensional generalization of the metric chr.inv.-tensor On the other hand, the term “rest-mass” is not applicable to for the light-like space and a reference frame located in it. massless particles — they are always on the move. Their relativistic The space inhabited by massless particles is a space-time area, masses are defined from energy equivalent E = mc2, measured in which corresponds with the four-dimensional light-like (isotropic) electron-volts. Subsequently, massless particles have no rest spin- cone set by the equation g dxαdxβ = 0. This cone exists in any energy η = n mnA . αβ 0 ~ mn point of the four-dimensional pseudo-Riemannian space with the Nevertheless, the Planck tensor found in spin-energy η enables alternating signature (+ ). quantization of relativistic masses of massless particles and angular −−− The four-dimensional velocity vector of the light-like reference velocities of the space rotation. Hence to obtain angular velocities frame of massless particles is of the space rotation for massless particles we need an expanded formula of their relativistic spin-energy η, which would not contain dxα dxα ˜bα = = , ˜b ˜bα = 0 , (4.259) the relativistic square root. dσ cdτ α Quantum Mechanics speaks of “spirality” of massless particles so its chr.inv.-projections in the reference frame of a regular “sub- — the projection of spin on the direction of impulse. The reason for light” observer are introducing such term is the fact that massless particles can not be at rest in respect of any regular observer, as they always travel ˜b 1 dxi 1 0 = 1, ˜bi = = ci, (4.260) at the light velocity in respect of such. Hence we can assume that √g00 ± c dτ c spin of any massless particle is tangential to its light-like trajectory (either co-directed or oppositely directed to it). while the other components of this isotropic vector (4.259) are Keeping in mind that the spin quantum number n of any mass- 0 1 1 i 1 ˜b = vic 1 , ˜bi = (ci vi) , (4.261) less particle is 1, we assume for them that √g c2 ± − c ± 00 mn ˜ η = ~ Amn , (4.256) where ci is the chr.inv.-vector of the light velocity. Let us consider properties of the light-like space of massless where A˜mn is the angular velocities chr.inv.-tensor of their space particles in details. The isotropy condition of the particles’ four- rotation (the light-like space). α Hence to obtain the relativistic spin-energy of a massless particle dimensional velocity bαb = 0 in chr.inv.-form becomes (4.256) we need to find components of the angular velocities i k 2 hik c c = c = const, (4.262) chr.inv.-tensor of the light-like space rotation. We are going to build the tensor similar to the four-dimensional tensor of the space where hik is the metric chr.inv.-tensor of a regular “sub-light” ob- rotation Aαβ (4.11), which describes rotation of the space of a frame server’s reference space. Components of the four-dimensional light- of reference, which travels in respect of the observer at an arbitrary like metric tensor h˜αβ (4.258), which three-dimensional components 4.10 Massless spin-particles 207 208 Chapter 4 Motion of spin-particle make up the light-like space’s metric chr.inv.-tensor h˜ik, are ∂c 1 vkhim vihkm ∗ m + cm cihkn ckhin − − ∂t 2c2 − × k k 1 k n (4.266) vkv 2vkc + 2 vkvnc c ∂c ∂c ∂v ∂v h˜00 = ± c , m n m n 2 n m n m . c2 1 w × ∂x − ∂x ± ∂x − ∂x c2 − (4.263) 1 ∂vm ∂vn 1 i i 1 k i The quantity + (Fnvm Fmvn), by definition, v c + vkc c 2 ∂xn ∂xm 2c2 0i c2 ik ik 1 i k − − h˜ = ± , h˜ = h + c c , is the chr.inv.-tensor of angular velocities of the observer’s space w c2 c 1 rotation A — the non-holonomity tensor of the non-isotropic − c2 mn space∗. where “plus” stands for the light-like space with the direct flow of ∂c ∂c The quantity 1 m n + 1 (F c F c ) by its struc- time (our world) and “minus” stands for the reverse-time (mirror) 2 ∂xn − ∂xm 2c2 n m − m n world. ture is similar to the tensor Amn, but instead of the linear velocity Now we have to deduce components of the rotor of the four- of the non-isotropic space rotation vi it has components of the n dimensional velocity vector of massless particles, found in the for- covariant chr.inv.-vector of the light velocity cm = hmnc . The vec- mula (4.257). After some algebra we obtain tor cm is physical observable quantity, because it was obtained by lowering indices in the chr.inv.-vector cn using the metric chr.inv.- 1 w ∗∂ci tensor h . We denote that tensor as A˘ , where the inward curved a˜ = 0 , a˜ = 1 F , mn mn 00 0i 2c2 − c2 ± i − ∂t cap means the quantity belongs to the isotropic space with the † (4.264) direct flow of time — the “upper” part of the light cone, which in a 1 ∂c ∂c 1 ∂v ∂v a˜ = i k i k . twisted space-time gets “round” shape. Then ik 2c ∂xk − ∂xi ± 2c ∂xk − ∂xi 1 ∂c ∂c 1 A˘ = m n + (F c F c ) . (4.267) Generally, to define the spin-energy of a massless particle mn 2 ∂xn − ∂xm 2c2 n m − m n (4.256) we need covariant spatial components of the tensor of its space rotation, namely — lower-indices components A˜ik. To deduce In a particular case, where gravitational potential is negligible them we take the formula for contravariant components A˜ik and (w 0) the tensor becomes ≈ lower their indices, as for any chr.inv.-quantity using the metric 1 ∂c ∂c chr.inv.-tensor of the observer’s reference space. A˘ = m n , (4.268) mn 2 ∂xn − ∂xm Substituting into so it is the chr.inv.-rotor of the light velocity. Therefore we will ˜ik ˜i0˜k0 ˜i0˜km ˜im˜k0 ˜im˜kn A = c h h a˜00 +h h a˜0m +h h a˜m0 +h h a˜mn (4.265) refer to A˘mn as the isotropic space rotor. the obtained components h˜αβ and a˜ , we arrive to ∗We will refer as a non-isotropic space to an area in the four-dimensional space- αβ time, where particles with non-zero rest-masses exist. This is the area of world- trajectories along which ds = 0. Subsequently, if the interval ds is real, then the ik im kn 1 ∂cm ∂cn 1 particles travel at sub-light velocities6 (regular particles); if it is imaginary, then the A˜ = h h + (Fncm Fmcn) 2 ∂xn − ∂xm 2c2 − ± particles travel at super-light velocities (tachyons). So, the space of both sub-light particles and super-light tachyons is non-isotropic by definition. im kn 1 ∂vm ∂vn 1 We will refer as the isotropic space to an area of the four-dimensional space- h h + (Fnvm Fmvn) + † ± 2 ∂xn − ∂xm 2c2 − time, inhabited by massless (light-like) particles. This area can be also called the light membrane. From geometric viewpoint the light membrane is the surface of the 1 n k im i km ∗∂cm isotropic cone, i. e. the set of its four-dimensional elements (world-lines of the light + vnc 1 c h c h c2 ± − ∂t − propagation). 4.10 Massless spin-particles 209 210 Chapter 4 Motion of spin-particle

The following example gives geometric illustration of the iso- cone bear non-zero rest-masses, they are “heavier” that massless tropic space rotor. As it is known, the necessary and sufficient particles on the light membrane. Hence the inner “content” of the condition of the equality Amn = 0 (the space holonomity condition) light cone is too inertial media. g0i is equality to zero of all components vi = c , i. e. the absence of Now we return to the formula for the relativistic spin-energy of − √g00 a massless particle η = mnA˜ (4.256). By lowering indices in the ˘ ~ mn the space rotation. The tensor Amn is defined only in the isotropic non-holonomity tensor of the isotropic space A˜ik (4.266), we obtain space, inhabited by massless particles. Outside the isotropic space it is senseless, because the “interior” of the light cone is inhabited 1 ∂ (cm vm) ∂ (ck vk) A˜ = A +A˘ + cm c ± ± by sub-light particles, while tachyons inhabit its “exterior”. ik ± ik ik 2c2 i ∂xk − ∂xm − Our subject are massless spin-particles (photons). From (4.268) ∂ (cm vm) ∂ (ci vi) ∗∂ck ∗∂ci it is seen that non-holonomity of the isotropic space is linked to ck ± ± + vi vk + (4.270) − ∂xi − ∂xm ∂t − ∂t rotor nature of the linear velocity of massless particles cm. Hence any photon is a spatial rotor of the isotropic space, while the pho- 1 n ∗∂ci ∗∂ck + vnv 1 ck ci . ton’s spin results from interaction between its inner rotor field and c2 ± ∂t − ∂t the external tensor field A˘ . mn Having A˜ contracted with the Planck tensor ik, we have To make the explanations even more illustrative, we depict areas ik ~ of existence of different kinds of particles. The light cone exists in ik 1 n ∗∂ci ∗∂ck α β η = η0 + n~ A˘ik + vnv 1 ck ci + every point of the space. The light cone equation gαβdx dx = 0 in c2 ± ∂t − ∂t chr.inv.-notation is ∗∂ck ∗∂ci ik 1 ik m ∂ (cm vm) + vi vk n~ + n~ c ci ± (4.271) c2τ 2 h xixk = 0 , h xixk = σ2. (4.269) ∂t − ∂t 2c2 ∂xk − − ik ik ∂ (c v ) ∂ (c v ) ∂ (c v ) On Minkowski’s diagram the light cone “interior” is filled with k ± k c m ± m i ± i , − ∂xm − k ∂xi − ∂xm the non-isotropic space, where sub-light particles exist. Outside there is also an area of the non-isotropic space, inhabited by super- where “plus” stands for our world and “minus” — for the mirror light particles (tachyons). The specific space of massless particles is world. a space-time membrane between these two non-isotropic areas. The 2 2 The quantity η0 = η 1 v /c for massless particles is zero, be- picture is mirror-symmetric: in the upper part of the cone there cause they travel at the light− velocity. Hence keeping in mind that is the sub-light space with the direct flow of time (our world), mn p η0 = n~ Amn, we obtain an additional condition imposed on the separated with the observer’s spatial section from the lower part non-holonomity tensor of the isotropic space A˜ik: in any point of — the sub-light space with the reverse flow of time (the mirror the trajectory of any massless particle the next condition world). In other word, the upper part is inhabited by real particles mn with positive masses and energies, while the lower part is inhabited ~ Amn = 2~ (A12 + A23 + A31) = 0 , (4.272) by their mirror “counterparts”, whose masses and energies are 1 2 3 negative (from our viewpoint). must be true. Or, in the other notation, Ω + Ω + Ω = 0. Therefore, rotation of the sub-light non-isotropic space “inside” Therefore, in an area, where the observer “sees” the massless the cone involves the surrounding light membrane (the isotropic particle, the angular velocity of rotation of the observer’s non- space). As a result, the light cone begins rotation described by the isotropic space equals zero. Other terms consisting the particle’s relativistic spin-energy (4.271) are due to possible non-stationarity tensor A˘mn — the isotropic space rotor. Of course we can assume of the light velocity ∗∂ci and other dependencies which include a reverse order of events, where rotation of the light cone involves ∂t “the content” of its inner part. But because particles “inside” the squares of the light velocity. 4.10 Massless spin-particles 211 212 Chapter 4 Motion of spin-particle

We analyze the obtained formula (4.271) to make two simplifi- Any photon’s spin quantum number is 1. Besides, its energy cation assumptions: E = ~ ω is positive in our world. Hence taking into account the live 1. Gravitational potential is negligible (w 0); forces integral (4.255), for observable our-world photons we have ≈ 2. The three-dimensional chr.inv.-velocity of light is stationary. i 1 m ik n ~ω = ~ i Ω˘ ∗ + cic ~ εkmnΩ˘ ∗ . (4.277) ∗ 2 In this case the quantities Aik and A˘ik, so the observer’s space c non-holonomity tensor and the isotropic space rotor, become We assume that the rotation pseudovector of the isotropic space i Ω∗ is directed along the axis z, while the light velocity is directed 1 ∂vk ∂vi 1 ∂ck ∂ci Aik = , A˘ik = , (4.273) along y. Then the relationship (4.277) obtained for photons becomes 2 ∂xi − ∂xk 2 ∂xi − ∂xk ˘ ~ ω = 2~Ω, or, after having the Planck constant cancelled, and the massless particle’s relativistic spin-energy (4.271) becomes ω 2πν Ω˘ = = = πν , (4.278) 2 2 ik 1 m ik η = n ~ A˘ik + cic ~ A˘km . (4.274) c2 so the frequency Ω˘ of the isotropic space rotation in the massless particle up within a constant coincides the particle’s own frequency Therefore this quantity η, describing action of the massless ν. Thanks to this formula, which results from the quantization law particle’s spin, is defined (aside for the spin) only by the isotropic for relativistic masses of massless particles, we can estimate the space rotor and in no way depends on the observer’s space non- isotropic space’s angular velocities, which correspond to photons of holonomity (the rotation). different energy levels. Table 2 gives the results. To make further deductions simpler, we transform η (4.274) as i 1 ikm ˘ 1 follows. Similar to the space rotation pseudovector Ω∗ = 2 ε Akm Kind of photons Frequency Ω, s− we introduce a pseudovector Radiowaves 103 – 1011 11 15 i 1 ikm Infra-red rays 10 – 1.2×10 Ω˘ ∗ = ε A˘km , (4.275) 15 15 2 Visible light 1.2×10 – 2.4×10 15 17 Ultraviolet rays 2.4×10 – 10 which can be formally interpreted as the pseudovector of rotation X-rays 1017 – 1019 angular velocity of the isotropic space. Gamma rays 1019 – 1023 and above n Subsequently, A˘km = εkmnΩ˘ ∗ . Then the formula for η (4.274) can be presented as follows Table 2. Rotation frequencies of the isotropic space, which correspond to photons i 1 m ik n η = n ~ i Ω˘ ∗ + cic ~ εkmnΩ˘ ∗ . (4.276) From Table 2 we see that angular velocities of the isotropic ∗ c2 space rotation in photons, taken in gamma range, are in the scale This means that the inner rotor (spin) of a massless particle of frequencies of the regular space rotation in electrons and other only reveals itself in interaction with the isotropic space rotor. The elementary particles (see Table 1). i result of the interaction is the scalar product ~ i Ω˘ ∗ , to which the ∗ massless particle’s spin is attributed. Hence massless particles are §4.11 Conclusions elementary light-like rotors of the isotropic space itself. Let us estimate rotations of the isotropic space for massless Here is what we have obtained in this Chapter. particles bearing different energies. At present we know for sure Spin of any particle is characterized by the four-dimensional that among massless particles are photons. antisymmetric tensor of the 2nd rank called as the Planck tensor. 4.11 Conclusions 213

