Acoustic Torsion - Fields, Vacuum, And the Mirror Universe - Four-dimensional Pseudo-Riemannian Space

8/4/2019 Acoustic Torsion - Fields, Vacuum, And the Mirror Universe - Four-dimensional Pseudo-Riemannian Space

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Fields, Vacuum, and the Mirror UniverseL. B. Borissova and D. D. Rabounski

This study builds theory o motion o charged particles with spin in our-dimensionalpseudo-Riemannian space. The theory employs the mathematical apparatus o chronometricinvariants by A. L. Zelmanov (physical observable values in the General Theory o Relativity).Numerous special efects in electrodynamics, physics o elementary particles, and cosmologyare discussed. The efects explain anomalous annihilation o orthopositronium and link withstop-light experiment.

This is an electronic version o the book: Borissova L. B. and Rabounski D.D. Fields,vacuum, and the mirror Universe. Editorial URSS, Moscow, 2001.ISBN 5836000484 PACS: 04.20.Cv, 04.70.s, 98.80.Hw, 36.10.Dr

ContentsPre ace 3

Chapter 1 Introduction

1.1 Geodesic motion o particles 51.2 Physical observable values 81.3 Dynamic equations o motion o ree particles 131.4 Introducing concept o non-geodesic motion o particles.

Problem statement 16

Chapter 2 Tensor algebra and the analysis

2.1 Tensors and tensor algebra 19

2.2 Scalar product o vectors 222.3 Vector product o vectors. Antisymmetric tensors andpseudotensors 24

2.4 Introducing absolute diferential and derivative to the direction 272.5 Divergence and rotor 292.6 Laplace and dAlembert operators 352.7 Conclusions 37

Chapter 3 Charged particle in pseudo-Riemannian space

3.1 Problem statement 383.2 Observable components o electromagnetic eld tensor. Invariants

o the eld 393.3 Chronometrically invariant Maxwell equations. Law o

conservation o electric charge. Lorentz condition 42

3.4 Four-dimensional dAlembert equations or electromagneticpotential and their observable components 473.5 Chronometrically invariant Lorentz orce. Energy-impulse tensor

o electromagnetic eld 503.6 Equations o motion o charged particle obtained using parallel

trans er method 543.7 Equations o motion, obtained using the least action principle

as a partial case o the previous equations 58

E-mail: [email protected]


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3.8 Geometric structure o electromagnetic our-dimensionalpotential 60

3.9 Building Minkowski equations as a partial case o the obtainedequations o motion 64

3.10 Structure o the space with stationary electromagnetic eld 66

3.11 Motion o charged particle in stationary electric eld 673.12 Motion o charged particle in stationary magnetic eld 753.13 Motion o charged particle in stationary electromagnetic eld 853.14 Conclusions 91

Chapter 4 Particle with spin in pseudo-Riemannian space

4.1 Problem statement 924.2 Spin-impulse o a particle in the equations o motion 954.3 Equations o motion o spin-particle 984.4 Physical conditions o spin-interaction 1034.5 Motion o elementary spin-particles 1054.6 Spin-particle in electromagnetic eld 1104.7 Motion in stationary magnetic eld 1144.8 Law o quantization o masses o elementary particles 122

4.9 Compton wavelength 1254.10 Massless spin-particle 1264.11 Conclusions 130

Chapter 5 Physical vacuum and the mirror Universe

5.1 Introduction 1315.2 Observable density o vacuum. T-classi cation o matter 1355.3 Physical properties o vacuum. Cosmology 1375.4 Concept o Inversional Explosion o the Universe 1425.5 Non-Newtonian gravitational orces 1445.6 Gravitational collapse 1455.7 In ational collapse 1495.8 Concept o the mirror Universe. Conditions o transition

through membrane rom our world into the mirror Universe 151

5.9 Conclusions 158Chapter 6 Annihilation and the mirror Universe

6.1 Isotope anomaly and T -anomaly o orthopositronium.The history and problem statement 159

6.2 Zero-space as home space or virtual particles. Interpretationo Feynman diagrams in General Relativity 161

6.3 Building mathematical concept o annihilation.Parapositronium and orthopositronium 165

6.4 Annihilation o orthopositronium: 2+1 split o 3-photonannihilation 167

6.5 Isotope anomaly o orthopositronium 1696.6 Conclusions 170

Appendix A Notation 172Appendix B Special expressions 174Bibliography 176Index 179

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Pre ace

This book is addressed to readers who want to have a look at the laws o micro and macro worldrom a single viewpoint. This is the English translation o our Theory o Non-Geodesic Motion o Particles , originally published in Russian in 1999, with some recent amendements.

The background behind the book is as ollows. In 1991 we initiated a study to nd out what kindso particles may theoretically inhabit our-dimensional space-time. As the instrument, we equippedourselves with mathematical apparatus o physical observable values (chronometric invariants) devel-oped by A. L. Zelmanov, a prominent cosmologist.

The study was completed by 1997 to reveal that aside or mass-bearing and massless (light-like)particles, those o third kind may exist. Their trajectories lay beyond regular space-time o GeneralRelativity. For a regular observer the trajectories are o zero our-dimensional length and zero three-dimensional observable length. Besides, along these tra jectories interval o observable time is also zero.Mathematically, that means such particles inhabit ully degenerated space-time with non-Riemanniangeometry. We called such space zero-space and such particles zero-particles.

For a regular observer their motion in zero-space is instant, i. e. zero-particles are carriers o long-range action. Through possible interaction with our worlds mass-bearing or massless particleszero-particles may instantly transmit signals to any point in our three-dimensional space.

Considering zero-particles in the rames o the wave-particle concept we obtained that or a reg-ular observer they are standing waves and the whole zero-space is a system o standing light-likewaves (zero-particles), i. e. a standing-light hologram. This result links with stop-light experiment(Cambridge, Massachusetts, January 2001).

Using methods o physical observable values we also showed that in basic our-dimensional space-time a mirror world may exist, where coordinate time has reverse ow in respect to the viewpoint o regular observers time.

We presented the results in 1997 in two pre-prints 1 .B. M. Levin, an expert in orthopositronium problem came across these publications. He contacted

us immediately and told us about critical situation around anomalies in annihilation o orthopositro-nium, which had been awaiting theoretical explanation or over a decade.

Rate o annihilation o orthopositronium (the value reciprocal to its li e span) is among the re er-ences set to veri y the basic laws o Quantum Electrodynamics. Hence any anomalies contradict withthese reliably proven laws. In 1987 Michigan group o researchers (Ann Arbor, Michigan, USA) usingadvanced precision equipment revealed that the measured rate o annihilation o orthopositroniumwas substantially higher compared to its theoretical value.

That implies that some atoms o orthopositronium annihilate not into three photons as requiredby laws o conservation, but into lesser number o photons, which breaks those laws. In the same 1987

Levin discovered what he called isotope anomaly in anomalous annihilation o orthopositronium(GatchinaSt.Petersburg, Russia). Any attempts to explain the anomalies by means o QuantumElectrodynamics over 10 years would ail. This made S. G. Karshenboim, a prominent expert in theeld, to resume that all capacities o standard Quantum Electrodynamics to explain the anomalieswere exhausted.

In our 1997 publications Levin saw a means o theoretical explanation o orthopositronium anoma-lies by methods o General Relativity and suggested a joint research efort in this area.

1 Borissova L.B. and Rabounski D. D. Movement o particles in our-dimensional space-time. Lomonossov Workshop,Moscow, 1997 ( in Russian ); Rabounski D. D. Three orms o existence o matter in our-dimensional space-time.Lomonossov Workshop, Moscow, 1997 ( in Russian ).

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PREFACE 4

Solving the problem we obtained that our world and the mirror Universe are separated with aspace-time membrane, which is a degenerated space-time (zero-space). We also arrived to physicalconditions under which exchanges may occur between our world and the mirror Universe. Thanks tothis approach and using methods o General Relativity we developed a geometric concept o virtualinteractions: it was mathematically proven that virtual particles are zero-particles that travel in zero-space and carry long-range action. Application o the results to annihilation o orthopositroniumshowed that two modes o decay are theoretically possible: (a) all three photons are emitted into ourUniverse; (b) one photon is emitted into our world, while two others go to the mirror Universe andbecome unavailable or observation.

All the above results stemmed exclusively rom application o Zelmanovs mathematical apparatuso physical observable values.

When tackling the problem we had to amend the existing theory with some new techniques. In theiramous The Classical Theory o Fields , which has already become a de- acto standard or a universityre erence book on General Relativity, L. D. Landau and E. M. Li shitz give an excellent account o theory o motion o particle in gravitational and electromagnetic elds. But the monograph does notcover motion o spin-particles, which leaves no room or explanations o orthopositronium experiments(as its para and ortho states difer by mutual orientation o electron and positron spins). Besides,Landau and Li shitz employed general covariant methods. The technique o physical observable values

(chronometric invariants) has not been yet developed by that time by Zelmanov, which should be alsotaken into account.There ore we aced the necessity to introduce methods o chronometric invariants into the existing

theory o motion o particles in gravitational and electromagnetic elds. Separate consideration wasgiven to motion o particles with inner mechanical momentum (spin). We also added a chapter withaccount o tensor algebra and analysis. This made our book a contemporary supplement to TheClassical Theory o Fields to be used as a re erence book in university curricula.