Its diagonal and space-time components are zeroes, while non- diagonal spatial components are ~ depending on the spatial direction of the spin and our choice± of a right or left-handed frame of reference. Chapter 5 The spin (the inner vortical field of the particle) interacts an external field of the space non-holonomity. As a result, the particle PHYSICAL VACUUM gains an additional impulse, which deviates the moving particle AND THE MIRROR UNIVERSE from geodesic line. This interaction energy is found from the scalar chr.inv.-equation of motion of the particle (the live forces theorem), so the equation must be taken into account when solving the vector chr.inv.-equations of motion. §5.1 Introduction Particular solution of the scalar chr.inv.-equation is the law of quantization of masses of elementary spin-particles, which unam- According to the recent data the average density of matter in our biguously links rest-masses of mass-bearing elementary particles 30 gram 3 Universe is 5–10×10− /cm . That of substance concentrated in with angular velocities of the observer’s space rotation, as well 31 gram 3 galaxies is even lower at 3×10− /cm , which seems to be due as between relativistic masses of photons and angular velocities to so-called “hidden masses”≈ in the galaxies. Besides, astronomical of rotation of their inner light-like space. Because an area, where observations show that most part of the cosmic mass is accumulated light-like particles exist, is the area of four-dimensional isotropic in compact objects, e. g. in stars, which total volume is incomparable trajectories, the terms “isotropic space” and “light-like space” can to that of the whole Universe (so called “island” distribution of be used as synonyms. substance). We can therefore assume that our Universe is predom- Please note that we have obtained the result using only geo- inantly empty. metric methods of the General Theory of Relativity, not Quantum For a long time the words “emptiness” and “vacuum” have been Mechanics’ methods. In future, this result may possibly become a considered synonyms. But since 1920’s geometric methods of the “bridge” between these two theories. General Theory of Relativity have showed that those are different states of matter. Distribution of matter in the Universe is characterized by the ♦ energy-momentum tensor, which is linked to the geometric structure of the space-time (the fundamental metric tensor) by the equations of gravitational field. In Einstein’s theory of gravitation, which is an application of mathematical methods of Riemannian geometry, the equations referred to as Einstein’s equations are 1 R g R = κ T + λg . (5.1) αβ − 2 αβ − αβ αβ These equations, aside for the energy-momentum tensor and the fundamental metric tensor, include other quantities, namely: ...β Rασ = Rαβσ is Ricci’s tensor∗ — the result of contraction of • ∙ Riemann-Christoffel’s curvature tensor Rαβγδ by two indices;

∗Gregorio Ricci-Curbastro (1853–1925), an Italian mathematician who was the 5.1 Introduction 215 216 Chapter 5 Vacuum and the mirror Universe

αβ R = g Rαβ is the scalar curvature; tensor describing distribution of matter is genetically linked to both • 8πG 27 the metric tensor and the Ricci tensor, and hence to the Riemann- κ = = 1.862 10 [ cm/gram ] is Einstein’s gravitational c2 × − Christoffel curvature tensor. Equality of the Riemann-Christoffel • 8 constant, and G = 6.672×10− is Gauss’ gravitational constant 3 tensor to zero is the necessary and sufficient condition for the given [ cm /gram sec2 ]. Note that some researchers prefer to use not × space-time to be flat. The Riemann-Christoffel tensor is not zero for κ = 8πG [4], but κ = 8πG [5]. To understand the reason we c2 c4 curved spaces only. It reveals itself as an increment of an arbitrary have to look at chr.inv.-projections of the energy-momentum vector V α in its parallel transfer along a closed contour T00 tensor Tαβ: = ρ is the chr.inv.-scalar of the observable g00 μ 1 ...μ α βγ cT i ΔV = Rαβγ V Δσ , (5.2) mass density, 0 = J i is the chr.inv.-vector of the observable −2 ∙ √g00 βγ momentum density, and c2T ik = U ik is the chr.inv.-tensor of where Δσ is the area within this contour. As a result, the initial α α α the observable momentum flux density [4]. The scalar chr.inv.- vector V and the vector V + ΔV have different directions. From quantitative viewpoint the difference is described by a quantity projection of the Einstein equations is G00 = κT00 + λ. As it g00 − g00 K, referred to as the four-dimensional curvature of the pseudo- 2 is known, the Ricci tensor has dimension [ 1/cm ], hence the Riemannian space along the given parallel transfer (see [4] for κT00 8πGρ Einstein tensor Gαβ and the quantity g = 2 has the details) 00 c tan ϕ same one. Consequently, it is evidently that the dimension K = lim , (5.3) Δσ 0 Δσ of the energy-momentum tensor Tαβ is that of mass density → 3 8πG [ gram/cm ]. This implies that when we use in the right where tan ϕ is the tangent of the angle between the vector V α and c4 part of the Einstein equations, we actually use not the energy- the projection of the vector V α + ΔV α on the area constructed by 2 momentum tensor itself, but the quantity c Tαβ, which scalar the transfer contour. For instance, we consider a surface and a and vector chr.inv.-projections are the observable energy den- “geodesic” triangle on it, produced by crossing of three geodesic 2 c3T i lines. We transfer a vector, defined in any arbitrary point of that sity c T00 = ρc2 and the observable energy flux 0 = c2J i; g00 √g00 triangle, parallel to itself along the sides of the triangle. The sum- λ [ 1/cm2 ] is so called the cosmological term, which describes mary rotation angle ϕ after the vector returns to the initial point is • ϕ = Σ π (where Σ is the sum of the inner angles of the triangle). non-Newtonian forces of attraction or repulsion, depending on − the sign before λ (λ > 0 stands for the repulsion, λ < 0 stands We assume the surface curvature K equal in all its points, then for the attraction). The term is referred to as “cosmological” tan ϕ ϕ one, because it is assumed that forces described by λ grow up K = lim = = const, (5.4) Δσ 0 Δσ σ proportional to distance and therefore reveal themselves in a → full scale at “cosmological” distances comparable to size of the where σ is the triangle’s area and ϕ = Kσ is called spherical excess. Universe. Because the non-Newtonian gravitational fields (λ- If ϕ = 0, then the curvature is K = 0, so the surface is flat. In this fields) have never been observed, for our Universe in general case the sum of all inner angles of the geodesic triangle is π (a 56 the cosmological term is λ < 10− 1/cm2 (as of today’s mea- | | flat space). If Σ > π (the transferred vector is rotated towards the surement accuracy). circuit), then there is positive spherical excess, so the curvature From the Einstein equations (5.1) we see, the energy-momentum K > 0. An example of such space is the surface of a sphere: a triangle on the surface is convex. If Σ < π (the transferred vector teacher of Tullio Levi-Civita in Padua in 1890’s. The left part of the field equations 1 is rotated counter the circuit), the spherical excess is negative and (5.1) is often referred to as the Einstein tensor Gαβ = Rαβ 2 gαβ R, in notation G = κ T + λg . − the curvature is K < 0. αβ − αβ αβ 5.1 Introduction 217 218 Chapter 5 Vacuum and the mirror Universe

30 gram 3 Einstein postulated that gravitation is the space-time curvature. substance in our Universe is so small 5–10×10− /cm , that He understood the curvature as not equality to zero of the Riemann- we can assume it near vacuum. The Einstein equations say that Christoffel tensor Rαβγδ = 0 (the same is assumed in Riemannian the energy-momentum tensor is functionally dependent from the geometry). This concept fully6 includes Newtonian gravitational con- metric tensor and the Ricci tensor (i. e. from the curvature tensor, cept, so Einstein’s four-dimensional gravitation-curvature for a reg- contracted by two indices). At such small numerical values of den- ular physical observer can reveal itself as follows: sity we can assume the energy-momentum tensor proportional to (a) Newtonian gravitation; the metric tensor Tαβ gαβ and hence proportional to the Ricci tensor. Therefore aside∼ for the field equations in vacuum (5.5) we (b) Rotation of the three-dimensional space (the spatial section); can consider the equations as follows (c) Deformation of the three-dimensional space; R = kg , k = const, (5.6) (d) The three-dimensional curvature, so that non-zero first de- αβ αβ rivatives of Christoffel’s symbols. i. e. where the energy-momentum tensor is different from the met- According to Mach Principle, on which the Einstein theory of ric tensor only by a constant. This case, including the absence of gravitation rests, “. . . the property of inertia is fully determined by masses (i. e. in vacuum) and some conditions close to it and related interaction of matter” [23], so the space-time curvature is produced to our Universe, were studied in details by Petrov [24]. Spaces for by a matter which fills it. Proceeding from that and from the which the energy-momentum tensor is proportional to the metric Einstein equations (5.1) we can give mathematical definitions of tensor (and, hence, to the Ricci tensor) he called Einstein spaces. emptiness and vacuum: Spaces with Rαβ = kgαβ (namely — Einstein spaces) are homogeneous in every their point, have no mass fluxes, while the density Emptiness is the state of a given space-time, for which the Ricci of matter which fills them (including any substances) is everywhere tensor is Rαβ = 0, that implies the absence of any substance constant. In this case Tαβ = 0 and of non-Newtonian gravitational fields λ = 0. The αβ αβ field equations (5.1) in emptiness∗ are as simple as Rαβ = 0; R = g Rαβ = kgαβ g = 4k , (5.7) Vacuum is the state in which any substance is absent T = 0, αβ while the Einstein tensor takes the form but λ = 0 and hence Rαβ = 0. Emptiness is a particular case of vacuum6 in the absence6 of λ-fields. The field equations in 1 G = R g R = kg , (5.8) vacuum are αβ αβ − 2 αβ − αβ 1 Rαβ gαβR = λgαβ . (5.5) where kg is the analog of the energy-momentum tensor for that − 2 αβ matter which fills Einstein spaces. The Einstein equations are applicable to the most varied cases To find out what kinds of matter fill Einstein spaces, Petrov of distribution of matter, aside for the cases where the density studied the algebraic structure of the energy-momentum tensor. is close to that of substance in atomic nuclei. It is hard to give This is what he did: the tensor Tαβ was compared to the metric accurate mathematical description to all cases of distribution of tensor in an arbitrary point; for this point the difference Tαβ ξgαβ matter because such problem is so general one and it can not − is calculated, where ξ are so-called eigenvalues of the matrix Tαβ; be approached per se. On the other hand, the average density of the difference is equalized to zero to find the values of ξ, which make the equality true. This problem is also referred to as the matrix If we put down the Einstein equations for an empty space R 1 g R = 0 in ∗ αβ − 2 αβ eigenvalues problem . The set of the matrix eigenvalues allows the mixed form Rβ 1 gβ R = 0, then after contraction (Rα 1 gαR = 0) we obtain ∗ α − 2 α α − 2 α R 1 4R = 0. So the scalar curvature in emptiness is R = 0. Hence the field equations Generally, the problem should be solved in a given point, but the obtained result − 2 ∗ (the Einstein equations) in the empty space are Rαβ = 0. is applicable to any point of the space. 5.1 Introduction 219 220 Chapter 5 Vacuum and the mirror Universe to define the algebraic kind of this matrix. For a sign-constant four eigenvalues are the same, so three space vectors and the time metric this problem had been already solved, but Petrov proposed vector of the “ortho-reference” of the tensor Tαβ are equal to each a method to bring any matrix to canonical form for the indefinite other∗. The matter which corresponds to the energy-momentum (sign-alternating) metric, which allowed using it in the pseudotensor of such structure has a constant density μ = const, equal to Riemannian space, in particular, to study the algebraic structure of coinciding eigenvalues of the energy-momentum tensor μ = ξ (the 3 the energy-momentum tensor. This can be illustrated as follows. dimension of μ is the same that of Tαβ, and it is [ gram/cm ]). The Eigenvalues of elements of the matrix Tαβ are similar to basic energy-momentum tensor itself in this case is† vectors of the metric tensor matrix, so the eigenvalues define a kind of the “skeleton” of the tensor Tαβ (the skeleton of matter); Tαβ = μgαβ . (5.9) but even if we know what is the skeleton, we may not know exactly what are the muscles. Nevertheless, the structure of such skeleton The field equations under λ = 0 are (the length and mutual direction of the basic vectors) we can judge 1 on the properties of matter, such as homogeneity or isotropy, and R g R = κμg , (5.10) αβ 2 αβ αβ their relation to the space curvature. − − As a result, Petrov had obtained that Einstein spaces have three and, under the cosmological term λ = 0, are basic algebraic kinds of the energy-momentum tensor and a few 6 subtypes. According to his algebraic classification of the energy- 1 Rαβ gαβR = κμgαβ + λgαβ . (5.11) momentum tensor and the curvature tensor, all Einstein spaces are − 2 − sub-divided into three basic kinds so-called Petrov’s classification . ∗ Gliner called such state of matter μ-vacuum [26, 27, 28], because Einstein spaces of the kind I are best intuitively comprehensible, the state is related to vacuum-like states of substance (T g , because the field of gravitation there is produced by a massive αβ αβ R = kg ), but is not exactly vacuum (in vacuum T = 0∼). At island (the “island” distribution of substance), while the space itself αβ αβ αβ the same time Gliner showed that spaces filled with μ-vacuum may be empty or filled with vacuum. The curvature of such space is are Einstein spaces, so three basic kinds of μ-vacuum exist, which created by the island mass and by vacuum. At the infinite distance correspond to three basic algebraic kinds of the energy-momentum from the island mass, in the absence of vacuum, this space remains tensor (and the curvature tensor). In other words, an Einstein space flat. Devoid of any island masses but filled with vacuum, the space of each kind (I, II, and III), if matter is present in them, is filled of the kind I also bears curvature (e. g. de Sitter’s space). An empty with μ-vacuum of the corresponding kind (I, II, or III). space of the kind I, i. e. the one devoid of any island masses or Actually, because being taken in the “ortho-reference” of the vacuum, is flat. energy-momentum tensor of μ-vacuum all three space vectors and Einstein spaces of the kind II and of the kind III are more the time vector are the same (all the four directions have the exotic, because they are curved by themselves. Their curvature same rights), μ-vacuum is the highest degree of isotropic matter. is not related to the island distribution of masses or the presence Besides, because Einstein spaces are homogeneous, so that the of vacuum. The kind II and the kind III are generally attributed to radiation fields, for instance, to gravitational waves. ∗If we introduce a local flat space, tangential to the given Riemannian space in A few years later Gliner [26, 27, 28] in his study of the algebraic a given point, then the eigenvalues ξ of the tensor Tαβ are the values in an ortho- structure of the energy-momentum tensor of vacuum-like states of reference, corresponded to this tensor, in contrast to the eigenvalues of the metric tensor gαβ in an ortho-reference, defined in this tangential space. matter (Tαβ gαβ, Rαβ = kgαβ) outlined its special kind for which all Gliner used the signature ( +++), hence he had T = μg . So because the ∼ † − αβ − αβ observable density is positive ρ = T00 = μ > 0, he had negative numerical values ∗Chr.inv.-interpretation of this algebraic classification of Einstein spaces (or, in g00 − (+ ) other word, of Petrov’s gravitational fields) had been obtained in 1970 by a co-author of the μ. In our book we use the signature −−− , because in this case three- of this book (Borissova, nee´ Grigoreva [25]). dimensional observable interval is positive. Hence we have μ > 0 and Tαβ = μgαβ . 5.1 Introduction 221 222 Chapter 5 Vacuum and the mirror Universe matter density in their every point is everywhere equal [24], then consideration. μ-vacuum that fills them does not only have a constant density, but As it is known, positive curvature Riemannian spaces are gener- is homogeneous as well. alization of a regular sphere, while the negative curvature ones are As we have seen, Einstein spaces can be filled with μ-vacuum, generalization of Lobachewski-Bolyai space, an imaginary-radius with regular vacuum Tαβ = 0 or with emptiness. Besides, there sphere. In Poincare´ interpretation negative curvature spaces reflect may exist isolated “islands” of mass, which also produce the space on the inner surface of a sphere. Using the methods of chronometric curvature. Therefore Einstein spaces of the kind I are the best invariants, Zelmanov showed that in the pseudo-Riemannian space illustration of our knowledge of our Universe as a whole. And thus (its metric is indefinite) the three-dimensional observable curvature to study geometry of our Universe and physical states of matter, is negative to the Riemannan four-dimensional curvature. Because which fills it, is the same that to study Einstein spaces of the kind I. we percept our planet as a sphere, the observable curvature is Petrov has proposed and proven a theorem (see §13 in [24]), we positive in our world. If any hypothetical beings inhabited the will refer to it as: “inner” surface of the Earth, they would percept it as concave and their world will be of negative curvature. PETROV’S THEOREM Such illustration inspired some researchers for the idea of pos- Any space of a constant curvature is an Einstein space. < So that > sible existence of our mirror twin, the mirror Universe inhabited . . . Einstein spaces of the kind II and of the kind III can not be by antipodes. Initially it was assumed that once our world has a constant curvature spaces. positive curvature, the mirror Universe must be a negative curvature space. But Synge showed (see [29], Chapter VII) that in a Hence constant curvature spaces are Einstein spaces of the kind I, positive curvature de Sitter space space-like geodesic trajectories according to the Petrov classification. If K = 0 an Einstein space of are open, while in a negative curvature de Sitter space they are the kind I is flat. This makes our study of vacuum and vacuum-like closed. In other word, a negative curvature de Sitter space is not a states of matter in the Universe even simpler, because by today we mirror reflection of its positive curvature counterpart. have well studied constant curvature spaces. These are de Sitter On the other hand, in the study [15] (see also §1.3 herein) we spaces, or, in other word, spaces with de Sitter’s metric. found another approach to the concept of the mirror Universe. This In any de Sitter space we have Tαβ = 0 and λ = 0, so it is spher- study considered motion of free particles with the reverse time 6 ically symmetric, filled with regular vacuum and does not contain flow. As a result it has been obtained that the observable scalar “islands” of substance. On the other hand we know that the average component of a particle’s four-dimensional impulse vector is its density of matter in our Universe is rather low. Looking at it negative relativistic mass. Noteworthy, the particles of “mirror” in general, we can neglect presence of occasional “islands” and masses were obtained as a formal result of projecting their four- inhomogeneities, which locally distort the spherical symmetry. dimensional impulse on a regular observer’s time line and the result Hence our space can be generally assumed as de Sitter space with was not related to changing sign of the space curvature: particles the radius equal to that of the Universe. with either the direct or reverse flow of time may either exist in Theoretically a de Sitter space may bear either a positive cur- positive or negative curvature spaces. vature K > 0 or a negative one K < 0. Analysis (see Synge’s book) These results obtained by geometric methods of the General shows that in de Sitter worlds with K < 0 time-like geodesic lines Theory of Relativity inevitably affect our view of matter and cos- are closed: a test-particle repeats its motion again and again along mology of our Universe. the same trajectory. This hints some ideas, which seem to be In §5.2 we are going to obtain the energy-momentum tensor of too “revolutionary” from viewpoint of today’s physics [29]. Con- vacuum and in the same time a formula for its observable density. sequently, most physicists (Synge, Gliner, Petrov, and others) have We will also introduce a classification of matter according to the left negative curvature de Sitter spaces beyond the scope of their obtained forms of the energy-momentum tensor (namely — T- 5.2 The observable density of vacuum 223 224 Chapter 5 Vacuum and the mirror Universe classification). In §5.3 we are going to look at physical properties As a result we can see that λ-fields and vacuum are practically of vacuum in Einstein spaces of the kind I; in particular, we will the same thing, so vacuum is a non-Newtonian field of gravitation. discuss physical properties of vacuum in de Sitter space and make We will call this point of the theory the physical definition of conclusions on the global structure of the Universe. Following this vacuum. Hence λ-fields are action of own potential of vacuum. approach in §5.4 we will set forth the concept of origin and develop- This means that the term λgαβ can not be lost in the field ment of the Universe as a result of the Inversion Explosion from equations in vacuum, whatever small it is, because it describes the pra-particle that possessed some specific properties. In §5.5 we vacuum, which is among the reasons that make the space curved. 1 will obtain a formula for non-Newtonian gravitational inertial force, Then the field equations Rαβ 2 gαβR = κ Tαβ + λgαβ can be put which is proportional to distance, §5.6 and §5.7 will focus on collapse down as follows − − in Schwarzschild space (gravitational collapse, a black hole) and in 1 Rαβ gαβR = κ Tαβ , (5.12) de Sitter space (inflational collapse, an inflanton). In §5.8 will be − 2 − shown that our Universe and the mirror Universe are worlds with in the right part of which the tensor e mirror time that co-exist in de Sitter space with four-dimensional λ negative curvature. Also we will set forth physical conditions, which Tαβ = Tαβ + T˘αβ = Tαβ gαβ (5.13) allow transition through the membrane which separates our world − κ and the mirror Universe. is the energy-momentume tensor which describes matter in general (both a substance and vacuum). The first term here is the energy- momentum tensor of the substance. The second term §5.2 The observable density of vacuum. T-classification of matter λ T˘αβ = gαβ (5.14) The Einstein equations (the field equations in Einstein’s theory of −κ gravitation) are functions which link the space curvature to distrib- is the analog to the energy-momentum tensor for vacuum. 1 ution of matter. Generally they are Rαβ gαβR = κ Tαβ + λgαβ. Therefore, because Einstein spaces may be filled with vacuum, − 2 − The left part, as it is known, describe geometry of the space, while their mathematical definition is better to be set forth in a more the right one describes matter filled into the space. The sign of the general form to take account for the presence of both the substance second term depends on that of λ. As we are going to see, the sign and vacuum (λ-fields): Tαβ gαβ. In particular, doing this helps to of λ, so behaviour of Newtonian gravitation (attraction or repulsion) avoid contradictions when considering∼ Einstein empty spaces. is directly linked to the sign of the vacuum density. Noteworthy, the obtainede formula for the energy-momentum Einstein spaces are defined by the condition T g , the field tensor of vacuum (5.14) is a direct consequence of the field equa- αβ ∼ αβ equations for them are Rαβ = kgαβ. Such field equations can exist tions in general form. in two cases: (a) where Tαβ = 0 (a substance); (b) where Tαβ = 0 If λ > 0 (the non-Newtonian forces of gravitation repulsion) the (vacuum). But because in Einstein6 spaces, filled with vacuum, the observable density of vacuum is negative energy-momentum tensor equals zero, it can not be proportional to T˘ λ λ the metric tensor. This fact contradicts the definition of Einstein ρ˘ = 00 = = | | < 0 , (5.15) spaces T g ). So what is the problem here? In the absence of g00 −κ − κ αβ ∼ αβ any substance (so actually in vacuum) the field equations become while if λ < 0 (the non-Newtonian forces of gravitation attraction) 1 Rαβ 2 gαβR = λgαβ, so the space curvature is produced by λ-fields the observable density of vacuum is, to the contrary, positive (non-Newtonian− fields of gravitation) rather then by substance. In ˘ the absence of both an substance and λ-fields we have R = 0, so T00 λ λ αβ ρ˘ = = = | | > 0 . (5.16) the space is empty but generally it is not flat. g00 −κ κ 5.2 The observable density of vacuum 225 226 Chapter 5 Vacuum and the mirror Universe