In conclusion we would like to express our sincere gratitude to Dr. Abram Zelmanov (19131987)and Pro . Kyril Stanyukovich (19161989). Many years o acquaintance and hours o riendly conver-sations with them have planted seeds o undamental ideas which by now grew up in our minds to bere ected on these pages.

We are grate ul to Dr. Kyril Dombrovski whose works greatly in uenced our outlooks.We highly appreciate contribution rom our colleague Dr. Boris Levin. With enthusiasm peculiar

to him he stimulated our writing o this book.Special thanks go to our amily or permanent support and among them to Gershin Kaganovski or

discussion o the manuscript. Many thanks go to Grigory Semyonov, a riend o ours, or preparing themanuscript in English. We also are grate ul to our publisher Domingo Marn Ricoy or his interest toour works. Specially we are thank ul to Dr. Basil K. Malyshev who powered us by his BA KO MA -TEXsystem 2 .

L. B. Borissova and D. D. Rabounski

2 http://www.tex.ac.uk/tex-archive/systems/win32/bakoma/


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Chapter 1

Introduction

1.1 Geodesic motion o particles

Numerous experiments aimed at proving the conclusions o the General Theory o Relativity have alsoproven that its basic space-time ( our-dimensional pseudo-Riemannian space) is the basis o our realworld geometry. This also implies that even by progress o experimental physics and astronomy, whichwill discover new efects in time and space, our-dimensional pseudo-Riemannian space will remainthe cornerstone or urther widening o the basic geometry o General Relativity and will become oneo its speci c cases. There ore, when building mathematical theory o motion o particles, we areconsidering their motion speci cally in the our-dimensional pseudo-Riemannian space.

A terminology note should be taken at this point. Generally, the basic space-time in General Rela-tivity is a Riemannian space 1 with our dimensions with sign-alternating Minkowskis label (+ )or (+ ++). The latter implies 3+1 split o coordinate axis in Riemannian space into three spatialcoordinate axis and the time axis. For convenience o calculations a Riemannian space with (+ )signature is considered, where time is real while space is imaginary. Some theories, largely GeneralRelativity, also employs ( + ++) label with imaginary time and real space. But Riemannian spacesmay as well have non-alternating signature, e.g. (+ + ++). There ore a Riemannian space with alter-nating label is commonly re erred to as pseudo-Riemannian space , to emphasize the split o coordinateaxis into two types. But even in this case all its geometric properties are still properties o Rieman-nian geometry and the pseudo notation is not absolutely proper rom mathematical viewpoint.Nevertheless we are going to use this notation as a long-established and traditionally understood one.

We consider motion o a particle in our-dimensional pseudo-Riemannian space. A particle afectedby gravitation only alls reely and moves along the shortest ( geodesic) line. Such motion is re erredto as ree or geodesic motion . I the particle is also afected by some additional non-gravitationalorces, the latter divert the particle rom its geodesic trajectory and the motion becomes non-geodesicmotion .

From geometric viewpoint motion o a particle in our-dimensional pseudo-Riemannian space isparallel trans er o some our-dimensional vector Q which describes motion o the particle and isthere ore tangential to the tra jectory at any o its points. Consequently, equations o motion o particleactually de ne parallel trans er o vector Q along its our-dimensional trajectory and are equationso absolute derivative o the vector by certain parameter , which exists along all the trajectory o particles motion and is not zero along the way,

DQ

d=

dQ

d+ Q

dx

d, , , = 0 , 1, 2, 3. (1.1)

Here DQ = dQ + Q dx is the absolute diferential (the absolute increment o vector Q ),which is diferent rom a regular diferential dQ by presence o Christofel symbols o 2nd kind

1 A metric space which geometry is de ned by metric ds 2 = g dx dx called to as Riemanns metric. Bernhard Rie-mann (18261866), a German mathematician, the ounder o Riemannian geometry (1854).

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CHAPTER 1. INTRODUCTION 6

(coherence coe cients o Riemannian space), which are calculated through Christofel symbols(coherence coe cients) o 1st kind , and are unctions o rst derivatives o undamental metrictensor g 2

= g , , , =

12

gx

+gx

gx

. (1.2)

When moving along a geodesic trajectory ( ree motion) parallel trans er occurs in the meaning o Levi-Civita 3 . Here the absolute derivative o our-dimensional vector o particle Q equals to zero

dQ

d+ Q

dx

d= 0 , (1.3)

and the square o the vector being trans erred is conserved along all trajectory Q Q = const . Suchequations are re erred to as equations o motion o ree particles.

Kinematic motion o particle is characterized by our-dimensional vector o acceleration (also re-erred to as kinematic vector)

Q =dx

d, (1.4)

in parallel trans er by Levi-Civita equations o our-dimensional trajectories o ree particle are ob-tained ( equations o geodesic lines )

d2xd2

+ dxd

dxd

= 0 . (1.5)

Necessary condition =0 along the trajectory o motion implies that derivation parameters are notthe same along trajectories o diferent kind. In pseudo-Riemannian space three kinds o trajectoriesare principally possible, each kind corresponds to its type o particles:

non-isotropic real trajectories , that lay within the light hyper-cone. Along such trajectoriesthe square o space-time interval ds2> 0, while the interval ds is real. These are trajectories o regular sub-light-speed particles with non-zero rest-mass and real relativistic mass;

non-isotropic imaginary trajectories , which lay outside the light hyper-cone. Along such trajectories the square o space-time interval ds2< 0, while the interval ds is imaginary. These aretrajectories o super-light-speed tachyon particles with imaginary relativistic mass [18, 19];

isotropic trajectories , which lay on the sur ace o light hyper-cone and are trajectories o particleswith zero rest-mass (massless light-like particles), which travel at the light speed. Along theisotropic trajectories the space-time interval is zero ds2=0, but the three-dimensional interval isnot zero.

As a derivation parameter to non-isotropic trajectories space-like interval ds is commonly used.But it can not be used in such capacity to trajectories o massless particles ds=0. There ore as earlyas in 19411944 A. L. Zelmanov in his doctorate thesis proposed another variable that does not turninto zero along isotropic trajectories, to be used as derivation parameter to isotropic trajectories [6],

d2 = gik +g0i g0k

g00dx i dxk , (1.6)

which is a three-dimensional physical observable interval [6]. L. D. Landau and E. M. Li shitz also ar-rived to the same conclusion independently (see Section 84 in their The Classical Theory o Fields [1]).

Substituting respective diferentiation parameters into generalized equations o geodesic lines (1.5),we arrive to equations o non-isotropic geodesic lines (trajectories o mass-bearing particles)

d2x

ds2+

dx

dsdx

ds= 0 , (1.7)

2 Coherence coe cients o Riemannian space (Christofel symbols) are named a ter German mathematician ElvinBruno Christofel (18291900), who obtained them in 1869. In the Special Relativity space-time (Minkowski space) onecan always set an inertial system o re erence, where the matrix o undamental metric tensor is a unit diagonal tensorand all Christofel symbols become zeroes.

3 Tullio Levi-Civita (18731941), an Italian mathematician, who was the rst to study such parallel trans er [3].


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and to equations o isotropic geodesic lines (light propagation equations)

d2x

d2+

dx

ddx

d= 0 . (1.8)

But in order to get the whole picture o motion o particle, we have to build dynamic equations o

motion, which contain physical properties o particle (mass, requency, energy, etc.).From geometric viewpoint dynamic equations o motion are equations o parallel trans er o our-

dimensional dynamic vector o particle along its trajectory (absolute derivative o dynamic vector byparameter, not equal to zero along the trajectory, is zero).

Motion o ree mass-bearing particles (non-isotropic geodesic trajectories) is characterized by our-dimensional impulse vector

P = m0dx

ds, (1.9)

where m0 is rest-mass o particle. Parallel trans er in the meaning o Levi-Civita o our-dimensionalimpulse vector P gives dynamic equations o motion o ree mass-bearing particles

dP

ds+ P

dx

ds= 0 , P P = m20 = const. (1.10)

Motion o massless light-like particles (isotropic geodesic lines) is characterized by our-dimensionalwave vector

K =c

dx

d, (1.11)

where is speci c cyclic requency o massless particle. Respectively, parallel trans er in the meaningo Levi-Civita o vector K gives dynamic equations o motion o ree massless particles

dK

d+ K

dx

d= 0 , K K = 0 . (1.12)

There ore we have got dynamic equations o motion or ree massless particles. These are presentedin our-dimensional general covariant orm. This orm has got its own advantage as well as a substantialdrawback. The advantage is invariance in all transitions rom one rame o re erence to another. Thedrawback is that in covariant orm the terms o the equations do not contain actual three-dimensionalvalues, which can be measured in experiments or observations ( physical observable values ). Thisimplies that in general covariant orm equations o motion o particle are merely an intermediatetheoretical result, not applicable to practice. There ore, in order to make results o any physicalmathematical theory usable in practice, we need to ormulate its equations with physical observablevalues. Namely, to calculate trajectories o certain particles we have to ormulate general covariantdynamic equations o motion with physical observable properties o these particles as well as throughobservable properties o an actual physical rame o re erence o the observer.