The latter fact, as we will see in the next §5.3, is of great But in the next §5.3 we will show that the state equation of importance, because a de Sitter space with λ < 0, being a constant- μ-vacuum has fully different form p = ρc2 (the state of inflation, − negative curvature space∗ filled with vacuum only (no any sub- expansion of the media in the case of its positive density). Hence stance present), best fits our observation data on our Universe in the pressure and density in atomic nuclei should not be constant general. as to prevent transition of their inner substance into a vacuum-like Therefore proceeding from studies by Petrov and Gliner and state. taking into account our note on existence of the energy-momentum Noteworthy, this T-classification, just like the field equations, is tensor and, hence, physical properties in vacuum (λ-fields), we can only about distribution of matter which affects the space curvature, set forth a “geometric” classification of states of matter according to but not about test-particles — material points which own masses its energy-momentum tensor. We will call this one T-classification and sizes are so small that their effect on the space curvature of matter: can be neglected. Therefore the energy-momentum tensor is not defined for particles, and they should be considered beyond this I. Emptiness: Tαβ = 0, λ = 0 (a space-time without matter), so T-classification. that the field equations are Rαβ = 0;

II. Vacuum: Tαβ = 0, λ = 0 (produced by λ-fields), so the field equations are G = λg6 ; §5.3 Physical properties of vacuum. Cosmology αβ − αβ III. μ-vacuum: Tαβ = μgαβ, μ = const (a vacuum-like state of the Einstein spaces are defined by the field equations like Rαβ = kgαβ, substance, filled into the space), in this case the field equations where k = const. With λ = 0 and Tαβ = μgαβ the space is filled with are G = κμg ; 6 αβ − αβ a matter, which the energy-momentum tensor is proportional to IV. Substance: Tαβ = 0, Tαβ gαβ (this state comprises both a the fundamental metric tensor, so this matter is μ-vacuum. As 6 6∼ regular substance and electromagnetic fields). we saw in the previous §5.2, for vacuum the energy-momentum Generally the energy-momentum tensor of substance (Kind IV tensor is also proportional to the metric tensor. This implies that in T-classification) is not proportional to the metric tensor. On the physical properties of vacuum and those of μ-vacuum are mostly other hand, there are states of substance in which the energy- the same, save for a scalar coefficient which defines the composition momentum tensor contains a term proportional to the metric ten- of the matter (λ-fields or a substance) and the absolute values of the sor, but because it also contains other terms so it is not μ-vacuum. acting forces. Therefore we are going to consider an Einstein space Such are, for instance, an ideal fluid filled with vacuum and μ-vacuum. In this case the field equations become p p 1 Tαβ = ρ UαUβ gαβ , (5.17) R g R = (κμ λ) g . (5.19) − c2 − c2 αβ − 2 αβ − − αβ where p is the fluid pressure, and also electromagnetic fields Putting them down in a mixed form and then contracting we arrive to the scalar curvature ρσ σ Tαβ = FρσF gαβ FασFβ∙ , (5.18) − ∙ R = 4 (κμ λ) , (5.20) ρσ − where FρσF is the first invariant of an electromagnetic field under 2 consideration (3.27), Fαβ is the Maxwell tensor. If p = ρc (a sub- substituting which into the initial equations (5.19) we obtain the stance inside atomic nuclei) and p = const, the energy-momentum field equations in their final form tensor of the ideal fluid seems to be proportional to the metric R = (κμ λ) g , (5.21) tensor. αβ − αβ We mean here the Riemannian four-dimensional curvature. where κμ λ = const = k. ∗ − 5.3 Properties of vacuum. Cosmology 227 228 Chapter 5 Vacuum and the mirror Universe

Let us look at physical properties of vacuum and μ-vacuum. We part (the viscosity of the 1st kind, which reveal itself in anisotropic i deduce chr.inv.-projections of the energy-momentum tensor: the deformations), where α = αi is the spur of the tensor αik. observable density of the matter ρ = T00 , the observable density of Formulating the strength tensor for μ-vacuum (5.24) in the g00 i reference frame, which accompanies μ-vacuum itself, we arrive to i c T0 ik 2 ik impulse J = , and the observable strength tensor U = c T . 2 √g00 Uik = phik = ρc hik , (5.30) For the energy-momentum tensor of μ-vacuum Tαβ = μgαβ − chr.inv.-projections are and similarly to the strength tensor of vacuum (5.27), we have T00 ˘ 2 ρ = = μ , (5.22) Uik =ph ˘ ik = ρc˘ hik . (5.31) g00 − This implies that vacuum and μ-vacuum are non-viscous media i c T (α = 0, β = 0) which equations of state∗ is J i = 0 = 0 , (5.23) ik ik √g00 p˘ = ρc˘ 2, p = ρc2. (5.32) − − U ik = c2T ik = μc2hik = ρc2hik. (5.24) Such state is referred to as inflation because at the positive − − density of the matter the pressure becomes negative, so the media For the energy-momentum tensor T˘ = λ g (5.14), which αβ −κ αβ expands. describes vacuum, chr.inv.-projections are These are physical properties of vacuum and μ-vacuum: they are homogeneous ρ = const, non-viscous α = β = 0, and non-emitting T˘ λ ik ik ρ˘ = 00 = , (5.25) J i = 0 medias filled in the state of inflation. g −κ 00 Having these general physical properties a base, let us turn c T˘i to analysis of vacuum, which fills constant curvature spaces, in J˘i = 0 = 0 , (5.26) particular, a de Sitter space, which is the closest approximation of √g 00 our Universe as a whole. λ In constant curvature spaces the Riemann-Christoffel tensor is U˘ ik = c2T˘ik = c2hik = ρc˘ 2hik. (5.27) κ − (see Chapter VII in the Synge book [29]) From here we see that vacuum (λ-fields) and μ-vacuum have a R = K (g g g g ) ,K = const. (5.33) αβγδ αγ βδ − αδ βγ constant density, so these are uniformly distributed matter. They Having the tensor contracted by two indices, we obtain a for- are also non-emitting medias, because the energy flux c2J i in them mula for the Ricci tensor, which subsequent contraction allows to is zero c3T˘i c3T i deduce the scalar curvature. As a result we have c2J˘i = 0 = 0 , c2J i = 0 = 0 . (5.28) √g00 √g00 R = 3Kg ,R = 12K. (5.34) αβ − αβ − In the reference frame, which accompanies the medium, the Assuming our Universe a constant curvature space, we obtain strength tensor equals (see the Zelmanov book [4]) the field equations formulated with the curvature U = p h α = ph β , (5.29) 3Kg = κ T + λg . (5.35) ik 0 ik − ik ik − ik αβ − αβ αβ where p is the equilibrium pressure, defined from the state equa- ∗The state equation of a distributed matter is the relationship between its 0 pressure and the density. For instance, p = 0 is the state equation of a dust media, tion, p is the true pressure, αik is the viscosity of the 2nd kind 2 1 2 p = ρc is the state equation of a matter in atomic nuclei, p = 3 ρc is the state (the viscous strength tensor), and β = α 1 αh is its anisotropic equation of an ultra-relativistic gas. ik ik − 3 ik 5.3 Properties of vacuum. Cosmology 229 230 Chapter 5 Vacuum and the mirror Universe

We put them down in Synge’s notation as (λ 3K) gαβ = κ Tαβ. expands. Besides, because in this case λ < 0, non-Newtonian forces Then the energy-momentum tensor of a substance− in constant of gravitation are those of attraction. Then the Universe can keep curvature spaces is expanding from nearly a point until the vacuum density becomes λ 3K so low that its expanding action becomes equal to the compressing Tαβ = − gαβ . (5.36) κ action of non-Newtonian λ-forces. From here we see that in constant curvature space the problem As seen, the question of the curvature sign is the most crucial of geometrization of matter solves by itself: the energy-momentum one for cosmology of our Universe. tensor (5.36) contains the metric tensor and constants only. But human perception is three-dimensional and a regular ob- De Sitter space is a constant curvature space, where Tαβ = 0 and server can not judge anything on sign of the four-dimensional λ = 0, so the one filled with vacuum (any substance is absent). Then curvature by means of direct observations. What can be done then? having6 the energy-momentum tensor of substance (5.36) equalized The way out of the situation is in the theory of chronometric to zero we obtain the same result as Synge did it: in de Sitter spaces invariants — a method to define physical observable quantities. λ = 3K. Among the goals that Zelmanov set for himself was to build the Taking into account this relationship, the formula for the ob- curvature tensor of an observer’s spatial section (the observable servable density of vacuum in de Sitter worlds becomes three-dimensional space — inhomogeneous, non-holonomic, deformed, and curved, in general case). The Zelmanov curvature λ 3K 3Kc2 ρ˘ = = = . (5.37) tensor possess all properties of the Riemann-Christoffel tensor in −κ − κ − 8πG the observer’s three-dimensional space and also, in the same time, Now we are approaching the key question: what is the sign possesses the property of chronometric invariance. of the four-dimensional curvature in our Universe? The reason Zelmanov decided to build such tensor using similarity with the to ask is not strictly curiosity. Depending from the answer the Riemann-Christoffel tensor, which results from non-commutativity de Sitter world cosmology we have built may fit the available data of the second derivatives from an arbitrary vector in a given space. of observations or may lead to results totally alien to commonly Deducing the difference of the second chr.inv.-derivatives from an accepted astronomical facts. arbitrary vector, he arrived to the equation As a matter of fact, given that the four-dimensional curvature is positive K > 0 the vacuum density must be negative and hence the 2Aik ∗∂Ql ...j ∗ i ∗ k Ql ∗ k ∗ i Ql = + Hlki Qj , (5.38) inflational pressure must be greater than zero — vacuum contracts. ∇ ∇ − ∇ ∇ c2 ∂t Then because of λ > 0, non-Newtonian forces of gravitation are which contains the chr.inv.-tensor those of repulsion. We will then witness struggle of two actions: at ∂Δj ∂Δj the positive inflational pressure of vacuum, which tend to compress H...j = ∗ il ∗ kl + ΔmΔj ΔmΔj , (5.39) the space, we will observe repulsion forces of non-Newtonian grav- lki ∂xk − ∂xi il km − kl im itation. The result will be as follows: at first, because λ-forces are which is similar to Schouten’s tensor from the theory of non- proportional to distance, their expanding effect would grow along holonomic manifolds . But in general case in the presence of the with growth of the radius of the Universe and the expansion would ∗ space rotation (A = 0), the tensor H...j is algebraically different accelerate. At second, if the Universe has ever been of size less than ik 6 lki the distance, at which the contracting pressure of vacuum is equal ∗Schouten had built the theory of non-holonomic manifolds for an arbitrary to the expanding action of λ-forces, the expansion would become dimension space, considering a m-dimensional sub-space in a n-dimensional space, impossible. where m < n [30]. In the theory of chronometric invariants we actually consider an observer’s (m = 3)-dimensional sub-space in the (n = 4)-dimensional pseudo- If to the contrary the four-dimensional curvature is negative Riemannian space. In the same time the theory of chronometric invariants is K < 0, the inflational pressure will be less than zero — vacuum applicable to any metric space, in general — see [4]. 5.3 Properties of vacuum. Cosmology 231 232 Chapter 5 Vacuum and the mirror Universe from the Riemann-Christoffel tensor. Therefore Zelmanov intro- Substituting the necessary components of the Riemann-Christoffel duced a new tensor tensor [4], and having indices lowered, we obtain

1 ∗∂Dij l l 1 1 Clkij = (Hlkij Hjkil + Hklji Hiljk) , (5.40) Xij = Di+Ai∙ (Djl+Ajl) + (∗ iFj+∗ j Fi) FiFj , (5.43) 4 − − ∂t − ∙ 2 ∇ ∇ −c2 which was not only chr.inv.-quantity, but it also possessed all algeb- 2 Yijk = ∗ i (Djk + Ajk) ∗ j (Dik + Aik) + 2 AijFk , (5.44) raic properties of the Riemann-Christoffel tensor. Therefore Clkij ∇ − ∇ c is the curvature tensor of the three-dimensional observable space Z = D D D D + A A A A + 2A A c2C . (5.45) of an observer, who accompanies his reference body. Having it iklj ik lj − il kj ik lj − il kj ij kl − iklj contracted, we obtain the chr.inv.-quantities From these Zelmanov formulas we see that spatial observable components of the Riemann-Christoffel tensor (5.45) are directly ...i im j lj Ckj = Ckij = h Ckimj ,C = Cj = h Clj , (5.41) linked to the chr.inv.-tensor of the three-dimensional observable ∙ curvature C . which also describe the curvature of the three-dimensional space. iklj Let us deduce a formula for the three-dimensional observable Because C , C , and C are chr.inv.-quantities, they are physical lkij kj curvature in a constant curvature space. In such space the Riemann- observable values for this observer. In particular, the C is the three- Christoffel tensor is as of (5.33), then dimensional observable curvature [4]. Concerning our analysis of physical properties of vacuum and R0i0k = Khikg00 , (5.46) cosmology, we need to know how the observable three-dimensional − K curvature C is linked to the four-dimensional curvature K in gen- R0ijk = √g00 (vj hik vkhij ) , (5.47) eral and in a de Sitter space in particular. We are going to tackle c − this problem step-by-step. 1 Rijkl = K hikhjl hilhkj + 2 vi (vlhkj vkhjl) + The Riemann-Christoffel four-dimensional curvature tensor is − c − (5.48) a tensor of the 4th-rank, hence it has n4 = 256 components, out of h 1 + 2 vj (vkhil vlhik) , which only 20 are significant. The rest components are either zeroes c − i or contain each other, because the Riemann-Christoffel tensor is: Having deduced its chr.inv.-projections (5.42), we obtain ik 2 ik ijk ijkl 2 ik jl il jk Symmetric by each pair of its indices Rαβγδ = Rγδαβ; X = c Kh ,Y = 0 ,Z = c K h h h h , (5.49) • − Antisymmetric in respect of transposition inside each pair of hence the spatial observable components with lower indices are • the indices R = R , R = R ; αβγδ − βαγδ αβγδ − αβδγ Its components are constructed with the property R = 0, Z = c2K (h h h h ) . (5.50) α(βγδ) ijkl ik jl − il jk • where round brackets stand for (β, γ, δ)-transpositions. Contracting this quantity step-by-step, we obtain Significant components of the Riemann-Christoffel tensor pro- i... 2 j 2 duce three chr.inv.-tensors Zjl = Z jil = 2c Khjl ,Z = Zj = 6c K. (5.51) ∙ i k ijk ik 2 R0∙ ∙0 ijk R0∙ ... ijkl 2 ijkl On the other hand, we know the formula for Zijkl in an arbitrary X = c ∙ ∙ ,Y = c ,Z = c R . (5.42) − g00 − √g00 − curvature space (5.45). The formula contains the three-dimensional observable curvature. Evidently it is true for K = const as well. The tensor Xik has 6 components, Y ijk has 9 components, while Then having the general formula (5.45) contracted, we have Zijkl has only 9 due to its symmetry. Components of the second k k k 2 tensor are constructed by the property Y = Y + Y + Y = 0. Zil = DikDl DilD + AikAl∙ + 2AikA l∙ c Cil , (5.52) (ijk) ijk jki kij − ∙ ∙ − 5.3 Properties of vacuum. Cosmology 233 234 Chapter 5 Vacuum and the mirror Universe

il ik 2 ik 2 Z = h Zil = DikD D AikA c C. (5.53) Universe is among the physical properties of vacuum, the negative − − − inflational pressure of vacuum also implies expansion of the Uni- In a constant curvature space we have Z = 6c2K (5.51), hence in verse as a whole. such space the relationship between the four-dimensional curvature Therefore the observable three-dimensional space of our Uni- K and the three-dimensional observable curvature C is verse (C > 0) is a three-dimensional expanding sphere, which is a 2 ik 2 ik 2 6c K = DikD D AikA c C. (5.54) sub-space of the four-dimensional space-time (K < 0, a space which − − − geometry is a generalized case of Lobachewski-Bolyai geometry). We see that in the absence of the space rotation and the de- Of course a de Sitter space is merely an approximation of our formation the four-dimensional curvature has the opposite sign in Universe. Astronomical data say that though “islands” of masses are respect of the three-dimensional observable curvature. In de Sitter occasional and hardly affect the global curvature, their effect on the spaces (because no the rotation or deformation) we have space curvature in their vicinities is significant (deviation of light 1 rays within gravitational fields and similar effects). But in our study K = C, (5.55) −6 of the Universe as a whole we can neglect occasional “islands” of so there the three-dimensional observable curvature is C = 6K. substance and local non-uniformities in the curvature. In such cases Now we are able to build a model for development of our− Uni- a de Sitter space with the negative four-dimensional curvature verse relying on two experimental facts: (a) the sign of the observ- (so, the observable three-dimensional curvature is positive) can be able density of matter, and (b) the sign of the observable three- assumed the background space of our Universe. dimensional curvature. At first, our everyday experience shows that the density of matter in our Universe is positive however sparse it may be. Then to §5.4 The concept of Inversion Explosion of the Universe ensure that the vacuum density (5.37) is positive, the cosmological term should be negative λ < 0 (non-Newtonian forces attract) and From the previous §5.3 we know that in a de Sitter space λ = 3K, hence the four-dimensional curvature should be negative K < 0. so that according to its physical sense λ-term is the same as the At second, as Ivanenko wrote in his preface to Weber’s book [23] curvature. For a three-dimensional spherical sub-space the observ- “Though the data of cosmological observations are evidently not able curvature C = 6K is − exact, but, for instance, McWittie [31] maintains that the best 1 results of observation of the Hubble red shift H 75–100 km/sec Mpc C = , (5.57) × R2 31 ≈3 and of average density of matter ρ 10− gram/cm support the idea of the non-disappearing cosmological≈ term λ < 0”. where R is the observable radius of the curvature (the sphere radi- As a result we can assume that the vacuum density in our us). Then the four-dimensional curvature of the de Sitter space is Universe is positive and the three-dimensional observable curva- 1 K = , (5.58) ture is C > 0. Hence the four-dimensional curvature is K < 0 and −6R2 hence the cosmological term is λ < 0. Then from (5.37) we obtain the observable density of vacuum in our Universe, formulated with i. e. the larger is the radius of the sphere, the less is the curvature the observable three-dimensional curvature K. According to astronomical estimates, our Universe emerged 10–20 billion years ago. Hence the distance covered by a photon λ 3K C 27 28 ρ˘ = = = > 0 , (5.56) since it was born at the dawn of the Universe is RH 10 –10 cm. −κ − κ 2κ This distance is referred to as the radius of the horizon≈ of events. so the inflational pressure of vacuum is negative p˘= ρc˘ 2 — vacuum Assuming our Universe as whole to be a de Sitter space with K < 0, expands. And because homogeneous distribution of− matter in the for the four-dimensional curvature and hence for λ-term λ = 3K we 5.4 Inversion Explosion of the Universe 235 236 Chapter 5 Vacuum and the mirror Universe have the estimate while that inside the di Bartini particle it is

1 56 K = 10− 1/cm2. (5.59) M 2 93 gram 3 −6R ≈ − ρ = = 9.03×10 /cm . (5.64) H 2π2r3 On the other hand, similar figures for the event horizon, the On the other hand, an outer space, being the inversion image curvature and λ-term are available from Roberto di Bartini [32, of the inner one, according to its properties can be assumed as 33], who studied relationships between physical constants from de Sitter space. So forth let us assume that a space with the cur- topological viewpoint. In his works the space radius of the Universe vature radius, equal to the di Bartini radius R = 5.895 1029 cm, is is interpreted as the longest distance, defined from topological con- × a de Sitter space with K < 0. Then the four-dimensional curvature text. According to di Bartini’s inversion relationship and λ-term are R ρ 1 61 = 1 , (5.60) K = = 4.8 10− 1/cm2, (5.65) r2 −6R2 − × the space radius R (the longest distance) is the inversion image of 1 61 55 − 1 cm2 the gravitational radius of electron ρ = 1.347×10− cm in respect of λ = 3K = 2 = 14.4×10 / , (5.66) 13 −2R − the radius of a spherical inversion r = 2.818×10− cm, which is the same that the classical radius of electron (according to di Bartini — so they are five orders of magnitude less than the observed esti- 56 the radius of the spherical inversion). The space radius (the largest mate, which equals λ < 10− . This can be explained because the | | radius of the event horizon) equals Universe keeps on expanding and in a distant future numerical values of the space curvature and the cosmological term will grow 29 R = 5.895×10 cm . (5.61) down to approach the figures in (5.65, 5.66), calculated for the longest distance (the space radius). The estimated density of vacu- From topological context di Bartini defined the space mass (the um in the de Sitter space within the space radius is mass within the space radius) and also the space density, which are 2 57 34 3K 3Kc gram 3 34 3 M = 3.986×10 gram , ρ = 9.87×10− /cm . (5.62) ρ˘ = = 7.7 10− gram/cm (5.67) − κ − 8πG ≈ × As a matter of fact, studies done by di Bartini say that the space so it is also less than the observed average density in the Universe of the Universe (from the classical radius of electron up to the 30 gram 3 (5–10×10− /cm ) and it is close to the density of matter within event horizon) is an external inversion image of the inner space of a 34 the space radius according to di Bartini 9.87 10− gram/cm3. certain particle with the size of electron (its radius can be estimated × To find how long will our Universe keep the expanding we have within the range from the classical radius of electron up to its to define the difference between the observed radius of the event gravitational radius). From other viewpoints the particle is different 57 horizon RH and the curvature radius R. Assuming the maximal from electron: its mass equals the space mass M = 3.986×10 gram, 28 radius of the event horizon in the Universe RH(max) equal to the while that of electron is m = 9.11 10− gram. × space radius (the outer inversion distance), which according to The space within that particle can not be represented as a 29 di Bartini is R = 5.895×10 cm (5.61), and comparing it with the de Sitter space. As a matter of fact, the vacuum density in a 27 28 observed radius of the event horizon (RH 10 –10 cm), we obtain de Sitter space with K < 0 and the curvature observable radius 29 ≈ 13 ΔR = RH(max) RH 5.8×10 cm, so the time left for the expansion r = 2.818×10− cm is − ≈