But de ning physical observable values is not a trivial problem. For instance, i or a our-dimensional vector Q (as ew as our components) we may heuristically assume that its three spatialcomponents orm a three-dimensional observable vector, while the temporal component is observablepotential o the vector eld (which generally does not prove they can be actually observed, though),a contravariant 2nd rank tensor Q (as many as 16 components) makes the problem much moreinde nite. For tensors o higher rank the problem o heuristic de nition o observable components isar more complicated. Besides there is an obstacle related to de nition o observable components o covariant tensors (in which indices are the lower ones) and o mixed type tensors, which have bothlower and upper indices.

There ore the most reasonable way out o the labyrinth o heuristic guesses is creating a strictmathematical theory to enable calculation o observable components or any tensor values. Suchtheory was created in 19411944 by Zelmanov and set orth in his dissertation thesis [6]. It shouldbe noted, though, that many researchers were working on theory o observable values in 1940s. Forexample, Landau and Li shitz in later editions o their well-known The Classical Theory o Fields [1]


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introduced observable time and observable three-dimensional interval, similar to those introduced byZelmanov. But the authors limited themselves only to this part o theory and did not arrive to generalmathematical methods to de ne physical observable values in pseudo-Riemannian space.

Over the next decades Zelmanov would improve his mathematical apparatus o physical observablevalues (the methods o chronometric invariants), setting orth the results in some later papers [7, 8,9, 10]. Similar results were obtained by C. Cattaneo, an Italian mathematician, independently romZelmanov. However Cattaneo published his rst study on the theme in 1958 [11, 12, 13, 14].

A systematic description o Zelmanovs mathematical methods o chronometric invariants is givenin our two pre-prints [15, 16]. There ore in the next Section o this Chapter we will give just a brie overview o the methods o theory o physical observable values, which is necessary or understandingthem and using in practice.

In Section 1.3 we will present the results o studying geodesic motion o particles using the tech-nique o chronometric invariants [15, 16]. In Section 1.4 will ocus on setting problem o buildingdynamic equations o particles along non-geodesic trajectories, i. e. under action o non-gravitationalexternal orces.

1.2 Physical observable values

This Section introduces Zelmanovs mathematical apparatus o chronometric invariants.To de ne mathematically which components o any our-dimensional values are physical observable

values, we consider a real rame o re erence o some observer, which includes coordinate nets , spannedover some physical body ( body o re erence ), at each point o which real clock is installed. The bodyo re erence being a real physical body possesses a certain gravitational potential, may rotate andde orm, making the space o re erence non-uni orm and anisotropic. Actually, the body o re erenceand attributed to it space o re erence may be considered as a set o real physical re erences, to whichobserver compares all results o his measurements. There ore, physical observable values should beobtained as a result o projecting our-dimensional values on time and space o observers real bodyo re erence.

From geometric viewpoint three-dimensional space is spatial section x0= ct= const . At any pointo the space-time a local spatial section (orthogonal space) can be done orthogonal to the line o time .I exists space-time enveloping curve to local spaces it is a spatial section everywhere orthogonal tolines o time. Such space is known as holonomic space . I no enveloping curve exists to such localspaces, i. e. there only exist spatial sections locally orthogonal to lines o time, such space is known asnon-holonomic .

We assume that the observer rests in respect to his physical re erences (body o re erence). Frameo re erence o such observer in any displacements accompanies the body o re erence and is calledaccompanying rame o re erence . Any coordinate nets that rest in respect to the same body o re erence are related through trans ormation

x0 = x0 x0 , x1 , x2 , x3 , x i = x i x1 , x 2 , x3 , x i

x 0= 0 , (1.13)

where the latter equation implies independence o spatial coordinates in tilde-marked net rom timeo non-marked net, which is equivalent to setting coordinate net o concrete and xed lines o timexi = const in any point o coordinate net. Trans ormation o coordinates is nothing but transitionrom one coordinate net to another within the same spatial section. Trans ormation o time implieschanging the whole set o clocks, i. e. transition rom to another spatial section (space o re erence).In practice that means replacement o one body o re erence along with all o its physical re erenceswith another body o re erences that has got its own physical re erences. But when using diferentre erences observer will obtain quite diferent results (observable values). There ore physical observablevalues must be invariant in respect to trans ormations o time, i. e. should be chronometrically invar-iant values .

Because trans ormations (1.13) de ne a set o xed lines o time, then chronometric invariants(physical observable values) are all values, invariant in respect to these trans ormations.


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In practice, to obtain physical observable values in accompanying rame o re erence we have tocalculate chronometrically invariant projections o our-dimensional values on time and space o a realphysical body o re erence and ormulate them with chronometrically invariant (physically observable)properties o the space o re erence.

We project our-dimensional values using operators that characterize properties o real space o re erence. Operator o projection on time b is a unit vector o our-dimensional velocity o observersrame o re erence ( our-dimensional velocity o body o re erence)

b =dx

ds, (1.14)

which is tangential to our-dimensional observers trajectory in its every point. Because any rame o re erence is described by its own tangential unit vector b , Zelmanov called the vector a monad . Theoperator o projection on space is de ned as our-dimensional symmetric tensor

h = g + b b , h = g + b b , (1.15)which mixed components are

h = g + b b . (1.16)Previous studies show that these values possess necessary properties o projection operators

[6, 10, 16]. Projection o tensor value on time is a result o its contraction with monad vector.Projection on space is contraction with tensor o projection on space.

In accompanying rame o re erence three-dimensional observers velocity in respect to the bodyo re erence is zero bi =0. Other components o the monad are

b0 =1

g00 , b0 = g0 b = g00 , bi = gi b = gi 0g00 . (1.17)

Respectively, in accompanying rame o re erence ( bi =0) components o tensor o projection onspace are

h00 = 0 , h00 = g00 +1

g00, h00 = 0 ,

h0i = 0 , h0i

= g0i

, hi0 =

i0 = 0 ,

h i0 = 0 , hi 0 = gi0 , h0i =gi 0g00

,

h ik = gik +g0i g0k

g00, hik = gik , hik = gik = ik .

(1.18)

Tensor h in three-dimensional space o an accompanying rame o re erence shows properties o undamental metric tensor

h i hk =

ik bk bi = ik , ik =

1 0 00 1 00 0 1

, (1.19)

where ik is a unit three-dimensional tensor4 . There ore, in accompanying rame o re erence three-

dimensional tensor hik

may li t or lower indices in chronometrically invariant values.Projections on time T and space L o a certain vector Q (1st rank tensor) in accompanying

rame o re erence ( bi =0) are

T = b Q = b0Q0 =Q0

g00 , (1.20)

L0 = h0 Q =

g0kg00

Qk , Li = h i Q = ik Q

k = Qk . (1.21)

4 Tensor ik is the three-dimensional part o our-dimensional unit tensor , which can be used to replace indices in

our-dimensional values.


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Below are some projections o 2nd rank tensor Q in accompanying rame o re erence

T = b b Q = b0b0Q00 =Q00g00

, (1.22)

L00 = h0 h0 Q

=

g0i g0k

g200

Q ik , Lik = h i hk Q

= Q ik . (1.23)

Experimental check o invariance o the obtained physical values in respect to trans ormations(1.13) says that physical observable values are projection o our-dimensional value on time and spatialcomponents o projection on space.

Hence, projecting our-dimensional coordinates x on time and space we obtain physical observ-able time

= g00 t + g0icg00 xi , (1.24)

and physical observable coordinates , which coincide with spatial coordinates x i . Similarly, projec-tion o elementary interval o our-dimensional coordinates dx gives elementary interval o physicalobservable time

d = g00 dt + g0icg00 dxi , (1.25)

and elementary interval o physical observable coordinates dx i . Respectively, physical observablevelocity o particle or an observer is three-dimensional chronometrically invariant vector

vi =dx i

d , (1.26)

which is diferent rom the value u i = dxi

dt , which is the vector o its three-dimensional coordinatevelocity.

Projecting undamental metric tensor on space we obtain that in accompanying rame o re erencephysical observable spatial metric tensor consists o spatial components o tensor o projection on space

h i hk g

= gik = h ik , hi hk g = gik bi bk = h ik . (1.27)

There ore the square o physical observable interval d isd2 = h ik dx i dxk . (1.28)

Four-dimensional space-time interval ormulated with physical observable values can be obtainedby substituting g rom (1.15)

ds2 = c2d 2 d2 . (1.29)But aside or projections on space and time our-dimensional values o 2nd rank and above also

have mixed components which have both upper and lower indices at the same time. How do we ndphysical observable values among them, i any? The best approach is to develop a generalized methodto calculate physical observable values based solely on their property o chronometric invariance andallowing to nd all observable values at the same time in any tensor. Such method was developed byZelmanov and set orth as a theorem.Zelmanov theorem

We assume that Q ik...p00 ... 0 are components o our-dimensional tensor Q...00 ... 0 o r -th rank, in which all

upper indices are not zero, while all m lower indices are zeroes. Then tensor values

T ik...p = ( g00 ) m2 Q ik...p00 ... 0 (1.30)

make up chronometrically invariant three-dimensional contravariant tensor o ( rm)-th rank. Hencetensor T ik...p is a result o m- old projection on time by indices , . . . and projection on space byrm indices , . . . o the initial tensor Q

...... .