3K 1 2 51 ΔR ρ˘ = = r = 3.39 10 gram/cm3, (5.63) t = 600 billion years. (5.68) − κ −2κ × c ≈ 5.4 Inversion Explosion of the Universe 237 238 Chapter 5 Vacuum and the mirror Universe

These calculations of the vacuum density and of other properties sion Explosion), remain unknown. . . but so do the reasons for the of the de Sitter space pave the way for conclusions on the origin “emerge” of the Universe in some other contemporary cosmological and evolution of our Universe and allow the only interpretation of concepts, for instance, in the concept of Big Bang from a singular the di Bartini inversion relationship. We will call it the cosmological point. concept of Inversion Explosion. This concept is based on our analysis of properties of the de Sitter space using geometric methods §5.5 Non-Newtonian gravitational forces of the General Theory of Relativity, and the di Bartini inversion relationship as a result of the contemporary knowledge of physical Einstein spaces of the kind I, including constant curvature spaces, constants. We can set forth the concept as follows: aside for having occasional “islands of matter” may be either empty In the beginning there existed a single pra-particle with a or filled with a homogeneous matter. But an empty Einstein space radius equal to the classical radius of electron and with a mass of the kind I (its curvature is K = 0) is dramatically different from equal to the mass of the entire Universe. not empty one (K = const = 0). 6 Then the inversion explosion had been occurred: a topo- To make our discourse more concrete, let us look at the most logical transition had inverted matter in the pra-particle in typical examples of empty and not-empty Einstein spaces of the respect of its surface into the outer world, which gave birth to kind I. our expanding Universe. At present, 10–20 billion years since If an island of mass is a ball (spherically symmetric distribution the explosion, the Universe is in the early stage of its evolution. of mass in the island) located into emptiness, then the curvature of The expansion will continue for almost 600 billion years. such space is derived from Newtonian field of gravitation, produced At the end of this period the expanding Universe will by the island, and such is not a constant curvature space. At an reach its curvature radius, at which non-Newtonian forces of gravitation, proportional to distance, will equalize the infla- infinite distance from the island the space becomes flat, i. e. a tional expanding pressure of vacuum. The expansion will dis- constant curvature space with K = 0. A typical example of the field continue and stability will be reached, which will latter until of gravitation, produced by a spherically symmetric island of mass the next inversion topological transition occurs. in emptiness is a field described by Schwarzschild’s metric Parameters of matter at stages of the evolution are calculated in r dr2 ds2 = 1 g c2dt2 r2 dθ2 + sin2 θ dϕ2 , (5.69) Table 3 — the pra-particle before the inversion explosion, the stage − r − rg − 1 r of the inversion expansion at the present time, and the stage after − the expansion. where r is the distance from the island, rg is the island’s gravitational radius. Evolution Age, Space Density, λ-term, No any space rotation or the deformation exist in a space with stage years radius, cm gram/cm3 1/cm2 Schwarzschild metric. Components of the chr.inv.-vector of gravi-

13 93 tational inertial force (1.38) can be deduced there as follows. Ac- Pra-particle 0 2.82×10− 9.03×10 ? 9 27 28 30 56 cording to the metric (5.69), the component g00 is Present time 10–20×10 10 –10 5–10×10− < 10− 9 29 34 60 After expansion 623 10 5.89 10 9.87 10− 1.44 10− rg × × × × g = 1 , (5.70) 00 − r Table 3. Parameters of matter at different stages of then, derivating the gravitational potential w = c2(1 g ) with the evolution of the Universe √ 00 respect to xi, we obtain − 2 The reasons for this topological transition, which led to the ∂w c ∂g00 i = i . (5.71) spherical inversion of matter from the pra-particle (after its Inver- ∂x −2√g00 ∂x 5.5 Non-Newtonian gravitational forces 239 240 Chapter 5 Vacuum and the mirror Universe

Having it substituted into the formula for gravitational inertial is that with Kottler’s metric [34] force (1.38), in the absence of the space rotation we have 2 2 2 ar b 2 2 dr 2 2 2 2 2 2 ds = 1+ + c dt r dθ + sin θ dϕ , c rg 1 c rg 3 r − ar2 b − F = ,F 1 = . (5.72) 1+ + 1 2 r 2 3 r − 2r 1 g − 2r − r ar b (5.75) 2 3 2 1 2 ar b i F = c − 2r ,F = c , Therefore, the vector F in a Schwarzschild space describes a 1 2 2 − 1 + ar + b − 3 − 2r Newtonian gravitational force, which is reciprocal to the square of 3 r the distance r from the gravitating mass. where both Newtonian and λ-forces exist: it is filled with vacuum If a space is filled with a spherically symmetric distribution of and includes islands of mass, the latter which produce Newtonian vacuum and it does not include any island of mass, its curvature will forces of gravitation. On the other hand, Kottler proposed his metric be everywhere the same. An example of such field is that described with two unknown constants a and b to define which some addi- by de Sitter’s metric tional constraints are required. Hence despite some of attractive 2 2 features of Kottler metric, only two its “ultimate” cases are of λr dr 2 ds2 = 1 c2dt2 r2 dθ2 + sin θ dϕ2 . (5.73) practical interest for us — Schwarzschild metric (Newtonian forces − 3 − λr2 − 1 of gravitation) and de Sitter metric (λ-forces — non-Newtonian − 3 forces of gravitation). Note that though any de Sitter space has not islands of mass, which produce Newtonian fields of gravitation. So, in a de Sitter §5.6 Gravitational collapse space we can consider motion of small test-particles, which own Newtonian fields are so weak that can be neglected. It is evidently, representing our Universe as either a de Sitter space Any de Sitter metric space is a constant curvature one, which (filled with vacuum without islands of mass) or a Schwarzschild becomes flat only in the absence of λ-fields. No any rotation or the space (an island of mass in emptiness) is a certain approximating deformation exist there, while components of the chr.inv.-vector of assumption. The real metric of our world in “something in the be- gravitational inertial force are tween”. Nevertheless, in some problems dealt with non-Newtonian gravitation (produced by vacuum), where influence of concentrated λc2 r λc2 F = ,F 1 = r , (5.74) masses can be neglected, de Sitter metric is optimal. And vice 1 2 3 1 λr 3 versa, in problems with gravitating fields of concentrated masses 3 − Schwarzschild metric is more reasonable. An illustrative example so the vector F i in a de Sitter space describes non-Newtonian of such “split” of the models is collapse — a state, where g00 = 0. gravitational forces, proportional to r: if λ < 0, those are attraction Gravitational potential w for an arbitrary metric is (1.38). Then forces, if λ > 0 those are repulsion forces. Therefore forces of non- w 2 2w w2 Newtonian gravitation (λ-forces) grow along with distance at which g = 1 = 1 + , (5.76) 00 − c2 − c2 c4 they act. 2 Therefore we can see the principal difference between empty so collapse g00 = 0 occurs at w = c . and non-empty Einstein spaces of the kind I: in empty one with an Commonly, gravitational collapse is considered — compression island of mass only Newtonian forces exist, while in the one filled of an island of mass under action of Newtonian gravitation until the with vacuum without islands of mass there are non-Newtonian mass island reaches its gravitational radius. Hence “strict” grav- gravitation forces only. An example of a “mixed” space of the kind I itational collapse occurs in a Schwarzschild metric space (5.69), 5.6 Gravitational collapse 241 242 Chapter 5 Vacuum and the mirror Universe

because only Newtonian field of a spherically symmetric island of so at the distance r = rg the interval of observable time equals zero mass in emptiness is present. dτ = 0: from viewpoint of an external observer the time on the At larger distances from the concentrated mass the gravitational surface of a Schwarzschild sphere stops∗. Inside the Schwarzschild field becomes weak and Newton’s law of gravitation becomes true. sphere the interval of observable time becomes imaginary. We can Hence in a weak field of Newtonian gravitation the field potential is also be sure that a regular observer who lives on the Earth surface, apparently stays outside its Schwarzschild sphere with radius of GM w = , (5.77) 0.443 cm and he can only look at process of gravitational collapse r from “outside”. where G is the Gauss gravitational constant, M is the mass of the If r = rg then the quantity island, which produced that gravitational field. Because in the weak field the third term in (5.76) is so small that it can be neglected, 1 g11 = (5.83) rg hence the formula for g00 becomes −1 − r 2GM g = 1 , (5.78) grows up to infinity. But the determinant of the metric tensor gαβ is 00 − c2r 2 so gravitational collapse in a Schwarzschild space occurs if g = r4 sin θ < 0 , (5.84) − 2GM = 1 , (5.79) so a space-time area inside a gravitational collapsar is generally not c2r degenerated, though the collapse is also possible in a zero-space. where the quantity At this point a note concerning photometric distance and metric 2GM rg = , (5.80) observable distance should be taken. c2 The quantity r is not a metric distance along the axis x1 = r, 1 which has the dimension of length, is referred to as the gravitational 2 rg − because the metric (5.69) has dr with the coefficient 1 r . radius of the mass island. Then g can be presented as follows − 00 The quantity r is a photometric distance defined as function of r g = 1 g . (5.81) illumination, produced by a stable source of light and reciprocal 00 − r to the square of distance. In other word, the r is the radius of a 2 From here we see that the collapse occurs in a Schwarzschild non-Euclidean sphere with the surface area 4πr [4]. space at r = rg. In such case all mass of the spherically symmetric According to the theory of chronometric invariants, an elemen- island (the source of the Newtonian field) becomes concentrated tary observable metric distance between two points in a Schwarz- within its gravitational radius. Therefore the surface of such mass 1 i At g00 = 0 (collapse) an interval of observable time (1.25) is dτ = v dx , island is referred to as a Schwarzschild sphere. Such objects are ∗ − c2 i g0i where vi = c is the linear velocity of the space rotation (1.37). Only assuming also called black holes, because within the gravitational radius an − √g00 escape velocity is above that of the light velocity so light can not be goi = 0 and vi = 0 the condition of collapse can be defined correctly: for an external observer the observable time flow on the surface of a collapsar stops dτ = 0, while a 2 2 i k emitted from such objects outside. four-dimensional interval is ds = dσ = gikdx dx . From here a single conclusion As it is easy to see from formula (5.69), in a Schwarzschild field can be made: on the surface of a collapsar− the space is holonomic, so the collapsar does not rotate. of gravitation the three-dimensional space does not rotate (g0i = 0), hence an interval of observable time (1.25) is As it was shown in the study [15], a fully degenerated space-time (so called zero- space, where ds = 0, dτ = 0, and dσ = 0 are true) collapses if it does not rotate. Here we proved a more general theorem: if g00 = 0 the space is holonomic irrespective of rg whether it is degenerated (g = 0, a zero-space) or for it g < 0 (the space-time of the dτ = √g00 dt = 1 dt , (5.82) − r General Theory of Relativity). r 5.6 Gravitational collapse 243 244 Chapter 5 Vacuum and the mirror Universe schild space is 1. The space of both metrics is holonomic, i. e. it does not rotate Aik = 0; dr2 dσ = + r2 dθ2 + sin2 θ dϕ2 . (5.85) 2. The external metric is stationary, the vector of gravitational v rg 1 GM u1 r inertial force is F = ; u − − r2 t 3. The internal metric is non-stationary, the vector of gravita- At θ = const and ϕ = const it is tional inertial force is zero. r r 2 2 dr Let us give more detailed analysis of the external and internal σ = h11 dr = (5.86) rg metrics. To make the analysis simpler we assume θ = const and Zr1 Zr1 1 p − r ϕ = const, so that out of all possible spatial directions we limit our q and it is not the same as the photometric distance r. study to radial directions only. Then the external metric is Let us define the space-time metric inside a Schwarzschild 2 2 rg 2 2 dr sphere. So forth, we formulate the external metric (5.69) for a ds = 1 c dt + r , (5.90) − r − g 1 radius r < rg. As a result we have r − 2 while for the internal metric we have 2 rg 2 2 dr 2 2 2 2 ds = 1 c dt + r r dθ + sin θ dϕ . (5.87) c2dt˜2 r − r − g 1 − 2 g 2 r ds = r 1 dr˜ . (5.91) − g 1 − ct˜ − ˜ Introducing notations r = ct˜ and ct =r ˜ we obtain ct − Now we will define the physical observable distance (5.86) to the 2 ˜2 2 c dt rg 2 2 ˜2 2 2 2 attracting mass (namely — the gravitational collapsar) ds = r 1 dr˜ c dt dθ + sin θ dϕ , (5.88) g 1 − ct˜ − − ct˜ − dr σ = = r (r rg) + rg ln √r + r rg + const (5.92) rg − − so the space-time metric inside the Schwarzschild sphere is similar Z 1 − r q p to the external metric, provided that the time coordinate and the q spatial coordinate r swap their roles: the photometric distance r along the radial direction r. From here we see: at r = rg the observ- outside the black hole is the coordinate time ct˜inside, while outside able distance is the black hole the coordinate time ct is the photometric distance r˜ σg = rg ln √rg + const, (5.93) inside. and it is a constant value. This means that a Schwarzschild sphere, From the first term of Schwarzschild inner metric (5.88) we see defined by a photometric radius rg, for an external observer is that it is not stationary and it exists within a limited period of time a sphere with the observable radius σg = rg ln √rg + const (5.93). r Therefore for an external observer any gravitational collapsar (a t˜= g . (5.89) c black hole) is a sphere with constant observable radius, on which surface his observable time stops. For the Sun, which gravitational radius is 3 km, the life span 5 Let us analyze a gravitational collapsar’s interiors. An interval of of such space would be approximately < 10− sec. For the Earth, observable time (5.82) inside a Schwarzschild sphere is imaginary which gravitational radius is a small as 0.443 cm, the life span of 11 for an external observer inner Schwarzschild metric would be even less at 1.5×10− sec. r Comparison of the metrics inside a gravitational collapsar (5.88) dτ = i g 1 dt , (5.94) and outside the collapsed body (5.69) implies that: r − r 5.6 Gravitational collapse 245 246 Chapter 5 Vacuum and the mirror Universe or, in the “interior” coordinates r = ct˜ and ct =r ˜ (from viewpoint of On any Schwarzschild sphere we have r = r , so dτ = 0 there. g dt an “inner” observer), Hence any particle, including a light-like one, will stop there. A 1 four-dimensional interval on the sphere is dτ˜ = dt˜ . (5.95) r g 1 ds2 = dσ2 < 0 , (5.99) ct˜ − − r Hence for the external observer the collapsar’s internal “imag- so it is space-like one. This implies that Schwarzschild spheres inary” time (5.94) stops at its surface, while the “inner” observer (gravitational collapsars) are filled with particles with imaginary sees the pace of his observable time on the surface grows infinitely. rest-mass. So forth, from viewpoint of the external observer, the physical observable distance inside the collapsar, according to the metric §5.7 Inflational collapse (5.87), equals There are no islands of mass in de Sitter spaces, hence fields of dr rg Newtonian gravitation are absent as well — gravitational collapse is σ = = r (r rg) + rg arctan 1 + const, (5.96) rg − − r − impossible. Nevertheless, the condition g = 0 is a strictly geometric Z 1 r 00 r − q definition of collapse, not necessarily related to Newtonian fields. q or, from viewpoint of the “inner” observer Subsequently, we can consider collapse in any arbitrary space. We are going to look at de Sitter metric (5.73), which describes rg a non-Newtonian field of gravitation in a constant curvature space σ˜ = 1 dr . (5.97) ct˜ − without islands of mass (a de Sitter space). In this case collapse may Z r occur due to non-Newtonian gravitational forces. From de Sitter ˜ From here we see: at r = ct = rg for the external observer the metric (5.73) we see that observable distance between any two points converges to a constant, λr2 while for the “inner” observer the observable distance grows down g00 = 1 , (5.100) to zero. − 3 In conclusion we will address the question of what happens to 2 so gravitational potential w = c (1 √g00) in a de Sitter space is particles, which fall from “outside” on a Schwarzschild sphere along − its radial direction. Its external metric is as follows λr2 w = c2 1 1 . (5.101) r dr 2 2 2 2 g − r − 3 ! ds = c dτ dσ , dτ = 1 dt , dσ = r . (5.98) − − r 1 g − r Because it is a potential of non-Newtonian gravitation, produced For real-mass particles ds2 > 0, for light-like particles ds2 = 0, for by vacuum, we will call it λ-potential. From here we see that the super-light tachyons ds2 < 0 (their masses are imaginary). In radial λ-potential is zero, if the de Sitter space is flat so that λ = 3K = 0. motion towards the black hole these conditions are: Because in any de Sitter space λ = 3K, hence 2 r 2 2 1 1. Mass-bearing real particles dτ < c2 1 g ; g00 = 1 Kr > 0 at distances r < ; dt − r • − √K 2 r 2 2 1 2. Light-like particles dτ = c2 1 g ; g00 = 1 Kr < 0 at distances r > ; dt − r • − √K 2 rg 2 2 1 3. Imaginary particles-tachyons dτ > c2 1 . g00 = 1 Kr = 0 (collapse) at distances r = . dt − r • − √K 5.7 Inflational collapse 247 248 Chapter 5 Vacuum and the mirror Universe