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An immediate result o the theorem is that or vector Q two values are physical observable, whichwere obtained earlier by projecting

b Q =Q0

g00 , hi Q

= Q i . (1.31)

For a symmetric 2nd rank tensor Q the three values are physical observable ones, namely

b b Q =Q00g00

, hi b Q =Q i0

g00 , hi h

k Q

= Q ik . (1.32)

The calculated physical observable values (chronometric invariants) have to be compared to there erences (the standards o measure) observed properties o the space o re erence which are speci cor any particular body o re erence. There ore we will now consider the basic properties o the spaceo re erence with which the nal equations o our theory are to be ormulated.

Physical observable properties o the space o re erence are obtained with the help o chronomet-rically invariant operators o diferentiation by time and spatial coordinates

t

=1

g

00

t

,

x i=

x i

g0ig

00

x 0

, (1.33)

which are not commutative, i. e. diference between 2nd derivatives with respect to time and spacecoordinates becomes not zero

2

x i t 2

tx i=

1c2

F i

t, (1.34)

2

x i x k 2

x k x i=

2c2

Aik

t. (1.35)

Here Aik is a three-dimensional antisymmetric chronometrically invariant tensor o angular veloc-ities o rotation o the re erences space

Aik =1

2

vk

xi

vi

xk +

1

2c2 (F i vk

F k vi ) , (1.36)

where vi stands or rotation velocity o space

vi = cg0i

g00 . (1.37)Tensor Aik being equal to zero is the necessary and su cient condition o holonomity o space

[6, 10]. In this case g0i =0 and vi =0. In non-holonomic space Aik =0 is always not zero. There ore,tensor Aik is also a tensor o the spaces non-holonomity 5 .

The value F i is a three-dimensional chronometrically invariant vector o gravitational inertial orce

F i =c2

c2 wwx i

vit

, w = c2 (1 g00 ) , (1.38)

where w stands or gravitational potential o the body o re erences space 6 . In quasi-Newtonianapproximation, i. e. in a weak gravitational eld at speeds much lower than the speed o light and inabsence o rotation o space F i becomes a non-relativistic orce

F i =wx i

. (1.39)

5 Special Relativity space-time (Minkowski space) in Galilean rame o re erence and some cases in General Relativityare examples o holonomic spaces A ik =0.

6 Values w and vi do not possess property o chronometric invariance o their own. Vector o gravitation inertial orceand tensor o angular velocity o spaces rotation, built using them, are chronometric invariants.


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Because the observers body o re erence is a real physical body, coordinate nets it bears arede ormed. Consequently the space o a real body o re erence is de ormed too. There ore comparisono observable values with physical re erences o the body o re erence has to take into account theeld o de ormation o the space o re erence, i. e. that the eld o tensor h ik is not stationary. Inpractice stationary de ormation o space is rather rare: eld o de ormation varies all the time whichshould be as well taken into account in measurements. This can be done by de ning in equations athree-dimensional symmetric chronometrically invariant tensor o de ormation velocities

D ik =12

h ikt

, D ik = 12

h ik

t, D = D kk =

ln ht

, h = det h ik . (1.40)

Given these de nitions we can generally ormulate any geometric object in Riemannian space withobservable parameters o the space o re erence. For instance, Christofel symbols that appear inequations o motion are not tensors [2]. Nevertheless, they can be as well ormulated with physicalobservable values [6, 15, 16]

000 = 1c3

1

1 wc2

wt

+ 1 wc2

vk F k , (1.41)

k00 = 1c2

1 wc2

2F k , (1.42)

00i =1c2

1

1 wc2

wx i

+ vk D ki + Aki +

1c2

vi F k , (1.43)

k0i =1c

1 wc2

D ki + Aki +

1c2

vi F k , (1.44)

0ij = 1c

1

1 wc2

D ij +1c2

vn vj (D ni + Ani ) + vi D

nj + A

nj +

1c2

vi vj F n +

+12

v ix j

+v jx i

12c2

(F i vj + F j vi ) nij vn ,(1.45)

kij =kij

1c2

vi D kj + A kj + vj D

ki + A

ki +

1c2

vi vj F k , (1.46)

where kij are chronometrically invariant Christofel symbols which are de ned similarly to regularChristofel symbols (1.2) but through physical observable metric tensor h ik and chronometricallyinvariant diferentiation operators

ijk = h

imjk,m =

12

h imh jmx k

+h kmx j

h jkx m

. (1.47)

We have discussed the basics o mathematical apparatus o chronometric invariants. Now hav-ing any equations obtained using general covariant methods we can calculate their chronometricallyinvariant projections on time and on space o any particular body o re erence and ormulate themwith its real physical observable properties. From here we arrive to equations containing only valuesmeasurable in practice.

Naturally, the rst possible application o this mathematical apparatus that comes to our mind iscalculation o chronometrically invariant dynamic equations o motion o ree particles and studyingthe results. Partial solution o this problem was obtained by Zelmanov [6, 10]. We presented thegeneral one in our previous works [15, 16]. The next Section will ocus on the results o our study.


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1.3 Dynamic equations o motion o ree particles

Absolute derivative o vector o motion o particle to scalar parameter is actually a our-dimensionalvector

N =dQ

d+ Q

dx

d. (1.48)

There ore chronometrically invariant (physical observable) components o equation o motion arede ned similarly to those o any our-dimensional vector (1.31)

N 0g00 =

g0 N

g00 =1

g00 g00 N 0 + g0i N i , (1.49)

N i = h i N = h i0N

0 + h ik N k . (1.50)

From geometric viewpoint this is a projection o vector N on time and spatial components o its projection on space in accompanying rame o re erence. In a similar way we can project generalcovariant dynamic equations o motion o ree mass-bearing particles (1.10) and o ree masslessparticles (1.12). The technique to calculate these projections is given in details in our previouspublications [15, 16]. As a result we arrive to chronometrically invariant dynamic equations o motiono ree mass-bearing particles

dmd

mc2

F i vi +mc2

D ik vi vk = 0 , (1.51)

d mvi

d + 2 m D ik + A

ik v

k mF i + m ink vn vk = 0 , (1.52)and o ree massless particles

dkd

kc2

F i ci +kc2

D ik ci ck = 0 , (1.53)

d kci

d + 2 k D ik + A

ik c

k kF i + k ink cn ck = 0 , (1.54)where m stands or relativistic mass o mass-bearing particle, k= c is wave number that character-

izes massless particle and ci

is three-dimensional chronometrically invariant vector o light velocity.As seen, contrary to general covariant dynamic equations o motion (1.10, 1.12), chronometricallyinvariant equations have a single derivation parameter or both mass-bearing particles and masslessparticles (which is physical observable time ).

These equations were rst obtained by Zelmanov [6]. But later it was ound that unction o timedtd they include is strictly positive [15, 16]. Physical time has direct ow d> 0. Flow o coordinatetime dt shows change o time coordinate o particle x0= ct in respect to observers clock. Hence thesign o the unction shows where the particle travels to in time in respect to observer.

Function o time dtd [15, 16] is obtained rom the condition that the square o our-dimensionalvelocity o particle is constant along its our-dimensional trajectory u u = g u u = const . Equationsin respect to dtd are the same or sub-light-speed mass-bearing particles, or massless particles andor super-light-speed mass-bearing particles and have two solutions, which are

dtd 1,2

=vi vi c2

c2 1 wc2

. (1.55)

As shown in [15, 16] time has direct ow i vi vic2> 0, time has reverse ow i vi vic2< 0, andow o time stops i vi vic2=0. There ore there exists a whole range o solutions or various types o particles and directions they travel in time in respect to observer. For instance, relativistic mass o particle, which is projection o its our-dimensional vector on time P 0g00 = m is positive i particle


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travels into uture and negative i it travels into past. Wave number o massless particle K 0g00 = k isalso positive or movement into uture and is negative or movement into past.

In the studies [15, 16] we also showed that chronometrically invariant dynamic equations o motiono ree mass-bearing particles with reverse ow o time dtd < 0 that travel rom uture into past are

dmd mc2 F i vi + mc2 D ik vi vk = 0 , (1.56)d mvi

d + mF i + m ink v

n vk = 0 . (1.57)

For ree massless particles that travel into past we arrive to

dkd

kc2

F i ci +kc2

D ik ci ck = 0 , (1.58)

d kci

d + kF i + k ink c

n ck = 0 . (1.59)

For super-light-speed mass-bearing particles equations o motion are similar to those or sub-light

speeds, save that relativistic mass m is multiplied by imaginary unit i.Equations o motion o particles into uture and into past are not symmetric due to diferentphysical conditions in case o direct and reverse time ows, and some terms in equations will be missing.