2 At curvature K < 0 the numerical value of g00 = 1 Kr is always means that the inflational collapsar (inflanton) is filled with vacuum greater than zero. Hence collapse is only possible in a− de Sitter space with the negative density and it is in the state of fragile balance with K > 0. between the compressing pressure of vacuum and the expanding In §5.3 we showed that the basic space of our Universe as a forces of non-Newtonian gravitation. whole has K < 0. But we can assume the presence of local inhomo- In the de Sitter space with K > 0 we have geneities with K > 0, which do not affect the space curvature in general. In particular, on such inhomogeneities collapse may occur. r2 dτ = √g dt = 1 Kr2 dt = 1 dt , (5.104) Therefore it is reasonable to consider a de Sitter space with K > 0 00 2 − s − rinf as a local space in the vicinities of some compact objects. p In de Sitter spaces the three-dimensional observable curvature so on the surface of the inflational sphere the observable time (+ ) C is linked to the four-dimensional curvature with relationship flow stops dτ = 0. The signature we have accepted −−− , i. e. the C = 6K (5.55). Then assuming the observable three-dimensional condition g00 > 0, is true at r < rinf . − space to be a sphere, we obtain C = 1 (5.57) and hence K = 1 Using the term the “inflational radius” we represent de Sitter R2 −6R2 (5.58), where R is the observable radius of the curvature. In the metric with K > 0 as follows case K < 0 the value of R is real, at K > 0 it becomes imaginary. 2 2 2 r 2 2 dr 2 2 2 2 Collapse in a de Sitter space is only possible at K > 0. In this ds = 1 2 c dt 2 r dθ + sin θ dϕ , (5.105) − rinf − 1 r − case the observable radius of the curvature is imaginary. We denote 2 − rinf R = iR∗, where R∗ is its absolute value. Then in the de Sitter space with K > 0 we have then components of the chr.inv.-vector of gravitational inertial force 1 (5.74) are K = , (5.102) 2 2 c r 1 2 r 6R∗ F1 = 2 2 ,F = c 2 . (5.106) 2 r rinf rinf and the collapse condition g00 = 1 Kr can be written as follows 1 2 − − rinf r = R∗√6 . (5.103) Let us deduce formulas for observable distances and the observable inflational radius in an inflanton. To make our calculations So at the distance r = R∗√6 in a de Sitter space with K > 0 the simpler we assume θ = const and ϕ = const, i. e. out of all spatial condition g00 = 0 is true, hence the observable time flow stops and directions only radial one will be considered. Then an arbitrary collapse occurs. three-dimensional observable interval is In other word, an area of a de Sitter space within the radius r = R √6 stays in collapse. Taking into account that vacuum, which dr r ∗ σ = h11 dr = = rinf arcsin + const, (5.107) 2 r fills any de Sitter space, stays in inflation, we will refer to such Z Z 1 Kr inf collapse as inflational collapse to differ it from gravitational one p − so the observable inflationalp radius is constant (which occurs in Schwarzschild spaces), while the value r = R∗√6 r (5.77) will be referred to as the inflational radius rinf . Then the inf dr π collapsed area of the de Sitter space within the inflational radius σinf = = rinf . (5.108) 0 1 Kr2 2 will be referred to as the inflational collapsar, or as inflanton. Z − Inside an inflanton we have K > 0, so the three-dimensional In a space with Schwarzschildp metric, which we looked at in observable curvature is C < 0. In this case the vacuum density is the previous §5.6, a collapsar is a collapsed compact mass, which negative (the inflational pressure is positive, vacuum compresses) produces the curvature of the space as a whole. A regular observer and λ > 0, so there are non-Newtonian forces of repulsion. This of a Schwarzschild space stays outside gravitational collapsar. 5.8 The concept of the mirror Universe 249 250 Chapter 5 Vacuum and the mirror Universe

In a de Sitter space a collapsar is vacuum, which fills the whole of negative masses; rather, our empirical experience says that they space. A collapse area there is comparable to a surface, which radius have never been observed. Both Newtonian theory of gravitation equals the radius of the space curvature. So, a regular observer of and Einstein’s the General Theory of Relativity predicted behaviour a de Sitter space stays under the surface of inflational collapsar and of negative masses, totally different from what electrodynamics he “watches” it from within. prescribes for negative charges. For two bodies, one of which bears To look beyond an inflational collapsar we present de Sitter positive mass and the other bears negative one, but equal to the first metric with K > 0 (5.105) for r > rinf . Considering radial directions, one in the absolute value, it would be expected that positive mass in coordinates of a regular observer (“inner” coordinates of the will attract the negative one, while the negative mass will repulse collapsar) we obtain the positive one, so that one will chase the other! If motion occurs along a line which links the centres of the two bodies, such system r2 dr2 2 2 2 will move with a constant acceleration. This problem had been ds = 2 1 c dt + 2 , (5.109) − rinf − r 1 studied by Bondi [35]. Assuming the gravitational mass of positron 2 − rinf to be negative (observations say that its inertial mass is positive) and or, from viewpoint of an observer, who is located outside the col- using Quantum Electrodynamics’ methods, Schiff had obtained that lapsar (in its “external” coordinates r = ct˜ and ct =r ˜), we have thre is a difference between the inertial mass of positron and its gravitational mass. The difference proved to be much greater than 2 ˜2 2˜2 2 c dt c t 2 the error margin in Eotv¨ os’¨ experiment, which showed equality ds = 2 2 2 1 dr˜ . (5.110) c t˜ − rinf − of gravitational and inertial masses [36]. As a result, Schiff had 2 1 rinf − concluded that a negative gravitational mass in positron can not exist (see Chapter 1 in the Weber book [23]). Besides, “co-habitation” of positive and negative masses in the §5.8 The concept of the mirror Universe. Conditions of transition same space-time area would cause ongoing annihilation. Possible through the membrane from our world into the mirror consequences of particles of a “mixed” kind, which bear both pos- Universe itive and negative masses, were also studied by Terletskii [37, 38]. Therefore this idea of the mirror Universe as a world of negative As we mentioned in §5.1, attempts to represent our Universe and masses and energies faced two obstacles: the mirror Universe as two spaces with positive and negative cur- (a) The experimentum crucis, which would point directly at ex- vature failed: even within de Sitter metric, which is among the change interactions between our Universe and the mirror simplest space-time metrics, trajectories in a positive curvature Universe; space are substantially different from those in its negative curvature twin (see Chapter VII in the Synge book [29]). (b) The absence of a theory, which would clearly explain sep- On the other hand, numerous researchers, beginning from Paul aration of the worlds with positive and negative masses as Dirac, intuitively predicted that the mirror Universe (as the anti- different space-time locations. pode to our Universe) must be sought not in a space with the In this §5.8 we are going to tackle the second (theoretical) part opposite curvature sign, but rather in a space, where particles bear of the problem. masses and energies with the opposite sign. That is, because masses Let us look at the term “mirror properties” as applied to the of particles in our Universe are positive, then those of particles in space-time metric. To solve this problem we write the square of the mirror Universe must be evidently negative. the space-time interval in chr.inv.-form, namely Joseph Weber [23] wrote that neither Newton’s law of gravita- ds2 = c2dτ 2 dσ2, (5.111) tion nor the relativistic theory of gravitation ruled out existence − 5.8 The concept of the mirror Universe 251 252 Chapter 5 Vacuum and the mirror Universe where Let us divide both parts of the formula for ds2 (5.115) by the 2 i k dσ = hik dx dx , (5.112) next quantities, according to the kind of particle’ trajectory: v2 w 1 w + v ui 1. c2dτ 2 1 if the space-time interval is real ds2 > 0; i i − c2 dτ = 1 2 dt 2 vi dx = 1 2 dt . (5.113) − c − c − c 2. c2dτ 2if the space-time interval equals zero ds2 = 0; 2 From here we see that an elementary spatial interval (5.112) 3. c2dτ 2 v 1 if the interval is imaginary ds2 < 0. is a square function of elementary spatial increments dxi. Spatial − c2 − coordinates xi are all equal, so there is no principal differences As a result in all cases we obtain the same square equation in between translational movement to the right or to the left, up or respect of the function of the “true coordinate time” t from the down. Therefore we will no longer consider mirror reflections in observer’s measured physical time τ, namely — the equation respect of spatial coordinates. dt 2 2v vi dt 1 1 Time is a different thing. Physical observable time τ for a regular i + v v vivk 1 = 0 , (5.117) 2 4 i k observer always flows from past into future, so that dτ > 0. But dτ − 2 w dτ w c − c 1 2 1 2 there are two cases where time stops. At first, it is possible in a − c − c regular space-time in the state of collapse. At second, this happens which has two solutions in a zero-space — the fully degenerated four-dimensional space- dt 1 1 time. Therefore the state of an observer, whose own observable = v vi + 1 , (5.118) w 2 i time stops, may be regarded transitional one, i. e. unavailable under dτ 1 1 c c2 regular conditions. − We will consider the problem of the mirror Universe for both dt 1 1 = v vi 1 . (5.119) dτ > 0 and dτ = 0. In the last case the analysis will be done separa- w 2 i dτ 2 1 c − tely for collapsed areas of the regular space-time and for the zero- − c2 space. We start the analysis from a regular case of dτ > 0. From the Having t integrated with respect to τ, we obtain formula for physical observable time (5.113) it is evidently, that this 1 v dxi dτ condition is true if t = i + const. (5.120) 2 w w w + v ui < c2. (5.114) c 1 ± 1 i Z − c2 Z − c2

In the absence of the space rotation (vi = 0) this formula becomes It can be easily integrated, if the space does not rotate and w < c2, which corresponds the space-time structure in the state of gravitational potential is w = 0. Then the integral is t = τ + const. collapse. Proper choice of the initial conditions can make integration± constant Then ds2 (5.111) can be expanded as follows zero. In this case we obtain

2 2 w 2 2 w i t = τ , τ > 0 , (5.121) ds = 1 2 c dt 2 1 2 vi dx dt ± − c − − c − (5.115) 1 that graphically represents two beams, which are mirror reflections h dxidxk + v v dxidxk, − ik c2 i k of each other in respect of τ > 0. We can say that the observer’s own time serves here as the mirror membrane, which mirror surface on the other hand separates two worlds: one with the direct flow of coordinate time∗ 2 2 2 2 2 2 2 v 2 i k Any observer’s measured physical time τ everywhere flows from past into ds = c dτ dσ = c dτ 1 , v = hikv v . (5.116) ∗ − − c2 future, so the condition dτ > 0 is true in any observer’s reference frame. 5.8 The concept of the mirror Universe 253 254 Chapter 5 Vacuum and the mirror Universe from past into future t = τ, and the other, the mirror one, with the then reverse flow of coordinate from future into past t = τ. dτ dτ − t = = , (5.125) Noteworthy, the world with the reverse flow of time is not like ± 2 ± r2 Z 1 λr Z 1 a videotape being rewound. The both worlds are quite equal, but 3 − r2 r − r inf for a regular observer the mentioned time coordinate in the mirror which implies that the closer is the measured photometric distance Universe are negative. The mirror surface of the membrane in this r to the inflational radius in the space, the faster is the coordinate case only reflects the time flow, but does not affect it. time flow. In the ultimate case at r r we have t . → inf → ∞ Now we assume that the space does not rotate vi = 0, but gravi- Therefore, in the absence of the space rotation but in the pres- tational potential is not zero w = 0. Then we have 6 ence of a gravitational field, the coordinate time flow is the faster dτ the stronger is the field potential. This is true both in a Newtonian t = + const. (5.122) gravitational field and in a field on non-Newtonian gravitation. ± 1 w Z − c2 Now we turn to a more general case, when both the space rotation and gravitational fields are present. Then the integral for If gravitational potential is weak (w c2), then we obtain that t takes the form (5.120), so coordinate time in a non-holonomic our integral (5.122) becomes (rotating) space includes: 1 1. “Rotational” time determined by the presence of the term t = τ + wdτ = (τ + Δt) , (5.123) v dxi, which has dimension of rotational momentum divided ± c2 ± i Z by an unit mass; where Δt is a correction to take into account that field w, which 2. Regular coordinate time, linked to the pace of the observer’s produces acceleration. This quantity w may define any scalar po- measured time. tential field — either a field of Newtonian potential or a field of From the integral for t (5.120) we see that the “rotational” non-Newtonian gravitation. coordinate time, produced by the space rotation, exists indepen- If a field of gravitation produced by the potential w is strong, dently from the observer because it does not depend from τ. For an then this integral will become as of (5.122) and it will depend on the observer who is at rest on the Earth surface (anywhere aside for the potential w: the stronger is the field w, the faster is the coordinate poles) it can be interpreted as the time flow determined by rotation 2 time flow (5.122). In the ultimate case, where w = c , we have t . of the planet. It always exists irrespectively of whether the observer 2 → ∞ On the other hand, at w = c collapse occurs dτ = 0. We will look at records it in this particular location or not. Regular coordinate time 2 that case in the below, but now we are still assuming w < c . is linked to our presence (it depends from our measured time τ) and Let us look at coordinate time in a Schwarzschild space and a to that field, which exists in the point of observation; in particular, de Sitter space. If the potential w describes a Newtonian gravita- to the field of Newtonian potential. tional field (the space with Schwarzschild metric), then Noteworthy, at vi = 0 time coordinate t at the initial moment of observation (when the6 observer’s measured time is τ = 0) is not dτ dτ 0 zero. t = = r , (5.124) ± 1 GM ± 1 g Presenting the integral for t (5.120) as follows Z − c2r Z − r 1 i which implies that the closer we approach the gravitational radius 2 vi dx dτ t = c ± , (5.126) of the mass, the bigger is the difference between coordinate time 1 w and the observer’s measured time. If w is the potential of a non- Z − c2 Newtonian field of gravitation (the space with de Sitter metric), we obtain that the formula under the integral sign is: 5.8 The concept of the mirror Universe 255 256 Chapter 5 Vacuum and the mirror Universe

(a) Positive quantity, if 1 v dxi > dτ ; dipole consisting of a positive mass +m and a negative mass m. But c2 i ∓ in projection of P α on the spatial section, both its projections− merge (b) Zero quantity, if 1 v dxi = dτ ; c2 i ± into a single one — the particle’s three-dimensional observable i i 1 i impulse p = mv . In other word, each observable particle with a (c) Negative quantity, if vi dx < dτ. c2 ∓ positive relativistic mass has its own mirror twin with the same Hence coordinate time t for a real observer stops, if the scalar negative mass: the particle and its mirror twin are only different product of the linear velocity of the space rotation and the observ- by the sign of mass, while three-dimensional impulses of both the i 2 able velocity of the object is viv = c . This happens, if numerical particles are positive. ± values of the both velocities equal to that of light, and they are Similarly, for the four-dimensional wave vector either co-directed or oppositely directed. α α i 2 α ω dx dx An area of the space-time, where the condition viv = c is K = = k , (5.131) true, so that coordinate time stops for a real observer, is the mirror± c dσ dσ membrane separating two areas of positive and negative time coor- which describes a massless particle, chr.inv.-projections are dinate — areas with the direct and reverse flow of time. K0 k It is also evidently that no regular observer, who is located in = k , Ki = ci. (5.132) g ± c an Earth regular laboratory, can accompany such space. √ 00 We will refer as the mirror space to an area of the space- This implies that any massless particle, as a four-dimensional time, where coordinate time takes negative numerical values. Let object, also exists in two states: in our world with the direct time us analyze properties of particles, which inhabit the mirror space flow it is a massless particle with a positive frequency, while in the in respect of those of particles located in the regular space, where world with the reverse time flow it is a massless particle with the time coordinate is positive. same negative frequency. The four-dimensional impulse vector of a mass-bearing particle, We define the material Universe as the four-dimensional space- which bears a non-zero rest-mass m0, is time, filled with a substance and fields. Then because any particle α is a four-dimensional dipole object, we can say that the material α dx P = m0 , (5.127) Universe as a combination of the basic space-time and particles is ds also a four-dimensional dipole object, which exists in two states: which chr.inv.-projections are as our Universe, inhabited by particles with positive masses and P dt m frequencies, and as its mirror twin — the mirror Universe, where 0 = m = m , P i = vi, (5.128) √g00 dτ ± c masses and frequencies of particles are negative, while three- dimensional observable impulse remains positive. On the other where “plus” stands for the direct flow of coordinate time, while hand, our Universe and the mirror Universe have the same back- “minus” stands for the reverse time flow in respect of the observer’s α ground four-dimensional space-time. measured time. The square of the vector P is For instance, analyzing properties of the Universe as a whole, we α α β 2 neglect action of Newtonian fields, produced by occasional islands PαP = gαβ P P = m0 , (5.129) of substance, so we assume the basic four-dimensional space of our while its length equals Universe to be a de Sitter space with the negative four-dimensional α curvature, while its three-dimensional observable curvature is pos- PαP = m0 . (5.130) itive (see §5.5 herein). Hence we can assume that our Universe p Therefore any particle with non-zero rest-mass, being a four- as a whole is an area in the de Sitter space with the negative four- dimensional structure, is projected on the observer’s time line as a dimensional curvature, where the time coordinate is positive as well 5.8 The concept of the mirror Universe 257 258 Chapter 5 Vacuum and the mirror Universe as masses and frequencies of particles located in the area. Besides, should be found there, because their rest-masses are real (in that vice versa, the mirror Universe is an area of the same de Sitter time they are regular particles), while their relativistic masses space, where the time coordinate is negative as well as masses and become imaginary after only as the particles become super-light frequencies of particles located in it. tachyons. The space-time membrane, which separates our Universe and On the surface of any collapsar the term “observable velocity” the mirror Universe in the basic space-time, does not allow them is void, because the observer’s measured time stops there dτ = 0. to “mix”, thus preventing total annihilation. This membrane will Components of the four-dimensional impulse vector of a particle be discussed at the end of this §5.8. found on the surface of a collapsar (5.136), can be formally written Let us turn to the dipole structure of the Universe for dτ = 0, as follows m c m so we will consider collapsed areas of the regular space-time and a P 0 = 0 ,P i = 0 ui. (5.139) fully degenerated space-time area (zero-space). u u As we have shown, the condition dτ = 0 is true in a regular But as a matter of fact that we can not observe such particle, (non-degenerated) space-time, where collapse occurs and the space because on the surface of a collapsar our observable time stops. is holonomic (it does not rotate). Then i On the other hand, the velocity ui = dx , found in this formula, is dt w dτ = 1 dt = 0 . (5.133) coordinate one and it does not depend from the observer’s measured − c2 time which stops there. Hence we can interpret the spatial vector 3 i m0 i m0c This condition is true for collapse of any kind, so for fields P = u u as the particle’s coordinate impulse, while u can be of gravitational potential w of any kind, including non-Newtonian interpreted as the particle’s energy. Here the energy has only one potential. At dτ = 0 (5.133) the four-dimensional metric is sign, so the surface of any collapsar as a four-dimensional area is not a dipole four-dimensional object, presented by two mirror 2 2 i k i k i k 2 ds = dσ = hik dx dx = gik dx dx = gik u u dt , (5.134) twins. The surface of any collapsar, irrespective of its Newtonian − − or non-Newtonian nature, exists in a single state. hence in this case the absolute value of the interval ds equals On the other hand, the surface of a collapsar (g00 = 0) can be regarded as a membrane, which separates four-dimensional areas of ds = idσ = i h uiuk dt = iudt , u2 = h uiuk, (5.135) | | ik ik the space-time before the collapse and after the collapse. Before the so that the four-dimensionalp impulse vector on the surface of a collapse we have g00 > 0, so the observer’s measured time τ is real. collapsar is After the collapse we have g00 < 0, thus τ becomes imaginary. When dxα the observer, penetrating into the collapsar, crosses the surface then P α = m , dσ = udt . (5.136) 0 dσ his measured time subjects to 90◦ “rotation”, swapping roles with Its square is his measured spatial coordinates. The term “light-like particle” has no sense in the surface of a P P α = g P αP β = m2 , (5.137) α αβ − 0 collapsar, because for light-like particles we have dσ = cdτ so on the hence the length of the vector P α (5.136) is imaginary surface (dτ = 0) for such particles

α i k PαP = im0 . (5.138) i k hik dx dx dσ cdτ u = hik u u = 2 = = = 0 . (5.140) p r dt dt dt The latter, in particular, implies that the surface of the collapsar p is inhabited by particles with imaginary rest-masses. But, at the The observer’s measured time also stops dτ = 0 in a fully degen- same time, this does not imply that super-light particles (tachyons) erated space-time (zero-space). There, by definition, the conditions 5.8 The concept of the mirror Universe 259 260 Chapter 5 Vacuum and the mirror Universe dτ = 0 and dσ = 0 are true. The degeneration conditions can be also in a generalized space-time, which permits total degeneration written as follows of the metric. 2 Conditions inside the membrane (t = const, so that dt = 0), in i 2 i k 2 w w + viu = c , gik u u = c 1 . (5.141) accordance with (5.144) are defined by the formula − c2 i 2 Particles found in the degenerated space-time (zero-particles) vi dx c dτ = 0 , (5.145) ± bear zero relativistic mass m = 0, but non-zero mass M (1.71) and non-zero constant-sign impulse which can be also written in the form m v vi = c2. (5.146) M = , pi = Mui. (5.142) i ± 1 1 (w + v ui) − c2 i This condition can be presented as follows Therefore, mirror twins are only found in regular matter — v vi = v vi cos (\v ; vi) = c2. (5.147) massless and mass-bearing particles, which are not in the state i i i ± of collapse. Collapsed objects in the regular space-time (including From here we see that it is true, if numerical values of the i black holes), which do not possess the property of mirror dipoles, velocities vi and v equal to that of light and are either co-directed are common objects for our Universe and the mirror Universe. (“plus”) or oppositely directed (“minus”). Zero-space objects, which neither possess the property of mirror Thus the membrane from physical viewpoint is a space which dipoles, lay beyond the basic space-time due to total degeneration experiences translational motion at the light velocity and at the of the metric. It is possible to enter “neutral zones” on the surfaces same time rotates also at the light velocity, so it is a world of light- of collapsed objects of the regular space and in the zero-space like spiral trajectories. It is possible, such space may be attributed from either our Universe (where coordinate time is positive) or to particles, which possess the spirality property (e. g. photons). the mirror Universe (where coordinate time is negative). Having dt = 0 substituted into the formula for ds2 we obtain the Now we need to discuss the question of the membrane which metric inside the membrane separates our Universe and the mirror Universe in the basic space- 2 i k time, thus preventing total annihilation of all particles with negative ds = gik dx dx , (5.148) and positive masses. which is the same as on the surface of a collapsar. Because it is a In our Universe dt > 0, in the mirror Universe dt < 0. Hence the particular case of a space-time metric with signature (+ ), then membrane is an area of the space-time, where dt = 0 so coordinate −−− ds2 is always positive. This implies that in an area of the four- time stops. It is an area, where dimensional space-time, which serves the membrane between our

dt 1 1 i Universe and the mirror Universe, the four-dimensional interval is = 2 viv 1 = 0 , (5.143) dτ 1 w c ± space-like. The difference from the space-like metric on the surface − c2 of a collapsar (5.134) is that there is no any rotation of the space so 1 which can be also presented as the physical condition that gik = hik, while in this case gik = hik + c2 vivk (1.18). Or, in other word,− inside the membrane we have− 1 1 dt = v dxi dτ = 0 . (5.144) w 2 i 2 i k i k 1 i k 1 c ± ds = gik dx dx = hik dx dx + vivk dx dx , (5.149) − c2 − c2 The latter notation is more versatile, because of being applicable so the four-dimensional metric there becomes space-like due to the not only in the space-time of the General Theory of Relativity, but space rotation, which makes the condition v dxi = c2dτ true. i ± 5.9 Conclusions 261 262 Chapter 5 Vacuum and the mirror Universe

As a result a regular mass-bearing particle (irrespective of the the basic space-time, where from viewpoint of a regular observer sign of its mass) can not in its “natural” form pass through the time coordinate is negative so all particles bear negative masses membrane: this area of the space-time is inhabited by light-like and energies. From viewpoint of an our-world observer the mirror particles which move along light-like spirals. Universe is a world with the reverse flow of time, where particles On the other hand the ultimate case of particles with m > 0 or travel from future into past in respect of us. m < 0 are particles with zero relativistic mass m = 0. From geometric The two worlds are separated with the membrane — an area of viewpoint the area, where such particles are found, is tangential to the space-time, inhabited by light-like particles which travel along areas inhabited by particles with either m > 0 or m < 0. This implies light-like spirals. In the scale of elementary particles such space can that zero-mass particles may have exchange interactions with either be attributed to particles which possess spirality (e. g. photons). our-world particles m > 0 or mirror-world particles m < 0. This membrane prevents mixing of positive-mass and negative- Particles with zero relativistic mass, by definition, exist in an mass particles, so it prevents their total annihilation. Exchange area of the space-time where ds2 = 0 and c2dτ 2 = dσ2 = 0. Equalizing interactions between the two worlds can be effected through par- ds2 to zero inside the membrane (5.148) we obtain ticles with zero relativistic masses (zero-particles) under physical conditions, which exist on the surfaces of collapsars in the fully 2 i k ds = gik dx dx = 0 , (5.150) degenerated space-time (zero-space). so this condition may be true in two cases: (1) All values of dxi are zeroes, so dxi = 0; ♦ (2) The three-dimensional metric is degenerated g˜ = det g = 0. || ik|| The first case may occur in the regular space-time under the ultimate conditions on the surface of a collapsar: when all the surface shrinks into a point, all dxi = 0 so the metric on the surface 2 i k i k according to ds = hik dx dx = gik dx dx (5.134) becomes zero. The second case− occurs on the surface of a collapsar located in 2 the zero-space: because the condition g dxidxk = 1 w c2dt2 is ik − c2 2 i k true there, then at w = c we have gik dx dx = 0 always. The first case is asymptotic, because it never occurs in reality. Hence we can expect that “middlemen” in exchanges between our Universe and the mirror Universe are those particles with zero relativistic mass, which inhabit the surfaces of collapsars located in the fully degenerated space-time. In other word, the “middlemen” are those zero-particles, which inhabit the surfaces of collapsars in the zero-space.