Besides, in our previous studies [15, 16] we considered motion o mass-bearing and massless par-ticles within the wave-particle concept, assuming that motion o any particles can be represented aspropagation o waves in approximation o geometric optics. In this case the dynamic vector o masslessparticles will be [1]

K =x

, (1.60)

where is wave phase (eikonal). In a similar way we consider dynamic vector o mass-bearing particles

P =hc

x

, (1.61)

where h is Plank constant. Wave phase equation (eikonal equation) in approximation o geometricoptics is the condition K K =0. Hence chronometrically invariant eikonal equation or masslessparticles will be

1c2

t

2

+ h ik

x i

x k= 0 , (1.62)

and or mass-bearing particles

1c2

t

2

+ h ik

x i

x k=

m20c2

h2. (1.63)

Substituting wave orm o dynamic vector into general covariant dynamic equations o motion(1.10, 1.12) and their projection on time and space we obtain wave orm o chronometrically invariantequations o motion. For mass-bearing particles the equations are

dd t + F i

x i Dik v

k x i

= 0 , (1.64)

dd

h ik

x k Dik + A

ik

1c2

t

vk hkm

x m

1c2

t

F i + hmn imk vk

x n= 0 ,

(1.65)

where plus in alternating terms stands or motion o particles rom past into uture (direct owo time), while minus stands or motion into past (reverse ow o time). Noteworthy, contrary to


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corpuscular orm o equations o motion (1.51, 1.52) and (1.56, 1.57) these equations are symmetric inrespect to direction o motion in time. For massless particles wave orm o chronometrically invariantequations o motion shows the only diference: instead o three-dimensional observable velocity o particle v i it includes three-dimensional chronometrically invariant vector o velocity o light ci .

The act that corpuscular equations o motion into past and into uture are not symmetric leadsto evident conclusion that in our-dimensional non-uni orm space-time there exists a undamentalasymmetry o directions in time. To understand physical sense o this undamental asymmetry inprevious study we introduced the mirror principle or observable efect o the mirror Universe [16].

Imagine a mirror in our-dimensional space-time which coincides with spatial section and there oreseparates past rom uture. Then particles and waves traveling rom past into uture (with positiverelativistic mass and requency) hit the mirror and bounce back in time, i. e. into past. And theirproperties take negative values. And vice versa, particles and waves traveling into past (with negativerelativistic mass and requency) bounce rom the mirror to give positive values to their properties andto begin traveling into uture. When bouncing rom the mirror the value dt changes its sign andequations o wave propagation into uture become equations o wave propagation into past (and viceversa). Noteworthy, when re ecting rom the mirror equations o wave propagation trans orm intoeach other completely without contracting or adding new terms. In other words, wave orm o matterundergoes ull re ection rom our mirror. To the contrary, corpuscular equations o motion do not trans orm completely in re ection rom our mirror. Spatial components o equations or mass-bearingand massless particles, traveling rom past into uture, have an additional term

2m D ik + A ik v

k , 2k D ik + A ik c

k , (1.66)

not ound in equations o motion rom uture into past. Equations o motion o particle into pastgain an additional term when re ecting rom the mirror. And vice versa, equations o motion intouture lose a term when particle hits the mirror. That implies that either in case o motion o particles(corpuscular equations) as well as in case o propagation o waves (wave equations) we come acrossnot a simple bouncing rom the mirror, but rather passing through the mirror itsel into anotherworld, i. e. into a mirror world .

In this mirror world all particles bear negative masses and requencies and travel ( rom viewpointo our worlds observer) rom uture into past. Wave orm o matter in our world does not afect events

in the mirror world, while wave orm o matter in the mirror world does not afect events in our world.To the contrary, corpuscular orm o matter (particles) in our world may produce signi cant efect onevents in the mirror world, while particles in the mirror world may afect events in our world. Ourworld is ully isolated rom the mirror world (no mutual efect between particles rom two worlds)under an evident condition D ik v

k = A ik vk , at which the additional term in corpuscular equations iszero. This becomes true, in particular, when D ik =0 and A ik =0, i. e. when dynamic de ormation androtation o the body o re erences space is totally absent.

So ar we have only considered motion o particles along non-isotropic trajectories, whereds2= c2dt2d2> 0, and that along isotropic (light-like) trajectories, where ds2=0 and c2dt2= d2=0.Besides, in our previous studies [15, 16] we considered trajectories o third kind, which, aside ords2=0, meet even more strict conditions c2dt2= d2=0

d = 1

1

c2w + v

iu i dt = 0 , (1.67)

d2 = h ik dx i dxk = 0 . (1.68)

We will re er to such trajectories as degenerated or zero trajectories , because rom viewpoint o aregular sub-light-speed observer interval o observable time and observable three-dimensional intervalare zero along them. We can as well show that along zero-trajectories the determinant o undamentalmetric tensor o Riemannian space is also zero g=0. In Riemannian space by de nition g< 0, i.e. themetric in strictly non-degenerated. We will re er to a space with ully degenerated metric as zero-space ,while particles that move along trajectories in such space will be re erred to as zero-particles .


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Physical conditions o degeneration are obtained rom (1.67, 1.68)

w + vi u i = c2 , (1.69)

gik u i uk = c2 1 wc2

2. (1.70)

Respectively, mass o zero-particles M , which include physical conditions o degeneration, is di -erent rom relativistic mass m o regular particles in non-degenerated space-time and is

M =m

1 1c2

w + vi u i, (1.71)

i. e. is a ratio between two values, each one equals to zero in case o degenerated metric, but the ratiois not zero 7 .

Corpuscular and wave orms o dynamic vector o zero-particles are

P =M c

dx

dt, P =

hc

x

. (1.72)

Then corpuscular orm o chronometrically invariant dynamic equations o motion in zero-space is

MD ik u i uk = 0 , (1.73)

ddt

Mu i + M ink un uk = 0 . (1.74)

Wave orm o the same equations is

D mk uk

x m= 0 , (1.75)

ddt

h ik

x k+ hmn imk u

k x n

= 0 . (1.76)

Equation o eikonal or zero-particles is

h ik

x i

x k= 0 , (1.77)

and is a standing wave equation (in ormation ring). There ore, rom viewpoint o a regular sub-light-speed observer all zero-space is lled with a system o standing light-like waves (zero-particles), i. e.with a standing-light hologram . Besides, in zero-space observable time has the same value or anytwo events (1.67). This implies that rom viewpoint o a regular observer velocity o zero-particles isin nite, i. e. zero-particles can instantly trans er in ormation rom one point o our regular world toanother, thus per orming long-range action [15, 16].

1.4 Introducing concept o nongeodesic motion o particles. Problemstatement

We obtained that ree motion o particle (along geodesic lines) leaves absolute derivative o dynamicvector o particle ( our-dimensional impulse vector) zero and its square is conserved along the trajec-tory o motion. In other words, parallel trans er is efected in the meaning o Levi-Civita.

In case o non- ree (non-geodesic) motion o particle absolute derivative o its our-dimensionalimpulse is not zero. But equal to zero is absolute derivative o sum o our-dimensional impulse o particle P and impulse vector L , which particle gains rom interaction with external elds which

7 This is similar to the case o massless particles, because given v 2 = c2 values m 0 =0 and 1 v2 /c 2 =0 are zero,but their ratio is m = m 0

1 v2 /c 2=0.


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deviates its motion rom geodesic line. Superposition o any number o vectors can be subjected toparallel trans er [2]. Hence, building dynamic equations o non-geodesic motion o particles rst o allrequires de nition o non-gravitational perturbation elds.

Naturally, external eld will only interact with particle and deviate it rom geodesic line i theparticle bears physical property o the same kind as the external eld does. As o today, we know o three undamental physical properties o particles, not related to any others. These are mass o particle,electric charge and spin . I undamental character o the ormer two was under no doubt, spin o electron over a ew years a ter experiments by O. Stern and W. Gerlach (1921) and their interpretationby S. Gaudsmith and G. Ulenbek (1925), was considered as its speci c moment o impulse caused byrotation around its own axis. But experiments done over the next decades, in particular, discoveryo spin in other elementary particles, proved that views o spin particles as rotating gyroscopes werewrong. Spin proved to be a undamental property o particles just like mass and charge, though it hasdimension o moment o impulse and in interactions reveals as speci c rotation moment o particle.

Gravitational eld by now has received geometric interpretation. In theory o chronometric invari-ants gravitational orce and gravitational potential (1.38) are obtained as unctions o only geometricproperties o the space itsel . There ore considering motion o particle in pseudo-Riemannian spacewe actually consider its motion in gravitational eld.

But we still do not know whether electromagnetic orce and potential can be expressed through

geometric properties o space. There ore electromagnetic eld at the moment has no geometric in-terpretation and is introduced into space-time as a separate tensor eld (Maxwell tensor eld). Bynow the basic equations o electromagnetic theory have been obtained in general covariant orm 8 . Inthis theory charged particle gains our-dimensional impulse e

c2A rom electromagnetic eld, where

A is our-dimensional potential o electromagnetic eld and e is particles charge [1, 4]. Adding thisextra impulse to speci c vector o impulse o particle and actuating parallel trans er we obtain generalcovariant dynamic equations o motion o charged particles.