§5.9 Conclusions

So we have shown that our Universe is the observable area of the basic space-time, where time coordinate is positive so all particles bear positive masses and energies. The mirror Universe is an area of 264 Notations

Electromagnetic fields Aα the four-dimensional potential of electromagnetic field ϕ the time chr.inv.-component of Aα (the physical observ- Appendix A able scalar potential of electromagnetic field) Ai spatial chr.inv.-components of Aα (the physical observ- NOTATIONS able vector-potential of electromagnetic field) F αβ the Maxwell tensor of electromagnetic field ik Ei, E∗ the three-dimensional chr.inv.-stress of electric fields i The theory of chronometric invariants Hik, H∗ the three-dimensional chr.inv.-stress of magnetic fields

bα the four-dimensional monad vector Riemannian space h the three-dimensional metric chr.inv.-tensor ik α τ the physical observable time x four-dimensional coordinates i dσ the spatial physical observable interval x three-dimensional coordinates vi the three-dimensional chr.inv.-velocity t the coordinate time ds the space-time interval Aik the three-dimensional antisymmetric chr.inv.-tensor of the space’s rotation (the non-holonomity tensor) gαβ the four-dimensional fundamental metric tensor δα the unit four-dimensional tensor vi the three-dimensional linear velocity of the space rota- β tations J the determinant of the Jacobi matrix (Jacobian) αβμν F i the three-dimensional chr.inv.-vector of gravitational e the four-dimensional completely antisymmetric unit inertial force tensor ikm w the gravitational potential e the three-dimensional completely antisymmetric unit ci the three-dimensional chr.inv.-velocity of light tensor Eαβμν the four-dimensional completely antisymmetric tensor Dik the three-dimensional chr.inv.-tensor of the rate of the ikm space deformations ε the completely antisymmetric chr.inv.-tensor α i Γμν the Christoffel symbols of the 2nd kind Δjk the Christoffel chr.inv.-symbols of the 2nd kind Γμν,ρ the Christoffel symbols of the 1st kind Motion of particles Rαβμν the Riemann-Christoffel curvature tensor Tαβ the energy-momentum tensor α u four-dimensional velocity J i the vector of the chr.inv.-density of momentum i u three-dimensional coordinate velocity U ik the chr.inv.-tensor of momentum flux (the stress tensor) α P the four-dimensional impulse vector Rαβ the Ricci tensor i p the three-dimensional impulse vector K the four-dimensional curvature α K the four-dimensional wave vector C the three-dimensional chr.inv.-curvature i k the three-dimensional wave vector λ the cosmological term (λ-term) ψ the wave phase (eikonal) S action L the Lagrange function (Lagrangian) ~αβ the four-dimensional antisymmetric Planck tensor αβ ~∗ the dual four-dimensional Planck pseudotensor 266 Special expressions

∗∂ ∂ 1 ∗∂ Chr.inv.-derivative with respect = + v ∂xi ∂xi c2 i ∂t to spatial coordinates The square of the physical ob- v2 = v vi = h vivk Appendix B i ik servable velocity

i 0i k The linear velocity of the space SPECIAL EXPRESSIONS v = cg √g00, vi = hik v − rotations

The square of vi. This is the σβ β proof: because of gασg = gα, α 2 i k ∂A v = hik v v then under α = β = 0 we have dAα = dxσ Ordinary differential of a vector σ0 0 σ g0σg = δ0 = 1, hence we become ∂x 2 2 00 v = c (1 g00g ) − DAα = Aαdxβ = dAα + Γα Aμdxβ Absolute differential of a contra- ∇β βμ variant vector The relation between the deter- g = h g00 minants of the metric tensors gαβ β μ β Absolute differential of a covar- − DAα = β Aαdx = dAα Γ Aμdx and hαβ ∇ − αβ iant vector p p p α ∂A d ∗∂ k ∗∂ Derivative with respect to physi- Aα = + Γα Aμ Absolute derivative of a contra- = + v ∇β ∂xβ βμ variant vector dτ ∂t ∂xk cal observable time ∂A d 1 d The 1st derivative with respect to A = α Γμ A Absolute derivative of a covariant = β α β αβ μ the space-time interval ∇ ∂x − vector ds c 1 v2 dτ c2 ∂F σα − F σα = + Γα F σμ + Γσ F αμ Absolute derivative of a contra- q β β βμ βμ d2 1 d2 1 ∇ ∂x variant 2nd rank tensor = + 2 2 2 2 2 The 2nd derivative with ∂F ds c v dτ (c2 v2) × F = σα Γμ F Γμ F Absolute derivative of a covariant − −  respect to the space-time β σα β αβ σμ σβ αμ 2nd rank tensor i  ∇ ∂x − − i k dv 1 ∗∂hik i k m d  interval Dikv v +vi + v v v  ∂Aα × dτ 2 ∂xm dτ Aα = + Γα Aσ Absolute divergence of a vector α ∂xα ασ  ∇ 1  h = g + v v i √ ik ik 2 i k i ∗∂q i ∗∂ ln h − c The metric chr.inv.-tensor ∗ i q = + q Chr.inv.-divergence of a vector  ∇ ∂xi ∂xi hik = gik, hk = δk − i i  1 i i i Physical chr.inv.-divergence ∗ i q = ∗ i q 2 Fi q iα kβ m iq ks m  ∇ ∇ − c g g Γαβ = h h Δqs , αβ D’Alembert’s general covariant i Zelmanov’s relations between e= g α β i i c i g0kΓ00  ∇ ∇ operator Dk + Ak∙ = Γ0k ,  the Christoffel regular symbols ∙ √g − g  00 00  and chr.inv.-characteristics of Δ = gik Laplace’s ordinary operator  − ∇i∇k c2Γk  the reference space F k = 00 ik Δ = h The Laplace chr.inv.-operator − g00  ∗ ∗ i ∗ k  ∇ ∇   ∗∂ 1 ∂ Chr.inv.-derivative with respect ∗∂A 1 ∗∂F ∗∂F = ik + k i = 0 Zelmanov’s 1st identity ∂t √g00 ∂t to time ∂t 2 ∂xi − ∂xk Special expressions 267

∗∂Akm ∗∂Ami ∗∂Aik i + k + m + ∂x ∂x ∂x Zelmanov’s 2nd identity 1  + (FiAkm + FkAmi + FmAik) = 0  2 Bibliography  d 2 d i k i k v = hikv v = 2Dikv v + dτ dτ Derivative from v2 with respect  k to physical observable time ∗∂hik i k m dv  1. Levi-Civita T. Nozione di parallelismo in una varieta` qualunque e + v v v + 2vk  ∂xm dτ consequente specificazione geometrica della curvatura Riemanniana.  Rend. Circolo mat. di Palermo, t. 42, 1917, 173–205. 0ikm  ikm 0ikm e 2. Terletskii Ya. P. Causality principle and the 2nd law of thermo- ε = √g00 E = , √h The completely antisymmetric dynamics. Doklady Acad. Nauk USSR, 1960, v. 133 (2), 329–332.  3. Feinberg G. Possibility of faster-than light particles. Physical Review, E0ikm  chr.inv.-tensor εikm = = e0ikm√h  1967, v. 159, 1089. √g  00 4. Zelmanov A. L. Chronometric invariants. Dissertation, 1944. First   published: CERN, EXT-2004-117, 236 pages. 5. Landau L. D. and Lifshitz E. M. The classical theory of fields. GITTL, Moscow, 1939 (referred with the 4th final revised edition, Butterworth–Heinemann, 1980, 428 pages). 6. Zelmanov A. L. Chronometric invariants and co-moving coordinates in the general relativity theory. Doklady Acad. Nauk USSR, 1956, v. 107 (6), 815–818. 7. Zelmanov A. L. To relativistic theory of anisotropic inhomogeneous Universe. Proceedings of the 6th Soviet Conference on Cosmogony, Nauka, Moscow, 1959, 144–174 (in Russian). 8. Zelmanov A. L. The problem of the deformation of the co-moving space in Einstein theory of gravitation. Doklady Acad. Nauk USSR, 1960, v. 135 (6), 1367–1370. 9. Zelmanov A. L. and Agakov V. G. Elements of the General Theory of Relativity. Nauka, Moscow, 1988, 236 pages (in Russian). 10. Cattaneo C. General Relativity: Relative standard mass, momentum, energy, and gravitational field in a general system of reference. Il Nuovo Cimento, 1958, v. 10 (2), 318–337. 11. Cattaneo C. On the energy equation for a gravitating test particle. Il Nuovo Cimento, 1959, v. 11 (5), 733–735. 12. Cattaneo C. Conservation laws in General Relativity. Il Nuovo Cimen- to, 1959, v. 13 (1), 237–240. 13. Cattaneo C. Problemes` d’interpretation´ en Relativite´ Gen´ erale.´ Colloques Internationaux du Centre National de la Recherche Scientifique, no. 170 “Fluides et champ gravitationel en Relativite´ Gen´ erale”,´ Editions´ du Centre National de la Recherche Scientifique, Paris, 1969, 227–235. Bibliography 269 270 Bibliography

14. Raschewski P. K. Riemannian geometry and tensor analysis. Nauka, 31. McWittie G. C., 1959, cited by Ivanenko’s preface to Russ. Ed. of [21]. Moscow, 1964, 664 pages (in Russian). 32. Oros di Bartini R. Some relation between physical constants. Doklady 15. Rabounski D. D. and Borissova L. B. Particles here and beyond the Acad. Nauk USSR, 1965, v. 163 (4), 861–864. Mirror. Editorial URSS, Moscow, 2001, 84 pages (e-print ed.: arXiv, 33. Oros di Bartini R. Relation between physical values. Problems of the gr-qc/0304018). theory of gravitation and elementary particles, Atomizdat, Moscow, 16. Terletskii Ya. P. and Rybakov J. P. Electrodynamics. High School, 1966, 249–266 (in Russian). Moscow, 1980, (in Russian). 34. Kottler F. Uber¨ die physikalischen Grundlagen der Einsteinschen 17. Papapetrou A. Spinning test-particles in General Relativity. I. Pro- Gravitationstheorie. Annalen der Physik, 1918, Ser. 4, Vol. 56. ceedings of the Royal Society A, 1951, v. 209, 248–258. 35. Bondi H. Negative mass in General Relativity. Review of Modern 18. Corinaldesi E. and Papapetrou A. Spinning test-particles in General Physics, 1957, v. 29 (3), 423–428. Relativity. II. Ibidem, 259–268. 36. Schiff L. I. Sign of gravitational mass of a positron. Physical Review 19. Del Prado J. and Pavlov N. V. Personal reports to A. L. Zelmanov, Letters, 1958, v. 1 (7), 254–255. 1968–1969. 37. Terletskii Ya. P. Cosmological consequences of the hypothesis of 20. Stanyukovich K. P. Gravitational field and elementary particles. negative masses. Modern problems of gravitation, Tbilisi Univ. Press, Nauka, Moscow, 1965 (in Russian). Tbilisi, 1967, 349–353 (in Russian). 21. Sudbery A. Quantum mechanics and the particles of nature. Cam- 38. Terletskii Ya. P. Paradoxes of the General Theory of Relativity. Pat- bridge University Press, Cambridge (UK), 1986. rice Lumumba Univ. Press, Moscow, 1965 (in Russian). 22. Stanyukovich K. P. On existence of the stable particles within the Metagalaxy. Problems of the theory of gravitation and elementary particles, Atomizdat, Moscow, 1966, 267–279 (in Russian). 23. Weber J. General Ralativity and gravitational waves. R. Marshak, New York, 1961 (referred with the 2nd edition, Foreign Literature, Moscow, 1962, 271 pages). 24. Petrov A. Z. Einstein spaces. Pergamon, London, 1969. 25. Grigoreva L. B. Chronometrically invariant representation of classification of Petrov’s gravitational fields. Doklady Acad. Nauk USSR, 1970, v. 192 (6), 1251. 26. Gliner E. B. Algebraic properties of energy-momemtum tensor and vacuum-like states of matter. Soviet Physics JETP–USSR, 1966, v. 49 (2), 543–548. 27. Gliner E. B. Vacuum-like state of a medium and Friedmann’s cosmology. Doklady Acad. Nauk USSR, 1970, v. 192 (4), 771–774. 28. Sakharov A. D. Initial stage of an explanding Universe and appearance of a nonuniform distribution of matter. Soviet Physics JETP–USSR, 1966, v. 22, 241. 29. Synge J. L. Relativity: the General Theory. North Holland, Amster- dam, 1960 (referred with the 2nd edition, Foreign Literature, Moscow, 1963, 432 pages). 30. Schouten J. A. und Struik D. J. Einfuhrung¨ in die neuren Methoden der Differentialgeometrie (Bd. I: Schouten J. A. Algebra und Ubertragungslehre;¨ Bd. II: Struik D. J. Geometrie). Zentralblatt fur¨ Mathematik, 1935, Bd. 11 und Bd. 19. 272 Index

Kottler’s metric 240 rotor 55

Lagrange’s function 155 scalar 32 Index Laplace’s operator 57 scalar product 37 law of mass-quantization 199–201 Schwarzschild’s metric 238 Levi-Civita T. 10 signature of space-time 8,155,220 accompanying observer 14 divergence 48, 51 — parallel transfer 10 de Sitter’s metric 239 action 95, 154 λ-term 215, 223, 233 de Sitter space 221, 229, 233, 234, antisymmetric tensor 40 eikonal (wave phase) 24 long-range action 27 235–240, 246–249 antisymmetric unit tensors 41 eikonal equation 24, 27 spatial section 14 magnetic “charge” 76 asymmetry of motion along time Einstein A. 214, 217 spin-impulse 149, 159 Mach’s Principle 217 axis 23, 26 Einstein’s constant 215 spirality 205 Einstein’s equations 214 Maxwell’s equations 69, 74–76 spur (trace) 37 metric fundamental tensor 9, 30 di Bartini R. 235–236 Einstein spaces 218–223, 238 Stanyukovich K. P. 77 metric observable tensor 17 — inversion relationship 235 Einstein’s tensor 215 state equation 228 Minkowski’s equations 82, 97 substance 223 Biot-Savart law 75 electromagnetic field tensor 63 mirror principle 25 Synge J. L. 221, 228, 249 bivector 33 elementary particles 199–204 mirror Universe 222 black hole 241 emptiness 214, 217, 225 monad vector 15 T-classification of matter 225 body of reference 13 energy-momentum tensor 86, 215, 223–226 multiplication of tensors 36 tensor 32 Terletskii Ya. P. 250 Cattaneo C. 13 equations of motion 9, 21 nongeodesic motion 28 time function 22 Christoffel E. B. 9 — charged particle 91–94 non-Newtonian gravitational for- time line 14 Christoffel’s symbols 9, 20, 34 — free particle 22–27 ces 223, 237, 239 trajectories 10 chronometric invariants 14 — spin-particle 164, 166 Compton wavelength 204 Galilean frame of references 41 operators of projecting on time unit tensor 16 conservation of electric charge 70 geodesic line 9 and space 15 continuity equation 72 geodesic (free) motion 9 vacuum 214, 217, 225 contraction of tensors 36 geometric object 32 Papapetrou A. 29 — physical properties 227 coordinate nets 13 Gliner E. B. 219 Pavlov N. V. 74 — μ-vacuum 220, 225, 227 coordinate velocity 155 gravitational collapse 240 Petrov A. Z. 218 vector product 40 current vector 72 gravitational inertial force 19 Petrov classification 219 viscous strengths tensors 227 curvature of space-time 216, 221– Petrov’s theorem 221 223 hologram 27 physical observable values 12–15 Weber J. 233 — scalar curvature 215 holonomity of space 14 Planck tensor 151–154 — physical observable cur- — non-holonomity tensor 19 Poynting vector 87 Zelmanov A. L. 11, 13, 22, 217, vature 231–234 horizon of events 234 del Prado J. 74 222, 230 pseudo-Riemannian space 8 Zelmanov’s theorem 221 d’Alembert’s operator 58 inflanton 247 pseudotensors 43 zero-particles 26 degenerated space-time 26 inflational collapse 247 deformation velocities tensor 20 Inversion Explosion 237 Ricci’s tensor 77 derivative 48 isotropic space 208, 213 Riemann B. 8 differential 9, 46 Riemann-Christoffel’s tensor 231 discriminant tensors 44–45 Jacobian 44 Riemannian space 8