The case o spin particles is ar more complicated. To calculate impulse that particle gains dueto its spin, we have to de ne the external eld that interacts with spin as a undamental propertyo particle. Initially this problem was approached using methods o quantum mechanics only (Diracequations, 1928). Methods o General Relativity were rst used by A. Papapetrou and E. Corinaldesi[20, 21] to study the problem. Their approach relied upon general view o particles as mechanicalmonopoles and dipoles. From this viewpoint a regular mass-bearing particle is a mechanical monopole .A particle that can be represented as two masses co-rotating around a common center o gravity is amechanical dipole . There ore, proceeding rom representation o spin particle as a rotating gyroscopewe can (to a certain extent) consider it as a mechanical dipole, which center o gravity lays over theparticles sur ace. Then Papapetrou and Corinaldesi considered motion o mechanic dipole in pseudo-Riemannian space with Schwarzschild metric, i. e. in every speci c case when rotation o space is zeroand its metric is stationary (tensor o de ormation velocities is zero).

No doubt the method proposed by Papapetrou is worth attention, but it has a signi cant drawback.Being developed in 1940s it ully relied upon view o a spin-particle as a swi tly rotating gyroscope,which does not match experimental data o the recent decades 9 .

There is another way to tackle the problem o motion o spin particles. In Riemannian spaceundamental metric tensor is symmetric g = g . Nevertheless we can build a space in which metrictensor will have arbitrary orm g = g (such space will have non-Riemannian geometry). Then anon-zero antisymmetric part can be ound in metric tensor 10 . Appropriate additions will also appear

8 Despite this, due to complicated calculations o energy-impulse tensor o electromagnetic eld in pseudo-Riemannianspace, speci c problems are commonly solved either or certain particular cases o General Relativity or in Galileanrame o re erence in a at Minkowski space (space-time o Special Relativity).

9 As a matter o act, considering electron as a ball with radius o r e =2.8 10 13 cm implies that linear speed o itsrotation on the sur ace is u= h2m 0 r e = 2 10

11 cm/s which is about 70 times as high as the light speed. But experimentsshow there are no such speeds in electron.

10 Generally, in any tensor o 2nd rank and above symmetric and antisymmetric parts can be distinguished. Forinstance, in 2nd rank undamental metric tensor g = 12 g + g +

12 g g = S + N , where S is sym-

metric part and N is antisymmetric part o tensor g . Because metric tensor o Riemannian space is symmetricg = g , its antisymmetric part is zero.


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in Christofel symbols (1.2) and in Riemann-Christofel curvature tensor R . These additionswill cause a vector trans erred along a closed contour not to return into the initial point, i. e. trajectoryo trans er becomes twisted like a spiral. Such space is re erred to as twisted space. In such space spinrotation o particle can be considered as trans er o rotation vector along its sur ace contour, whichgenerates local eld o space twist.

But this method has got signi cant drawbacks as well. First, with g

= g

unctions o compo-nents with diferent order o indices may be varied. The unctions have to be xed somehow in to orderto set a concrete eld o twist, which dramatically narrows the range o possible solutions, enablingonly building equations or a range o speci c cases. Second, the method ully relies upon assumptiono spins physical nature as a local eld o twist produced by trans er o vector o particles rotationalong a contour. This, in its turn, again implies the view o spin particle as a rotating gyroscope witha limited radius (like in Papapetrous method), which does not match experimental data.

Nevertheless, there is little doubt in that additional impulse, which a spin-particle gains, canbe represented with methods o General Relativity. Adding it to the speci c dynamic vector o aparticle (efect o gravitation) and accomplishing parallel trans er, we obtain general covariant dynamicequations o motion o a spin-particle.

Once we have general covariant dynamic equations o motion o spin and electric charged particlesobtained, we should project them on space and time o accompanying rame o re erence and express

through physical observable properties o the space o re erence. As a result we arrive to chronomet-rically invariant (physical observable) dynamic equations o nongeodesic motion o particles.There ore, the problem we are going to tackle in this book alls apart into a ew stages. First, we

should build chronometrically invariant theory o electromagnetic eld in pseudo-Riemannian spaceand arrive to chronometrically invariant dynamic equations o motion o charged particle. This prob-lem will be solved in Chapter 3.

Then, we have to create a theory o motion o a spin-particle. We will approach the problem inits most general orm, assuming spin a undamental property o matter (like mass or electric charge).In Chapter 4 detailed study will show that eld o non-holonomity o space interacts with spin givingparticle additional impulse.

In Chapter 5 we are going to discuss observable projections o Einstein equations. Proceeding romthem we will study properties o physical vacuum and how they are dealt in cosmology.

In Chapter 6 we prove that ully degenerated space-time (zero-space) is an area inhabited by

virtual particles, and build geometric concept o annihilation o particles using methods o GeneralRelativity. Within the concept, we explain anomalous rate o annihilation o othopositronium.

But be ore turning to these studies we would like to have a look into our-dimensional tensoranalysis in terms o physical observable values (chronometric invariants). Original publications byZelmanov gave a very ragmented account o the subject, which prevented a reader not amiliar withthis mathematical apparatus rom learning it on their own. There ore we recommend our Chapter 2to readers who are going to use mathematical apparatus o chronometric invariants in their theoreticstudies. For general understanding o our book, though, reading this Chapter may be not necessary.


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Chapter 2

Tensor algebra and the analysis

2.1 Tensors and tensor algebra

We assume a space (not necessarily a metric one) with an arbitrary rame o re erence x . In somepart o the space, there exists an object G de ned by n unctions f n o coordinates x . We know the

trans ormation rule to calculate these n unctions in any other rame o re erence x

in this space.Given all this G is a geometric object , which in the rame o re erence x has axial components f n (x ),while in any other rame o re erence x it has components f n (x ).

We assume a tensor object ( tensor ) o zero rank is any geometric object , trans ormable accordingto the rule

=x

x, (2.1)

where the index takes in turn numbers o all coordinate axis (such notation is re erred to as by-component notation or tensor notation ). Zero rank tensor has got a single component and is alsoknown as scalar . Scalar in space is a point to which a certain number is attributed.

Consequently, scalar eld 11 is a set o points in space, which have some common property. Forinstance, mass o a material point is a scalar, while distribution o mass in gas makes up a scalar eld.

Contravariant tensor o 1st rank is a geometric object A with components trans ormable according

to the rule A = A x

x . (2.2)

From geometric viewpoint it is a n-dimensional vector. For instance, vector o displacement dxis a contravariant tensor o 1st rank.

Contravariant tensor o 2nd rank A is a geometric object with components trans ormable ac-cording to the rule

A = A x

x x

x . (2.3)

From geometric viewpoint this is an area (parallelogram) constrained by two vectors. There ore2nd rank contravariant tensor is sometimes re erred to as bivector .

Similarly, contravariant tensors o higher ranks are

A... = A... x

x x

x . (2.4)

Vector eld or eld o tensors o higher rank is also space distribution o these values. For instance,because mechanical stress characterizes both magnitude and direction, its distribution in a physicalbody can be presented as a vector eld.

Covariant tensor o 1st rank A is a geometric object, trans ormable according to the rule

A = Ax

x. (2.5)

11 Algebraic notations o a tensor and o a tensor eld are the same: eld o a tensor is represented as a tensor at apoint in space, but its presence at other points in this part o the space is assumed.

19


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CHAPTER 2. TENSOR ALGEBRA AND THE ANALYSIS 20

In particular, gradient o scalar eld o any invariant , i. e. the value A = x , is a covariant tensoro 1st rank. That is, because or a regular invariant we have = , then

x

= x

x

x=

x

x

x. (2.6)

Covariant tensor o 2nd rank A is a geometric object the trans ormation rule or which is

A = Ax

xx

x. (2.7)

Similarly, covariant tensors o higher ranks are

A... = A... x

x x

x. (2.8)

Mixed tensors are tensors o 2nd rank and above with both upper and lower indices. For instance,mixed symmetric tensor A is a geometric object trans ormable according to the rule

A = A

x

x x

x. (2.9)

Tensor objects exist both in metric and non-metric spaces, where distance between any two pointscan not be measured.

A tensor has an components, where a is dimension o the tensor and n is the rank. For instance,our-dimensional tensor o zero rank has 1 component, 1st rank tensor has 4 components, 2nd ranktensor has 16 components and so on. But indices, i. e. axial components, are ound not in tensorsonly, but in other geometric objects as well. There ore, i we come across a value in by-componentnotation, this is not necessarily a tensor value.

In practice, to know whether a given object is a tensor or not, we have to know the equation orthis object in a certain rame o re erence and to trans orm it to any other rame o re erence. Forinstance: are coe cients o coherence o space, i. e. Christofel symbols, tensors?

To know this, we have to calculate the values in another (tilde-marked) rame o re erence

= g , , , =12

g x +

g x

g x (2.10)

proceeding rom values in non-marked rame o re erence.Now we are going to calculate the terms in brackets (1.10). Fundamental metric tensor, just like

any other covariant 2nd rank tensor, is trans ormable to tilde-marked rame o re erence accordingto the rule

g = g x

xx

x. (2.11)

Because g depends upon non-tilde-marked coordinates, its derivative by tilde-marked coordinates(which are also unctions o non-tilde-marked ones) is calculated according to the rule

g x

=g x

x

x. (2.12)

Then the rst term in brackets (2.10) taking into account the rule o trans ormation o undamentalmetric tensor, is

g x

=g x

x

xx

xx

x+ g

x

x 2x

x x+

x

x 2x

x x. (2.13)

Similarly, calculating the rest o the terms o tilde-marked Christofel symbols (2.10), a ter trans-position o ree indices we arrive to

, = , x

xx

xx

x+ g

x

x 2x

x x, (2.14)


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=

x

x x

xx

x+

x

x 2x

x x. (2.15)

We see that coe cients o space coherence (Christofel symbols) are trans ormed not in the waytensors are, hence they are not tensors.

Tensors can be represented as matrices. But in practice, such orm may be illustrative or ten-sors o 1st and 2nd rank (single-row and at matrices, respectively). For instance, elementary our-dimensional displacement tensor is

dx = dx0 , dx 1 , dx 2 , dx 3 , (2.16)

and our-dimensional undamental metric tensor is

g =

g00 g01 g02 g03g10 g11 g12 g13g20 g21 g22 g23g30 g31 g32 g33

. (2.17)

Tensor o 3rd rank is a three-dimensional matrix. Representing tensors o higher ranks as matricesis even more problematic.

We now turn to tensor algebra a part o tensor calculus that ocuses on algebraic operationsover tensors.

Only same-type tensors o the same rank with indices in the same position can be added orsubtracted. Adding up two same-type n-rank tensors gives a new tensor o the same type and rankwith components being sums o respective components o the tensors added up. For instance, sum o two vectors and sum o two mixed 2nd rank tensors are

A + B = D , A + B = D

. (2.18)

Multiplication is permitted not only or same-type, but or any tensors o any ranks. External multiplication o n-rank and m-rank tensors gives an ( n+ m)-rank tensor

A B = D , A B = D . (2.19)

Contraction is multiplication o same-rank tensors when indices are the same. Contraction o tensors by all indices gives a scalar value

A B = C , A B = D . (2.20)

O ten multiplication o tensors implies contraction by not all indices. Such multiplication is re erredto as internal multiplication which implies contraction o some indices inside the multiplication

A B = D , A B = D

. (2.21)

Using internal multiplication o geometric objects we can nd whether they are tensors or not.There is a so-called theorem o ractions.

Theorem o ractions

I B is a tensor and its internal multiplication with a geometric object A (, ) is tensor D (, )

A (, ) B = D (, ) , (2.22)

then this object A (, ) is also a tensor [10].According to it, i internal multiplication o an object A with tensor B gives tensor D

A B = D , (2.23)

then object A is a tensor. Or, i internal multiplication o some object A and tensor B givestensor D

A B = D , (2.24)


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then object A is a tensor.Geometric properties o metric space are de ned by its undamental metric tensor g , which may

lower or li t indices in objects o metric space 12 . For example,

g A = A , g g A = A . (2.25)

In Riemannian space mixed undamental metric tensor g equals to unit tensor g = g g = .Diagonal components o our-dimensional unit tensor are ones, while the rest are zeroes. Using theunit tensor we can replace indices

A = A ,

A

= A . (2.26)

Contraction o 2nd rank tensor with undamental metric tensor gives a scalar value known as spur o tensor or trace o tensor

g A = A . (2.27)

For example, spur o undamental metric tensor in our-dimensional Riemannian space equals tothe number o coordinate axis

g g = g = g00 + g

11 + g

22 + g

33 = 4 . (2.28)

Physical observable metric tensor h ik (1.27) in three-dimensional space has properties o undamen-tal metric tensor. There ore it can lower, li t or replace indices in chronometrically invariant values.Namely, we can calculate squares o our-dimensional objects. Respectively, spur o three-dimensionaltensor is obtained by means o its contraction with observable metric tensor.

For instance, spur o tensor o velocities o space de ormation D ik (1.40) is a scalar

h ik D ik = D mm , (2.29)

that stands or absolute value o the speed o relative expansion o elementary volume o space.O course our brie account can not ully cover such a vast eld like tensor algebra. Moreover, there

is even no need in doing that here. Detailed accounts o tensor algebra can be ound in numerousmathematical books not related to General Relativity. Besides, many speci c techniques o this

science, which occupy substantial part o mathematical textbooks, are not used in theoretical physics.There ore our goal was to give only a basic introduction into tensors and tensor algebra, necessary orunderstanding this book. For the same reasons we have not covered issues like weight o tensors ormany others not used in calculations given in the below.

2.2 Scalar product o vectors

Scalar product o two vectors A and B in our-dimensional pseudo-Riemannian space is value

g A B = A B = A0B 0 + Ai B i . (2.30)

Scalar product is contraction because multiplication o vectors at the same time contracts allindices. There ore scalar product o two vectors (1st rank tensors) is always a scalar value (zerorank tensor).

I both vectors are the same, their scalar product

g A A = A A = A0A0 + Ai Ai (2.31)

is the square o vector A . Consequently length o a vector A is a scalar

A = |A | = g A A . (2.32)12 In Riemannian space metric has quadratic orm ds 2 = g dx dx , and respectively undamental metric tensor is a

2nd rank tensor g .


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Because our-dimensional pseudo-Riemannian space by its de nition has inde nite metric (i. e.sign-alternating signature), then length o our-dimensional vector may be real, imaginary or zero.According to this, vectors with non-zero (real or imaginary) length are re erred to as non-isotropicvectors . Vectors with zero length are re erred to as isotropic vectors . Isotropic vectors are tangentialto trajectories o propagation o light-like particles (isotropic trajectories).

In three-dimensional Euclidean space scalar product o two vectors is a scalar value with moduleequal to product o lengths o the two vectors multiplied by cosine o the angle between them

Ai B i = Ai B i = cos Ai ; B i . (2.33)

Theoretically at every point o Riemannian space a tangential at space can be set, which basicvectors will be tangential to basic vectors o Riemannian space at the tangential points. Then metrico tangential at space will be metric o Riemannian space at this point. There ore this statement isalso true in Riemannian space i we consider the angle between coordinate lines and replace Roman(three-dimensional) indices with Greek ones.

From here we can see that scalar product o two multiplicated vectors is zero i the vectors areorthogonal. In other words, scalar product rom geometric viewpoint is projection o one vectoronto another. I multiplicated vectors are the same, vector is projected onto itsel and the result o projection is its lengths square.

We will denote chronometrically invariant (physical observable) components o arbitrary vectorsA and B as

a =A0

g00 , ai = Ai , (2.34)

b =B0

g00 , bi = B i . (2.35)

Then the other components are

A0 =a + 1c vi a

i

1 wc2, Ai = a i

ac

vi , (2.36)

B0

=

b + 1c vi bi

1 wc2 , Bi = bi b

cvi . (2.37)

Substituting values o observable components into ormulas or A B and A A we arrive to

A B = ab a i bi = ab h ik a i bk , (2.38)A A = a2 a i a i = a2 h ik a i ak . (2.39)

From here we see that the square o vectors length is diference between squares o lengths o its projections onto time and space. I both projections are equal, the vectors length is zero and itis isotropic. Hence isotropic vector equally belongs to time and space. Equality o time and spaceprojections also implies that the vector is orthogonal to itsel . I temporal projection is longer, itbecomes real. I spatial projection is longer the vector becomes imaginary.

Scalar product o our-dimensional vector with itsel can be instanced by square o length o space-time interval

ds2 = g dx dx = dx dx = dx0dx0 + dx i dx i . (2.40)

In terms o physical observable values it can be represented as

ds2 = c2d 2 dx i dx i = c2d 2 h ik dx i dxk = c2d 2 d2 . (2.41)Length o interval ds= g dx dx may be real, imaginary or zero depending upon whether

ds is time-like c2d 2>d 2 (sub-light real trajectories), space-like c2d 2


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2.3 Vector product o vectors. Antisymmetric tensors and pseudotensors

Vector product o two vectors A and B is a 2nd rank tensor V obtained rom their externalmultiplication according to the rule

V = A ; B =1

2A B

A B =

1

2

A A

B

B . (2.42)

As seen, here the order in which vectors are multiplied does matter, i. e. the order in which we writedown tensor indices. There ore tensors obtained as vector products are antisymmetric tensors . In anantisymmetric tensor V = V indices being moved reserve their places as dots g V = V

,

thus showing rom where an index was moved. In symmetric tensors there is no need o reservingplaces or moved indices, because the order in which they appear does not matter. In particular,undamental metric tensor is symmetric tensor g = g , while tensor o space curvature R issymmetric in respect to transposition by pair o indices and is antisymmetric inside each pair o indices. Evidently, only tensor o 2nd rank or above may be symmetric or antisymmetric.

All diagonal components o any antisymmetric tensor by its de nition are zeroes. For examlpe, inantisymmetric 2nd rank tensor we have

V = [ A ; B ] =1

2(A B

A B ) = 0 . (2.43)

In three-dimensional Euclidean space absolute value o vector product o two vectors is de ned asthe area o the parallelogram they make and equals to product o modules o the two vectors multipliedby sine o the angle between them

V ik = Ai B k = sin Ai ; B k . (2.44)

This implies that vector product o two vectors (antisymmetric 2nd rank tensor) is a pad orientedin space according to directions o the orming vectors.

Contraction o an antisymmetric tensor V with any symmetric tensor A = A A is zero due toits properties V =0 and V = V

V A A = V 00 A0A0 + V 0i A0Ai + V i 0Ai A0 + V ik Ai Ak = 0 . (2.45)

According to theory o chronometric invariants physical observable components o antisymmetric2nd rank tensor V are values

V i0g00 =

V i 0g00 =

12

abi bai , (2.46)

V ik =12

a i bk ak bi , (2.47)expressed through observable components o its orming vectors A and B . Because in an anti-symmetric tensor all diagonal components are zeroes, the third observable component V 00g00 (1.32) isalso zero.

Physical observable components V ik (projections o V upon spatial section o our-dimensional

space-time) are the analog o vector product in three-dimensional space, while the value V i

0

g00, which

is space-time (mixed) projection o tensor V , has no analogs among components o a regular three-dimensional vector product.

Square o antisymmetric 2nd rank tensor, ormulated with observable components o ormingvectors, is

V V =12

a i a i bk bk a i bi ak bk + aba i bi 12

a2bi bi b2a i a i . (2.48)The latter two terms in the ormula contain values a (2.34) and b (2.35), which are projections o

multiplied vectors A and B onto time and there ore have no analogs in vector product in three-dimensional Euclidean space.


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Antisymmetry o tensor eld is de ned by re erence antisymmetric tensor. In Galilean rame o re erence 13 such re erences are Levi-Civita tensors: or our-dimensional values this is our-dimensional completely antisymmetric unit tensor e and or three-dimensional values this is three-dimensional completely antisymmetric unit tensor eikm . Components o these tensors, which have all indicesdiferent, are either +1 or 1 depending upon the number o transpositions o indices. All othercomponents, i. e. those having at least two coinciding indices, are zeroes. Moreover, or the signaturewe are using (+ ) all non-zero components bear the sign opposite to their respective covariantcomponents 14 . For example, in Minkowski space

g g g g e = g00 g11 g22 g33 e0123 = e0123 ,gi gk gm e = g11 g22 g33 e123 = e123

(2.49)

due to signature conditions g00 =1 and g11 = g22 = g33 = 1. There ore, components o tensor e aree0123 = +1 , e1023 = 1, e1203 = +1 , e1230 = 1,e0123 = 1, e1023 = +1 , e1203 = 1, e1230 = +1 ,

(2.50)

and components o tensor eikm are

e123 = +1 , e213 = 1, e231 = +1 , e123 = 1, e213 = +1 , e231 = 1. (2.51)Because the sign o the rst component is arbitrary, we can assume e0123 = 1 and e123 = 1.Subsequently, other components will change too. In general, our-dimensional tensor e is related

to three-dimensional tensor eikm as e0ikm = eikm .Multiplying our-dimensional antisymmetric unit tensor e by itsel we obtain a regular 8th

rank tensor with non-zero components, which are presented in the matrix

e e =

. (2.52)

Other properties o tensor e are obtained rom the previous one by means o contraction o indices

e e =

, (2.53)

e e = 2

= 2 , (2.54)

e e = 6 , e e = 6 = 24. (2.55)Multiplying three-dimensional antisymmetric unit tensor eikm by itsel we obtain a regular 6th

rank tensor

eikm

erst =

ir is it

kr

ks

ktmr ms mt . (2.56)

Other properties o tensor eikm can be expressed as

eikm ersm = ir iskr ks

= is kr ir ks , (2.57)

13 Galilean rame o re erence is the one that does not rotate, is not subject to de ormation and alls reely in a atspace-time (Minkowski space). Here lines o time are linear and so are three-dimensional coordinate axis.

14 In case o signature ( + ++) this is only true or our-dimensional tensor e . Components o three-dimensionaltensor eikm will have same signs as the respective components o eikm .


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eikm erkm = 2 ir , eikm eikm = 2 ii = 6 . (2.58)

Completely antisymmetric unit tensor de nes or a tensor object its respective pseudotensor markedwith asteriks.

For instance, our-dimensional scalar, vector and tensors o 2nd, 3rd, and 4th ranks have respectiveour-dimensional pseudotensors o the ollowing ranks

V = e V, V = e V , V =12

e V ,

V =16

e V , V =124

e V ,(2.59)

where 1st rank pseudotensor V is sometimes called pseudovector , while zero-rank pseudotensorV is called pseudoscalar . Tensor and its respective pseudotensor are re erred to as dual to eachother to emphasize their common genesis. Similarly, three-dimensional tensors have respective three-dimensional pseudotensors

V ikm = eikm V, V ik = eikm V m ,

V i =1

2eikm V km , V =

1

6eikm V ikm .

(2.60)

Pseudotensors are called such because contrary to regular tensors, they do not change being re-ected in respect to one o the axis. For instance, being re ected in respect to abscises axis x1= x1 ,x2= x2 , x3= x3 . Re ected component o antisymmetric tensor V ik , orthogonal to x1 axis, is V 23 = V 23 ,while its dual component o pseudovector V i is

V 1 =12

e1km V km =12

e123 V 23 + e132 V 32 = V 23 ,

V 1 =12

e1km V km =12

ek1m V km =12

e213 V 23 + e312 V 32 = V 23 .(2.61)

Because our-dimensional antisymmetric tensor o 2nd rank and its dual pseudotensor are o thesame rank, their contraction is pseudoscalar

V V = V e V = e B = B . (2.62)

Square o pseudotensor V and square o pseudovector V i , expressed through their dual anti-symmetric tensors o 2nd rank are

V V = e V e V = 24V V , (2.63)V i V i = eikm V km eipq V pq = 6 V km V km . (2.64)

In non-uni orm and anisotropic pseudo-Riemannian space we can not set a Galilean rame o re er-ence and the re erence o antisymmetry o tensor eld will depend upon non-uni ormity and anisotropyo the space itsel , which are de ned by undamental metric tensor. Here re erence antisymmetric ten-sor is a our-dimensional completely antisymmetric discriminant tensor

E = e

g, E = e g . (2.65)

Here is the proo . Trans ormation o a unit completely antisymmetric tensor rom Galilean (non-tilde-marked) rame o re erence into an arbitrary (tilde-marked) rame o re erence is

e =x

xx

xx

xx

xe = Je , (2.66)

where J =det x

x is called the Jacobian o trans ormation (the determinant o Jacobi matrix)


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J = det

x 0 x0

x 0 x1

x 0 x2

x 0 x3

x 1 x0

x 1 x1

x 1 x2

x 1 x3

x 2 x0

x 2 x1

x 2 x2

x 2 x3

x3

x0 x3

x1 x3

x2 x3

x3

. (2.67)

Because metric tensor g is trans ormable according to the rule

g =x

xx

xg , (2.68)

then its determinant in tilde-marked rame o re erence is

g = detx

xx

xg = J 2g . (2.69)

Because in Galilean (non-tilde-marked) rame o re erence

g = det g = det

1 0 0 0

0 1 0 00 0 1 00 0 0 1= 1, (2.70)

then J 2= g2 . Expressing e in an arbitrary rame o re erence as E and writing down metrictensor in a regular non-tilde-marked orm, we obtain E = e g (2.65). In a similar way weobtain trans ormation rules or components E , because or them g = gJ 2 , where J = det x

x .But discriminant tensor E is not a physical observable value. Physical observable re erence o

antisymmetry o tensor elds is three-dimensional chronometrically invariant discriminant tensor

= h h h

b E

= b E , (2.71)

= h h h

b

E = b E , (2.72)

which in an accompanying rame o re erence ( bi =0), taking into account that g = h g00 , willtake the ormikm = b0E 0ikm = g00 E 0ikm = e

ikm

h , (2.73)

ikm = b0E 0ikm =E 0ikmg00 = eikm

h . (2.74)With its help we can trans orm chronometrically invariant (physical observable) pseudotensors.

For instance, rom chronometrically invariant antisymmetric tensor o spaces rotation Aik (1.36) weobtain observable pseudovector o angular velocity o rotation o space i = 12

ikm Akm .

2.4 Introducing absolute diferential and derivative to the direction

In geometry a diferential o a unction is its variation between in nitely close points with coor-dinates x and x + dx . Respectively, absolute diferential in n-dimensional space is variation o n-dimensional values between in nitely close points o n-dimensional coordinates in this space. Forcontinuous unctions, we commonly deal with in practice, variations between in nitely close pointsare in nitesimal. But in order to de ne in nitesimal variation o a tensor value we can not use simplediference between its values in points x and x + dx , because tensor algebra does not de ne theratio between values o tensors at diferent points in space. This ratio can be de ned only using ruleso trans ormation o tensors rom one rame o re erence into another. As a consequence, diferentialoperators and the results o their application to tensors must be tensors themselves.


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For instance, absolute diferential o a tensor value is a tensor o the same rank as the value itsel .For a scalar it is a scalar

D =x

dx . (2.75)

In accompanying rame o re erence ( bi =0) it is

D = t

d + x i

dx i . (2.76)

We can see that aside or three-dimensional observable diferential there is an additional term thattakes into account dependence o absolute displacement D rom ow o physical observable time d .

Absolute diferential o contravariant vector A , ormulated with operator o absolute derivation(nabla) is

DA = A dx =A

x dx + A

dx = dA + A dx , (2.77)

where A is absolute derivative A by coordinate x and d stands or regular diferential

A =A

x + A

, (2.78)

d = x

dx . (2.79)

Notation o absolute diferential with physical observable values is equivalent to calc

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